[    {        "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": "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. 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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": "0705.1432v3.Effective_temperature_and_Gilbert_damping_of_a_current_driven_localized_spin.pdf",        "content": "arXiv:0705.1432v3  [cond-mat.mes-hall]  4 Feb 2008Effective temperature and Gilbert damping of a current-driv en localized spin\nAlvaro S. N´ u˜ nez∗\nDepartamento de F´ ısica, Facultad de Ciencias Fisicas y Mat ematicas,\nUniversidad de Chile, Casilla 487-3, Codigo postal 837-041 5, Santiago, Chile\nR.A. Duine†\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: October 31, 2018)\nStarting from a model that consists of a semiclassical spin c oupled to two leads we present a\nmicroscopic derivation of the Langevin equation for the dir ection of the spin. For slowly-changing\ndirection it takes on the form of the stochastic Landau-Lifs chitz-Gilbert equation. We give ex-\npressions for the Gilbert damping parameter and the strengt h of the fluctuations, including their\nbias-voltage dependence. At nonzero bias-voltage the fluct uations and damping are not related by\nthe fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible\nto introduce a voltage-dependent effective temperature tha t characterizes the fluctuations in the\ndirection of the spin, and its transport-steady-state prob ability distribution function.\nPACS numbers: 72.25.Pn, 72.15.Gd\nI. INTRODUCTION\nOne of the major challenges in the theoretical descrip-\ntion of various spintronics phenomena1, such as current-\ninduced magnetization reversal2,3,4,5and domain-wall\nmotion6,7,8,9,10,11,12, is their inherent nonequilibrium\ncharacter. In addition to the dynamics of the collective\ndegreeoffreedom, themagnetization, thenonequilibrium\nbehavior manifests itself in the quasi-particle degrees of\nfreedomthataredrivenoutofequilibriumbythe nonzero\nbias voltage. Due to this, the fluctuation-dissipation\ntheorem13,14cannot be applied to the quasi-particles.\nThis, in part, has led to controversysurrounding the the-\nory of current-induced domain wall motion15,16.\nEffective equations of motion for order-parameter\ndynamics that do obey the equilibrium fluctuation-\ndissipation theorem often take the form of Langevin\nequations, or their corresponding Fokker-Planck\nequations13,14,17. In the context of spintronics the rele-\nvant equation is the stochastic Landau-Lifschitz-Gilbert\nequationforthe magnetizationdirection18,19,20,21,22,23,24.\nIn this paper we derive the generalization of this equa-\ntion to the nonzero-current situation, for a simple\nmicroscopic model consisting of a single spin coupled\nto two leads via an onsite Kondo coupling. This model\nis intended as a toy-model for a magnetic impurity\nin a tunnel junction25,26,27. Alternatively, one may\nthink of a nanomagnet consisting of a collection of\nspins that are locked by strong exchange coupling.\nThe use of this simple model is primarily motivated\nby the fact that it enables us to obtain analytical\nresults. Because the microscopic starting point for\ndiscussing more realistic situations has a similar form,\nhowever, we believe that our main results apply quali-\ntatively to more complicated situations as well. Similar\nmodels have been used previously to explicitly study\nthe violation of the fluctuation-dissipation relation28, 1 1.1 1.2 1.3\n 0 0.1 0.2 0.3 0.4 0.5α/α0\n|e|V/µ\n 0 0.5 1 1.5 2 2.5 3\n 0  2  4  6  8  10Teff/T\n|e|V/(kB T)\nFIG. 1: Effective temperature as a function of bias voltage.\nThedashedlineshows thelarge bias-voltage asymptoticres ult\nkBTeff≃ |e|V/4 +kBT/2. The inset shows the bias-voltage\ndependence of the Gilbert damping parameter normalized to\nthe zero-bias result.\nand the voltage-dependence of the Gilbert damping\nparameter27. Starting from this model, we derive an\neffective stochastic equation for the dynamics of the\nspin direction using the functional-integral description\nof the Keldysh-Kadanoff-Baymnonequilibrium theory29.\n(For similar approaches to spin and magnetization\ndynamics, see also the work by Rebei and Simionato30,\nNussinov et al.31and Duine et al.32.) This formalism\nleads in a natural way to the path-integral formulation\nof stochastic differential equations33,34. One of the\nattractive features of this formalism is that dissipation\nand fluctuations enter the theory separately. This allows\nus to calculate the strength of the fluctuations even\nwhen the fluctuation-dissipation theorem is not valid.\nWe find that the dynamics of the direction of the spin\nis described by a Langevin equation with a damping ker-2\n,T\nSµL,T µR\nFIG. 2: Model system of a spin Sconnected to two tight-\nbinding model half-infinite leads. The chemical potential o f\nthe left lead is µLand different from the chemical potential\nof the right lead µR. The temperature Tof both leads is for\nsimplicity taken to be equal.\nnel and a stochastic magnetic field. We give explicit\nexpressions for the damping kernel and the correlation\nfunction of the stochastic magnetic field that are valid\nin the entire frequency domain. In general, they are not\nrelated by the fluctuation-dissipation theorem. In the\nlow-frequency limit the Langevin equation takes on the\nform ofthe stochasticLandau-Lifschitz-Gilbertequation.\nMoreover, in that limit it is always possible to introduce\nan effective temperature that characterizes the fluctua-\ntions and the equilibrium probability distribution for the\nspin direction. In Fig. 1 we present our main results,\nnamely the bias-voltage dependence of the effective tem-\nperature and the Gilbert damping parameter. We find\nthat the Gilbert damping constant initially varies lin-\nearly with the bias voltage, in agreement with the re-\nsult of Katsura et al.27. The voltage-dependence of the\nGilbert damping parameter is determined by the den-\nsity of states evaluated at an energy equal to the sum\nof the Fermi energy and the bias voltage. The effective\ntemperature is for small bias voltage equal to the actual\ntemperature, whereas for large bias voltage it is inde-\npendent of the temperature and proportional to the bias\nvoltage. This bias-dependence of the effective tempera-\nture is traced back to shot noise35.\nEffective temperatures for magnetization dynam-\nics have been introduced before on phenomenolog-\nical grounds in the context of thermally-assisted\ncurrent-driven magnetization reversal in magnetic\nnanopillars36,37,38. A current-dependent effective tem-\nperature enters in the theoretical description of these\nsystems because the current effectively lowers the energy\nbarrier thermal fluctuations have to overcome. In addi-\ntion to this effect, the presence of nonzero current alters\nthe magnetization noise due to spin current shot noise35.\nCovington et al.39interpret their experiment in terms of\ncurrent-dependent noise although this interpretation is\nstill under debate30. Foroset al.35also predict, using a\ndifferent model and different methods, a crossover from\nthermal to shot-noise dominated magnetization noise for\nincreasing bias voltage. Our main result in Fig. 1 is an\nexplicit example of this crossover for a specific model.\nThe remainderofthe paperis organizedasfollows. We\nstart in Sec. II by deriving the general Langevin equation\nforthe dynamicsofthe magneticimpurity coupledtotwo\nleads. In Sec. III and IV we discuss the low-frequency\nlimit in the absence and presence of a current, respec-\ntively. We end in Sec. V with our conclusions.II. DERIVATION OF THE LANGEVIN\nEQUATION\nWe use a model that consists of a spin Son a site\nthat is coupled via hopping to two semi-infinite leads, as\nshown in Fig. 2. The full probability distribution for the\ndirection ˆΩ of the spin on the unit sphere is written as a\ncoherent-state path integral over all electron Grassmann\nfield evolutions ψ∗(t) andψ(t), and unit-sphere paths\nS(t), that evolve from −∞totand back on the so-called\nKeldysh contour Ct. It is given by29\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]δ/bracketleftBig\n|S|2−1/bracketrightBig\nd[ψ∗]d[ψ]\n×exp/braceleftbiggi\n/planckover2pi1S[ψ∗,ψ,S]/bracerightbigg\n, (1)\nwhere the delta functional enforces the length constraint\nof the spin. In the above functional integral an inte-\ngration over boundary conditions at t=−∞, weighted\nby an appropriate initial density matrix, is implicitly in-\ncluded in the measure. We have not included boundary\nconditions on the electron fields, because, as we shall see,\nthe electron correlation functions that enter the theory\nafter integrating out the electrons are in practice conve-\nniently determined assumingthat the electronsareeither\nin equilibrium or in the transport steady state.\nThe action S[ψ∗,ψ,S] is the sum of four parts,\nS[ψ∗,ψ,S] =SL/bracketleftBig/parenleftbig\nψL/parenrightbig∗,ψL/bracketrightBig\n+SR/bracketleftBig/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+SC/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+S0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,S/bracketrightBig\n. (2)\nWe describe the leads using one-dimensional non-\ninteracting electron tight-binding models with the action\nSL/R/bracketleftBig/parenleftBig\nψL/R/parenrightBig∗\n,ψL/R/bracketrightBig\n=\n/integraldisplay\nCtdt′\n\n/summationdisplay\nj,σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\ni/planckover2pi1∂\n∂t′ψL/R\nj,σ(t′)\n+J/summationdisplay\n/an}bracketle{tj,j′/an}bracketri}ht;σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\nψL/R\nj′,σ(t′)\n\n, (3)\nwhere the sum in the second term of this action is\nover nearest neighbors only and proportional to the\nnearest-neighbor hopping amplitude Jin the two leads.\n(Throughout this paper the electron spin indices are de-\nnoted byσ,σ′∈ {↑,↓}, and the site indices by j,j′.) The\ncoupling between system and leads is determined by the\naction\nSC[/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR] =/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψL\n∂L,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψL\n∂L,σ(t′)/bracketrightBig\n+3\n/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψR\n∂R,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψR\n∂R,σ(t′)/bracketrightBig\n,\n(4)\nwhere∂L and∂R denote the end sites of the semi-infinite\nleft and right lead, and the fields/parenleftbig\nψ0(t)/parenrightbig∗andψ0(t) de-\nscribe the electrons in the single-site system. The hop-\nping amplitude between the single-site system and the\nleads is denoted by JC. Finally, the action for the sys-\ntem reads\nS0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ∗,S/bracketrightBig\n=/integraldisplay\nCtdt′\n/summationdisplay\nσ/parenleftbig\nψ0\nσ(t′)/parenrightbig∗i/planckover2pi1∂\n∂t′ψ0\nσ(t′)\n−/planckover2pi1SA(S(t′))·dS(t′)\ndt′+h·S(t′)\n+∆/summationdisplay\nσ,σ′/parenleftbig\nψ0\nσ(t′)/parenrightbig∗τσ,σ′·S(t′)ψ0\nσ′(t′)\n.(5)\nThe second term in this action is the usual Berry phase\nfor spin quantization40, withA(S) the vector potential\nof a magnetic monopole\nǫαβγ∂Aγ\n∂Sβ=Sα, (6)\nwhere a sum over repeated Greek indices α,β,γ∈\n{x,y,z}is implied throughout the paper, and ǫαβγis\nthe anti-symmetric Levi-Civita tensor. The third term in\nthe action in Eq. (5) describes the coupling of the spin to\nan external magnetic field, up to dimensionful prefactors\ngiven by h. (Note that hhas the dimensions of energy.)\nThe last term in the action models the s−dexchange\ncoupling of the spin with the spin of the conduction elec-\ntrons in the single-site system and is proportional to the\nexchange coupling constant ∆ >0. The spin of the con-\nduction electronsisrepresentedbythe vectorofthe Pauli\nmatrices that is denoted by τ.\nNext, we proceed to integrate out the electrons using\nsecond-order perturbation theory in ∆. This results in\nan effective action for the spin given by\nSeff[S] =/integraldisplay\nCtdt′/bracketleftbigg\nS/planckover2pi1A(S(t′))·dS(t′)\ndt′+h·S(t′)\n−∆2/integraldisplay\nCtdt′′Π(t′,t′′)S(t′)·S(t′′)/bracketrightbigg\n. (7)\nThis perturbation theory is valid as long as the electron\nband width is much larger than the exchange interac-\ntion with the spin, i.e., J,JC≫∆. The Keldysh quasi-\nparticleresponsefunctionisgivenintermsoftheKeldysh\nGreen’s functions by\nΠ(t,t′) =−i\n/planckover2pi1G(t,t′)G(t′,t), (8)\nwhere the Keldysh Green’s function is defined by\niG(t,t′) =/angbracketleftBig\nψ0\n↑(t)/parenleftbig\nψ0\n↑(t′)/parenrightbig∗/angbracketrightBig\n=/angbracketleftBig\nψ0\n↓(t)/parenleftbig\nψ0\n↓(t′)/parenrightbig∗/angbracketrightBig\n.(9)We willgiveexplicit expressionsforthe responsefunction\nand the Green’s function later on. For now, we will only\nmake use of the fact that a general function A(t,t′) with\nits argumentson the Keldysh contour is decomposed into\nits analytic pieces by means of\nA(t,t′) =θ(t,t′)A>(t,t′)+θ(t′,t)A<(t,t′),(10)\nwhereθ(t,t′) is the Heaviside step function on the\nKeldysh contour. There can be also a singular piece\nAδδ(t,t′), but suchageneraldecompositionisnot needed\nhere. Also needed are the advanced and retarded com-\nponents, denoted respectively by the superscript ( −) and\n(+), and defined by\nA(±)(t,t′)≡ ±θ(±(t−t′))/bracketleftbig\nA>(t,t′)−A<(t,t′)/bracketrightbig\n,(11)\nand, finally, the Keldysh component\nAK(t,t′)≡A>(t,t′)+A<(t,t′), (12)\nwhich, as we shall see, determines the strength of the\nfluctuations.\nNext we write the forward and backward paths of the\nspin on the Keldysh contour, denoted respectively by\nS(t+) andS(t−), as a classical path Ω(t) plus fluctua-\ntionsδΩ(t), by means of\nS(t±) =Ω(t)±δΩ(t)\n2. (13)\nMoreover,it turns out to be convenient to write the delta\nfunctional, which implements the length constraintofthe\nspin, as a path integral over a Lagrange multiplier Λ( t)\ndefined on the Keldysh contour. Hence we have for the\nprobability distribution in first instance that\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]d[Λ]exp/braceleftbiggi\n/planckover2pi1Seff[S]+i\n/planckover2pi1SΛ[S,Λ]/bracerightbigg\n,\n(14)\nwith\nSΛ[S,Λ] =/integraldisplay\nCtdt′Λ(t′)/bracketleftBig\n|S(t′)|2−1/bracketrightBig\n.(15)\nWe then also have to split the Lagrange multiplier into\nclassical and fluctuating parts according to\nΛ(t±) =λ(t)±δλ(t)\n2. (16)\nNote that the coordinate transformations in\nEqs. (13) and (16) have a Jacobian of one. Before\nwe proceed, we note that in principle we are required\nto expand the action up to all orders in δΩ. Also note\nthat for some forward and backward paths S(t+) and\nS(t−) on the unit sphere the classical path Ωis not\nnecessarily on the unit sphere. In order to circumvent\nthese problems we note that the Berry phase term in\nEq. (5) is proportional to the area on the unit sphere\nenclosed by the forward and backward paths. Hence, in4\nthe semi-classical limit S→ ∞27,40paths whose forward\nand backward components differ substantially will be\nsuppressed in the path integral. Therefore, we take this\nlimit from now on which allows us to expand the action\nin terms of fluctuations δΩ(t) up to quadratic order. We\nwill see that the classical path Ω(t) is now on the unit\nsphere. We note that this semi-classical approximation\nis not related to the second-order perturbation theory\nused to derive the effective action.\nSplitting the paths in classical and fluctuation parts\ngives for the probability distribution\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[λ]d[δλ]exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,δλ]/bracerightbigg\n,\n(17)\nwith the action, that is now projected on the real-time\naxis,\nS[Ω,δΩ,λ,δλ] =/integraldisplay\ndt/braceleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)\n+δλ(t)/bracketleftbig\n|Ω(t)|2−1+|δΩ(t)|2/4/bracketrightbig/bracerightbigg\n−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n−∆2\n2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftbig\nδΩα(t)ΠK(t,t′)δΩα(t′)/bracketrightbig\n. (18)\nFrom this action we observe that the integration over\nδλ(t) immediately leads to the constraint\n|Ω(t)|2= 1−|δΩ(t)|2\n4, (19)\nas expected. Implementing this constraint leads to terms\nof order O(δΩ3) or higher in the above action which we\nare allowed to neglect because of the semi-classical limit.\nFrom now on we can therefore take the path integration\noverΩ(t) on the unit sphere.\nThephysicalmeaningofthetermslinearandquadratic\ninδΩ(t) becomes clear after a so-called Hubbard-\nStratonovich transformation which amounts to rewrit-\ning the action that is quadratic in the fluctuations as\na path integral over an auxiliary field η(t). Performing\nthis transformation leads to\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[η]d[λ]\n×exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,η]/bracerightbigg\n,(20)\nwhere the path integration over Ωis now on the unit\nsphere. The action that weighs these paths is given by\nS[Ω,δΩ,λ,η] =/integraldisplay\ndt/bracketleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)+δΩα(t)ηα(t)/bracketrightbigg−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n+1\n2∆2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftBig\nηα(t)/parenleftbig\nΠK/parenrightbig−1(t,t′)ηα(t′)/bracketrightBig\n.(21)\nNote that the inverse in the last term is defined as/integraltext\ndt′′ΠK(t,t′′)/parenleftbig\nΠK/parenrightbig−1(t′′,t′) =δ(t−t′).\nPerforming now the path integral over δΩ(t), we ob-\nserve that the spin direction Ω(t) is constraint to obey\nthe Langevin equation\n/planckover2pi1SǫαβγdΩβ(t)\ndtΩγ(t) =hα+2λ(t)Ωα(t)\n+ηα(t)+/integraldisplay∞\n−∞dt′K(t,t′)Ωα(t′),(22)\nwith the so-called damping or friction kernel given by\nK(t,t′) =−∆2/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\n.(23)\nNote that the Heaviside step functions in Eq. (11) appear\nprecisely such that the Langevin equation is causal. The\nstochastic magnetic field is seen from Eq. (21) to have\nthe correlations\n∝an}bracketle{tηα(t)∝an}bracketri}ht= 0 ;\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht=iδαβ/planckover2pi1∆2ΠK(t,t′).(24)\nUsing the fact that Ω(t) is a unit vector within our semi-\nclassical approximation, the Langevin equation for the\ndirection of the spin ˆΩ(t) is written as\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×/bracketleftbigg\nh+η(t)+/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′)/bracketrightbigg\n,\n(25)\nwhich has the form of a Landau-Lifschitz equation with\na stochastic magnetic field and a damping kernel. In\nthe next sections we will see that for slowly-varying spin\ndirection we get the usual form of the Gilbert damping\nterm.\nSo far, we have not given explicit expressions for the\nresponsefunctionsΠ(±),K(t,t′). Todeterminethesefunc-\ntions, we assume that the left and right leads are in\nthermal equilibrium at chemical potentials µLandµR,\nrespectively. Although not necessary for our theoretical\napproachwe assume, for simplicity, that the temperature\nTof the two leads is the same. The Green’s functions for\nthe system are then given by41,42\n−iG<(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}N(ǫ−µk) ;\niG>(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}[1−N(ǫ−µk)] ;\nG≶,K(t−t′) =/integraldisplaydǫ\n(2π)e−iǫ(t−t′)//planckover2pi1G≶,K(ǫ),(26)\nwithN(ǫ) ={exp[ǫ/(kBT)]+1}−1the Fermi-Dirac dis-\ntribution function with kBBoltzmann’s constant, and\nA(ǫ) =i/bracketleftBig\nG(+)(ǫ)−G(−)(ǫ)/bracketrightBig\n, (27)5\nthe spectral function. Note that Eq. (26) has a particu-\nlarlysimpleformbecausewearedealingwithasingle-site\nsystem. The retarded and advanced Green’s functions\nare determined by\n/bracketleftBig\nǫ±−2/planckover2pi1Σ(±)(ǫ)/bracketrightBig\nG(±)(ǫ) = 1, (28)\nwithǫ±=ǫ±i0, and the retarded self-energy due to one\nlead follows, for a one-dimensional tight-binding model,\nas\n/planckover2pi1Σ(+)(ǫ) =−J2\nC\nJeik(ǫ)a, (29)\nwithk(ǫ) = arccos[ −ǫ/(2J)]/athe wave vector in the\nleads at energy ǫ, andathe lattice constant. The ad-\nvanced self-energy due to one lead is given by the com-\nplex conjugate of the retarded one.\nBefore proceeding we give a brief physical description\nof the above results. (More details can be found in\nRefs. [41] and [42].) They arise by adiabatically elim-\ninating (“integrating out”) the leads from the system,\nassuming that they are in equilibrium at their respective\nchemical potentials. This procedure reduces the problem\nto a single-site one, with self-energy corrections for the\non-site electron that describe the broadening of the on-\nsite spectral function from a delta function at the (bare)\non-site energy to the spectral function in Eq. (27). More-\nover, the self-energy corrections also describe the non-\nequilibrium occupation of the single site via Eq. (26)\nFor the transport steady-state we have that\nΠ(±),K(t,t′) depends only on the difference of the time\narguments. Using Eq. (8) and Eqs. (10), (11), and (12)\nwe find that the Fourier transforms are given by\nΠ(±)(ǫ)≡/integraldisplay\nd(t−t′)eiǫ(t−t′)//planckover2pi1Π(±)(t,t′)\n=/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)1\nǫ±+ǫ′−ǫ′′\n×/bracketleftbig\nG<(ǫ′)G>(ǫ′′)−G>(ǫ′)G<(ǫ′′)/bracketrightbig\n,(30)\nand\nΠK(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)δ(ǫ+ǫ′−ǫ′′)\n×/bracketleftbig\nG>(ǫ′)G<(ǫ′′)+G<(ǫ′)G>(ǫ′′)/bracketrightbig\n.(31)\nIn the next two sections we determine the spin dynamics\nin the low-frequency limit, using these expressions to-\ngether with the expressions for G≶(ǫ). We consider first\nthe equilibrium case.\nIII. EQUILIBRIUM SITUATION\nIn equilibrium the chemical potentials of the two leads\nare equal so that we have µL=µR≡µ. Combining re-\nsults from the previous section, we find for the retardedand advanced response functions (the subscript “0” de-\nnotes equilibrium quantities) that\nΠ(±)\n0(ǫ) =/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)\n×[N(ǫ′−µ)−N(ǫ′′−µ)]\nǫ±+ǫ′−ǫ′′.(32)\nThe Keldysh component of the response function is in\nequilibrium given by\nΠK\n0(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)δ(ǫ−ǫ′+ǫ′′)\n{[1−N(ǫ′−µ)]N(ǫ′′−µ)+N(ǫ′−µ)[1−N(ǫ′′−µ)]}.(33)\nThe imaginary part of the retarded and advanced re-\nsponse functions are related to the Keldysh component\nby means of\nΠK\n0(ǫ) =±2i[2NB(ǫ)+1]Im/bracketleftBig\nΠ(±)\n0(ǫ)/bracketrightBig\n,(34)\nwithNB(ǫ) ={exp[ǫ/(kBT)]−1}−1the Bose distribu-\ntion function. This is, in fact, the fluctuation-dissipation\ntheorem which relates the dissipation, determined as we\nshall see by the imaginary part of the retarded and\nadvanced components of the response function, to the\nstrength of the fluctuations, determined by the Keldysh\ncomponent.\nFor low energies, corresponding to slow dynamics, we\nhave that\nΠ(±)\n0(ǫ)≃Π(±)\n0(0)∓i\n4πA2(µ)ǫ . (35)\nWith this result the damping term in the Langevin equa-\ntion in Eq. (25) becomes\n/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′) =−/planckover2pi1∆2A2(µ)\n2πdˆΩ(t)\ndt,(36)\nwhere we have not included the energy-independent part\nofEq. (35) because it does not contribute to the equation\nof motion for ˆΩ(t). In the low-energy limit the Keldysh\ncomponent of the response function is given by\nΠK\n0(ǫ) =A2(µ)\niπkBT . (37)\nPutting all these results together we find that the dy-\nnamics of the spin direction is, as long as the two leads\nare in equilibrium at the same temperature and chemical\npotential,determinedbythestochasticLandau-Lifschitz-\nGilbert equation\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×[h+η(t)]−/planckover2pi1α0ˆΩ×dˆΩ(t)\ndt,(38)\nwith the equilibrium Gilbert damping parameter\nα0=∆2A2(µ)\n2π. (39)6\nUsing Eqs. (24), (37), and (39) we find that the strength\nof the Gaussian stochastic magnetic field is determined\nby\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht= 2α0/planckover2pi1kBTδ(t−t′)δαβ.(40)\nNote that these delta-function type noise correlations\nare derived by approximating the time dependence of\nΠK(t,t′) by a delta function in the difference of the time\nvariables. This means that the noisy magnetic field η(t)\ncorresponds to a Stratonovich stochastic process13,14,17.\nThe stationary probability distribution function gen-\nerated by the Langevin equation in Eqs. (38) and (40) is\ngiven by the Boltzmann distribution18,19,20,21,22,23,24\nP[ˆΩ,t→ ∞]∝exp/braceleftBigg\n−E(ˆΩ)\nkBT/bracerightBigg\n, (41)\nwith\nE[ˆΩ] =−h·ˆΩ, (42)\nthe energy of the spin in the external field. It turns\nout that Eq. (41) holds for any effective field h=\n−∂E[ˆΩ]/∂ˆΩ, and in particular for the case that E[ˆΩ] is\nquadratic in the components of ˆΩ as is often used to\nmodel magnetic anisotropy.\nIt is important to realize that the equilibrium prob-\nability distribution has precisely this form because of\nthe fluctuation-dissipation theorem, which ensures that\ndissipation and fluctuations cooperate to achieve ther-\nmal equilibrium13,14. Finally, it should be noted that\nthis derivation of the stochastic Landau-Lifschitz-Gilbert\nequation from a microscopic starting point circumvents\nconcerns regarding the phenomenological form of damp-\ning and fluctuation-dissipation theorem, which is subject\nof considerable debate22,23.\nIV. NONZERO BIAS VOLTAGE\nIn this section we consider the situation that the chem-\nical potential of the left lead is given by µL=µ+|e|V,\nwith|e|V >0 the bias voltage in units of energy, and\nµ=µRthe chemical potential of the right lead. Using\nthe general expressions given for the response functions\nderived in Sec. II, it is easy to see that the imaginary\npart of the retarded and advanced components of the\nresponse functions are no longer related to the Keldysh\ncomponent by means of the fluctuation-dissipation theo-\nrem in Eq. (34). See also the work by Mitra and Millis28\nfor a discussion of this point. As in the previous section,\nwe proceed to determine the low-frequency behavior of\nthe response functions.\nUsing Eqs. (26), (27), and (30) we find that the re-\ntarded and advanced components of the response func-\ntion are given by\nΠ(±)(ǫ) =∓i\n8π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\nǫ .(43)In this expression we have omitted the energy-\nindependent part andthe contribution followingfrom the\nprincipal-value part of the energy integral because, as we\nhave seen previously, these do not contribute to the final\nequation of motion for the direction of the spin. Follow-\ning the same steps as in the previous section, we find\nthat the damping kernel in the general Langevin equa-\ntion in Eq. (25) reduces to a Gilbert damping term with\na voltage-dependent damping parameter given by\nα(V) =∆2\n4π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\n≃α0/bracketleftbigg\n1+O/parenleftbigg|e|V\nµ/parenrightbigg/bracketrightbigg\n. (44)\nThis result is physically understood by noting that the\nGilbert damping is determined by the dissipative part of\nthe response function Π(+)(ǫ). In this simple model, this\ndissipative part gets contributions from processes that\ncorrespond to an electron leaving or entering the system,\nto or from the leads, respectively. The dissipative part\nis in general proportional to the density of states at the\nFermi energy. Since the Fermi energy of left and right\nlead is equal to µ+|e|Vandµ, respectively, the Gilbert\ndamping has two respective contributions corresponding\nto the two terms in Eq. (44).\nNote that the result that the Gilbert damping param-\neter initially varies linearly with the voltage is in agree-\nment with the results of Katsura et al.27, although these\nauthorsconsideraslightlydifferentmodel. Inthe insetof\nFig. 1 we show the Gilbert damping parameter as a func-\ntion of voltage. The parameters taken are ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100.\nAlthoughwe cannolongermakeuseofthefluctuation-\ndissipation theorem, we are nevertheless able to deter-\nmine the fluctuations by calculating the low-energy be-\nhavioroftheKeldyshcomponentoftheresponsefunction\nin the nonzero-voltage situation. It is given by\nΠK(ǫ) =−i\n2/integraldisplaydǫ′\n(2π)A2(ǫ′){[N(µL−ǫ′)+N(µR−ǫ′)]\n×[N(ǫ′−µL)+N(ǫ′−µR)]}. (45)\nWe define an effective temperature by means of\nkBTeff(T,V)≡iΠK(ǫ)∆2\n2α(V). (46)\nThis definition is motivated by the fact that, as we\nmention below, the spin direction obeys the stochas-\ntic Landau-Lifschitz-Gilbert equation with voltage-\ndependentdampingandfluctuationscharacterizedbythe\nabove effective temperature43. From the expression for\nα(V) and ΠK(ǫ) we see that in the limit of zero bias\nvoltage we recover the equilibrium result Teff=T. In\nthe situation that |e|Vis substantially larger than kBT,\nwhichis usuallyapproachedin experiments, wehavethat\nkBTeff(T,V)≃|e|V\n4+kBT\n2, (47)7\nwhich in the limit that |e|V≫kBTbecomes indepen-\ndent of the actual temperature of the leads. In Fig. 1\nthe effective temperature as a function of bias voltage is\nshown, using the expression for ΠK(ǫ) given in Eq. (45).\nThe parameters are the same as before, i.e., ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100. Clearly the ef-\nfective temperature changes from Teff=Tat zero bias\nvoltagetotheasymptoticexpressioninEq.(47)shownby\nthe dashed line in Fig. 1. The crossover between actual\ntemperatureandvoltageasameasureforthefluctuations\nis reminiscent of the theory of shot noise in mesoscopic\nconductors44. This is not surprising, since in the single-\nsite model we use the noise in the equation of motion ul-\ntimately arises because of fluctuations in the number of\nelectronsin thesingle-sitesystem, andisthereforeclosely\nrelated to shot noise in the current through the system.\nForoset al.35calculate the magnetization noise arising\nfrom spin currentshot noisein the limit that |e|V≫kBT\nand|e|V≪kBT. In these limits our results are similar\nto theirs.\nWith the above definition of the effective temperature\nwefind that in the nonzerobiasvoltagesituationthe spin\ndirection obeys the stochastic Landau-Lifschitz-Gilbert\nequation, identical in form to the equilibrium case in\nEqs. (38) and (40), with the Gilbert damping parame-\nter and temperature replaced according to\nα0→α(V) ;\nT→Teff(T,V). (48)\nMoreover, the transport-steady-state probability distri-\nbution for the direction of the spinis a Boltzmann distri-\nbution with the effective temperature characterizing the\nfluctuations.\nV. DISCUSSION AND CONCLUSIONS\nWe have presented a microscopic derivation of the\nstochastic Landau-Lifschitz-Gilbert equation for a semi-\nclassical single spin under bias. We found that the\nGilbert damping parameter is voltage dependent and to\nlowest order acquires a correction linear in the bias volt-\nage, in agreement with a previous study for a slightly\ndifferent model27. In addition, we have calculated the\nstrength of the fluctuations directly without using the\nfluctuation-dissipation theorem and found that, in the\nlow-frequency regime, the fluctuations are characterized\nby a voltage and temperature dependent effective tem-\nperature.\nTo arrive at these results we have performed a low\nfrequency expansion of the various correlation functions\nthat enter the theory. Such an approximation is valid as\nlong as the dynamics is much slower than the times set\nby the other energy scales in the system such as temper-\nature and the Fermi energy. Moreover, in order for the\nleads to remain in equilibrium as the spin changes direc-\ntion, the processes in the leads that lead to equilibrationhave to be much faster than the precession period of the\nmagnetizationspin. Both these criteria are satisfied in\nexperiments with magnetic materials. In principle how-\never, the full Langevinequationderivedin Sec. II alsode-\nscribes dynamics beyond this low-frequency approxima-\ntion. The introduction of the effective temperature relies\non the low-frequency approximation though, and for ar-\nbitrary frequencies such a temperature can no longer be\nuniquely defined28.\nAn effective temperature for magnetization dynam-\nics has been introduced before on phenomenological\ngrounds36,37,38. Interestingly, the phenomenological ex-\npression of Urazhdin et al.36, found by experimentally\nstudying thermal activation of current-driven magneti-\nzation reversal in magnetic trilayers, has the same form\nas our expression for the effective temperature in the\nlarge bias-voltage limit [Eq. (47)] that we derived micro-\nscopically. Zhang and Li37, and Apalkov and Visscher38,\nhave, on phenomenological grounds, also introduced an\neffective temperature to study thermally-assisted spin-\ntransfer-torque-induced magnetization switching. In\ntheir formulation, however, the effective temperature is\nproportional to the real temperature because the current\neffectively modifies the energy barrier for magnetization\nreversal.\nForoset al.35consider spin current shot noise in the\nlargebias-voltagelimit andfind forsufficiently largevolt-\nage that the magnetization noise is dominated by shot\nnoise. Moreover, they also consider the low bias-voltage\nlimit and predict a crossover for thermal to shot-noise\ndominated magnetization fluctuations. Our main result\nin Fig. 1 provides an explicit example of this crossover\nfor a simple model system obtained by methods that are\neasily generalized to more complicated models. In the\nexperiments of Krivorotov et al.45the temperature de-\npendence of the dwell time of parallel and anti-parallel\nstates of a current-driven spin valve was measured. At\nlow temperatures kBT/lessorsimilar|e|Vthe dwell times are no\nlonger well-described by a constant temperature, which\ncould be a signature of the crossover from thermal noise\nto spin current shot noise. However, Krivorotov et al.\ninterpret this effect as due to ohmic heating, which is\nnot taken into account in the model presented in this\npaper, nor in the work by Foros et al.35. Moreover, in\nrealistic materials phonons provide an additional heat\nbath for the magnetization, with an effective tempera-\nture that may depend in a completely different manner\non the bias voltage than the electron heat-bath effec-\ntive temperature. Nonetheless, we believe that spin cur-\nrent shot noise may be observable in future experiments\nand that it may become important for applications as\ntechnological progress enables further miniaturization of\nmagnetic materials. Moreover, the formalism presented\nhereisanimportantstepinunderstandingmagnetization\nnoise from a microscopic viewpoint as its generalization\nto more complicated models is in principle straightfor-\nward. Possible interesting generalizations include mak-\ning one of the leads ferromagnetic (see also Ref. [46]).8\nSince spin transfer torques will occur on the single spin\nas a spin-polarized current from the lead interacts with\nthe single-spin system, the resulting model would be a\ntoy model for microscopically studying the attenuation\nof spin transfer torques and current-driven magnetiza-\ntion reversal by shot noise. Another simple and use-\nful generalization would be enlarging the system to in-\nclude more than one spin. The formalism presented here\nwould allow for a straightforward microscopic calcula-\ntion of Gilbert damping and adiabatic and nonadiabatic\nspin transfer torques which are currently attracting a lot\nof interest in the context of current-driven domain wall\nmotion6,7,8,9,10,11,12. The application of our theory in thepresentpaperis, in additiontoitsintrinsicphysicalinter-\nest, chosen mainly because of the feasibility of analytical\nresults. Theapplicationsmentionedabovearemorecom-\nplicated and analytical results may be no longer obtain-\nable. In conclusion, we reserve extensions of the theory\npresented here for future work.\nIt is a great pleasure to thank Allan MacDonald\nfor helpful conversations. This work was supported in\npart by the National Science Foundation under grants\nDMR-0606489, DMR-0210383, and PHY99-07949. ASN\nis partially funded by Proyecto Fondecyt de Iniciacion\n11070008 and Proyecto Bicentenario de Ciencia y Tec-\nnolog´ ıa, ACT027.\n∗Electronic address: alvaro.nunez@ucv.cl;\nURL:http://www.ph.utexas.edu/ ~alnunez\n†Electronic address: duine@phys.uu.nl;\nURL:http://www.phys.uu.nl/ ~duine\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n2J.C. Slonczewski, J. Mag. Mag. Mat. 159, L1 (1996).\n3L. Berger, Phys. Rev. B 54, 9353 (1996).\n4M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck,\nV. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998).\n5E. B. Myers, D. C. Ralph, J. A. Katine, R. 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Apalkov and P. B. Visscher, Phys. Rev. B 72,\n180405(R) (2005).\n39M. Covington, M. AlHajDarwish, Y. Ding, N. J. Gokemei-\njer, and M. A. Seigler, Phys. Rev. B 69, 184406 (2004).\n40See, for example, A. Auerbach, Interacting Electrons and\nQuantum Magnetism (Springer-Verlag, New York, 1994).\n41C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James,\nJ. Phys. C: Solid State Physics 5, 21 (1972).\n42S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, 1995).\n43See also: Liliana Arrachea and Leticia F. Cugliandolo, Eu-\nrophys. Lett. 70, 642 (2005) for the introduction of an\neffective temperature in a driven electronic system.\n44M. J. M. de Jong and C. W. J. Beenakker in Mesoscopic9\nElectron Transport , edited by L.L. Sohn, L.P. Kouwen-\nhoven, and G. Schoen, NATO ASI Series Vol. 345 (Kluwer\nAcademic Publishers, Dordrecht, 1997), pp. 225-258.\n45I. N. Krivorotov, N. C. Emley, A. G. F. Garcia, J. C.Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 93, 166603 (2004) .\n46New. J. Phys. 10, 013017 (2008)."    },    {        "title": "0705.1990v1.Identification_of_the_dominant_precession_damping_mechanism_in_Fe__Co__and_Ni_by_first_principles_calculations.pdf",        "content": "arXiv:0705.1990v1  [cond-mat.str-el]  14 May 2007Identification of the dominant precession damping mechanis m in Fe, Co, and Ni by\nfirst-principles calculations\nK. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1\n1National Institute of Standards and Technology, Gaithersb urg, MD 20899-8412\n2Physics Department, Montana State University, Bozeman, MT 59717\n(Dated: October 23, 2018)\nThe Landau-Lifshitz equation reliably describes magnetiz ation dynamics using a phenomenolog-\nical treatment of damping. This paper presents first-princi ples calculations of the damping param-\neters for Fe, Co, and Ni that quantitatively agree with exist ing ferromagnetic resonance measure-\nments. This agreement establishes the dominant damping mec hanism for these systems and takes\na significant step toward predicting and tailoring the dampi ng constants of new materials.\nMagnetic damping determines the performance of\nmagnetic devices including hard drives, magnetic ran-\ndom access memories, magnetic logic devices, and mag-\nnetic field sensors. The behavior of these devices can be\nmodeled using the Landau-Lifshitz (LL) equation [1]\n˙m=−|γ|m×Heff−λ\nm2m×(m×Heff),(1)\nor the essentially equivalent Gilbert (LLG) form [2, 3].\nThe first term describes precession of the magnetization\nmabouttheeffectivefield Heffwhereγ=gµ0µB/¯histhe\ngyromagnetic ratio. The second term is a phenomeno-\nlogical treatment of damping with the adjustable rate λ.\nTheLL(G) equationadequatelydescribesdynamicsmea-\nsured by techniques as varied as ferromagnetic resonance\n(FMR) [4], magneto-optical Kerr effect [5], x-ray absorp-\ntion spectroscopy [6], and spin-current driven rotation\nwith the addition of a spin-torque term [7, 8].\nAccess to a range of damping rates in metallic mate-\nrials is desirable when constructing devices for different\napplications. Ideally, one would like the ability to de-\nsign materials with any desired damping rate. Empiri-\ncally, dopingNiFe alloyswith transitionmetals[9]orrare\nearths [10] has produced compounds with damping rates\nin the range of α= 0.01 to 0.8. A recent investigation\nof adding vanadium to iron resulted in an alloy with a\ndamping rate slightly lower than that for pure iron [11],\nthe systemwiththe lowestpreviouslyknownvalue. How-\never, the damping rate of a new material cannot be pre-\ndicted because there has not yet been a first-principles\ncalculation of damping that quantitatively agrees with\nexperiment. The challenging pursuit of new materials\nwith specific or lowered damping rates is further com-\nplicated by the expectation that, as device size contin-\nues to be scaled down, material parameters, such as λ,\nshould change [12]. A detailed understanding of the im-\nportant damping mechanisms in metallic ferromagnets\nand the ability to predictively calculate damping rates\nwould greatly facilitate the design of new materials ap-\npropriate for a variety of applications.\nThe temperature dependence of damping in the tran-\nsition metals has been carefully characterized through\nmeasurement of small angle dynamics by FMR [13].While one might na¨ ıvely expect damping to increase\nmonotonically with temperature, as it does for Fe, both\nCo and Ni also exhibit a dramatic rise in damping at low\ntemperature as the temperature decreases. These ob-\nservations indicate that two primary mechanisms are in-\nvolved. Subsequent experiments [14, 15] partition these\nnon-monotonic damping curves into a conductivity-like\nterm that decreases with temperature and a resistivity-\nliketermthatincreaseswithtemperature. Thetwoterms\nwere found to give nearly equal weight to the damping\ncurve of Ni and have temperature dependencies similar\nto those of the conductivity and resistivity, suggesting\ntwo distinct roles for electron-lattice scattering.\nThe torque-correlation model of Kambersky [16] ap-\npears to qualitatively match the data. However, like\nmost of the various models presented by Kambersky\n[16, 17, 18, 19] and others [20], it has not been quan-\ntitatively evaluated in a rigorous fashion. This has left\nthe community to speculate, based on rough estimates or\nless, astowhichdampingmechanismsareimportant. We\nresolve this matter in the present work by reporting first-\nprinciples calculations of the Landau-Lifshitz damping\nconstant according to Kambersky’s torque-correlation\nexpression. Quantitative comparison of the present cal-\nculations to the measured FMR values [13] positively\nidentifies this damping pathway as the dominant effect\nin the transition metal systems. In addition to present-\ning these primary conclusions, we also describe the re-\nlationship between the torque-correlation model and the\nmore widely understood breathing Fermi surface model\n[18, 21], showing that the results of both models agree\nquantitatively in the low scattering rate limit.\nThe breathing Fermi surface model of Kambersky pre-\ndicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn/integraldisplaydk3\n(2π)3η(ǫn,k)/parenleftbigg∂ǫn,k\n∂θ/parenrightbigg2τ\n¯h.(2)\nThis model offers a qualitative explanation for the low\ntemperature conductivity-like contribution to the mea-\nsureddamping. The modeldescribesdamping ofuniform\nprecession as due to variations ∂ǫn,k/∂θin the energies\nǫn,kof the single-particle states with respect to the spin2\ndirection θ. The states are labeled with a wavevector\nkand band index n. As the magnetization precesses,\nthespin-orbitinteractionchangestheenergyofelectronic\nstatespushingsomeoccupiedstatesabovetheFermilevel\nand some unoccupied states below the Fermi level. Thus,\nelectron-hole pairs are generated near the Fermi level\neven in the absence of changes in the electronic popula-\ntions. The ηfunction in Eq. (2) is the negative derivative\nof the Fermi function and picks out only states near the\nFermi level to contribute to the damping. gis the Land´ e\ng-factor and µBis the Bohr magneton. The electron-hole\npairs created by the precession exist for some lifetime τ\nbeforerelaxingthroughlattice scattering. The amountof\nenergy and angular momentum dissipated to the lattice\ndepends on how far from equilibrium the system gets,\nthus damping by this mechanism increases linearly with\nthe electron lifetime as seen in Eq.2. Since the electron\nlifetime is expected to decrease as the temperature in-\ncreases, this model predicts that damping diminishes as\nthe temperature is raised.\nBecause the predicted damping rate is linear in the\nscattering time the damping rate cannot be calculated\nmore accurately than the scattering time is known. For\nthis reason it is not possible to make quantitative com-\nparisonsbetween calculationsof the breathing Fermi sur-\nface and measurements. Further, while the breathing\nFermi surface model can explain the dramatic temper-\nature dependence observed in the conductivity-like por-\ntion of the data it fails to capture the physics driving the\nresistivity-like term. This is a significant limitation from\na practical perspective because the resistivity-like term\ndominates damping at room temperature and above and\nis the only contribution observed in iron [13] and NiFe\nalloys [22]. For these reasons it is necessary to turn to\nmore complete models of damping.\nKambersky’s torque-correlation model predicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn,m/integraldisplaydk3\n(2π)3/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k) (3)\nand we will show that it both incorporates the physics of\nthe breathing Fermi surface model and also accounts for\ntheresistivity-like terms. The matrix elements Γ−\nnm(k) =\n/angbracketleftn,k|[σ−, Hso]|m,k/angbracketrightmeasure transitions between states\nin bands nandminduced by the spin-orbit torque.\nThese transitionsconservewavevector kbecausethey de-\nscribe the annihilation of a uniform precession magnon,\nwhich carries no linear momentum. The nature of these\nscattering events, which are weighted by the spectral\noverlapWnm(k) = (1/π)/integraltext\ndω1η(ω1)Ank(ω1)Amk(ω1),\nwill be discussedin moredetail below. The electronspec-\ntral functions Ankare Lorentzians centered around the\nband energies ǫnkand broadenedbyinteractionswith the\nlattice. The width ofthe spectralfunction ¯ h/τprovidesa\nphenomenological account for the role of electron-lattice\nscattering in the damping process. The ηfunction is theh/τ(eV)\n108109λ (1/s)\n0.001α0.001 0.01 0.1 1\nFe\n1081091010λ (1/s)\n101310141015\n 1/τ (1/s)0.0010.010.1αNi108109λ (1/s)\n0.0010.01αCo\nFIG. 1: Calculated Landau-Lifshitz damping constant for Fe ,\nCo, and Ni. Thick solid curves give the total damping param-\neter while dotted curves give the intraband and dashed lines\nthe interband contributions. The top axis is the full-width -\nhalf-maximum of the electron spectral functions.\nsame as in Eq. (2) and enforces the requirement of spec-\ntral overlap at the Fermi level.\nEquation (3) captures two different types of scatter-\ning events: scattering within a single band, m=n, for\nwhich the initial and final states are the same, and scat-\ntering between two different bands, m/negationslash=n. As explained\nin [16] the overlap of the spectral functions is propor-\ntional (inverse) to the electron scattering time for intra-\nband (interband) scattering. From this observation the\nqualitative conclusion is made that the intraband contri-\nbutions matchthe conductivity-like terms while the inter-\nband contributions give the resistivity-like terms. While\nthis seems promising, evaluation of Eq. (3) is more com-\nputationally intensive than that of the breathing Fermi\nsurface model and until now only a few estimates for Ni\nand Fe have been made [19].3\nTABLE I: Calculated and measured [13] damping parameters. V alues for λ, the Landau-Lifshitz form, are reported in 109s−1,\nvalues of α, the Gilbert form, are dimensionless. The last two columns l ist calculated damping due to the intraband contribution\nfrom Eq. (3) and from the breathing Fermi surface model [12], respectively. Values for λ/τare given in 1022s−2. Published\nnumbers from [13] and [12] have been multiplied by 4 πto convert from the cgs unit system to SI.\nαcalcλcalcλmeasλcalc/λmeas(λ/τ)intra(λ/τ)BFS\nbcc Fe/angbracketleft001/angbracketright0.0013 0.54 0.88 0.61 1.01 0.968\nbcc Fe/angbracketleft111/angbracketright0.0013 0.54 – – 1.35 1.29\nhcp Co/angbracketleft0001/angbracketright0.0011 0.37 0.9 0.41 0.786 0.704\nfcc Ni/angbracketleft111/angbracketright0.017 2.1 2.9 0.72 6.67 6.66\nfcc Ni/angbracketleft001/angbracketright0.018 2.2 – – 8.61 8.42\nWe have performed first-principles calculations of the\ntorque-correlation model Eq. (3) with realistic band\nstructures for Fe, Co, and Ni. Prior to evaluating Eq. (3)\nthe eigenstates and energies of each metal were found us-\ning the linear augmented plane wave method [23] in the\nlocal spin density approximation (LSDA) [24, 25, 26].\nDetails of the calculations for these materials are de-\nscribed in [27]. The exchange field was fixed in the cho-\nsen equilibrium magnetization direction. Calculations of\nEq. (3) presented in this paper are converged to within\na standard deviation of 3 %, which required sampling\n(160)3k-points for Fe, (120)3for Ni, and (100)2k-points\ninthe basalplaneby57alongthe c-axisforCo. Electron-\nlattice interactions were treated phenomenologically as a\nbroadeningofthe spectralfunctions. The Fermi distribu-\ntion was smeared with an artificial temperature. Results\ndid not vary significantly with reasonable choices of this\ntemperature since the broadening of the Fermi distribu-\ntion was considerably less than that of the bands. The\ndamping rate was calculated for a range of scattering\nrates (spectral widths) just as damping has been mea-\nsured over a range of temperatures.\nThe results ofthese calculations arepresented in Fig. 1\nand are decomposed into the intraband and interband\nterms. The downward sloping line in Fig. 1 represents\nthe intraband contribution to damping. Damping con-\nstants were recently calculated using the breathing Fermi\nsurface model [12, 21] by evaluating the derivative of\nthe electronic energy with respect to the spin direc-\ntion according to Eq. (2). The results of the breathing\nFermi surface prediction are indistinguishable from the\nintraband terms of the present calculation even though\nthe computational approaches differed significantly; the\nagreement is quantified in Table I.\nThe breathing Fermi surface model could not be quan-\ntitatively compared to the experimental results because\nthe temperature dependence of the scattering rate has\nnot been determined sufficiently accurately. While the\npresent calculations also require knowledge of the scat-\ntering rate to determine the damping rate the non-monotonic dependence of damping on the scattering rate\nproduces a unique minimum damping rate. In the same\nmanner that the calculated curves of Fig. 1 have a mini-\nmumwithrespecttoscatteringrate,themeasureddamp-\ning curves exhibit minima with respect to temperature.\nWhatever the relation between temperature and scatter-\ning rate, the calculated minima may be compared di-\nrectly and quantitatively to the measured minima. Ta-\nble I makes this comparison. The agreement between\nmeasured and calculated values shows that the torque-\ncorrelationmodelaccountsforthedominantcontribution\nto damping in these systems.\nOur calculated values are smaller than the measured\nvalues. Using measured gvalues instead of setting g=\n2 would increase our results by a factor of ( g/2)2, or\nabout 10 % for Fe and 20 % for Co and Ni. Other pos-\nsible reasons for the difference include a simplified treat-\nment ofelectron-latticescatteringin which the scattering\nrates for all states were assumed equal, the mean-field\napproximation for the exchange interaction, errors asso-\nciatedwith thelocalspindensityapproximation(LSDA),\nand numerical convergence (discussed below). Other\ndamping mechanisms may also make small contributions\n[28, 29, 30].\nSince the manipulations involved with the equation of\nmotion techniques employed in deriving Eq. (3) obscure\nthe underlying physics we now discuss the two scatter-\ning processes and connect the intraband terms to the\nbreathing Fermi surface model. The intraband terms in\nEq. (3) describe scattering from one state to itself by\nthe torque operator, which is similar to a spin-flip oper-\nator. A spin-flip operation between some state and itself\nis only non-zero because the spin-orbit interaction mixes\nsmall amounts of the opposite spin direction into each\nstate. Since the initial and final states are the same, the\noperation is naturally spin conserving. The matrix ele-\nments do not describe a real transition, but rather pro-\nvide a measure of the energy of the electron-hole pairs\nthat are generated as the spin direction changes. The\nelectron-hole pairs are subsequently annihilated by a real4\nelectron-lattice scattering event.\nTo connect the derivatives ∂ǫ/∂θin Eq. (2) and the\ntorque matrix elements in Eq. (3) we imagine first point-\ning the magnetization in some direction ˆ z. The only\nenergy that changes with the magnetization direction is\nthe spin-orbit energy Hso. As the spin of a single parti-\ncle state |/angbracketrightrotates along ˆθabout ˆxits spin-orbit energy\nis given by ǫ(θ) =/angbracketleft|eiσxθHsoe−iσxθ|/angbracketright. The derivative\nwith respect to θis∂ǫ(θ)/∂θ=i/angbracketleft|eiσxθ[σx, Hso]e−iσxθ|/angbracketright.\nEvaluating this derivative at the pole ( θ= 0) gives\n∂ǫ/∂θ=i/angbracketleft|[σx, Hso]|/angbracketright. Similarly, rotating the spin along\nˆθabout ˆyleads to ∂ǫ/∂θ=i/angbracketleft|[σy, Hso]|/angbracketright. The torque\nmatrix elements in Eq. (3) are Γ−=/angbracketleft|[σ−, Hso]|/angbracketright=\n/angbracketleft|[σx, Hso]|/angbracketright−i/angbracketleft|[σy, Hso]|/angbracketright. Using the relations between\nthe commutators and derivatives just found the torque is\nΓ−=−i(∂ǫ/∂θ)x−(∂ǫ/∂θ)ywhere the subscripts in-\ndicate the rotation axis. Squaring the torque matrix\nelements gives |Γ−|2= (∂ǫ/∂θ)2\nx+ (∂ǫ/∂θ)2\ny. For high\nsymmetry directions ( ∂ǫ/∂θ)x= (∂ǫ/∂θ)yand we de-\nduce|Γ−|2= 2(∂ǫ/∂θ)2demonstrating that the intra-\nband terms of the torque-correlation model describe the\nsame physics as the breathing Fermi surface.\nThe monotonically increasing curves in Fig. 1 indi-\ncate the interband contribution to damping. Uniform\nmode magnons, which have negligible energy, may in-\nduce quasi-elastic transitions between states with differ-\nent energies. This occurs when lattice scattering broad-\nens bands sufficiently so that they overlap at the Fermi\nlevel. Thesewavevectorconservingtransitions, whichare\ndriven by the precessing exchange field, occur primarily\nbetween states with significantly different spin character.\nThe process may roughly be thought of as the decay of\na uniform precession magnon into a single electron spin-\nflip excitation. These events occur more frequently as\nthe band overlaps increase. For this reason the interband\nterms, which qualitatively match the resistivity-like con-\ntributions in the experimental data, dominate damping\nat room temperature and above.\nWe have calculated the Landau-Lifshitz damping pa-\nrameterfortheitinerantferromagnetsFe, Co, andNiasa\nfunction ofthe electron-latticescatteringrate. Theintra-\nband and interband components match qualitatively to\nconductivity- andresistivity-like terms observed in FMR\nmeasurements. A quantitative comparison was made be-\ntweentheminimaldampingratescalculatedasafunction\nof scattering rate and measured with respect to temper-\nature. This comparison demonstrates that our calcula-\ntions account for the dominant contribution to damping\nin these systemsand identify the primarydamping mech-\nanism. At room temperature and above damping occurs\noverwhelmingly through the interband transitions. The\ncontribution of these terms depends in part on the band\ngap spectrum around the Fermi level, which could be\nadjusted through doping.\nK.G. and Y.U.I. acknowledge the support of the Officeof Naval Research through grant N00014-03-1-0692 and\nthroughgrantN00014-06-1-1016. We wouldlike tothank\nR.D. McMichael and T.J. Silva for valuable discussions.\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] D. Twisselmann and R. McMichael, J. Appl. Phys. 93,\n6903 (2003).\n[5] T. Gerrits, J. Hohlfeld, O. Gielkens, K. Veenstra, K. Bal ,\nT. Rasing, and H. van den Berg, J. Appl. Phys. 89, 7648\n(2001).\n[6] W. Bailey, L. Cheng, D. Keavney, C. Kao, E. Vescovo,\nand D. Arena, Phys. Rev. B 70, 172403 (2004).\n[7] I. Krivorotov, D. Berkov, N. Gorn, N. Emley, J. Sankey,\nD. Ralph, and R. Buhrman, Phys. Rev. B (2007).\n[8] M. Stiles and J. Miltat, Spin dynamics in confined mag-\nnetic structures III (Springer, Berlin, 2006).\n[9] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro,\nJ. W.F. Egelhoff, B. Maranville, D.Pulugurtha, A. Chen,\nand L. Conners, J. Appl. Phys. 101, 033911 (2007).\n[10] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTrans. Mag. 37, 1749 (2001).\n[11] C. Scheck,L.Cheng, I.Barsukov, Z.Frait, andW.Bailey ,\nPhys. Rev. Lett. 98, 117601 (2007).\n[12] D. Steiauf and M. Faehnle, Phys. Rev. B 72, 064450\n(2005).\n[13] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n[14] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys.\n50, 7726 (1979).\n[15] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16,\n660 (1980).\n[16] V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n[17] B. Heinrich, D. Fraitova, and V. Kambersky,\nPhys. Stat. Sol. 23, 501 (1967).\n[18] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[19] V. Kambersky, Czech. J. Phys. B 34, 1111 (1984).\n[20] V. Korenman and R. Prange, Phys. Rev. B 6, 2769\n(1972).\n[21] J. Kunes and V. Kambersky, Phys. Rev. B 65, 212411\n(2002).\n[22] S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczews ki,\nP. Trouilloud, and R. Koch, Phys. Rev. B 66, 214416\n(2002).\n[23] L. Mattheiss and D. Hamann, Phys. Rev. B 33, 823\n(1986).\n[24] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[25] W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965).\n[26] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).\n[27] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[28] R. McMichael and A. Kunz, J. Appl. Phys. 91, 8650\n(2002).\n[29] E. Rossi, O. G. Heinonen, and A. H. MacDonald,\nPhys. Rev. B 72, 174412 (2005).\n[30] Y. Tserkovnyak, G. Fiete, and B. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004)."    },    {        "title": "0706.0529v1.Generation_of_microwave_radiation_in_planar_spin_transfer_devices.pdf",        "content": "arXiv:0706.0529v1  [cond-mat.mtrl-sci]  4 Jun 2007Generation of microwave radiation in planar spin-transfer devices\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: May, 2006)\nCurrentinducedprecession states inspin-transfer device sarestudiedinthecaseoflarge easyplane\nanisotropy (present in most experimental setups). It is sho wn that the effective one-dimensional pla-\nnar description provides a simple qualitative understandi ng of the emergence and evolution of such\nstates. Switching boundaries are found analytically for th e collinear device and the spin-flip tran-\nsistor. The latter can generate microwave oscillations at z ero external magnetic field without either\nspecial functional form of spin-transfer torque, or “field- like” terms, if Gilbert constant corresponds\nto the overdamped planar regime.\nPACS numbers: 85.75.-d, 75.40.Gb, 72.25.Ba, 72.25.Mk\nSpin-polarized currents are able to change the mag-\nnetic configuration of nanostructures through the spin-\ntransfer effect proposed more than a decade ago [1, 2].\nIntensive research is currently directed at understanding\nthe basic physics of this non-equilibrium interaction and\ndesigning magnetic nanodevices with all-electric control.\nInitial spin-transfer experiments emphasized the cur-\nrent induced switching between two static configurations\n[3]. Presently, the research focus is broadening to in-\nclude the states with continuous magnetization preces-\nsion powered by the energy of the current source [2, 4, 5].\nSpin-transfer devices with precession states (PS) serve as\nnano-generators of microwave oscillations with remark-\nable properties, e.g. current tunable frequency and ex-\ntremelynarrowlinewidth [6, 7, 8, 9]. Aparticularissueof\ntechnological importance is the search for systems sup-\nporting PS at zero magnetic field. Here several strate-\ngies are pursued: (i) engineering unusual angle depen-\ndence of spin-transfer torque [10, 11, 12], (ii) relying on\nthe presence of the “field-like” component of the spin\ntorque [13], (iii) choosing the “magnetic fan” geometry\n[14, 15, 16, 17].\nPS are more difficult to describe then the fixed equi-\nlibria: the amplitude of precession can be large and non-\nlinear effects are strong. As a result, information about\nthem if often obtained from numeric simulations. Here\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.westudy PS in planardevices [18] usingthe effective one-\ndimensional approximation [19, 20, 21] which is relevant\nfor the majority of experimental setups. It is shown that\nplanarapproximationprovidesaveryintuitivepictureal-\nlowing to predict the emergence of precession and subse-\nquent transformations between different types of PS. We\nshow that PS in devices with in-plane spin polarization\nof the current can exist at zero magnetic field without\nthe unusual properties (i),(ii) of the spin-transfer torque.\nA conventional spin-transfer device with a fixed polar-\nizerandafreelayer(Fig. 1)isconsidered. Themacrospin\nmagnetization of the free layer M=Mnhas a constant\nabsolute value Mand a direction given by a unit vector\nn(t). The LLG equation [2, 5] 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,αis the Gilbert damping constant, sis the spin-\npolarizer unit vector. The spin transfer strength u(n) is\nproportional to the electric current I[5, 20]. In general,\nit is a function of the angle between the polarizer and the\nfree layer u(n) =f[(n·s)]I, with the function f[(n·s)]\nbeing material and device specific [22, 23, 24].\nThe LLG equation can be written in terms of the polar\nangles(θ,φ) ofvector n. Planardevicesarecharacterized\nby the energy form E= (K⊥/2)cos2θ+Er(θ,φ) with\nK⊥≫ |Er|. The first term provides the dominating easy\nplane anisotropy and ensures that the low energy motion\nhappens close to the θ=π/2 plane. The residual energy\nErhas an arbitrary form. The smallness of δθ=θ(t)−\nπ/2 allows to derive a single effective equation on the\nin-plane angle φ(t) by performing the expansion in small\nparameter |Er|/K⊥[21]. For time-independent current\nand polarizer direction sone obtains:\n1\nω⊥¨φ+αeff˙φ=−γ\nM∂Eeff\n∂φ, (2)\nwhereω⊥=γK⊥/M. General expressions for αeff(φ)2Eeff\n0 π −π−φ m φm(1)(2)(3)(4)\n0 π −π(1)(2)(4)\nFIG.2: (Color online)Evolutionofeffectiveenergyprofilea nd\nstablesolutions withspin-transferstrength(graphsares hifted\nup asubecomes more negative) for a device with collinear\npolarizer. Left: low-field 0 < h <˜ω||regime. Right: high-\nfieldh >˜ω||regime. Evolutionstage(3)ismissinginthehigh-\nfieldregime duetotheabsenceofthesecondenergyminimum.\nThe red parts of the energy graphs mark the αeff<0 regions.\nFilled and empty circle gives represent the effective partic le.\nandEeff(φ) for arbitrary function Er(θ,φ) and polar-\nizer direction sare given in Ref. 21. In a special case\nfrequently found in practice the polarizer sis directed\nin the easy plane at the angle φs, and the residual en-\nergy satisfies ( ∂Er/∂θ)θ=π/2= 0, i.e. does not shift the\nenergy minima away from the plane. We will also use\nthe simplest form f[(n·s)] = const for the spin transfer\nstrength. A more realistic function can be employed if\nneeded. With these assumptions [21]:\nαeff=α+2ucos(φs−φ)\nω⊥, (3)\nEeff=Er(π/2,φ)−Mu2\n2γω⊥cos2(φs−φ).\nEquation (2) has the form of Newton’s equation of mo-\ntion for a particle in external potential Eeff(φ) with a\nvariable viscous friction coefficient αeff(φ). The advan-\ntage of such a description is that the motion of the effec-\ntive particle can be qualitatively understood by applying\nthe usualenergyconservationand dissipation arguments.\nIn the absence of current, the effective friction is a posi-\ntive constant, so after an initial transient motion the sys-\ntem always ends up in one of the minima of Er(π/2,φ).\nWhen current is present, effective friction and energy are\nmodified. Such a modification reflects the physical pos-\nsibility of extracting energy from the current source, and\nleads to the emergence of the qualitatively new dynamic\nregime of persistent oscillations. These oscillations of φ\ncorrespond to the motion of nalong the highly elongated\n(δθ≪1) closed orbits (see examples in Fig. 3, inset),\ni.e. constitute the limiting form of the precession states[2, 5, 7, 25] in spin-transfer systems.\nTo illustrate the advantages of the effective parti-\ncle description, consider a specific example of PS in\nthe nanopillar experiment [7] where Eris an easy axis\nanisotropy energy with magnetic field Hdirected along\nthat axis, Er(φ) = (K||/2)sin2φ−HMcosφ. The po-\nlarizersis directed along the same axis with φs= 0\n(collinear polarizer). With the definitions ω||=γKa/M,\nh=γH, the effective energy becomes [21]\nγ\nMEeff=˜ω||(u)\n2sin2φ−hcosφ , (4)\nwith ˜ω||=ω||+u2/ω⊥. Effective energy profiles are\nshown in Fig. 2. For low fields, |h|<˜ω||(u), the minima\natφ= 0,πare separated by maxima at ±φm(h).\nAccording to Eq. (3), the effective friction can become\nnegative at φ= 0 orφ=πat the critical value of spin-\ntransfer strength |u|=u1=αω⊥/2. If this value is\nexceeded, the position of the system in the energy min-\nimum becomes unstable. Indeed, the stability of any\nequilibrium in one dimension depends on whether it is\na minimum or a maximum of Eeffand on the sign of\nαeffat the equilibrium point. Out of four possible com-\nbinations, only an energy minimum with αeff>0 is\nstable. A little above the threshold, αeffis negative in a\nsmall vicinity of the minimum where the system in now\ncharacterized by negative dissipation. In this situation\nany small fluctuation away from the equilibrium initiates\ngrowing oscillations. As the oscillations amplitude ex-\nceeds the size of the αeff<0 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 stable cycle solution emerges (Fig. 2, profile (2)).\nThe requirement of zero total dissipation means\nthat an integral over the oscillation period satisfies/integraltext\nαeff(φ)(˙φ)2dt= 0. In typical collinear systems [25]\nGilbert damping satisfies α≈0.01≪/radicalbig\nω||/ω⊥≈0.1≪\n1, hence the oscillator (2),(4) operates in the lightly\ndamped regime. In zeroth order approximation the fric-\ntion term in (2) can be neglected, and a first integral\n˙φ2/(2ω⊥)+Eeff=E0exists. Zero dissipation condition\ncan be then approximated by\n/integraldisplayφ2\nφ1αeff(φ)/radicalBig\nE0−Eeff(φ)dφ= 0,(5)\nwithφ1,2(u) being the turning points of the effective par-\nticle trajectory, and E0=Eeff(φ1) =Eeff(φ2). Since\nthe integrand of (5) is a known function, the formula\nprovides an expression for the precession amplitude.\nConsider now the low positive field regime 0 < h <˜ω||.\nAtu=−u1the parallel configuration becomes unstable\nand a cycle emerges near the φ= 0 minimum. As u\nis made more negative, the oscillation amplitude grows3\nuntil eventually it reaches the point of energy maximum\natu=−u2. Equivalently, the effective particle starting\nat the energy maximum −φmis able to reach the other\nmaximum at + φm(Fig. 2, left, (3)). Above this thresh-\nold the particleinevitably goes overthe potential hill and\nfalls into the φ=πminimum which remains stable since\nαeff(π)>0 holds for negative u. In other words, the cy-\ncle solutionwith oscillationsaroung φ= 0 ceasestoexist.\nAt even more negative uthe third threshold is reached\nwhen the effective particle can complete the full rotation\nstarting from the energy maximum (Fig. 2, profile (4)).\nBelowu=−u3a new PS with full rotation emerges. In\nthe high-field regime h >˜ω||the evolution of the pre-\ncession cycle is similar (Fig. 2, right), but stage (3) is\nmissing since there is no second minimum. The thresh-\noldu=−u2separates the finite oscillations regime and\nthe full-rotation regime.\nThresholds uican be obtained analytically from (5) by\nsubstituting the critical turning points φ1,2listed above:\nu2=αω⊥hφm+ω||sinφm\nω||φm+hsinφm(h < ω||), (6)\nu2=αω⊥h\nω||(h > ω||), (7)\nu3=αω⊥h(φm−π/2)+ω||sinφm\nω||(φm−π/2)+hsinφm(h < ω||).(8)\nThe corresponding switching diagram is shown in Fig. 3\n(cf. numerically obtained Fig. 12 in Ref. 25). It shows\nthat different hysteresis patterns are possible depending\non the trajectory in the parameter space.\nPS in the low field regime was discussed analytically in\nan unpublished work [26]. However, since a conventional\ndescription with two polar angles was used, the calcula-\ntions were much less transparent. Numeric studies of the\nPS were performed in Refs. [7, 25] after the experimental\nobservation [7] of the current induced transition between\ntwo PS in the high field regime. They had shown that\nindeed the low-current precession state PS 1has a finite\namplitude of φ-oscillations, while the high-current state\nPS2exhibits full rotations of φ(Fig. 3, inset).\nNext, we consider the cycle solutions in a device called\na spin-flip transistor [18, 27]. It differs form the setup\nstudied above in the polarizer direction, which is now\nperpendicular to the easy axis with φs=π/2. No exter-\nnal magnetic field is applied. In this case [21]\nαeff=α+2usinφ\nω⊥, (9)\nγ\nMEeff=¯ω||(u)\n2sin2φ , (10)\nwith ¯ω||=ω||−u2/ω⊥. As the spin-transfer strength\ngrows, the behavior of the system changes qualitatively\nwhen ¯ω||orαeff|±π/2changesignsatthethresholds ¯ u1=\n±√ω||ω⊥and ¯u2=±αω⊥/2. In accord with previous in-uPS1s u1 u2\nu3PS2 PS1PS2h\nω\n-ωω (u)∼\n||\nz\nnφ\nxPS1PS2\nFIG. 3: (Color online) Switching diagram of a device with\ncollinear polarizer. The u-axis direction is reversed for the\npurpose of comparison with Refs. 7, 25. The parts of the\ndiagram not shown can be recovered by a 180-degree rotation\nof the picture. Stable directions in each region are given by\nsmall arrows, the precession states are marked as PS 1,2. The\nlarge arrow shows the polarizer direction. Inset: schemati c\ntrajectories of the PS 1,2states on the unit sphere.\nvestigations [16, 28] at |u|>¯u1theφ= 0,πenergy min-\nima are destabilized and the parallel state φ= sgn[u]φs\nbecomes stable. Surprisingly, for α > α ∗= 2/radicalbig\nω||/ω⊥a\nwindow ¯u1< u <¯u2of stability of antiparallel configu-\nration,φ=−sgn[u]φs, opens (Fig. 4). As discussed in\nRef. [21], the stabilization of the antiparallel state hap-\npens as the spin-transfer torque is increased in spite of\nthe fact that this torque repels the system from that di-\nrection. At u= ¯u2theantiparallelstateturnsintoacycle\n(Fig. 4, low right panel) which we will study here. Above\nthe ¯u2threshold the amplitude of oscillations grows until\nthey reach the energy maximum at u= ¯u3and the cy-\ncle solution disappears. Although αis not small, ¯ u3can\nstill be determined from Eq. (5) because αeffis small\nwhenuis close to u2. Calculating the integral in (5)\nwithφ1,2=−π,0 we get\n¯u3=2\nπαω⊥≈1.27¯u2. (11)\nThe usage of approximations (5),(11) is legitimate for\nα>∼2α∗whereαeff(¯u3)≪/radicalbig¯ω||(¯u3)/ω⊥holds. For\nsmaller values of αnumeric calculations are required.\nThey show the existence of a stable cycle down to α=\n0.8α∗where the stabilization of the antiparallel state is\nimpossible. For α≪α∗andu>∼¯u1the strong negative\ndissipation regime is realized, |αeff| ≫/radicalbig¯ω||/ω⊥. Nu-\nmeric results show that the amplitude of the oscillations4\nα*u\nαEeff\n0+π/2 π −π −π/2 0+π/2 π −π −π/2PS1s\nu1u2u3\nFIG. 4: (Color online) Switching diagram of a spin-flip tran-\nsistor. The u <0 part of the diagram can be obtained by\nreflection with respect to the horizontal axis. In each regio n\nstable directions are given by small arrows, precession sta te\nis marked by PS 1. The large arrow shows the polarizer di-\nrection. Threshold ¯ u3(α) is sketched as a dashed line where\napproximation (11) is not valid. Lower panels: the evolutio n\nof effective energy and trajectories (graphs are shifted up w ith\ngrowing u) atα << α ∗(left) and α > α ∗(right). The red\npart of the energy graph marks the αeff<0 region. Effective\nparticle is shown by filled and empty circles.\ninduced bynegativedissipationissobigthat theeffective\nparticle always reaches the energy maximum and drops\ninto the stable parallel state (Fig 4, low left panel). We\nconclude that the line ¯ u3(α) crosses the u= ¯u1line at\nsome point and terminates there.\nAs for the full-rotation PS, one can show analytically\nthat it does not exists in the small dissipation limit at\nα>∼2α∗. Numerical simulations do not find it in the\nα < α∗,u >¯u1regime either.\nIn conclusion, we have shown that the planar effective\ndescription can be very useful for studying precession so-\nlutions in the spin transfer systems. It was already used\nto describe the “magnetic fan” device with current spin\npolarization perpendicular to the easy plane [20]. Here\nthe switching diagrams were obtained for the spin polar-\nizers directed collinearly and perpendicular to the easy\ndirection within the plane. In collinear case we found an-\nalytic formulas for the earlier numeric results, while the\nstudy of precession solutions in the perpendicular case\n(spin-flip transistor) at large damping is new. The lat-\nter shows the possibility of generating microwave oscil-lations in the absence of external magnetic field with-\nout the need to engineer special angle dependence of the\nspin-transfer torque or “field-like” terms. The inequality\nα >2/radicalbig\nω||/ω⊥required for the existence of such oscil-\nlations can be satisfied by either reducing the in-plane\nanisotropy, or increasing αdue to spin-pumping effect\n[29]. Most importantly, the effective planar description\nallows for qualitative understanding of the precession cy-\ncles and makes it easy to predict their emergence, sub-\nsequent evolution, and transitions between different pre-\ncession cycle types. E.g., in the systems with one re-\ngion of negative effective dissipation, such as considered\nhere, it shows that no more then two precession states,\none with finite oscillations and another with full rota-\ntions, can exist. Numerical approaches, if needed, are\nthen based on a firm qualitative foundation. In addition,\nnumerical calculations in one dimension are easier then\nin the conventional description with two polar angles.\nThe authorthanks G. E.W. Bauerand M.D. Stiles for\ndiscussion. Research at Leiden University was supported\nby the Dutch Science Foundation NWO/FOM. Part of\nthis work was performed at Aspen Center for Physics.\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] J. A. Katine et al., Phys. Rev. Lett., 84, 3149 (2000).\n[4] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[5] Ya. B. Bazaliy et al., arXiv:cond-mat/0009034 (2000);\nPhys. Rev. B, 69, 094421 (2004).\n[6] M. Tsoi et al., Nature, 406, 46 (2000).\n[7] S. I. Kiselev et al., Nature, 425, 380 (2003).\n[8] S. Kaka et al., Nature 437, 389 (2005);\n[9] M. R. Pufall et al., Phys.Rev. Lett. 97, 087206 (2006);\n[10] J. Manschot et al., Appl. Phys. Lett. 85, 3250 (2004).\n[11] M. Gmitra et al, Phys. Rev. Lett., 96, 207205 (2006).\n[12] O. Boulle et al., Nature Physics, May 2007.\n[13] T. Devolder et al., J. Appl. Phys., 101, 063916 (2007)\n[14] A. D. Kent et al., Appl. Phys. Lett., 84, 3897 (2004)\n[15] K. J. Lee et al., Appl. Phys. Lett., 86, 022505 (2005).\n[16] X. Wang et al., Phys. Rev. B, 73, 054436 (2006).\n[17] D. Houssameddine et al., Nature Materials, April 2007.\n[18] A. Brataas et al., Phys. Rep., 427, 157 (2006).\n[19] C. Garcia-Cervera et al., J. Appl. Phys., 90, 370 (2001).\n[20] Ya. B. Bazaliy et al., arXiv:0705.0406v1 (2007), to be\npublished in J. Nanoscience and Nanotechnology.\n[21] Ya. B. Bazaliy, arXiv:0705.0508 (2007).\n[22] J. C. Slonczewski, JMMM, 247, 324 (2002).\n[23] A. A. Kovalev et al., Phys. Rev. B, 66, 224424 (2002)\n[24] J. Xiao et al., Phys. Rev. B, 70, 172405 (2004).\n[25] J. Xiao et al., Phys. Rev. B, 72, 014446 (2005)\n[26] T. Valet, unpublished preprint (2004).\n[27] A. Brataas et al., Phys. Rev. Lett. 84, 2481 (2000);\n[28] H. Morise et al., Phys. Rev. B, 71, 014439 (2005).\n[29] Ya. Tserkovnyak et al., Rev. Mod. Phys., 77, 1375\n(2005)."    },    {        "title": "0706.1736v1.Gilbert_and_Landau_Lifshitz_damping_in_the_presense_of_spin_torque.pdf",        "content": "Gilbert and Landau-Lifshitz  damping in the presense of spin-torque \n \nNeil Sm ith \nSan J ose Reserch Center, Hitachi Global Strage Tec hnologies, Sa n Jose, CA 95135 \n(Dated 6/12/07) \n \nA recent arti cle by Stiles  et al.  (cond-mat/0702020) argued  in favor of th e Land au-Lifshitz dampin g term in  the \nmicromagnetic equations of  motion over that of  the more commo nly accepted  Gilbert d amping for m. Much of \ntheir argument r evolved  around  spin-torque dr iven domai n wall motion in n arrow magnetic wires, since th e \npresence of spin -torques can mor e acut ely draw a distinct ion b etween the two form s of dam ping. In this ar ticle, \nthe author  uses simple argum ents and exam ples to offer  an alterna tive po int of  view favoring  Gilb ert.  \n \nI. PREL IMINARIES \n \n       The Gilbert1 (G) or La ndau-Li fshitz2 (LL) equations of  \nmotion for unit magnetization vect or  \nare formally descri bed by the gene ric form sMt t /),( ),(ˆ rM rm≡\n \n) derivativeal (variationˆ/ /1)] (ˆ[ /ˆ\neffeff totdamp tot\nNC\nm HH H HH Hm m\n∂∂−≡+≡+×γ−=\nE Mdt d\ns      (1) \n \nwhere  the satu ration  magnetizatio n, and  sM γ is th e \ngyromagnetic ratio (tak en here to  be a po sitive constant). \nThe total (p hysical) field  by h as con tributions fro m the \nusual \"effecti ve field\" term , pl us t hat of a   \n\"nonconservative-field\"  that is supp osed not to be \nderivable from the -gradient of the (internal) free-energy \ndensity functional . Although also non conservative \nby defi nition, the \"dam ping-field\"  is p rimarily a \nmath ematica l vehicle for  describing a physical d amping \ntorque , and i s properly treated separat ely. \nFor most of the rem ainder of th is article, an y sp atial \ndepe ndence of  will be im plicitly unde rstood.  effH\nNCH\nmˆ\n)ˆ(mE\ndampH\ndampˆHm×sM\n),(ˆtrm\n       As was described by Brown,3 the Gilb ert eq uations of \nmotion m ay be de rived using st andard techni ques of \nLagra ngian mechanics .4 In particular, a phenomenological \ndamping of the motion in included via the use of a R ayleigh \ndissipation function : )/ˆ( dtdmℜ\ndt d dt d M/dtdM\nGG\nss\n/ˆ)/ ( )/ˆ(/ /1ˆ )2/ (\nG\ndamp2\nm m Hm\nγα−= ∂∂ℜ−≡γα=ℜ\n     (2) \n \nwhere dimensionless  is th e Gilb ert damping parameter. \nBy d efinition,Gα\n4 2\ndampˆ / /ˆ 2G/dtd M dtds G m m H γα=⋅−=ℜ  \nis th e in stantaneous rate o f energy lost fro m the \nmagnetizatio n syste m to its thermal environment (e.g ., to \nthe lattice) d ue to the viscous  \"friction\" re present ed by the \ndamping field .   dt d/ˆG\ndampm H−∝\n        The Lag rangian method is well su ited to include \nnonconservative fields , which can be  generally \ndefined using the principles of virtual work:0NC≠H\n3,4 \n \n)ˆ ( /1 ) ˆ() ˆ( ˆ\nNC NCNC NC NC\nmN H HmHm m H\n× =⇔×=⇒δ⋅×=δ⋅ =δ\ns ss s\nM MNM M W θ       (3) \n \nThe latter exp ressio n is useful in cases (e.g.,  spin-torques) \nwhere the torque density funct ional  is specified. \nTreatin g  as fi xed, the  (virtual) dis placem ent )ˆ(mN\nsM mˆδ is of \nthe fo rm m m ˆ ˆ×δ=δθ , and only the orthogonal compone nts \nof the torque mNmN ˆ ˆ××↔  are physically signi ficant.  \n       Combining (1) an d (2) gives the Gilbert equations: \n \n)/ˆ ˆ( ) ˆ( /ˆtot dtd dt dG mm Hm m ×α+×γ−=           (4) \n \nAs is well known, the G eq uations o f (4) may be rearra nged \ninto their equivalent (and perhaps m ore common) f orm:  \n \n)] ˆ(ˆ ˆ[\n1/ˆtot tot2\nGHmm Hm m ××α+×\nα+γ−=G dt d      (5) \n \n       With re gard to the LL e quations, the form  of  is \nnot uniquely  defined in problems where LL\ndampH\n0NC≠H , whic h \nhave only c ome to the forefront with the  recent interest in \nspin-torque phen omena. Two d efinitions  conside red are \n \n) ˆ(eff damp LLLLHm H ×α≡ ,                       (6a) \n                        (6b) ) ˆ(tot damp LLLLHm H ×α≡\n \nThe fi rst de finition of (6a) is the historical/conventional \nform  of LL, and is that em ployed by Stiles et al.5 Howe ver, \nin this a uthor's view, the re is no a-priori reason, other than \nhistorical,  to not replace  as in (6b). Doing so \nyields a form  of LL that reta ins it \"usual\" e quivalence (i.e., \nto first o rder in tot eff H H→\nα) to G w hether or not 0NC≠H , as is \nseen by com paring (5) and (6b). The form o f (6b) treats \nboth  and  on an equal footin g. effHNCH       Nonet heless, to facilita te a com parative discussi on with \nthe analysis of Stiles et al. ,5  (6a) will he ncefort h be used to \ndefine what will be re ferred to below as the LL eq uatio ns \nof motion: \n \n)] ˆ(ˆ ˆ[ /ˆeff tot LL Hmm Hm m ××α+×γ−=dt d        (7) \n \nIn cases of pre sent interest where  , the difference \nbetwee n G in (4) (or (5)) and the f orm of LL give n in ( 7) \nare first orde r in the dam ping param eter, and  thus o f a more \nfundam ental nature. T hese differences a re the subj ect of the  \nremainder of t his article. 0NC≠H\n \nII. SPIN-TOR QUE  EXAM PLES \n \n       Two distinct situations where spin-torque effects have \ngarnere d substantial intere st are those  of CPP-GMR \nnanopillars, and spin-torque driven dom ain wall motion i n \nnanowires as was conside red in R ef. 5. The spi n-torque \nfunctio n  is taken t o have a \npredominant \"adiabatic\" c omponent , alon g with a \nsmall \"nona diabatic\" com pone nt   described \nphenomenolog ically  by the relation )ˆ( )ˆ( )ˆ(nad ad ST m NmNmN + =\n)ˆ(admN\n)ˆ(nadm N\nad nadˆNm N ×β−≡ , \nwith . In t he case of a narrow nanowire along the -\naxis, with m agnetization  and electron curre nt density  \n, the torque function  and associate d \nfield (see ( 2)) are descri bed by1<<β xˆ\n)(ˆxm\nx J ˆe eJ= )ˆ(STmN\n)ˆ(STmH5 \n \n)/ˆ /ˆ ˆ() 2/ ()/ˆ()2/ ()ˆ(\nSTad\ndxd dxd eM PJdxde PJ\ns ee\nm mm Hm mN\nβ+× −==\nhh   (8) \n \nwhere P is the  spin-polarization of t he electron curre nt.  \n       To check if  is conse rvative, one ca n \"discretize\"  \nthe spatial derivatives app earing in (8) in the form  STH\nx dxdi iixx∆ −→−+=2/)ˆ ˆ( /ˆ1 1m m m , whe re )(ˆ ˆi i xmm≡  \nand , not unlik e the com mon m icrom agnetics  \napproxim ation. For a c onservative H-field where \n, the set of  Cartesian tensorsi i x xx−≡∆+1\ni i Em H ∂∂∝ / 33×6 \nj i j iuv\nji E H mm mH ∂∂∂∝∂∂≡ /2/t\n will be sym metric, i.e., \nvu\nijuv\nji H Htt\n= , under sim ultaneo us reversal of s patial indices \n and vect or in dices ji, z yxvu or,, ,= . For the adiabatic \nterm in (6), it can be readily  shown that the  uv\njiHt\n are in \ngene ral asymmetric , i.e., always antisym metric in vect or \nindices (du e to cross pr oduct) , but asy mmetr ic in  spatial \nindices , being antisym metric he re only for  \nlocally uniform  magnetization . The  \nnonadiabatic term  yields an -inde pendent 1 ,±=iji\ni im m ˆ ˆ1=±\nmˆuv\njiHt\n that is  \nalways antisym metric, i.e., symmetric in ve ctor i ndices, but antisym metric in spatial indic es . The concl usion \nhere t hat  is in ge neral nonconservative a grees with \nthat f ound in Ref. 5, by way of a rathe r diffe rent argument.  1 ,±=iji\nSTH\n       Anot her well known example is a nanopillar stack wit h \nonly two fe rrom agnetic (FM ) layers, the \" refere nce\" layer \nhaving a m agnetization  rigidly fixe d in time, and  a \ndynamically  varia ble \"free \" lay er refˆm\n)(ˆ)( ˆfree t tm m= . As  \ndescri bed by  Sloncze wski,7 the (adiaba tic) spin-t orque \ndensity function a nd field is given by: )ˆ(STmH\n \n]ˆ )ˆ ˆ[() 4/ ()ˆ ˆ(]ˆ ˆˆ[) 4/ ()ˆ ˆ(\nref reffree refref free ref ad\nST\nm m mm m Hm mm m m N\nβ+××⋅−=×× ⋅−=\ntMe PJ gte PJ g\ns ee\nhh\n   (9) \n \nwhere  is the free layer thickness, and freet )ˆ ˆ(refm m⋅ g  is a \nfunctio n of order unity, the de tails of which are not relevant  \nto the present discu ssion. From  the the -tenso r, or by  \nsimple inspection, t he adiabatic  term  in (7) is \nmanifestly  nonc onservative . However, app roxim ating uvHt\nm m ˆ ˆref×\n)ˆ ˆ(refm m⋅ g ~ consta nt, the conse rvative nonadiabatic ter m \nresem bles a magnetic field d escribe d by the -gradient  of \nan Zeem an-like ene rgy function mˆ\nm m ˆ ˆref nad ⋅∝ E . The  \nremaining discussion will restrict attention to \nnonconservative contrib utions. \n \nIII. STATIO NARY SOLU TIONS OF G AND LL \n \n       With , stationary  (i.e, ST NC H H→ 0 /ˆ=dt dm ) \nsolutions  of G-equatio ns (4) satisfy  the c onditions  that 0ˆm\n \nST STST\n0 eff 0 0G\ndamp eff 0\nˆ ˆ 0 ˆ0 /ˆ ;0) ( ˆ\nHm Hm Hmm H H H m\n×−=×⇒≠×=∝ =+× dt d      (10) \n \nThe clear and physically intuitiv e interpreta tion of (10) is \nthat stationary  state  satisfies a condition of zero \nphysical tor que, 0ˆm\n0 ˆtot 0=×Hm , includin g bot h \nconservative ( ) an d nonconservative s pin-torque \n( ) fields.  Being visco us in nature, the G dam ping \ntorque  inde pendently vanishes..  effH\nSTH\n0 /ˆ ˆG\ndamp 0 ≡∝× dt dm Hm\n       Previous measurem ents6 of the angular depe ndence of \nspin-torque critical curre nts  in CPP-GMR \nnanopillar syste ms by this  author and colleagues  \ndemonstrated t he existence of such stationary states with \nnon-collinear )ˆ ˆ(refcritm m⋅eJ\n0 ˆ ˆ0 ref≠×m m  and crit0e eJ J<< . In t his \nsituation, it follows from (9) an d (10) that the stationa ry \nstate  satisfies 0ˆm 0 ˆ ˆeff 0 0 ST ≠×=×− Hm Hm . It is  \nnoted that the last result i mplies that  is not a (therm al) 0ˆmequilibrium  state which m inimizes the free energy , \ni.e., )ˆ(mE\n0 ) ˆ()ˆ ()ˆ/(ˆ/eff 0 0 ≠δ⋅×∝×δ⋅∂∂=δδ θ θ Hm m m m E E  \nfor arbitra ry .  θδ\n       In the present described circum stance of  stationary   \nwith , the LL equatio ns of (7) differ from G \nin a fundam ental respect. Setting  in (7) yield s  0ˆm\n0 ˆST 0≠×Hm\n0 /ˆ=dt dm\n \n) ˆ( ˆ ) ( ˆeff 0 0 eff 0 LL ST Hm m H H m ××α−=+×         (11) \n \nLike (10), (11) im plies  that 0 ˆeff 0≠×Hm  whe n \n. However, (11)  also  imply a static , nonzero \nphysical tor que , alon g with a static,  \nnonzero damping tor que  (see (6a) ) to \ncancel it out . In sim ple mechanical term s, the latte r \namounts to non-visco us \"static-frictio n\". It has n o anal ogue \nwith G in a ny circum stance, or with LL in conventional  \nsituations with  and  equilibrium \nfor which LL  dam ping wa s origi nally  develo ped as a  \nphenomenolog ical dam ping f orm. It furth er contra dicts th e \nviscous (o r -depe ndent) nature o f the damping \nmechanism s desc ribed by  physical (rathe r than  \nphenomenolog ical) base d theoretical m odels0 ˆST 0≠×Hm\n0 ˆtot 0≠×Hm\n0 ˆLL\ndamp 0 ≠×Hm\n0ST NC =↔H H ↔0ˆm\ndtd/ˆm\n,8,9. \n       The above arguments ignored therm al fluctuatio ns of \n. However, thermal fluctuations mˆ10 scale approxim ately a s \n, while ( 10) or (11) are scale-inva riant \nwith 2\neff 0 ) /( Hm⋅ kT\nH.  In t he simple CPP nanopillar exam ple of (9), one \ncan (conce ptually at least)  continually increase both eJ \nand a n applied  field  contri bution to  to scale up appHeffH\nST 0ˆHm×  and eff 0ˆHm×  while approxim ately keeping \na fixe d statio nary state  (satis fying 0ˆm 0 ˆ ˆref 0≠×mm  with \nfixed ).  However, unique to LL eq uatio ns (11) based \non (6a) is t he additional requi rement that the static dam ping \nmechanism  be able to produce an refˆm\neff dampˆLLHm H ×∝  \nwhich sim ilarly scales (without li mit). This author finds thi s \na physically unreasonable proposition.  \n \nIV. ENERGY A CCOUN TING  \n \nIf one ignores/forgets t he Lagrangian formulation3 of the \nGilbert e quatio ns (4), one may derive t he followin g energy \nrelationships, substitutin g the right side of (4) for evaluating \nvecto r products of form : dtd/ˆmH⋅\n \n)/ˆ ˆ( ) ˆ()/ˆ ˆ( ) ˆ(/ˆ )/ˆˆ/ /( /1\neff effeff effeff\nNCNC\ndtddtddtd dtd E dtdE Ms\nmm H Hm Hmm H Hm Hm H mm\n×⋅α−×⋅γ−=×⋅α−×⋅γ=⋅−≡⋅∂∂=\n   (12a) \n )/ˆ ˆ( ) ˆ(/ˆ / /1\nNC NCNC NC\neff dtddtd dt dWMs\nmm H Hm Hm H\n×⋅α+×⋅γ−=⋅≡   (12b) \n \n)/ˆ ˆ() () ˆ(/ˆ /ˆ\nNC efftot2\ndt ddt d dt d\nmm H HHm m m\n×⋅+γ=×γ−⋅=          (12c) \n \nSubtractin g (12b) from (12a), and usin g (12c) one  finds \n \ndt d M dt dWdt d M dt dW dtdE\nss\n/ˆ //ˆ / / / :G\nG\nNCG NC\ndamp2\nm Hm\n⋅ + =γα− =\n         (13) \n \nThe re sult o f (13) is essentially a state ment of energy \nconservation. Nam ely, that the rate of change of the internal \nfree e nergy (density) of the magnetic sy stem  is give by  the \nwork done on the system  by the (exte rnal) no nconservative \nforces/fields , minus the loss of energy (t o the lattice) \ndue t o dam ping. The G damping term  in ( 13) is ( not \nsurprisi ngly) t he sam e as expected from  (2). It is a strictly  \nlossy, negative-definite  contributio n to .  NCH\ndtdE/\n       Over a finite interval of motion from  time  to , the \nchange 1t2t\n)ˆ( )ˆ(1 2 m m E EE −=∆  is, from (12b ) and (13):  \n \n∫⋅γα− =∆2\n1G NCˆ)/ˆ / )ˆ( (t\ntsdtddt d dt MEmm m H       (14) \n \nSince  is nonc onservative, t he work NCHNCW∆  is pat h-\ndepe ndent, and so use of (14) requires indepe ndent \nknowledg e of the solution  of (4). Sin ce \n itself depe nds on ) (ˆ2 1 ttt≤≤m\n)(ˆtmGα, the  term's contribution  \nto (14) also can vary with . Regardless, NCH\nGα 0>∆E  can \nonly result in the case of a positive  amount of work \n done by .  ∫⋅ =∆2\n1NC NC )/ˆ (t\nts dtdt d M W m HNCH\n       Working out the results  analogous to (12 a,b) for the LL \nequatio ns of (6a) and (7), one finds \n \n) ˆ() ˆ( /ˆ/ˆ / / :LL\ntot eff dampdamp\nLLLL\nNC\nHm Hm m Hm H\n×⋅×αγ−=⋅⋅ + =\ndtddtd M dt dW dtdEs\n      (15) \n \nThe form of (15) is the sam e as the latter result in (13). \nHowever, unlike G, the LL damping term in (15) is not \nmanifestly negative-definite, except when 0NC=H .  \n       The results of (13)-(15) apply equally to situations \nwhere one inte grates over the spatial distribution of  \nto evaluate the total syste m free energy, rat her tha n (local ) \nfree e nergy density . Total time derivatives  may be \nreplace d by partial deri vative s  whe re appropriate. ),(ˆtrm\ndtd/\nt∂∂/      Dropping terms of order  (and sim plifying notation \n), (7) is easily transfor med to a Gilbert-like form : 2\nLLα\nα→αLL\n \n)ˆˆ( )] ˆ( [ˆˆ:LLNC totdtd\ndtd mm Hm Hmm×α+×α−×γ−=  (16) \n \nwhic h differs from G in  (4) by the term  ) ˆ(NCHm×α  \nwhich is first order in both  and . For the \"wire \nproblem\" described by  (8), the equation s of m otion bec ome αNCH\n \n)ˆ ˆ(ˆ ˆˆ ˆ:LL)ˆ ˆ(ˆ ˆˆ ˆ\neffeff :G\ndxdv\ndtd\ndxdv\ndtddxdv\ndtd\ndxdv\ndtd\nm mm Hmm mm mm Hmm m\nαβ+α+×α+×γ−=+αβ+×α+×γ−=+\n (17) \n \nwhere , and terms of or der eM PJ vs e2/γ=h βα are \ndropped for LL. A s noted previously,5,9,11 (17) permits \n\"translational\" solutions ) (ˆ)(ˆeq vtx x,t −=m m  whe n α=β  \n(G) or  (LL), with  the static , equilibrium \n(minimum E) solution of . Evaluatin g \n by takin g  from (8), and \n with , one finds \nthat 0=β )(ˆeqxm\n0) ˆ(eff eq =×H m\ndtd M dt dWs /ˆ /ST ST m H⋅ =STH\n) (ˆ /ˆeq vtx v dtd −′−→m m dqd q /ˆ )(ˆ m m≡′\n2\neq2) (ˆ)/ ( /ST vtx Mv dt dWs −′γβ= m . In transl ational \ncases where  is exactly  collinear  to , only \nthe nonadiabatic term  does work on t he -system.  dtd/ˆm dx d/ˆm\nmˆ\n       Interestingly, the energy interpretation of these \ntranslational solutions is very diffe rent fo r G or LL. For G, \nthe positive rate of work  when dt dW /ST α=β  exactly \nbalances t he negative damping l oss as  given in (13), the \nlatter alway s nonzero and scaling as . For LL by  \ncontrast, the wo rk done by  vanishes when 2v\nSTH 0=β , \nmatching t he damping l oss whic h, from (6a)  or (15), is \nalways zero since  regardle ss of 0) ˆ(eff eq =×H m v. If \n is a sharp domain wall, )(ˆeqxm ) (ˆ /ˆeq vtx v dtd −′−=m m  \nrepresents, from a spatially local  perspective at a fixed  point \nx, an abrupt, irreversi ble, non-equilibrium  reor ientation of \n at/near tim e  whe n the wall core passes by. The  \nprediction of LL/(6a) that t his magnetization re versal c ould \ntake place locally (at arbitrarily large v), with the com plete \nabsence of the spin-orbit couple d, ele ctron scatteri ng \nprocessesmˆ vxt /≈\n8 that lead to spin-latti ce dam ping/r elaxation in all \nother known circum stances (e. g., external field-driven \ndomain wall motion) is, in the vi ew of this author, a rather \ndubious, nonphysical aspect of (6a) when 0ST≠H . \n       Stiles et al.5 repo rt that micromagnetic com putation s \nusing G in the case  sho w (non-translational)  \ntime/distance l imited dom ain wall displacement, resulting  in a net positive  increase \n0=βE∆. They claim  that 1) \"spi n \ntrans fer to rques do not cha nge the ene rgy of the sy stem\", \nand that 2) \"Gilbert dam ping to rque is the only  torque th at \nchanges the e nergy\". Acce pting  as accurate, it is \nthis aut hor's view that t he elementary physics/ mathematics \nleading t o (13) and (14) demonstrably  prove t hat bot h of \nthese claim s must be incorrect  (err or in the first per haps \nleading t o the misinterpretation of the second). On a related \npoint, the res ults of (1 3) and (1 5) shows that excludi ng \nwork  or 0>∆E\ndt dW /ST STW∆ , only LL- damping may possibly  \nlead to a positive contri bution to  or dtdE/ E∆ when \n0ST≠H , in a pparent contradictio n to the claim  in R ef. 5 \nthat LL damping \"uniquely and  irre versibly  reduces \nmagnetic free energy whe n spin-transfer torque is prese nt\". \n \nACK NOWLEDGM ENTS \n \nThe aut hor would like to ackno wledge em ail discussions on \nthese or related topics with W. Sa slow and R. Duine, as \nwell as an  extende d series of friendly discussio ns with  \nMark Stiles. Obviously, the latter have not (as of yet) \nachieve d a mutually agree d viewpoint on thi s subj ect.  \n \nREFERENCES \n \n1 T. L. Gilber t, Armour Research Report, M ay 1956; IEEE Tran s. \nMagn., 40, 3343  (2004).  \n2 L. Landau  and E. Lifshit z, Phys. Z. Sow jet 8, 153 (1935). \n3 W. F. Brown, Micromagnetics  (Krieger , New Y ork 1978). \n4 H. Gol dstein, Classical Me chanics , (Addison  Wesley, Reading \nMassachusetts, 1 950). \n5M. D. Stiles, W. M. Saslow, M. J. Donahue,  and A . Zangwill, \narXiv:cond-m at/0702020. \n6 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, IEEE \nTrans. M agn. 41, 2935 (2005) ; N. Sm ith, J Appl. Ph ys. 99, \n08Q703 (2006). \n7 J. C. Slonczewski, J.  Magn. Magn. Mater. 159, L1 (1996); J. \nMagn. Magn . Mater. 247, 324 (2 002) \n8 V. Kambersky, Can. J. Phys. 48, 2906 (1970) ; V. Kam bersky and \nC. E. Patton , Phys. Rev . B 11, 2668 (1975). \n9 R. Duine, A. S. Nunez, J.  Sinova, and A. H. MacDonald, \narXiv:cond-m at/0703414. \n10 N. Sm ith, J. A ppl. Ph ys. 90, 57 68 (2001). \n11 S. E Barnes and S . Maekaw a, Phys. R ev. Lett. 95, 10720 4 \n(2005). \n "    },    {        "title": "0706.3160v3.Spin_pumping_by_a_field_driven_domain_wall.pdf",        "content": "arXiv:0706.3160v3  [cond-mat.mes-hall]  9 Jan 2008Spin pumping by a field-driven domain wall\nR.A. Duine\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: October 26, 2018)\nWe present the theory of spin pumping by a field-driven domain wall for the situation that spin\nis not fully conserved. We calculate the pumped current in a m etallic ferromagnet to first order\nin the time derivative of the magnetization direction. Irre spective of the microscopic details, the\nresult can be expressed in terms of the conductivities of the majority and minority electrons and\nthe dissipative spin transfer torque parameter β. The general expression is evaluated for the specific\ncase of a field-driven domain wall and for that case depends st rongly on the ratio of βand the\nGilbert damping constant. These results may provide an expe rimental method to determine this\nratio, which plays a crucial role for current-driven domain -wall motion.\nPACS numbers: 72.25.Pn, 72.15.Gd\nI. INTRODUCTION\nAdiabatic quantum pumping of electrons in quantum\ndots1,2has recently been demonstrated experimentally\nforboth charge3and spin4. Currently, the activityin this\nfieldismostlyconcentratedontheeffectsofinteractions5,\ndissipation6, and non-adiabaticity7. Complementary to\nthese developments, the emission of spin current by a\nprecessing ferromagnet — called spin pumping — has\nbeen studied theoretically and experimentally in single-\ndomain magnetic nanostructures8,9,10. One of the differ-\nences between spin pumping in single-domain ferromag-\nnets and quantum pumping in quantum dots is that in\nthelatterthehamiltonianoftheelectronicquasi-particles\nis manipulated directly, usually by varying the gate volt-\nage of the dot. In the case of ferromagnets, however,\nit is the order parameter — the magnetization direction\n— that is driven by an external (magnetic) field. The\ncoupling between the order parameter and the current-\ncarrying electrons in turn pumps the spin current11. The\nopposite effect, i.e., the manipulation of magnetization\nwith spin current, is called spin transfer12,13,14,15.\nRecently, the possibility of manipulating with cur-\nrent the position of a magnetic domain wall via\nspin transfer torques has attracted a great deal of\ntheoretical16,17,18,19,20,21,22,23,24,25,26,27,28,29,30and\nexperimental31,32,33,34,35,36,37,38interest. Although the\nsubject is still controversial18,21, it is by now established\nthat in the long-wavelength limit the equation of motion\nfor the magnetization direction Ω, which in the absence\nof current describes damped precession around the\neffective field −δEMM[Ω]/(/planckover2pi1δΩ), is given by\n/parenleftbigg∂\n∂t+vs·∇/parenrightbigg\nΩ−Ω×/parenleftbigg\n−δEMM[Ω]\n/planckover2pi1δΩ/parenrightbigg\n=−Ω×/parenleftbigg\nαG∂\n∂t+βvs·∇/parenrightbigg\nΩ, (1)\nand contains, to lowest order in spatial derivatives of the\nmagnetization direction, two contributions due the pres-\nence of electric current.The first is the reactivespin transfertorque16,17, which\ncorresponds to the term proportional to ∇Ωon the left-\nhand side of the above equation. It is characterized by\nthe velocity vsthat is linear in the curent and related to\nthe external electric field Eby\nvs=(σ↓−σ↑)E\n|e|ρs, (2)\nwhereσ↑andσ↓denote the conductivities of the major-\nity and minority electrons, respectively, and ρsis their\ndensity difference. (The elementary charge is denoted by\n|e|.) The second term in Eq. (1) due to the current is the\ndissipative spin transfer torque39that is proportional to\nβ19,20,21. Both this parameter, and the Gilbert damping\nparameter αG, have their microscopic origin in processes\nin the hamiltonian that break conservation of spin, such\nas spin-orbit interactions.\nIt turns out that the phenomenology of current-driven\ndomain-wallmotion depends crucially on the value of the\nratioβ/αG. For example, for β= 0 the domain wall\nis intrinsically pinned18, meaning that there is a criti-\ncal current even in the absence of inhomogeneities. For\nβ/αG= 1ontheotherhand, thedomainwallmoveswith\nvelocity vs. Although theoretical studies indicate that\ngenerically β/ne}ationslash=αG26,27,28,30, it is not well-understood\nwhat the relative importance of spin-dependent disorder\nand spin-orbit effects in the bandstructure is, and a pre-\ncise theoretical prediction of β/αGfor a specific mate-\nrial has not been attempted yet. Moreover, the determi-\nnation of the ratio β/αGfrom experiments on current-\ndriven domain wall motion has turned out to be hard\nbecause of extrinsic pinning of the domain and nonzero-\ntemperature29,38effects.\nIn this paper we present the theory of the current\npumped by a field-driven domain wall for the situation\nthat spin is not conserved. In particular, we show that\na field-driven domain wall in a metallic ferromagnet gen-\nerates a charge current that depends strongly on the ra-\ntioβ/αG. This charge current arises from the fact that\na time-dependent magnetization generates a spin cur-\nrent, similar to the spin-pumping mechanism proposed2\nby Tserkovnyak et al.8for nanostructures containing fer-\nromagnetic elements. Since the symmetry between ma-\njority and minority electrons is by definition broken in\na ferromagnet, this spin current necessarily implies a\ncharge current. In view of this, we prefer to use the term\n“spin pumping” also for the case that spin is not fully\nconserved, and defining the spin current as a conserved\ncurrent is no longer possible.\nThe generation of spin and charge currents by a mov-\ning domain wall via electromotiveforces is discussed very\nrecently by Barnes and Maekawa40. We note here also\nthe work by Ohe et al.41, who consider the case of the\nRashba model, and the very recent work by Saslow42,\nYanget al.43, and Tserkovnyak and Mecklenburg44. In\naddition to these recent papers, we mention the much\nearlier work by Berger, which discusses the current in-\nduced by a domain wall in terms of an analogue of the\nJosephson effect45.\nBarnes and Maekawa40consider the case that spin\nis fully conserved. In this situation it is convenient to\nperform a time and position dependent rotation in spin\nspace, such that the spin quantization axis is locally par-\nallel to the magnetization direction. As a result of spin\nconservation, the hamiltonian in this rotated frame con-\ntains nowonly time-independent scalarand exchangepo-\ntential terms. The kinetic-energy term of the hamilto-\nnian, however, will acquire additional contributions that\nhave the form of a covariant derivative. Perturbation\ntheory in these terms then amounts to performing a gra-\ndient expansion in the magnetization direction17. Hence,\nthe fact that Barnes and Maekawa consider the case\nthat spin is fully conserved is demonstrated mathemat-\nically by noting that in Eq. (5) of Ref. [40] there are\nno time-dependent potential-energy terms. Generaliz-\ning this approach to the case of spin-dependent disorder\nor spin-orbit coupling turns out to be difficult. Never-\ntheless, Kohno and Shibata were able to determine the\nGilbert damping and dissipative spin transfer torques us-\ning the above-mentioned method46. Since Barnes and\nMaekawa40consider the situation that spin is fully con-\nserved, they effectively are dealing with the case that\nαG=β= 0. This is because both the Gilbert damping\nparameter αGandthedissipativespintransfertorquepa-\nrameterβarise from processes in the microscopic hamil-\ntonian that do not conserve spin26,27,28,30. Hence, for the\ncase that αG=β= 0 our results agree with the results\nof Barnes and Maekawa40.\nThe remainder of this paper is organized as follows. In\nSec. II we derive a general expression for the electric cur-\nrent induced by a time-dependent magnetizationtexture.\nThis general expression is then evaluated in Sec. III for\na simple model of field-driven domain wall motion. We\nend in Sec. IV with a short discussion, and present our\nconclusions and outlook.II. ELECTRIC CURRENT\nQuite generally, the expectation value of the charge\ncurrent density, defined by j=−cδH/δAwithcthe\nspeed oflight, Hthe hamiltonian, and Athe electromag-\nnetic vector potential, is given as a functional derivative\nof the effective action\n/an}b∇acketle{tj(x,τ)/an}b∇acket∇i}ht=cδSeff\nδA(x,τ), (3)\nwithτthe imaginary-time variable that runs from 0 to\n/planckover2pi1/(kBT). (Planck’s constant is denoted by /planckover2pi1andkBTis\nthe thermal energy.) First, we assume that spin is con-\nserved meaning that the hamiltonian is invariant under\nrotations in spin space. The part of the effective action\nfor the magnetization direction that depends on the elec-\ntromagnetic vector potential is then given by17\nSeff=/integraldisplay\ndτ/integraldisplay\ndx/angbracketleftbig\njz\ns,α(x,τ)/angbracketrightbig˜Aα′(Ω(x,τ))∇αΩβ(x,τ),\n(4)\nwhere a summation over Cartesian indices α,α′,α′′∈\n{x,y,z}is implied throughout this paper. In this ex-\npression,\njα\ns,α′(x,τ) =/planckover2pi12\n4mi/bracketleftbig\nφ†(x,τ)τα∇α′φ(x,τ)\n−/parenleftbig\n∇α′φ†(x,τ)/parenrightbig\nταφ(x,τ)/bracketrightbig\n+|e|/planckover2pi1\n2mcAα′φ†(x,τ)ταφ(x,τ),(5)\nis the spin current, given here in terms of the Grassman\ncoherent state spinor φ†= (φ∗\n↑,φ∗\n↓). Furthermore, ταare\nthe Pauli matrices, and mis the electron mass. (Note\nthat since we are, for the moment, considering the situa-\ntion that spin is conserved there are no problems regard-\ning the definition of the spin current.) The expectation\nvalue/an}b∇acketle{t···/an}b∇acket∇i}htis taken with respect to the current-carrying\ncollinear state of the ferromagnet. Finally, ˜Aα(Ω) is the\nvector potential of a magnetic monopole in spin space\n[not to be confused with the electromagnetic vector po-\ntentialA(x,τ)] that obeys ǫα,α′,α′′∂˜Aα′/∂Ωα′′= Ωαand\nis well-known from the path-integral formulation for spin\nsystems47. Eq. (4) is most easily understood as arising\nfrom the Berry phase picked up by the spin of the elec-\ntrons as they drift adiabatically through a non-collinear\nmagnetization texture16,17. Variation of this term with\nrespect to the magnetization direction gives the reactive\nspin transfer torque in Eq. (1).\nThe expectation value of the spin current is given by\n/angbracketleftbig\njz\ns,α(x,τ)/angbracketrightbig\n=/integraldisplay\ndτ′/integraldisplay\ndx′Πz\nα,α′(x−x′;τ−τ′)Aα′(x′,τ′)\n/planckover2pi1c.\n(6)\nThe zero-momentum low-frequency part of the response\nfunction Πz\nα,α′(x−x′;τ−τ′)≡/angbracketleftbig\njz\ns,α(x,τ)jα′(x′,τ′)/angbracketrightbig\n0,\nwith/an}b∇acketle{t···/an}b∇acket∇i}ht0the equilibrium expectation value, is deter-\nmined by noting that for the vector potential A(x,τ) =3\n−cEe−iωτ/ωthe above equation [Eq. (6)] should in\nthe zero-frequency limit reduce to Ohm’s law /an}b∇acketle{tjz\ns/an}b∇acket∇i}ht0=\n−/planckover2pi1(σ↑−σ↓)E/(2|e|). Using this result together with\nEqs. (3-6), we find, after a Wick rotation τ→itto\nreal time, that\n/an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1\n2|e|V(σ↑−σ↓)∂\n∂t/integraldisplay\ndx˜Aα′(Ω(x,t))∇αΩα′(x,t),\n(7)\nwithVthe volume of the system. We note that the time-\nderivative of the Berry phase term is also encountered\nby Barnes and Maekawa in discussing the electromotive\nforce in a ferromagnet40. Such Berry phase terms are\nknown to occur in adiabatic quantum pumping48.\nWe now generalize this result to the situation where\nspin is no longer conserved, for example due to spin-\norbit interactions or spin-dependent impurity scatter-\ning. Linearizing around the collinear state by means of\nΩ≃(δΩx,δΩy,1−δΩ2\nx/2−δΩ2\ny/2) we find that the part\nof the effective action that contains the electromagnetic\nvector potential reads30\nSeff=/integraldisplay\ndτ/integraldisplay\ndx/integraldisplay\ndτ′/integraldisplay\ndx′/integraldisplay\ndτ′′/integraldisplay\ndx′′[δΩa(x,τ)\n×Kab(x,x′,x′′;τ,τ′,τ′′)·A(x′′,τ′′)δΩb(x′,τ′)],(8)\nwhere a summation over transverse indices a,b∈ {x,y}\nis implied. The spin-wave photon interaction vertex\nKab(x,x′,x′′;τ,τ′,τ′′) =\n∆2\n8/planckover2pi1c/an}b∇acketle{tφ†(x,τ)τaφ(x,τ)φ†(x′,τ′)τbφ(x′,τ′)j(x′′,τ′′)/an}b∇acket∇i}ht0,\n(9)\ngiven in terms of the exchange splitting ∆, is also en-\ncountered in a microscopic treatment of spin transfer\ntorques30. The reactive part of this interaction vertex\ndetermines the reactive spin transfer torque and, via\nEqs. (3) and (8), reproduces Eq. (7). The zero-frequency\nlong-wavelength limit of the dissipative part of the spin-\nwave photon interaction vertex determines the dissipa-\ntive spin transfer torque. (Note that in this approach\nthe definition of the spin current does not enter in deter-\nmining the spin transfer torques.) Although Eq. (9) may\nbe evaluated for a given microscopic model within some\napproximation scheme30, we need here only that varia-\ntion of the action in Eq. (8) reproduces both the reactive\nand dissipative spin torques in Eq. (1). The final result\nfor the electric current density is then given by\n/an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1\n2|e|V(σ↑−σ↓)/bracketleftbigg\nβ/integraldisplay\ndx∂Ω(x,t)\n∂t·∇αΩ(x,t)\n+∂\n∂t/integraldisplay\ndx˜Aα′(Ω(x,t))∇αΩα′(x,t)/bracketrightbigg\n.(10)\nThe above equation is essentially the result of a linear-\nresponse calculation in ∂Ω/∂t, and is the central result\nof this paper. We emphasize that the way in which thetransport coefficients σ↑andσ↓and theβ-parameter en-\nter does not rely on the specific details of the underlying\nmicroscopic model. Note that the above result reduces\nto that of Barnes and Maekawa (Eq.(9) of Ref. [40]) if we\ntakeβ= 0.\nIII. FIELD-DRIVEN DOMAIN WALL MOTION\nTo bring out the qualitative physics, we evaluate\nthe result in Eq. (10) using a simple model for field-\ndriven domain wall motion in a magnetic wire of length\nL. In polar coordinates θandφ, defined by Ω=\n(sinθcosφ,sinθsinφ,cosθ), we choose the micromag-\nnetic energy functional\nEMM[θ,φ] =ρs/integraldisplay\ndx/braceleftbiggJ\n2/bracketleftBig\n(∇θ)2+sin2θ(∇φ)2/bracketrightBig\n+K⊥\n2sin2θsin2φ−Kz\n2cos2θ+gBcosθ/bracerightbigg\n,(11)\nwhereJis the spin stiffness, and K⊥andKzare\nanisotropy constants larger than zero. The external field\nin the negative z-direction leads to an energy splitting\n2gB >0. We solve the equation of motion in Eq. (1)\nwithin the variational ansatz18,49\nθ(x,t) =θ0(x,t)≡2tan−1/bracketleftBig\ne−(rdw(t)−x)/λ/bracketrightBig\n,(12)\ntogether with φ(x,t) =φ0(t), that describes a rigid do-\nmain wall with width λ=/radicalbig\nJ/Kzat position rdw(t).\nThe chirality of the domain wall is determined by the\nangleφ0(t) and the magnetization direction is assumed\nto depend only on xwhich is taken in the long direction\nof the wire.\nThe equations ofmotion for the variationalparameters\nare given by18,29,49\n˙φ0(t)+αG/parenleftbigg˙rdw(t)\nλ/parenrightbigg\n=gB\n/planckover2pi1;\n/parenleftbigg˙rdw(t)\nλ/parenrightbigg\n−αG˙φ0(t) =K⊥\n2/planckover2pi1sin2φ0(t).(13)\nNote that the velocity vsis absent from these equations\nsince we consider the generation of electric current by\na field-driven domain wall. The above equations pro-\nvide a description of the field-driven domain wall and,\nin particular, of Walker breakdown49. That is, for an\nexternal field smaller than the Walker breakdown field\nBw≡αGK⊥/(2g) the domain wall moves with a con-\nstant velocity. For fields B > B wthe domain wall under-\ngoesoscillatorymotion, whichinitially makestheaverage\nvelocity smaller.\nSolving the equations of motion results in\n˙φ0=1\n(1+α2\nG)Re\n/radicalBigg/parenleftbigggB\n/planckover2pi1/parenrightbigg2\n−/parenleftbiggαGK⊥\n2/planckover2pi1/parenrightbigg2\n;\n˙rdw\nλ=gB\nαG/planckover2pi1−˙φ0\nαG, (14)4\n 0 0.5 1 1.5 2\n 0  0.5  1  1.5  2 |j/j0|\nB/Bwβ=0.015\nβ=0.01\nβ=0.005\nβ=0\nFIG. 1: Current generated by a field-driven domain wall in\nunits of j0= 2L/[|e|(σ↑−σ↓)αGK⊥], forαG= 0.01 and\nvarious values of β. The result is plotted as a function of\nmagnetic field in units of the Walker breakdown field Bw≡\nαGK⊥/(2g).\nwhere the ···indicates taking the time-averaged value.\nInsertingthe variational ansatzintoEq. (10) leads in first\ninstance to\n/an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1\n|e|L(σ↑−σ↓)/bracketleftbiggβ˙rdw(t)\nλ+˙φ0(t)/bracketrightbigg\n,(15)\nwhich, using Eq. (14), becomes\n/an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1\n|e|L(σ↑−σ↓)\n\nβgB\nαG/planckover2pi1\n+/parenleftBigg\n1−β\nαG\n1+α2\nG/parenrightBigg\nRe\n/radicalBigg/parenleftbigggB\n/planckover2pi1/parenrightbigg2\n−/parenleftbiggαGK⊥\n2/planckover2pi1/parenrightbigg2\n\n\n.(16)\nAs shown in Fig. 1, this result depends strongly on the\nratioβ/αG. In particular, for β > α Ga local maximum\nappears in the current as a function of magnetic field.\nSinceαGis determined independently from ferromag-\nnetic resonanceexperiments, measurementofthe slopeof\nthecurrentforsmallmagneticfieldsenablesexperimental\ndetermination of β. We note that within the present ap-\nproximation the current does not depend on the domain\nwall width λ. Furthermore, in the limit of zero Gilbert\ndampingand β, thedissipationlesslimit, wehavethatthe\ncurrent density is equal to /an}b∇acketle{tjx/an}b∇acket∇i}ht= (σ↓−σ↑)gB/(|e|L).\nThis is the result of Barnes and Maekawa40that corre-\nsponds to the situation that αG=β= 0, as discussed\nin the Introduction. We point out that, within our ap-\nproximation for the description of domain-wall motion,\nputtingβ=αGin Eq. (16) gives the same result as us-\ning Eqs. (13) and (15) with αG=β= 0. That the\nsituation discussed by Barnes and Maekawa40is indeed\nthat ofαG=β= 0 is seen by comparing their result\n[Eqs. (8) and (9) of Ref. [40], and the paragraph follow-\ning Eq. (9)] with our results in Eqs. (10) and (13).IV. DISCUSSION AND CONCLUSIONS\nOur result in Eq. (16) is a simple expression for the\npumped current as a function of magnetic field for a\nfield-driven domain wall. A possible disadvantage in us-\ning Eq. (16), however, is that in deriving this result we\nassumed a specific model to describe the motion of the\ndomain wall. This model does in first instance not in-\nclude extrinsic pinning and nonzero temperature. Both\nextrinsic pinning18and nonzerotemperature29can be in-\ncluded in the rigid-domain wall description. However,\nit is in some circumstances perhaps more convenient to\ndirectly use the result in Eq. (15) together with the ex-\nperimental determination of ˙ rdw(t). Since the only way\nin which the parameter βenters this equation is as a\nprefactor of ˙ rdw(t), this should be sufficient to determine\nits value from experiment. We note, however, that the\nprecision with which the ratio β/αGcan be determined\ndepends on how accurately the magnetization dynamics,\nand, in particular, the motion of the domain wall, is im-\naged experimentally. With respect to this, we note that\nthe various curves in Fig. 1 are qualitatively different for\ndifferent values of β/αG. In particular, the results for\nβ/αG>1 andβ/αG<1 differ substantially, and could\nmost likely be experimentally distinguished. In view of\nthis discussion, future research will in part be directed\ntowards evaluating Eq. (10) for more complicated mod-\nels of field-driven domain-wall motion, which will benefit\nthe experimental determination of β/αG.\nA typical current density is estimated as follows. For\nthe experiments of Beach et al.50we have that L∼20\nµm, andλ∼20 nm. The domain velocities measured\nin this experiment are ˙ rdw∼40−100 m/s. Taking as\na typical conductivity σ↑∼106Ω−1m−1we find, using\nequation Eq. (15) with β∼0.01, typical electric current\ndensities of the order of /an}b∇acketle{tjx/an}b∇acket∇i}ht ∼103−104A m−2. This re-\nsult depends somewhat on the polarization of the electric\ncurrent in the ferromagnetic metal, which we have taken\nequal to 50% −100% in this rough estimate. Although\nmuch smaller than typical current densities required to\nmove the domain wall via spin transfer torques, electri-\ncal current densities of this order appear to be detectable\nexperimentally.\nIn conclusion, we have presented a theory of spin\npumping without spin conservation, and, in particular,\nproposed a way to gain experimental access to the pa-\nrameterβ/αGthat is of great importance for the physics\nof current-driven domain wall motion. We note that the\nmechanism for current generation discussed in this pa-\nper is quite distinct from the generation of eddy cur-\nrents by a moving magnetic domain51. In addition to\nimproving upon the model used for describing domain-\nwall motion, we intend to investigate in future work\nwhetherthe dampingtermsinEq.(1), orpossiblehigher-\norder terms in frequency and momentum52, have a nat-\nural interpretation in terms of spin pumping, similar to\nthe spin-pumping-enhanced Gilbert damping in single-\ndomain ferromagnets8.5\nIt is a great pleasure to thank Gerrit Bauer, Maxim\nMostovoy, and Henk Stoof for useful comments and dis-cussions.\n1C. Bruder and H. Schoeller, Phys. Rev. Lett. 72, 1076\n(1994).\n2P.W. Brouwer, Phys. Rev. B 58, R10135 (1998).\n3M. Switkes, C. M. Marcus, K. Campman, A. C. Gossard,\nScience283, 1905 (1999).\n4Susan K. Watson, R. M. Potok, C. M. Marcus, and V.\nPhys. Rev. Lett. 91, 258301 (2003).\n5P. Sharma and C. Chamon, Phys. Rev. Lett. 87, 096401\n(2001).\n6D. Cohen, Phys. Rev. B 68, 201303 (2003).\n7Michael Strass, Peter H¨ anggi, and Sigmund Kohler, Phys.\nRev. Lett. 95, 130601 (2005).\n8Y. Tserkovnyak, A. 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Both the reactive and\ndissipative spin transfer torques in Eq. (1) are adiabatic i n\nthe sense that they arise to lowest order in a gradient ex-\npansion of the magnetization direction.\n40S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n41Jun-ichiro Ohe, Akihito Takeuchi, Gen Tatara,\narXiv:0705.0277v2 [cond-mat.mes-hall].\n42W.M. Saslow, Phys. Rev. B 76, 184434 (2007).\n43ShengyuanA.Yang, DiXiao, QianNiu, arXiv:0709.1117v2\n[cond-mat.mtrl-sci].\n44Y. Tserkovnyak and M. Mecklenburg, arXiv:0710.5193v1\n[cond-mat.mes-hall].\n45L. Berger, Phys. Rev. B 33, 1572 (1986).\n46H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 63710\n(2007).\n47See, for example, A. Auerbach, Interacting Electrons and\nQuantum Magnetism (Springer-Verlag, New York, 1994).\n48Huan-Quing Zhou, S. Y. Cho, and R. H. McKenzie, Phys.\nRev. Lett. 91, 186803 (2003).\n49N.L. Schryer and L.R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n50G. S. D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J. L.\nErskine, Nature Materials 4, 741 (2005).\n51F. Colaiori, G. Durin, and S. Zapperi, arXiv:0706.2122v1.\n52See for example Eq. (27) of Ref. [26] for possible higher-\norder terms in the presence of current."    },    {        "title": "0706.4270v2.Coherent_Magnetization_Precession_in_GaMnAs_induced_by_Ultrafast_Optical_Excitation.pdf",        "content": "1 Coherent Magnetization Precession in GaMnAs induced  \nby Ultrafast  Optical Excitation  \n \nJ. Qi, Y. Xu, N. Tolk  \nDepartment of Physics and Astronomy, Vanderbilt University, Nashville, TN, 37235  \nX. Liu, J. K. Furdyna  \nDepartment of Physics, University of Notre Dam e, Notre Dame, IN 46556  \nI. E. Perakis  \nDepartment of Physics, University of Crete, Heraklion, Greece  \n \nWe use femtosecond optical pulses to induce, control and monitor magnetization \nprecession in ferromagnetic Ga 0.965 Mn0.035 As. At temperatures below ~40 K we  \nobserve coherent oscillations of the local Mn spins, triggered by an ultrafast \nphotoinduced reorientation of the in -plane easy axis. The amplitude saturation of the \noscillations above a certain pump intensity indicates that the easy axis remains \nunchanged  above ~T C/2. We find that the observed magnetization precession damping \n(Gilbert damping) is strongly dependent on pump laser intensity, but largely \nindependent on ambient temperature. We provide a physical interpretation of the \nobserved light -induced col lective Mn -spin relaxation and precession.  \n \n \nThe magnetic semiconductor GaMnAs has received considerable attention in \nrecent years, largely because of its potential role in the development of spin -based \ndevices1,2. In this itinerant ferromagnet, the collec tive magnetic order arises from the \ninteraction between mobile valence band holes and localized Mn spins. Therefore, the \nmagnetic properties are sensitive to external excitations that change the carrier density \nand distribution. Ultrafast pump -probe magnet o-optical spectroscopy is an ideal \ntechnique for controlling and characterizing the magnetization dynamics in the \nmagnetic materials, and has been applied to the GaMnAs system by several groups3,4. \nAlthough optically induced precessional motion of magne tization has been studied in 2 other magnetic systems5, magnetization precession in ferromagnetic GaMnAs has \nbeen observed only recently4 and has yet to be adequately understood.  \n \nIn this paper, we  report comprehensive temperature and photoexcitation intens ity \ndependent measurements of photoinduced magnetization precession in Ga 1-xMnxAs (x \n= 0.035) with no externally imposed magnetic field. By comparing and contrasting \nthe temperature and intensity dependence of the precession frequency, damping, and \namplitu de, we identify the importance of light -induced nonlinear effects and obtain \nnew information on the relevant physica l mechanisms. Our measurements of the \nphotoinduced magnetization show coherent oscillations, arising from the precession \nof collective Mn sp ins. Amplitude of the magnetization precession saturates above \ncertain pump intensity is a strong indication that direction of the magnetic easy axis \nremains unchanged at temperatures above about half the Curie temperature (T C). The \nprecession  is explained  by invoking an ultrafast change in the orientation of the \nin-plane easy axis, due to an impulsive change in the magnetic anisotropy induced by \nthe laser pulse. We also find that the Gilbert damping coefficient, which characterizes \nthe Mn -spin relaxation, depends only weakly on the ambient temperature but changes \ndramatically with pump intensity . Our results suggest a general model for \nphotoinduced precessional motion and relaxation of magnetization in the GaMnAs \nsystem under compressive strain.  \n \nTime -resolved magneto -optical Kerr effect (MOKE) measurements were \nperformed on a 300 nm thick ferromagnetic Ga 1-xMnxAs (x = 0.035) sample, with \nbackground hole density p ≈1020 cm-3 and T C ≈70 K. The sample was grown by low \ntemperature molecular beam epitaxy  on a G aAs(001) substrate, and was therefore \nunder compressive strain. The pump -probe experiment employed a Coherent MIRA \n900 Ti:Sapphire laser, which produced ~150 -fs-wide pulses in the 720 nm (1.719 eV) \nto 890 nm (1.393 eV) wavelength range with a repetition ra te of 76 MHz. The pump \nbeam was incident normal to the sample, while the probe was at an angle of about 30o \nto the surface normal. The polarization of the pump beam could be adjusted to be 3 linear, right -circular ( σ+), or left -circular ( σ-) polarization. Th e probe beam was \nlinearly polarized. This configuration produced a combination of polar and \nlongitudinal MOKE, with the former dominating6. The temporal Kerr rotation signal \nwas detected using a balanced photodiode bridge, in combination with a lock -in \namplifier. Both pump and probe beams were focused onto the sample with a spot \ndiameter of about 100 µm, with an intensity ratio of 15:1. The pump light typically \nhad a pulse energy of 0.065 nJ, and a fluence of 0.85 µJ/cm2. \n \nFigure 1(a) shows our time -resolve d Kerr rotation (TRKR) measurements at \ntemperature of 20 K. The amplitude of the temporal Kerr rotation  signal was found to \nbe symmetric with respect to right or left -circularly polarized photo -excitation. In \nparticular, we observe a superimposed oscillato ry behavior at temperatures less than \n~40 K, indicating magnetization precession. It is important to note that these \noscillations were observed not only with σ+ and σ- polarized but also with linearly \npolarized pump light . The phase difference among the os cillations for σ+ or σ- pump \nexcitation is less than ~5o. This negligible phase difference implies that the oscillatory \nbehavior is not due to the non -thermal circular polarization -dependent carrier spin \ndynamics5. After the initial few picoseconds, where equilibration between the \nelectronic and lattice systems occurs, the oscillations can be fitted well by the \nfollowing equation (see Fig. 1(b)):  \n) cos()/ exp( )( 0j w t q + − = t t AtK           (1),  \nwhere A0, w, t, and j are the amplitude, precession frequency, decay time, and initial \nphase of  the oscillation, respectively. Some fitted parameters are shown in figure 2 as \na function of pump intensity and temperature.  \n \nOn a sub -picosecond time scale, the photoexcited electrons/holes scatter  and  \nthermalize with the Fermi sea via electron -electron  interactions. Following this initial \nnon-thermal temporal regime, the properties of the GaMnAs system can be \ncharacterized approximately by time -dependent carrier and lattice temperatures heT/ 4 and lT. Subsequently, the carrier system transfers its energy to the lattice within a \nfew picoseconds via the electron -phonon interaction. This leads to a \nquasi -equilibration of heT/and lT, which then relax back to the equilibrium \ntempera ture via a slow (ns) thermal diffusion process. Mn precession was also \nobserved in Ref. 4 and was attributed to the change of uniaxial anisotropy due to the \nincrease in hole concentration7. In our experiment, for a typical pump intensity of \n0.065 nJ/pulse,  we estimate that the relative increase in hole concentration is ~0.1%. \nThe resulting transient increase in local temperature and hole concentration leads to \nan impulsive change  in the in -plane magnetic anisotropies and in the easy axis \norientation . As a r esult, the effective magnetic field experienced by the Mn spins  \nchanges, thereby triggering the observed precession.  \n \nIt is known that the magnetic anisotropy parameters (i.e., uniaxial anisotropy \nconstant K1u and cubic anisotropy constant K1c), which dete rmine the direction of the \neasy axis in the GaMnAs system, are functions of temperature and hole \nconcentration4,7,8. Thus when the GaMnAs system is excited by an optical pulse, a \ntransient change in local hole concentration Δp and local temperature ΔT, reflecting \nvariation of both the carrier temperature )(/t TheΔ  and the lattice temperature ()lTtΔ, \ncan lead to changes in the magnetic anisotropy parameters. Below the Curie \ntemperature, the direction of the in -plane magnetic eas y axes (given by the angle f) \ndepends on the interplay between K1u and -K1c. After the optical excitation, the new \nangle of the in -plane easy axes is given by 100\n100((),())\n()((),())u\ncKTTtppt\ntKTTtpptff +Δ+Δ=+Δ+Δ , \nwhere T0 and p0 are the initial (ambient) temperature and hole conce ntration7,8. \nTherefore, the in -plane easy axes may quickly assume a new direction following \nphotoexcitation if ΔT(t) and Δp(t) are sufficiently strong. This transient change in the \nmagnetic easy axis, due to the change in the minimum of the magnetic free e nergy as \nfunction of Mn spin induced by the photoexcitation, triggers a precessional motion of \nthe magnetization around the new effective magnetic field.  5  \nWithin the mean field treatment of the p-d magnetic exchange interaction,  the \nMn spins precess aroun d the effective magnetic field Mn\neffH, which is determined by \nthe sum of the anisotropy field Mn\nanisH  and the hole -spin mean field JNholem. The \ndynamics of the hole magnetization  m is determined by its precession around the  \nmean field JNMnM due to the Mn spins , and by its rapid relaxation due to the strong \nspin-orbit interaction in the valence band with a rate ΓSO of the order of tens of \nfemtoseconds9,10. Here, m (M) is the hole (Mn) magnetization , J is the exchange \nconstant , and Nhole (NMn) is the number of holes (Mn -spins)2. For small fluctuations of \nthe magnetization orientation around the easy axis, the magnetization  dynamics  can \nbe described by the Landau -Lifshitz -Gilbert (LLG) equation2, which is appropriate to \napply to  our experimental data at low -pump intensities (e.g., 0.065 nJ/pulse). In the \nadiabatic limit, where the hole spins precess and relax much faster than the Mn spins, \nwe can eliminate the hole spins by transforming to the rotating frame11. In this way \nwe obt ain an effective LLG equation for the Mn magnetization M, whose precession \nis governed by the anisotropy field MnanisH  and  the effective Gilbe rt damping \ncoefficient including  the damping a0 due to e.g. spin -lattice interactions and the \ncontribution due to the p-d exchange interaction, which depends on the hole \nconcentration, the ratio of hole spin relaxation energy over exchange interaction \nenergy, and the ratio m/M of the collective hole and Mn spins[9,10].  \n \n    The LLG equation predict s an oscillatory behavior of the magnetization due to \nthe precession of the local Mn moments around the magnetic anisotropy field \nMnanisH (T0+??(t), p0+?p(t)). The precession frequency is proportional to this anisotropy \nfield, which is given by the gradient of magnetic free energy and is proportional to the \nanisotropy constants  K1u and K1c.2 The magnitude of MnanisH decreases as the ambient \ntemperature T0 or the transient temperature DT increases, primarily due to the  \ndecrease in K1c8. This leads to the decrease of the precession  frequency as the ambient 6 temperature or the pump intensity increases, and is consistent with the be havior \nobserved in Fig. 2. It should be pointed out that, in the Fourier transform of each \ntemporal signal of the oscillations, only one oscillatory mode was observed (see also \nin Fig. 1(b) and Fig. 1(c)). This indicates that only a single uniform -precessi on \nmagnon was excited in our experiment, and the scattering among uniform -precession \nmagnons can be neglected when interpreting damping of the Mn spin precession.  \n \nAs can be seen in Fig. 2,  the amplitude of the oscillations increas es as the \nambient tempera ture T0 decreas es or as the pump intensity increases. This result is in \naccord with the fact that the relative change DT/T0 and Δp/p0, which determines the \nmagnitude of f(t) and the photo -induced tilt in the easy axis, increase as T0 decreases \nor as the pu mp intensity increases. It is important to note that in our experiment the \namplitude of the oscillations saturates as the pump intensity exceeds about 4 I0 \n(I0=0.065nJ/pulse) at T0=10 K. Thus, the observed saturation may indicate that the \nmagnetic easy axi s is stabilized at pump intensity larger than 4 I0. We estimated that \nthe increase of local hole concentration Δp/p0 is about 0.4%, and the local temperature \nincrease ΔT /T0 is about 160% using the value of specific heat of 1mJ/g/K for GaAs12 \nfor pump inte nsity ~4 I0. This results in the transient local temperature T0+ΔT close to \nTC/2. Because the magnetic easy axis is already along the [110] direction for T0+ΔT \nclose to or higher than T C/28, our observed phenomenon is in agreement with the \nprevious reported  results.  \n \nFinally we turn to the oscillation damping, which is intimately related to \ncollective localized -spin lifetimes, and consequently to spintronic device development. \nFigure 3 shows the fitted Gilbert damping coefficient a obtained by using the LLG \nequation as a function of pump intensity and ambient temperature, respectively. It can \nbe seen that in Figure 3(a) a changes weakly with the ambient temperature and has an \naverage value ~0.135 for a fixed pump intensity of 0.065 nJ/pulse. However, Figure \n3(b) shows that a increases nonlinearly as pump intensity increases. To interpret these \nresults, we note that, as discussed above, the p-d kinetic -exchange coupling between 7 the local Mn moments and the itinerant carrier spins contributes significantly to \nGilbert damping10. In particular, a increases with increasing ratio m/M. The ratio \nm/M is known to increase nonlinearly with increasing temperature13 and should \ntherefore depend nonlinearly on the photoexcitation. The Gilbert damping  coefficient \ndue to the e xchange interaction also increases as hole density p and hole spin \nrelaxation rate SO\nMnJMNΓ\n increase. Here, ΓSO and Δp/p (<0.01) are relatively small. \nThus we can conclude that the damping coefficient due to the p-d exchange \ninteraction should increase with increasing ambient temperature ( T0) or increasing \npump intensity  (ΔT and Δp). On the other hand, we also note that damping may arise \nfrom an extrinsic inhomogeneous MnanisH  broadening attributed to a local temperatu re \ngradient due to inhomogeneities in the laser beam intensity profile and the detailed \nstructure of the sample. This extrinsic damping effect is expected to decrease as the \nambient temperature increases or the pump intensity decreases.  Thus, the data in F ig. \n3(a), which shows a only weakly dependent on ambient temperature, may result from \nthe competition between these two mechanisms, both of which, however, predict the \nresult in Fig. 3(b) that the damping coefficient increases nonlinearly with the increase  \nof pump intensity. It is important to note that the LLG equation is valid only at low \npump intensities. At high pump intensities, an alternative theoretical approach must be \nintroduced[14]. So our new experimental results in the time domain are not access ible \nwith static FMR experiments, and provide new information on the physical factors \nthat contribute to the damping effect.  \n \nIn conclusion, we have studied the photoinduced magnetization dynamics in \nGa0.965 Mn0.035 As by time -resolved MOKE with zero externa l magnetic fields. At \ntemperatures below ~40 K, a precessional motion associated with correlated local Mn  \nmoments was observed. This precession is attributed to an ultrafast reorientation of \nthe in -plane magnetic easy axis from an impulsive change in the m agnetic anisotropy \ndue to photoexcitation. Our results indicate that the magnetic easy axis does not 8 change at temperatures above about T C/2. We find the Gilbert damping coefficient is \nindependent of ambient temperature but depends nonlinearly on the pump intensity, \nWe attribute this nonlinearity to the hole -Mn spin exchange interaction and the \nextrinsic anisotropy field broadening due to temperature gradients in the sample. Our \nresults show that ultrafast optical excitation provides a way to control the am plitude, \nprecession frequency and damping of the oscillations arising from coherent localized \nMn spins in the GaMnAs system.  \n \nThis work was supported by ARO Grant W911NF -05-1-0436 (VU), NSF Grant \nDMR06 -03752 (ND), and by the EU STREP program HYSWITCH (Cret e).  \n \n1 H. Ohno, Science 281, 951 (1998)  \n2 J. Jungwirth, J.Sinova, J.Masek, J.Kucera, and A.H. MacDonald, Rev. Mod. Phys. \n78, 809 (2006)  \n3 J. Wang, C. Sun, Y. Hashimoto, J. Kono, G.A. Kh odaparast, L. Cywinski, L.J. Sham, \nG.D. Sanders, C.J. Stanton  and H. Munekata , J. Phys.:Condens. Matter 18, R501 \n(2006)  \n4 A. Oiwa, H. Takechi, and H. Munekata, Journal of Superconductivity 18, 9 (2005)  \n5 A.V. Kimel, A. Kirilyuk, F. Hansteen, R.V. Pisarev, and Th. Rasing , \nJ.Phys.:Condens. Matter 19, 043201(2007)  \n6 V. A. Kotov and A.  K. Zvezdin. Modern Magnetooptics and Magneto Optical \nMaterials . (Institute of Physics, London, 1997)  \n7 T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 (2001)  \n8 U.Wel p, V.K.Vlasko -Vlasov, X.Liu, J.K.Furdyna, and T.Wojtowicz, Phys. Rev. Lett. \n90, 167206 (2003)  \n9 J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys. Rev. Lett. 96, 057402 (2006) \nand unpublished.  \n10J.Sinova, T.Jungwirth, X.Liu, Y.Sasaki, J.K.Furdiyna, W.A .Atkinson, and \nA.H.MacDonald, Phys.  Rev. B 69, 085209  (2004)  \n11 Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 (2004)  9 12 J. S. Blakemore, J.  Appl.  Phys. 53, R123  (1982)  \n13 S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys. Rev. B 67, 155201(2003)  \n14 H. Suhl, IEEE Trans.Magn. 34, 1834(1998)  10 0 200 400 600 800-0.4-0.20.00.20.4FFT (a.u.)Kerr rotation (mdeg)\nLinear\nσ- σ+T=20K Kerr rotation (mdeg)\nDelay time (ps)0 200 400 600 800-0.010.000.01(b)\n   \nDelay time (ps)\n(a)\n0 5 10 15 20 25 30024(c)\n  \nFrequency (GHz)\n \nFigure 1. (a) Kerr rotation measurements for Ga 1-xMnxAs (x = 0.035) excited by \nlinearly -polarized and circularly -polarized light ( σ+ and σ-) at a temperature of 20 K. \nThe photon energy was 1 .56 eV. Oscillations due to magnetization precession are \nsuperimposed on the decay curves. (b) Oscillation data  (open circles) extracted from \n(a). The solid line is the fitted result. (c) Fourier transformation profile for the \noscillation data in (b).  \n10 20 30 40102030\nPump Intensity( I0) Temperature (K)\n \n ω (GHz)204060\n  \n A0 (µdeg) I=I0\n50100150200\nT0=10K  \n \n \n246810142128 \n \n \n \nFigure 2 Amplitude A0 and angular frequency w as a function of temperature T0 at \nconstant pump intensity I=I0; and as a function of pump intensity (in units of I0) at T0 \n= 10 K. I0 = 0.065 nJ/pulse.  11 0.5 1.0 1.5 2.00.100.150.200.25Damping Coefficient α\n  \nPump Intensity (I0)(b)\nT0=10K10 20 30 400.090.120.150.18\n  Damping Coefficient α\nTemperature (K)(a)\nI=I0\n \nFigure 3 (a) Gilbert damping coefficient a as a function of temperature (T 0) at a \nconstant pump intensity I=I0. I0=0.065 nJ/pulse; (b) Gilbert damping coefficient a as a \nfunction of pump intensity in units of I0 at T0= 10 K.  "    },    {        "title": "0708.0463v1.Strong_spin_orbit_induced_Gilbert_damping_and_g_shift_in_iron_platinum_nanoparticles.pdf",        "content": "arXiv:0708.0463v1  [cond-mat.other]  3 Aug 2007Strong spin-orbit induced Gilbert damping and g-shift in ir on-platinum nanoparticles\nJ¨ urgen K¨ otzler, Detlef G¨ orlitz, and Frank Wiekhorst\nInstitut f¨ ur Angewandte Physik und Zentrum f¨ ur Mikrostruk turforschung,\nUniversit¨ at Hamburg, Jungiusstrasse 11, D-20355 Hamburg , Germany\n(Dated: October 30, 2018)\nThe shape of ferromagnetic resonance spectra of highly disp ersed, chemically disordered\nFe0.2Pt0.8nanospheres is perfectly described by the solution of the La ndau-Lifshitz-Gilbert (LLG)\nequation excluding effects by crystalline anisotropy and su perparamagnetic fluctuations. Upon\ndecreasing temperature , the LLG damping α(T) and a negative g-shift,g(T)−g0, increase pro-\nportional to the particle magnetic moments determined from the Langevin analysis of the magneti-\nzation isotherms. These novel features are explained by the scattering of the q→0 magnon from\nan electron-hole (e/h) pair mediated by the spin-orbit coup ling, while the sd-exchange can be ruled\nout. The large saturation values, α(0) = 0.76 andg(0)/g0−1 =−0.37, indicate the dominance of\nan overdamped 1 meV e/h-pair which seems to originate from th e discrete levels of the itinerant\nelectrons in the dp= 3nmnanoparticles.\nPACS numbers: 76.50.+g, 78.67.Bf, 76.30.-v, 76.60.Es\nI. INTRODUCTION\nTheongoingdownscalingofmagneto-electronicdevices\nmaintainstheyetintenseresearchofspindynamicsinfer-\nromagnetic structures with restricted dimensions. The\neffect of surfaces, interfaces, and disorder in ultrathin\nfilms1, multilayers, and nanowires2has been examined\nand discussed in great detail. On structures confined to\nthe nm-scale in all three dimensions, like ferromagnetic\nnanoparticles, the impact of anisotropy3and particle-\nparticle interactions4on the Ne´ el-Brown type dynamics,\nwhich controls the switching of the longitudinal magneti-\nzation, is now also well understood. On the other hand,\nthe dynamics of the transverse magnetization, which e.g.\ndetermines the externally induced, ultrafast magnetic\nswitching in ferromagnetic nanoparticles, is still a top-\nical issue. Such fast switching requires a large, i.e. a\ncritical value of the LLG damping parameter α5. This\ndamping has been studied by conventional6,7and, more\nrecently, by advanced8ferromagnetic resonance (FMR)\ntechniques, revealing enhanced values of αup to the or-\nder of one.\nBy now, the LLG damping of bulkferromagnets is al-\nmost quantitatively explained by the scattering of the\nq= 0 magnon by conduction electron-hole (e/h) pairs\ndue to the spin-orbit coupling Ω so9. According to recent\nab initio bandstructure calculations10, the rather small\nvalues for α≈Ω2\nsoτresult from the small (Drude) re-\nlaxation time τof the electrons. For nanoparticles, the\nDrude scattering and also the wave-vector conservation\nare ill-defined, and ab initio many-body approaches to\nthe spin dynamics should be more appropriate. Numeri-\ncal workby Cehovin et al.11considersthe modification of\nthe FMR spectrum by the discrete level structure of the\nitinerant electrons in the particle. However, the effect of\nthe resulting electron-hole excitation, ǫp∼v−1\np, wherevp\nis the nanoparticle volume on the intrinsic damping has\nnot yet been considered.\nHere we present FMR-spectra recorded atω/2π=9.1 GHz on Fe0.2Pt0.8nanospheres, the struc-\ntural and magnetic properties of which are summarized\nin Sect. II. In Sect. III the measured FMR-shapes will be\nexamined by solutions of the LLG-equation of motion for\nthe particle moments. Several effects, in particular those\npredicted for crystalline anisotropy12and superparam-\nagnetic (SPM) fluctuations of the particle moments13\nwill be considered. In Sect. IV, the central results of this\nstudy, i.e. the LLG-damping α(T) reaching values of\nalmost one and a large g-shift,g(T)−g0, are presented.\nSince both α(T) and g(T) increase proportional to the\nparticle magnetization, they can be related to spin-orbit\ndamping torques, which, due to the large values of αand\n∆gare rather strongly correlated. It will be discussed\nwhich features of the e/h-excitations are responsible for\nthese correlations in a nanoparticle. A summary and the\nconclusions are given in the final section.\nII. NANOPARTICLE CHARACTERIZATION\nThe nanoparticleassemblyhasbeen prepared14follow-\ning the wet-chemical route by Sun et al.15. In order\nto minimize the effect of particle-particle interactions,\nthe nanoparticles were highly dispersed14. Transmis-\nsion electron microscopy (TEM) revealed nearly spher-\nical shapes with mean diameter dp= 3.1nmand a\nrather small width of the log-normal size distribution,\nσd= 0.17(3). Wide angle X-ray diffraction provided the\nchemically disordered fccstructure with a lattice con-\nstant a 0=0.3861 nm14.\nThemeanmagneticmomentsofthenanospheres µp(T)\nhave been extracted from fits of the magnetization\nisotherms M(H,T), measured by a SQUID magnetome-\nter (QUANTUM DESIGN, MPMS2) in units emu/g=\n1.1·1020µB/g, to\nML(H,T) =Npµp(T)L(µpH\nkBT). (1)2\nHere are L(y) = coth( y)−y−1withy=µp(T)H/kBT\nthe Langevin function and Npthe number of nanoparti-\ncles per gram. The fits shown in Fig.1(a) demanded for\na small paramagnetic background, M−ML=χb(T)H,\nwith a strong Curie-liketemperature variationof the sus-\nceptibility χb, signalizing the presence of paramagnetic\nimpurities. According to the inset of Fig. 1(b) this 1 /T-\nlaw turns out to agree with the temperature dependence\nof the intensity of a weak, narrow magnetic resonance\nwithgi= 4.3 depicted in Figs. 2 and 3. Such narrow\nline with the same g-factor has been observed by Berger\net al.16on partially crystallized iron-containing borate\nglass and could be traced to isolated Fe3+ions.\nThe results for µp(T) depicted in Fig.1(b) show the\nmoments to saturate at µp(T→0) = (910 ±30)µB.\nThis yields a mean moment per atom in the fccunit\ncell ofµ(0) =µpa3\n0/4vp= 0.7µBcorresponding to a\nspontaneous magnetization Ms(0) = 5.5kOe. Accord-\ning to previous work by Menshikov et al.17on chemi-\ncally disordered FexPt1−xthis corresponds to an iron-\nconcentration of x=0.20. Upon rising temperature the\nmoments decrease rapidly, which above 40 K can be\nrather well parameterized by the empirical power law,\nµp(T≥40K)∼(1−T/TC)βrevealing β= 2 and for\nthe Curie temperature TC= (320±20)K. This is con-\nsistent with TC= (310±10)KforFe0.2Pt0.8emerging\nfrom a slight extrapolation of results for TC(x≥0.25)\nofFexPt1−x18. No quantitative argument is at hand\nfor the exponent β= 2, which is much larger than the\nmean field value βMF= 1/2. We believe that β= 2\nmay arise from a reduced thermal stability of the magne-\ntization due to strong fluctuations of the ferromagnetic\nexchange between FeandPtin the disordered struc-\nture and also to additional effects of the antiferromag-\nneticFe−FeandPt−Ptexchange interactions. In\nthis context, it may be interesting to note that for low\nFeconcentrations, x≤0.3, bulkFexPt1−xexhibits fer-\nromagnetism only in the disordered structure17, while\nstructural ordering leads to para- or antiferromagnetism.\nRecent first-principle calculations of the electronic struc-\nture produced clear evidence for the stabilizing effect of\ndisorder on the ferromagnetism in FePt19.\nFrom the Langevin fits in Fig.1(a) we obtain for the\nparticle density Np= 3.5·1017g−1. Basing on the well\nknown mass densities of Fe0.2Pt0.8and the organic ma-\ntrix, we find by a little calculation20for the volume con-\ncentration of the particles cp= 0.013 and, hence, for the\nmean inter-particle distance, dpp=dp/c1/3\np= 13.5nm.\nThis implies forthe maximum (i.e. T=0) dipolar interac-\ntion between nearest particles, µ2\np(0)/4πµ0d3\npp= 0.20K,\nso that at the present temperatures, T≥20K, the sam-\nple should act as an ensemble of independent ferromag-\nnetic nanospheres. Since also the blocking temperature,\nTb= 9K, as determined from the maximum of the ac-\nsusceptibility at 0.1 Hz in zero magnetic field20, turned\nout to be low, the Langevin-analysis in Fig.1(a) is valid.FIG. 1: Fig. 1. (a) Magnetization isotherms of the\nnanospheres fitted to the Langevin model plus a small para-\nmagnetic background χb·H; (b) temperature dependences\nof the magnetic moments of the nanoparticles µpand of the\nbackground susceptibility χb( inset units are emu/g kOe )\nfitted to the indicated relations with TC= (320±20)K. The\ninset shows also data from the intensity of the paramagnetic\nresonance at 1.45 kOe, see Figs. 2 and 3.\nIII. RESONANCE SHAPE\nMagnetic resonances at a fixed X-band frequency of\n9.095 GHz have been recorded by a home-made mi-\ncrowave reflectometer equipped with field modulation to\nenhancethesensitivity. Adouble-walledquartztubecon-\ntaining the sample powder has been inserted to a mul-\ntipurpose, gold-plated VARIAN cavity (model V-4531).\nKeeping the cavity at room temperature, the sample\ncouldbeeithercooledbymeansofacontinuousflowcryo-\nstat (Oxford Instruments, model ESR 900) down to 15 K\norheatedupto500Kbyanexternal Pt-resistancewire20.\nAt all temperatures, the incident microwave power was\nvaried in order to assure the linear response.\nSomeexamplesofthespectrarecordedbelowthe Curie3\nFIG. 2: (a) Derivative of the microwave (f=9.095 GHz) ab-\nsorption spectrum recorded at T=52 K i.e. close to magnetic\nsaturation of the nanospheres. The solid and dashed curves\nare based on fits to Eqs.(4) and (8), respectively, which both\nignore SPM fluctuations and assume either a g-shift and zero\nanisotropy field HA(’∆g-FM’) or ∆ g= 0 and a randomly\ndistributed HA=0.5 kOe (’a-FM’), Eq. (9). Also shown are\nfits to predictions by Eq.(11), which account for SPM fluctu-\nations, with HA=0.5 kOe and ∆ g= 0 (’a-SPM’) and, using\nEq.(11), for ∆ g/negationslash= 0 and HA=0 (’∆g-SPM’). The weak, nar-\nrow resonance at 1.45 kOe is attributed to the paramagnetic\nbackground with g= 4.3±0.1 indicating Fe3+ 16impurities.\ntemperature are shown in Figs.2 and 3. The spectra have\nbeen measured from -9.5 kOe to +9.5 kOe and proved to\nbe independent of the sign of H and free of any hystere-\nsis. Thiscanbeexpectedduetothecompletelyreversible\nbehavior of the magnetization isotherms above 20 K and\nthe low blocking temperature of the particles. Lower-\ning the temperature, we observe a downward shift of the\nmainresonanceaccompaniedbyastrongbroadening. On\nthe other hand, the position and width of the weak nar-\nrow line at (1 .50±0.05)kOecorresponding to gi∼=4.3\nremain independent of temperature. This can be at-\ntributed to the previously detected paramagnetic Fe3+-\nimpurities16and is supported by the integrated intensity\nof this impurity resonance Ii(T) evaluated from the am-\nplitudedifferenceofthederivativepeaks. Sincetheinten-\nsity of a paramagnetic resonance is given by the param-\nagnetic susceptibility, Ii(T)∼/integraltext\ndH χ′′\nxx(H,T)∼χi(T)\ncan be compared directly to the background suscepti-\nbilityχb(T), see inset to Fig.1(b). The good agree-\nment between both temperature dependencies suggests\nto attribute χbto these Fe3+-impurities. An analysis of\nthe fitted Curie-constant, Ci= 5emuK/g kOe , yields\nNi= 164.1017g−1for theFe3+- concentration, which\ncorresponds to 50 Fe3+per 1150 atoms of a nanosphere.\nWith regard to the main intensity, we want to extract\na maximum possible information, in particular, on theFIG. 3: Fig. 3. Derivative spectra at some representative\ntemperatures and fits to Eq.(4). The LLG-damping, g-shift,\nand intensities are depicted in Fig. 4.\nintrinsic magnetic damping in nanoparticles. Unlike the\nconventional analysis of resonance fields and linewidths,\nas applied e.g. to Ni6and Co7nanoparticles, our ob-\njective is a complete shape analysis in order to disen-\ntangle effects by the crystalline anisotropy12, by SPM\nfluctuations13, by an electronic g-shift21, and by differ-\nent forms ofthe damping torque /vectorR22. Additional difficul-\nties may enter the analysis due to non-spherical particle\nshapes, size distributions and particle interaction, all of\nwhich, however, can be safely excluded for the present\nnanoparticle assembly.\nThe starting point of most FMR analyses is the phe-\nnomenological equation of motion for a particle moment\n( see e.g. Ref. 13 )\nd\ndt/vector µp=γ/vectorHeff×/vector µp−/vectorR , (2)\nusing either the original Landau-Lifshitz (LL) damping\nwith damping frequency λL\n/vectorRL=λL\nMs(/vectorHeff×/vector µp)×/vector sp ,(3)\northeGilbert-dampingwiththe Gilbertdampingparam-\neterαG,\n/vectorRG=αGd/vector µp\ndt×/vector sp , (4)\nwhere/vector sp=/vector µp/µpdenotes the direction of the particle\nmoment. In Eq.(2), the gyromagneticratio, γ=g0µB/¯h,\nis determined by the regular g-factorg0of the precessing\nmoments. It shouldperhapsbe notedthat the validityof\nthe micromagnetic approximationunderlying Eq. (2) has\nbeen questioned23for volumes smaller than (2 λsw(T))3,4\nwhereλsw= 2a0TC/Tis the smallest wavelengthof ther-\nmally excited spin waves. For the present particles, this\nestimate leads to a fairly large temperature of ∼0.7TC\nup to which micromagnetics should hold.\nAt first, we ignore the anisotropy being small in cubic\nFexPt1−x24,25, so that for the present nanospheres the\neffective field is identical to the applied field, /vectorHeff=\n/vectorH. Then, the solutions of Eq. (2) for the susceptibility\nof the two normal, i.e. circularly polarized modes, of\nNpindependent nanoparticles per gram take the simple\nforms\nχL\n±(H) =Npµpγ1∓iα\nγH(1∓iα)∓ω(5)\nfor/vectorR=/vectorRLwithα=λL/γMsand for the Gilbert torque\n/vectorRG\nχG\n±(H) =Npµpγ1\nγH∓ω(1+iαG).(6)\nFor the LL damping, the experimental, transverse sus-\nceptibility, χxx=1\n2(χ++χ−), takes the form\nχL\nxx(H) =NpµpγγH(1+α2)−iαω\n(γH)2(1+α2)−ω2−2iαωγH.\n(7)\nAs thesameshape isobtainedforthe Gilbert torquewith\nα=αG, the damping is frequently denoted as LLG pa-\nrameter. However, the gyromagnetic ratio in Eq.(7) has\ntobereplacedby γ/(1+α2), whichonlyfor α≪1implies\nalso the same resonance field Hr. Upon increasing the\ndamping up to α≈0.7 (the regime of interest here), the\nresonance field HrofχG\nxx(H) , determined by dχ′′/dH=\n0,remains constant, HG\nr≈ω/γ, whileHL\nrdecreases\nrapidly. After renormalization γ/(1+α2) the resonance\nfields and also the shapes of χL(H) andχG(H) become\nidentical. This effect should be observed when determin-\ning theg-factor from the resonance fields of broad lines.\nIt becomes even more important if the downward shift of\nHris attributed to anisotropy, as done recently for the\nratherbroadFMR absorptionof FexPt1−xnanoparticles\nwith larger Fe-content, x≥0.326.\nIn order to check here for both damping torques,\nwe selected the shape measured at a low temperature,\nT=52 K, where the linewidth proved to be large (see\nFig. 2) and the magnetic moment µp(T) was close to sat-\nuration (Fig. 1(b)). None of both damping terms could\nexplain both, the observed resonance field Hrand the\nlinewidth ∆ H=αω/γ, and, hence, the lineshape. By\nusing/vectorRG, the shift of Hrfromω/γ= 3.00kOewas not\nreproduced by HG\nr=ω/γ, while for /vectorRLthe resonance\nfieldHL\nr, demanded by the line width, became signifi-\ncantly smaller than Hr.\nThis result suggested to consider as next the effect of a\ncrystallineanisotropyfield /vectorHAonthetransversesuscepti-\nbility, which has been calculated by Netzelmann from thefree energy of a ferromagnetic grain12. Specializing his\ngeneral ansatz to a uniaxial /vectorHAoriented at angles ( θ,φ)\nwithrespecttothedc-field /vectorH||/vector ezandthemicrowavefield,\none obtains by minimizing\nF(θ,φ,ϑ,ϕ) =−µp[Hcosϑ+\n1\n2HA(sinϑsinθ−cos(ϕ−φ)+cosθcosϑ)2] (8)\nthe equilibrium orientation ( ϑ0,ϕ0) of the moment /vector µpof\na spherical grain. After performing the trivial average\noverφ, one finds for the transverse susceptibility of a\nparticle with orientation θ\nχL\nxx(θ,H) =γµp\n2×\n(Fϑ0ϑ0+Fϕ0ϕ0/tan2ϑ0)(1+α2)−iαµpω(1+cos2ϑ0)\n(1+α2)(γHeff)2−ω2−iαωγ∆H.\n(9)\nHereH2\neff= (Fϑ0ϑ0Fϕ0ϕ0−F2\nϑ0ϕ0/(µpsinϑ0)2) and\n∆H= (Fϑ0ϑ0+Fϕ0ϕ0/sin2ϑ0)/µpare given by the sec-\nond derivatives of F at the equilibrium orientation of /vector µp.\nFor the randomly distributed /vectorHAofNpindependent par-\nticles per gram one has\nχL\nxx(H) =/integraldisplayπ/2\n0d(cosθ)χL\nxx(θ,H).(10)\nIn a strict sense, this result should be valid at fields\nlarger than the so called thermal fluctuation field HT=\nkBT/µp(T), see e.g. Ref. 13, which for the present case\namounts to HT= 1.0kOe. Hence, in Fig. 2 we fit-\nted the data starting at high fields, reaching there an\nalmost perfect agreement with the curve a-FM. The fit\nyields a rather small HA= 0.5kOewhich implies a small\nanisotropyenergyper atom, EA=1\n2µp(0)HA= 1.0µeV.\nThis number is smaller than the calculated value for bulk\nfcc FePt ,EA= 4.0µeV25, most probably due to the\nlowerFe-concentration (x=0.20) and the strong struc-\ntural disorder in our nanospheres. We emphasize, that\nthe main defect of this a-FM fit curve arises from the\nfinite value of dχ′′\nxx/dHatH= 0. By means of Eq. (9)\none finds χ′′\nxx(H→0,θ)∼HAH/ω2, which remains fi-\nnite even after averaging over all orientations according\ntoθ(Eq. (10)).\nThefinite value ofthe derivativeof χ′′\nxx(H→0)should\ndisappear if superparamagnetic (SPM) fluctuations of\nthe particles are taken into account. Classical work27\npredicted the anisotropy field to be reduced by SPM,\nHA(y) =HA·(1/L(y)−3/y), which for y=H/HT≪1\nimpliesHA(y) =HA·y/5 and, therefore, χ′′\nxx(H→0)∼\nH2. A statistical theory for χL\nxx(H,T) which considers\nthe effect of SPM fluctuations exists only to first order in\nHA/H21. The result of this linear model (LM) in HA/H\nwhich generalizes Eq. (4), can be cast in the form\nχLM\n±(θ,H) =NpµpL(y)γ(1+A∓iαA)\nγ(1+B∓iαB)H∓ω.\n(11)5\nThe additional parameters are given in Ref. 21 and con-\ntain, depending on the symmetry of HA, higher-order\nLangevin functions Lj(y) and their derivatives. Observ-\ning the validity of the LM for H≫HA= 0.5kOe, we\nfitted the data in Fig. 2 to Eq. 11 with χLM\nxx(θ,H) =\n1\n2(χLM\n++χLM\n−) at larger fields. There one has also H≫\nHT= 1.0kOeand the fit, denoted as a-SPM, agrees\nwith the ferromagnetic result (a-FM). However, increas-\ning deviations appear below fields of 4 kOe. By varying\nHAandα, we tried to improve the fit near the resonance\nHr= 2.3kOeand obtained unsatisfying results. For low\nanisotropy, HA≤3kOe, the resonance field could be\nreproduced only by significantly lower values of α, which\nare inconsistent with the measured width and shape. For\nHA>3kOe, a small shift of Hroccurs, but at the\nsame time the lineshape became distorted, tending to a\ntwo-peak structure also found in previous simulations13.\nEven at the lowest temperature, T= 22K, where the\nthermal field drops to HT= 0.4kOe, no signatures of\nsuch inhomogeneous broadening appear (see Fig. 3). Fi-\nnally, it should be mentioned that all above attempts to\nincorporate the anisotropy in the discussion of the line-\nshape were based on the simplest non-trivial, i.e. uni-\naxial symmetry, which for FePtwas also considered by\nthe theory25. For cubic anisotropy, the same qualitative\ndiscrepancies were found in our simulations20. This in-\nsensitivitywith respecttothe symmetry of HAoriginates\nfrom the orientational averaging in the range of the HA-\nvalues of relevance here.\nAs a finite anisotropy failed to reproduce Hr,∆H, and\nalso the shape, we tried a novel ansatz for the magnetic\nresonanceofnanoparticlesbyintroducingacomplexLLG\nparameter,\nˆα(T) =α(T)−i β(T). (12)\nAccording to Eq. (4) this is equivalent to a negative g-\nshift,g(T)−g0=−β(T)g0, which is intended to com-\npensate the too large downward shift of HL\nrdemanded\nbyχL\nxx(H) due to the large linewidth. In fact, insert-\ning this ansatz in Eq. (5), the fit, denoted as ∆ g-FM\nin Fig. 2, provides a convincing description of the line-\nshape down to zero magnetic field. It may be interest-\ning to note that the resulting parameters, α= 0.56 and\nβ= 0.27, revealed the same shape as obtained by using\nthe Gilbert-susceptibilities, Eq. (6).\nIn spite of the agreement of the ∆ g-FM model with\nthe data, we also tried to include here SPM fluctuations\nby using ˆ α(T,H) = ˆα(T)(1/L(y)−1/y)21forHA= 0.\nThe result, designated as ∆ g-SPM in Fig. 2 agrees with\nthe ∆g-FM curve for H≫HTwhere ˆα(T,H) = ˆα(T),\nbut again significant deviations occur at lower fields.\nThey indicate that SPM fluctuations do not play any\nrole here, and this conclusion is also confirmed by the\nresults at higher temperatures. There, the thermal fluc-\ntuation field, HT=kBT/µp(T), increases to values\nlarger than the maximum measuring field, H= 10kOe,\nso that SPM fluctuations should cause a strong ther-/s48 /s51/s48/s48/s32/s75/s48/s55/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s56/s49/s46/s48/s32\n/s32\n/s32/s97/s41\n/s48/s46/s49/s56/s32/s43/s32/s48/s46/s53/s56/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48/s48/s46/s48/s48/s46/s50\n/s48/s46/s51/s57/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\n/s32/s84/s32/s40/s32/s75/s32/s41/s103/s32/s47/s32/s103\n/s48\n/s32/s32\n/s98/s41/s32\n/s32/s73/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s126/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\nFIG. 4: Temperature variation (a) of the LLG-damping αand\n(b) of the relative g-shifts with g0= 2.16 (following from the\nresonance fields at T > T C). Within the error margins, α(T)\nand ∆g(T) and also the fitted intensity of the LLG-shape\n(see inset) display the same temperature dependence as the\nparticle moments in Fig. 1(b).\nmal, homogeneous broadening of the resonance due to\nˆα(H≫HT) = ˆα·2HT/H. However, upon increasing\ntemperature, the fitted linewidths, (Fig. 3) and damping\nparameters (Fig. 4) display the reverse behavior.\nIV. COMPLEX DAMPING\nIn order to shed more light on the magnetization dy-\nnamics of the nanospheres we examined the tempera-\nture variation of the FMR spectra. Figure 3 shows\nsome examples recorded below the Curie temperature,\nTC= 320K, together with fits to the a−FM model\noutlined in the last section. Above TC, the resonance\nfields and the linewidths are temperature independent\nrevealing a mean g−factor,g0= 2.16±0.02, and a\ndamping parameter ∆ H/Hr=α0= 0.18±0.01. Since\ng0is consistent with a recent report on g-values of\nFexPt1−xforx≥0.4321, we suspect that this resonance\narises from small FexPt1−x-clusters in the inhomoge-\nneousFe0.2Pt0.8structure. Fluctuations of g0and of\nlocal fields may be responsible for the rather large width.6\nThis interpretation is supported by the observation that\naboveTCthe lineshape is closer to a Gaussian than to\nthe Lorentzian following from Eq.(7) for small α.\nThe temperature variation for both components of the\ncomplex damping, obtained from the fits below TCto\nEq. (7), are shown in Fig. 4. Clearly, they obey the same\npowerlaw as the moments, µp(T), displayed in Fig. 1(b),\nwhich implies\nˆα(T) = (α−i β)ms(T)+α0.(13)\nHerems(T)=µp(T)/µp(0),α=0.58, and β=-∆g(0)/g0=\n0.39 denote the reduced spontaneous magnetization and\nthe saturation values for the complex damping, respec-\ntively. It should be emphasized that the fitted inten-\nsityI(T) of the spectra, shown by the inset to Fig. 4(b),\nexhibits the same temperature variation I(T)∼µp(T).\nThis behavior is predicted by the ferromagnetic model,\nEq. (7), and is a further indication for the absence of\nSPM effects on the magnetic resonance. If the reso-\nnance were dominated by SPM fluctuations, the inten-\nsity should decrease like the SPM Curie-susceptibility,\nISPM(T)∼µ2\np(T)/T, following from Eq. (11), being\nmuch stronger than the observed I(T).\nAt the beginning of a physical discussion of ˆ α(T), we\nshould point out that the almost perfect fits of the line-\nshape to Eq. (7) indicate that the complex damping is\nrelated to an intrinsic mechanism and that eventual in-\nhomogeneous effects by distributions of particle sizes and\nshapes in the assembly, as well as by structural disorder\nareratherunlikely. Sinceageneraltheoryofthemagneti-\nzation dynamics in nanoparticles is not yet available, we\nstart with the current knowledge on the LLG-damping\ninbulkand thin film ferromagnets, as recently reviewed\nby B. Heinrich5. Based on experimental work on the\narchetypal metallic ferromagnets and on recent ab initio\nband structure calculations10there is now ratherfirm ev-\nidence that the damping ofthe q=0-magnonis associated\nwith the torques /vectorTso=/vector ms×/summationtext\nj(ξj/vectorLj×/vectorS) on the spin /vectorS\ndue to the spin-orbit interaction ξjat the lattice sites j.\nThe action of the torque is limited by the finite lifetime\nτof an e/h excitation, the finite energy ǫof which may\ncause a phase, i.e. a g-shift. As a result of this magnon\n- e/h-pair scattering, the temperature dependent part of\nthe LLG damping parameter becomes\nˆα(T)−α0=λL(T)\nγMs(T)\n=(Ωso·ms(T))2\nτ−1+i ǫ/¯h·1\nγMs(T). (14)\nForintraband scattering, ǫ≪¯h/τ, the aforementioned\nnumerical work10revealed Ω so= 0.8·1011s−1and 0.3·\n1011s−1as effective spin-orbit coupling in fcc Niand\nbcc Fe, respectively. Hence, the narrow unshifted (∆ g=\n0)bulkFMR lines in pure crystals, where α≤10−2,\nare related to intraband scattering with ǫ≪¯h/τand toelectronic (momentum) relaxation times τsmaller than\n10−13s.\nBasing on Eq. (14), we discuss at first the temperature\nvariation,whichimpliesalineardependence, ˆ α(T)−α0∼\nms(T). Obviously, both, the real and imaginary part of\nˆα(T)−α0, agree perfectly with the fits to the data in\nFig. 4, if the relaxation time τremains constant. It may\nbe interestingtonote herethat the observedtemperature\nvariation of the complex damping λL(T) is not predicted\nby the classical model28incorporating the sd-exchange\ncoupling Jsd. According to this model, which has been\nadvanced recently to ferromagnets with small spin-orbit\ninteraction29and ferromagnetic multilayers30,Jsdtrans-\nfers spin from the localized 3d-moments to the delocal-\nized s-electron spins within their spin-flip time τsf. From\nthe mean field treatment of their equations of motion by\nTurov31, we find a form analogous to Eq. (14)\nαsd(T) =Ω2\nsdχs\nτ−1\nsf+i/tildewideΩsd·1\nγMs(T)(15)\nwhere Ω sd=Jsd/¯his the exchange frequency , χsthe\nPauli-susceptibility of the s-electrons and /tildewideΩsd/Ωsd=\n(1 + Ω sdχs/γMd). The same form follows from more\ndetailed considerations of the involved scattering process\n(see e.g. Ref. 5). As a matter of fact, the LLG-damping\nαsd=λsd/γMdcannot account for the observed tem-\nperature dependence, because Ω sdandχsare constants.\nThe variation of the spin-torques with the spontaneous\nmagnetization ms(T) drops out in this model, since the\nsd-scattering involves transitions between the 3d spin-up\nand -down bands due to the splitting by the exchange\nfieldJsdms(T).\nBy passing from the bulk to the nanoparticle ferro-\nmagnet, we use Eq. (14) to discuss our results for the\ncomplex ˆ α(T), Eq. (13). Recently, for Conanoparticles\nwith diameters 1-4 nm, the existence of a discrete level\nstructure near ǫFhas been evidenced32, which suggests\nto associate the e/h-energy ǫwith the level difference ǫp\nat the Fermi energy. From Eqs. (13),(14) we obtain rela-\ntions between ǫand the lifetime of the e/h-pair and the\nexperimental parameters αandβ:\nτ−1=α\nβǫ\n¯h, (16a)\nǫ\n¯h=β\nα2+β2Ω2\nso\nγMs(0). (16b)\nDue toα/β= 1.5, Eq. (16a) reveals a strongly over-\ndamped excitation, which is a rather well-founded con-\nclusion. The evaluation of ǫ, on the other hand, de-\npends on an estimate for the effective spin-orbit cou-\npling, Ω so=ηLχ1/2\neξso/¯hwhereηLrepresents the ma-\ntrix element of the orbital angular momentum between7\nthe e/h states5. The spin-orbit coupling of the minor-\nityFe-spins in FePthas been calculated by Sakuma24,\nξso= 45meV, while the density of states D(ǫF)≈1/(eV\natom)24,33yields a rather high susceptibility of the elec-\ntrons,ηLχe=µ2\nBD(ǫF) = 4.5·10−5. Assuming ηL=1,\nboth results lead to Ω so≈3.5·1011s−1, which is by\none order of magnitude larger than the values for Fe\nandNimentioned above. One reason for this enhance-\nment and for a large matrix element, ηL=1, may be the\nstrong hybridization between 3 dand 4d−Ptorbitals24in\nFexPt1−x. By inserting this result into Eq. (16b) we find\nǫ= 0.8 meV. In fact, this value is comparable to an esti-\nmate for the level difference at ǫF32,ǫp= (D(ǫF)·Np)−1\nwhich for ourparticles with Np= (2π/3)(dp/a0)3= 1060\natoms yields ǫp= 0.9 meV. Regarding the several in-\nvolved approximations, we believe that this good agree-\nment between the two results on the energy of the e/h\nexcitation, ǫ≈ǫp, maybe accidental. However,wethink,\nthat this analysisprovidesa fairlystrongevidence for the\nmagnon-scattering by this excitation, i.e. for the gap in\nthe electronic states due to confinement of the itinerant\nelectrons to the nanoparticle.\nV. SUMMARY AND CONCLUSIONS\nThe analysis of magnetization isotherms explored\nthe mean magnetic moments of Fe0.2Pt0.8nanospheres\n(dp= 3.1nm) suspended in an organic matrix, their\ntemperature variation up to the Curie temperature TC,\nthe large mean particle-particle distance Dpp≫dpand\nthe presence of Fe3+impurities. Above TC, the res-\nonance field Hrof the 9.1GHzmicrowave absorption\nyielded a temperature independent mean g-factor,g0=\n2.16, consistent with a previous report21for paramag-\nneticFexPt1−xclusters. There, the lineshape proved\nto be closer to a gaussian with rather large linewidth,\n∆H/Hr= 0.18, which may be associated with fluctua-\ntions of g0and local fields both due to the chemically\ndisordered fccstructure of the nanospheres.\nBelow the Curie temperature, a detailed discussion of\nthe shape of the magnetic resonance spectra revealed a\nnumber of novel and unexpected features.\n(i) Starting at zero magnetic field, the shapes could be\ndescribed almost perfectly up to highest field of 10 kOe\nby the solution of the LLG equation of motion for inde-\npendentferromagneticsphereswithnegligibleanisotropy.\nSignatures of SPM fluctuations on the damping, which\nhave been predicted to occur below the thermal field\nHT=kBT/µp(T), could not be realized.\n(ii) Upon decreasing temperature, the LLG damping in-\ncreases proportional to µp(T), i.e. to the spontaneous\nmagnetization of the particles, reaching a rather largevalueα= 0.7 forT≪TC. We suspect that this high\nintrinsic damping may be responsible for the absence of\nthe predicted SPM effects on the FMR, since the under-\nlying statistical theory13has been developed for α≪1.\nThis conjecturemayfurther be based onthe fact that the\nlarge intrinsic damping field ∆ H=α·ω/γ= 2.1kOe\ncauses a rapid relaxation of the transverse magnetization\n(q= 0 magnon) as compared to the effect of statistical\nfluctuations of HTadded to Heffin the equation of mo-\ntion, Eq.(2)13.\n(iii) Along with the strong damping, the lineshape analy-\nsis revealed a significant reduction of the g-factor, which\nalso proved to be proportional to µp(T). Any attempts\nto account for this shift by introducing uniaxial or cubic\nanisotropy fields failed, since low values of /vectorHAhad no\neffects on the resonance field due to the orientational av-\neraging. On the other hand, larger /vectorHA’s, by which some\nsmall shifts of Hrcould be obtained, produced severe\ndistortions of the calculated lineshape.\nThe central results of this work are the temperature\nvariation and the large magnitudes of both α(T) and\n∆g(T). They were discussed by using the model of the\nspin-orbit induced scattering of the q= 0 magnon by\nan e/h excitation ǫ, well established for bulk ferromag-\nnets, where strong intraband scattering with ǫ≪¯h/τ\nproved to dominate5. In nanoparticles, the continuous\nǫ(/vectork)-spectrum of a bulk ferromagnet is expected to be\nsplit into discrete levels due to the finite number of lat-\ntice sites creating an e/h excitation ǫp. According to\nthe measured ratio between damping and g-shift, this\ne/h pair proved to be overdamped, ¯ h/τp= 1.5ǫp. Based\non the free electron approximation for ǫp32and the den-\nsity of states D(ǫF) from band-structure calculations\nforFexPt1−x24,33, one obtains a rough estimate ǫp≈\n0.9 meV for the present nanoparticles. Using a reason-\nable estimate of the effective spin-orbit coupling to the\nminority Fe-spins, this value could be well reproduced\nby the measured LLG damping, α= 0.59. Therefore we\nconclude that the noveland unexpected results of the dy-\nnamics of the transverse magnetization reported here are\ndue to the presence of a broade/h excitation with energy\nǫp≈1meV. Deeper quantitative conclusions, however,\nmust await more detailed information on the real elec-\ntronic structure of nanoparticles near ǫF, which are also\nrequired to explain the overdamping of the e/h-pairs, as\nit is inferred from our data.\nThe authors are indebted to E. Shevchenko and H.\nWeller (Hamburg) for the synthesis and the structural\ncharacterizationof the nanoparticles. One of the authors\n(J. 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Mat. 163, 331 (1996)."    },    {        "title": "0708.3323v1.Enhancement_of_the_Gilbert_damping_constant_due_to_spin_pumping_in_noncollinear_ferromagnet_nonmagnet_ferromagnet_trilayer_systems.pdf",        "content": "arXiv:0708.3323v1  [cond-mat.mes-hall]  24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear\nferromagnet / non-magnet / ferromagnet trilayer systems\nTomohiro Taniguchi1,2, Hiroshi Imamura2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\n(Dated: October 29, 2018)\nWe analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-\ncollinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert\ndamping constant depends both on the precession angle of the magnetization of the free layer and\non the direction of the magntization of the fixed layer. We find the condition to be satisfied to\nrealize strong enhancement of the Gilbert damping constant .\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThere is currently great interest in the dynamics of\nmagnetic multilayers because of their potential applica-\ntions in non-volatile magnetic random access memory\n(MRAM) and microwave devices. In the field of MRAM,\nmuch effort has been devoted to decreasing power con-\nsumption through the use of current-induced magnetiza-\ntion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally,\nCIMR is observed as the current perpendicular to plane-\ntype giant magnetoresistivity (CPP-GMR) of a nano pil-\nlar, in which the spin-polarized current injected from the\nfixed layer exerts a torque on the magnetization of the\nfree layer. The torque induced by the spin current is\nutilized to generate microwaves.\nThe dynamics of the magnetization Min a ferromag-\nnet under an effective magnetic field Beffis described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=−γM×Beff+α0M\n|M|×dM\ndt,(1)\nwhereγandα0are the gyromagnetic ratio and the\nGilbert damping constant intrinsic to the ferromagnet,\nrespectively. The Gilbert damping constant is an im-\nportant parameter for spin electronics since the critical\ncurrent density of CIMR is proportional to the Gilbert\ndamping constant [8, 9] and fast-switching time magne-\ntization reversal is achieved for a large Gilbert damp-\ning constant [10]. Several mechanisms intrinsic to ferro-\nmagnetic materials, such as phonon drag [11] and spin-\norbit coupling [12], have been proposed to account for\nthe origin of the Gilbert damping constant. In addition\nto these intrinsic mechanisms, Mizukami et al.[13, 14]\nand Tserkovnyak et al.[15, 16] showed that the Gilbert\ndampingconstantinanon-magnet(N) /ferromagnet(F)\n/ non-magnet(N) trilayersystem is enhanced due to spin\npumping. Tserkovnyak et al.[17] also studied spin pump-\ning in a collinear F/N/F trilayer system and showed that\nenhancement of the Gilbert damping constant depends\non the precession angle of the magnetization of the free\nlayer.\nOn the other hand, several groups who studied CIMR\nin a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat-\nically shown. The magnetization of the F 1layer (m1) pre-\ncesses around the z-axis with angle θand angular velocity ω.\nThe magnetization of the F 2layer (m2) is fixed with tilted\nangleρ. The precession of the magnetization in the F 1layer\npumpsspin current Ipump\nsintotheNandF 2layer, andcreates\nthe spin accumulation µNin the N layer. The spin accumu-\nlation induces the backflow spin current Iback(i)\ns(i= 1,2).\nmagnetization of the free layer is aligned to be perpen-\ndicular to that of the fixed layer have reported the reduc-\ntion of the critical current density [5, 6, 7]. Therefore, it\nis intriguing to ask how the Gilbert damping constant is\naffected by spin pumping in non-collinear F/N/F trilayer\nsystems.\nIn this paper, we analyze the enhancement of the\nGilbert damping constant due to spin pumping in non-\ncollinear F/N/F trilayer systems such as that shown in\nFig. 1. Following Refs. [15, 16, 17, 18], we calculate the\nspin current induced by the precession of the magnetiza-\ntion of the free layer and the enhancement of the Gilbert\ndamping constant. We show that the Gilbert damping\nconstant depends not only on the precession angle θof\nthe magnetization of a free layer but also on the angle ρ\nbetweenthemagnetizationsofthefixedlayerandthepre-\ncession axis. The Gilbert damping constant is strongly\nenhanced if angles θandρsatisfy the condition θ=ρor\nθ=π−ρ.\nThe system we consider is schematically shown in Fig.\n1. A non-magnetic layer is sandwiched between two fer-\nromagnetic layers, F 1and F 2. We introduce the unit2\nvectormito represent the direction of the magnetiza-\ntion of the i-th ferromagnetic layer. The equilibrium\ndirection of the magnetization m1of the left free fer-\nromagnetic layer F 1is taken to exist along the z-axis.\nWhen an oscillatingmagnetic field is applied, the magne-\ntization of the F 1layer precesses around the z-axis with\nangleθ. The precession of the vector m1is expressed\nasm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the\nangular velocity of the magnetization. The direction of\nthe magnetization of the F 2layer,m2, is assumed to be\nfixed and the angle between m2and thez-axis is repre-\nsented byρ. The collinear alignment discussed in Ref.\n[17] corresponds to the case of ρ= 0,π.\nBefore studying spin pumping in non-collinear sys-\ntems, we shall give a brief review of the theory of\nspin pumping in a collinear F/N/F trilayer system [17].\nSpin pumping is the inverse process of CIMR where the\nspin current induces the precession of the magnetization.\nContrary to CIMR, spin pumping is the generation of\nthe spin current induced by the precession of the mag-\nnetization. The spin current due to the precession of the\nmagnetization in the F 1layer is given by\nIpump\ns=/planckover2pi1\n4πg↑↓m1×dm1\ndt, (2)\nwhereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the\nDirac constant. Spins are pumped from the F 1layer\ninto the N layer and the spin accumulation µNis cre-\nated in the N layer. Spins also accumulate in the F 1\nand F 2layers. In the ferromagnetic layers the trans-\nverse component of the spin accumulation is assumed to\nbe absorbed within the spin coherence length defined as\nλtra=π/|k↑\nFi−k↓\nFi|, wherek↑,↓\nFiis the spin-dependent\nFermi wave number of the i-th ferromagnet. For fer-\nromagnetic metals such as Fe, Co and Ni, the spin co-\nherence length is a few angstroms [20]. Hence, the spin\naccumulation in the i-th ferromagnetic layer is aligned to\nbe parallel to the magnetization, i.e., µFi=µFimi. The\nlongitudinal component of the spin accumulation decays\non the scale of spin diffusion length, λFi\nsd, which is of the\norder of 10 nm for typical ferromagnetic metals [21].\nThe difference in the spin accumulation of ferromag-\nnetic and non-magnetic layers, ∆ µi=µN−µFimi(i=\n1,2), induces a backflow spin current, Iback(i)\ns, flowing\ninto both the F 1and F 2layers. The backflow spin cur-\nrentIback(i)\nsis obtained using circuit theory [18] as\nIback(i)\ns=1\n4π/braceleftbigg2g↑↑g↓↓\ng↑↑+g↓↓(mi·∆µi)mi\n+g↑↓mi×(∆µi×mi)/bracerightbig\n,(3)\nwhereg↑↑andg↓↓are the spin-up and spin-down con-\nductances, respectively. The total spin current flowing\nout of the F 1layer is given by Iexch\ns=Ipump\ns−Iback(1)\ns\n[17]. The spin accumulation µFiin the F ilayer is ob-\ntained by solving the diffusion equation. We assume\nthat spin-flip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current\nwithin the N layer, Iexch\ns=Iback(2)\ns. The torque τ1\nacting on the magnetization of the F 1layer is given by\nτ1=Iexch\ns−(m1·Iexch\ns)m1=m1×(Iexch\ns×m1). For\nthe collinear system, we have\nτ1=g↑↓\n8π/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\nm1×dm1\ndt,(4)\nwhereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless\nparameter introduced in Ref. [17]. The Gilbert damping\nconstant in the LLG equation is enhanced due to the\ntorqueτ1asα0→α0+α′with\nα′=gLµBg↑↓\n8πM1dF1S/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\n,(5)\nwheregLis the Land´ e g-factor,µBis the Bohr magneton,\ndF1is the thickness of the F 1layer andSis the cross-\nsection of the F 1layer.\nNext, we move on to the non-collinear F/N/F trilayer\nsystem with ρ=π/2, in which the magnetization of the\nF2layer is aligned to be perpendicular to the z-axis. Fol-\nlowing a similar procedure, the LLG equation for the\nmagnetization M1in the F 1layer is expressed as\ndM1\ndt=−γeffM1×Beff+γeff\nγ(α0+α′)M1\n|M1|×dM1\ndt,(6)\nwhereγeffandα′are the effective gyromagnetic ratio\nand the enhancement of the Gilbert damping constant,\nrespectively. The effective gyromagnetic ratio is given by\nγeff=γ/parenleftbigg\n1−gLµBg↑↓νcotθcosψsinωt\n8πMdF1Sǫ/parenrightbigg−1\n,(7)\nwhere cosψ= sinθcosωt=m1·m2and\nǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8)\nThe enhancement of the Gilbert damping constant is ex-\npressed as\nα′=gLµBg↑↓\n8πMdF1S/parenleftbigg\n1−νcot2θcos2ψ\nǫ/parenrightbigg\n.(9)\nItshouldbenotedthat, fornon-collinearsystems, both\nthegyromagneticratioandtheGilbert dampingconstant\nare modified by spin pumping, contrary to what occurs\nin collinear systems. The modification of the gyromag-\nnetic ratio and the Gilbert damping constant due to spin\npumping can be explained by considering the pumping\nspincurrentandthe backflowspincurrent[SeeFigs. 2(a)\nand 2(b)]. The direction of the magnetic moment car-\nried by the pumping spin current Ipump\nsis parallel to\nthe torque of the Gilbert damping for both collinear and\nnon-collinear systems. The Gilbert damping constant is\nenhanced by the pumping spin current Ipump\ns. On the\notherhand, the directionofthe magneticmoment carried3\nFIG. 2: (Color online) (a) Top view of Fig. 1. The dotted\ncircle in F 1represents the precession of magnetization M1\nand the arrow pointing to the center of this circle represent s\nthe torque of the Gilbert damping. The arrows in Ipump\nsand\nIback(1)\nsrepresent the magnetic moment of spin currents. (b)\nThe back flow Iback(1)\nshas components aligned with the di-\nrection of the precession and the Gilbert damping.\nby the backflow spin current Iback(1)\nsdepends on the di-\nrection of the magnetization of the F 2layer. As shown in\nEq. (3), the backflowspin current in the F 2layerIback(2)\ns\nhas a projection on m2. Since we assume that the spin\ncurrent is constant within the N layer, the backflow spin\ncurrent in the F 1layerIback(1)\nsalso has a projection on\nm2. For the collinear system, both Ipump\nsandIback(1)\ns\nare perpendicular to the precession torque because m2\nis parallel to the precession axis. However, for the non-\ncollinear system, the vector Iback(1)\nshas a projection on\nthe precession torque, as shown in Fig. 2(b). Therefore,\nthe angular momentum injected by Iback(1)\nsmodifies the\ngyromagnetic ratio as well as the Gilbert damping in the\nnon-collinear system.\nLet us estimate the effective gyromagnetic ratio using\nrealistic parameters. According to Ref. [17], the con-\nductancesg↑↓andg∗for a Py/Cu interface are given\nbyg↑↓/S= 15[nm−2] andν≃0.33, respectively. The\nLand´ eg-factor is taken to be gL= 2.1, magnetization is\n4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut-\ning these parameters into Eqs. (7) and (8), one can see\nthat|γeff/γ−1| ≃0.001. Therefore, the LLG equation\ncan be rewritten as\ndM1\ndt≃ −γM1×Beff+(α0+α′)M1\n|M1|×dM1\ndt.(10)\nThe estimated value of α′is of the order of 0.001. How-\never, we cannot neglect α′since it is of the same order\nas the intrinsic Gilbert damping constant α0[22, 23].\nExperimentally, the Gilbert damping constant is mea-\nsuredasthe width ofthe ferromagneticresonance(FMR)\nabsorptionspectrum. LetusassumethattheF 1layerhas\nno anisotropy and that an external field Bext=B0ˆzisapplied along the z-axis. We also assume that the small-\nangle precession of the magnetization around the z-axis\nis excited by the oscillating magnetic field B1applied in\nthexy-plane. The FMR absorption spectrum is obtained\nas follows [24]:\nP=1\nT/integraldisplayT\n0dtαγMΩ2B2\n1\n(γB0−Ω)2+(αγB0)2,(11)\nwhere Ω is the angular velocity of the oscillating mag-\nnetic field, T= 2π/Ω andα=α0+α′. Sinceαis very\nsmall, the absorption spectrum can be approximately ex-\npressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak\nproportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant. In Fig. 3(a), the time-averaged value\n∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot-\nted by the solid line as a function of the precession an-\ngleθ. The dotted line represents the enhancement of\nthe Gilbert damping constant α′for the collinear system\ngiven by Eq. (5). The time-averaged value of the en-\nhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes\nits maximum value at θ= 0,πfor the collinear system\n(ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the\nnon-collinear system in which ρ=π/2 takes its maxi-\nmum value at θ=π/2.\nAs shown in Fig. 2(b), the backflow spin current\ngives a negative contribution to the enhancement of the\nGilbert damping constant. This contribution is given by\nthe projection of the vector Iback(1)\nsonto the direction\nof the torque of the Gilbert damping, which is repre-\nsented by the vector m1×˙m1. Therefore, the condition\nto realize the maximum value of the enhancement of the\nGilbert damping is satisfied if the projection of Iback(1)\ns\nontom1×˙m1takes the minimum value; i.e., θ=ρor\nθ=π−ρ.\nWe can extend the above analysis to the non-collinear\nsystemwith anarbitraryvalue of ρ. After performingthe\nappropriate algebra, one can easily show that the LLG\nequation for the magnetization of the F 1layer is given\nby Eq. (6) with\nγeff=γ/bracketleftBigg\n1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ)\n8πMdS˜ǫ/bracketrightBigg−1\n(12)\nα′=gLµBg↑↓\n8πMdS/braceleftBigg\n1−ν(cotθcos˜ψ−cscθcosρ)2\n˜ǫ/bracerightBigg\n,\n(13)\nwhere cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2\nand\n˜ǫ=1−ν2cos2˜ψ\n−ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}.\n(14)\nSubstituting the realistic parameters into Eqs. (12) and\n(14), we can show that the effective gyromagnetic ratio4\n\u0013\n\u0003\u0013\u0011\u0013\u0013\u0015\u001a\u0003\u0013\u0011\u0013\u0013\u0016\u0015\u0003\u0013\u0011\u0013\u0013\u0016\u001a\n\u000bD\f\n\u000bE\f\u0013\u0011\u0013\u0013\u0016\u001a\n\u0013\u0011\u0013\u0013\u0016\u0015\n\u0013\u0011\u0013\u0013\u0015\u001a/c50/c0f/c12 /c50\n/c51/c1c/c41\n/c1e\n\u0013 /c50/c0f/c12 /c50\n/c52\u0013/c50/c0f/c12/c50\n/c51/c1c/c41\n/c1e\nFIG. 3: (Color online) (a) The time-averaged value of the en-\nhancement of the Gilbert damping constant α′is plotted as a\nfunction of the precession angle θ. The solid line corresponds\nto the collinear system derived from Eq. (9). The dashed\nline corresponds to the non-collinear system derived from E q.\n(5). (b) The time-averaged value of the enhancement of the\nGilbert damping constant α′of the non-collinear system is\nplotted as a function of the precession angle θand the an-\ngleρbetween the magnetizations of the fixed layer and the\nprecession axis.γeffcan be replaced by γin Eq. (6) and that the LLG\nequation reduces to Eq. (10). Figure 3(b) shows the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert\ndamping constant is strongly enhanced if angles θandρ\nsatisfy the condition that θ=ρorθ=π−ρ.\nIn summary, we have examined the effect of spin\npumping on the dynamics of the magnetization of mag-\nnetic multilayers and calculated the enhancement of the\nGilbertdampingconstantofnon-collinearF/N/Ftrilayer\nsystems due to spin pumping. The enhancement of the\nGilbert damping constant depends not only on the pre-\ncession angle θof the magnetization of a free layer but\nalso on the angle ρbetween the magnetizations of the\nfixed layerand the precession axis, as shown in Fig. 3(b).\nWe have shown that the θ- andρ-dependence of the en-\nhancement of the Gilbert damping constant can be ex-\nplained by analyzing the backflow spin current. The con-\ndition to be satisfied to realizestrongenhancement of the\nGilbert damping constant is θ=ρorθ=π−ρ.\nThe authors would like to acknowledge the valuable\ndiscussions we had with Y. Tserkovnyak, S. Yakata, Y.\nAndo, S. Maekawa, S. Takahashi and J. Ieda. This work\nwas supported by CREST and by a NEDO Grant.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[4] A. Deac, K. J. Lee, Y. Liu, O. Redon, M. Li, P. Wang,\nJ. P. Nozi´ eres, and B. Dieny, J. Magn. Magn. Mater.\n290-291 , 42 (2005).\n[5] A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys.\nLett.84, 3897 (2004).\n[6] K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86,\n022505 (2005).\n[7] T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl.\nPhys. Lett. 89, 172504 (2005).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys.\nRev. B67, 174402 (2003).\n[10] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett.\n92, 088302 (2004).\n[11] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[13] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.239, 42 (2002).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[15] Y. Tserkovnyak and A. Brataas and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n[19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19,\n183201 (2007).\n[22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[23] F. Schreiber, J. Pflaum, Th. M¨ uhge, and J. Pelzl, Solid\nState Commun. 93, 965 (1995).\n[24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO-\nNANCE (Israel Program for Scientific Translations Ltd.,\nJersalem, 1964)."    },    {        "title": "0708.3827v3.Linear_frictional_forces_cause_orbits_to_neither_circularize_nor_precess.pdf",        "content": "arXiv:0708.3827v3  [physics.class-ph]  27 Mar 2008Linear frictional forces cause orbits to neither\ncircularize nor precess\nP.M. Hamilton1,2and M. Crescimanno1\n1Dept. of Physics and Astronomy, Youngstown State University\n2Dept. of Physics, University of Maryland\nE-mail:pmham@umd.edu, mcrescim@cc.ysu.edu\nPACS numbers: 46.40Ff, 45.20.Jj, 45.50.Pk, 34.60.+z\nSubmitted to: J. Phys. A: Math. Gen.Linear frictional forces cause orbits to neither circulari ze nor precess 2\nAbstract: For the undamped Kepler potential the lack of precession has histo rically\nbeenunderstoodintermsoftheRunge-Lenzsymmetry. Forthed ampedKeplerproblem\nthis result may be understood in terms of the generalization of Poiss on structure to\ndampedsystems suggestedrecentlybyTarasov[1]. Inthisgenera lizedalgebraicstructure\nthe orbit-averaged Runge-Lenz vector remains a constant in the linearly damped Kepler\nproblem to leading order in the damping coefficient. Beyond Kepler, we prove that,\nfor any potential proportional to a power of the radius, the orbit shape and precession\nangle remain constant to leading order in the linear friction coefficient .\n1. Introduction\nWhat happens to orbits subject to linear frictional drag? In typica l physical settings,\nsuch as Rydberg atoms or stellar binaries, the effective frictional f orces are nonlinear\nand, typically, lead to the circularization of the orbit. Orbital evolut ion under linear\nfriction is special in that, as we show below, the eccentricity and the apsides do not\nchange to leading order in the damping. The purpose of this note is to understand this\nelementary result from the underlying dynamical symmetry of the K epler problem, thus\ndemonstrating theutility ofa Hamiltoniannotionin itsnon-Hamiltoniang eneralization.\nIn many astrophysical situations, the secular evolution due to fric tion of orbits\nin a two body system is towards circular orbits. In an orbit in a centra l field the\nangular momentum scales with the momentum while the energy genera lly scales with\nthe momentum-squared. Friction, assumed to be spatially isotropic and homogeneous\nbut time odd, typically scales the momentum. This means generally tha t the resultant\nsecular evolution in central force systems is that in which the energ y is minimized at\nfixed angular momentum. This is clearly the circular orbit. The velocity dependence\nof the frictional force is quite relevant, in particular as reference d against the velocity\ndispersion of the (undamped) motionin that central potential. Clea rly, under the action\nof such dissipative forces, a consequence of symmetry is that the flow in orbital shape\n(not size!) has two fixed points, circular orbits and strictly radial (in fall) orbits.\nFew physical problems have received more scrutiny than bounded o rbits in the two-\nand few- body system. Among these, the two-body Kepler problem is arguably the most\nexperimentally relevant and best studied example, having been illumina ted by intense\ntheoretical inquiry spanning hundreds of years leading to importan t insights even in\nrelatively recent times[2,3,4,5,6,7,8,9,10,11,12,13,14,15, 16,17,18,19,20,21,22].\nWe do not present a systematic review or histiography of this celebr ated problem\n(though we thankfully acknowledge also [23, 24, 25, 28, 26, 27, 29, 30] which we have\nfound quite useful for our study). We do not aim to contribute to t he vast literature\non astrophysically and microphysically relevant models of friction in or bital problems\n(though the interested reader may find references [31, 32, 33, 34, 35, 36, 37, 38, 39, 40,\n41, 42, 43] a useful launching point for such review).\nInstead, theourpurposehereistoaccomodatefromthedynamic al symmetry group\npoint-of-view the result that linear frictional damping (to leading or der) preserves theLinear frictional forces cause orbits to neither circulari ze nor precess 3\norbit’s shape. Although Hamiltonian systems may lose dynamical symm etry completely\nwhen dissipative forces are included, it can be shown that some stru cture may remain\nunder a modified symplectic form. After a brief introduction to the m ethod by which\nTarasov extends symplectic structure of Hamiltonian mechanics to dissipative systems,\nwe apply it to the determination of the time averages of dynamical qu antities. Damping\ninvariably introduces new dynamical timescales and the time averagin g we implement is\novertimesshortcomparedwiththesetimescales(butstilllongcomp aredwiththeorbital\ntimescales in the undamped problem). Tarasov’s construction reve als the relevance of\nthe dynamical symmetry algebra to the damped Kepler problem.\nWe then compare this aproach to the classic “variations of constan ts” method of\norbit parameter evolution by describing an improvement that follows from our study.\nThe elementary method can be generalized to non-Kepler homogene ous potentials and\nalsodetermines orbitalshapeevolutionforlinearlydampedKepler or bitsbeyondleading\norder.\n2. Dynamical Symmetry and Tarasov’s Construction\nIn the undamped Kepler problem the lack of precession is generally un derstood as a\nconsequence of a dynamical symmetry, the celebrated so(4) symmetry formed from the\ntwo commuting so(3), one from the angular momentum /vectorL=/vector r×/vector pthe other from\nthe Runge-Lenz vector, /vectorS=/vectorL×/vector p+k/vector r\n|/vector r|([6, 7, 8]) being the maximal set of local,\nalgebraicallyindependent operatorsthatcommutewiththeHamilton ian,H=/vector p2\n2+V(r),\nwithV(r) =krαfork <0 andα=−1. (though see [22] for a more precise and general\nstatement of the connection between algebra and orbits in a centr al field).\n{Li,Lj}= 2ǫijkLk{Li,Sj}= 2ǫijkSk{Si,Sj}=−2HǫijkSk (2.1)\nThe length of /vectorSis proportional to the eccentricity (and points along the semi-majo r\naxis of the orbit, in the direction to the periastron from the focus) . Defining /vectorLand\n/vectorShas utility beyond their being constants in the 2-body Kepler problem , for example,\nparameterizing the secular evolution of orbits under various Hamilto nian perturbations\n[29, 44]. This so(4) is one of the maximal compact factor groups of the so(4,2) (the\nconformal group) extended symmetry formed by /vectorL,/vectorA,H, the generalization of the\nscaling operator R=/vector r·/vector pand the Virial operator V=/vector p2\n2−r\n2∂rV(r) ([17, 20])\nThe other central potential posessing an easily recognizable dyna mical symmetry\nis the multi-dimensional harmonic oscillator ( Vas given with k >0,α= 2). As is\nwell known, the isotropic D-dimensional harmonic oscillator’s naive O(D) symmetry\nis part of a larger U(D) dynamical symmetry. For D= 2 harmonic oscillator, note\nthat the U(2) symmetry does enlarge further to a so(3,2) when including R,Vand\ntheir generalization (the virial subalgebra Equation (4.16) through Equation (4.19) of\neach oscillator alone and closes to a sl(2,R) subgroup of the so(3,2)). Note further\nthat it is this later algebra that is isomorphic to the dimensionally reduc edso(4,2) of\nthe 3-d Kepler problem, by which we mean the reduction of that algeb ra to generatorsLinear frictional forces cause orbits to neither circulari ze nor precess 4\nassociatedwiththeorbitalplaneonly. These considerationscanals obeunderstoodfrom\nthe KS construction[26, 27] of the Kepler problem, in which a four-d imensional isotropic\nharmonic oscillator is the starting point. In that construction the u(4) =su(4)×u(1)\nis, of itself, not preserved by the KS construction. Instead, it is t heu(2,2) subgroup of\nthe four identical, independent oscillator’s sp(8,R) symmetry in which the overall u(1)\ncan be isolated as the angular momentum constraint of the KS const ruction[28]. The\nresidual symmetry su(2,2)∼so(4,2) is that of the 3-d Kepler problem. The analytical\nconnection between the Kepler problem and the isotropic harmonic o scillator has deep\nhistorical roots, going back to Newton and Hooke (see [45] and ref erences therein).\nFinally, the geometric construction of the undamped Kepler problem as geodesic flow\non (spatial) a 3-manifolds of constant curvature relates the so(4) dynamical symmetry\nto the isometry group generated by Killing vector fields on the spatia l slice[19, 21, 23].\nThese various connections between the Kepler problem and the isot ropic harmonic\noscillator do not lead to a simple structural connection between the associated damped\nproblems.\nTo leading order in the damping, Kepler orbits subject to linear frictio nal force do\nnot change shape or precess as they decay. It would be satisfying to understand this\nelementary result as a consequence of the preservation of the dy namical algebra under\nlinear friction. Although this is reminiscent of the damped N-dimension al harmonic\noscillator, there is no simple way to relate the damped problems. Since the subgroup\nassociated with the shape and precession (through the /vectorS) is rank one it is suggestive\nthat the entire group structure is preserved to leading order in th e linear friction.\nA recent paper by Tarasov[1] suggests a straightforward gener alization of the\nPoisson structure to systems with dissipative forces. There are m any other approaches\nto addressing structural questions of dissipative systems (for o ne example, see [46, 47]).\nWe find the approach of [1] to be most useful for addressing quest ions of the dynamical\nsymmetries that survive including dissipation. For completeness we n ow briefly review\nTarasov’s construction, and apply it to dissipation in the central fie ld problem in the\nfollowing section.\nTo preserve as much of the algebraic structure as possible, Taras ov constructs a\none-parameter family of two forms (that define a generalized Poiss on structure) that -in\na sense- interpolate between different dampings. In the zero damp ing limit it smoothly\nmatches onto the canonical symplectic form. Dimensionally, any dam ping parameter\nintroduces a new time scale into the problem, thus this new interpolat ing two form must\nalso be explicitely time-dependent. Tarasov requires this family of tw o forms to have\nthe following useful properties\n(1) Non-degeneracy: The two form ω=ωij(t)dxi∧dxjis antisymmetric and non-\ndegenerate along the entire flow. The xiare the 2 N(local) phase space co-ordinates.\nIn positive terms, the inverse ωijωjk=δi\njexists almost globally ‡.\n‡Since we do not formulate this entirely in the exterior calculus, we mus t allow for higher codimension\nsingularites that may not be resolvable in the dissipative system.Linear frictional forces cause orbits to neither circulari ze nor precess 5\n(2) Jacobi Identity: the two form is used to define a new Poisson br acket{A,B}T=\nωij∂iA∂jBthat forms an associative algebra. Explicitely it satisfies.\n{A,{B,C}T}T+{B,{C,A}T}T+{C,{A,B}T}T= 0 (2.2)\nHere we use the subscript ’ T’ to distinguish this bracket from the Poisson bracket of the\nundamped problem.\n(3) Derivation property of time translation: with respect to this ne w bracket the\ntime derivative of the new Poisson bracket satisfies the derivation p roperty (also called\nthe Liebnitz rule)\nd\ndt{A,B}T={dA\ndt,B}T+{A,dB\ndt}T (2.3)\nThese requirements are remarkable for several reasons. First, property (1) indicates\nthat (2) and (3) are possible. The deeper relevance of property ( 1) is that we can\nregard the two-form as (essentially a) global metric on the phase s pace. Property (2)\nindicates local mechanical observables in this ’dissipation deformed’ algebra form a lie\nalgebra. Property (3) is key to the utility of Tarasov’s constructio n for understanding\nconstants of motion in dissipative systems. It stipulates that time d evelopment in the\ndissipative system, while no longer just {,H}(or even {,H}T), must be compatible\nwith the structure of the symplectic algebra in the new bracket and thus the (new)\nbracket of time independent quantities in the dissipative system are themselves time\nindependent. Thus, just as in the Hamiltonian case, time independen t quantities form\na closed subalgebra. Note that for a Hamiltonian system property ( 3) is automatic\nsince in that case time translation is an inner automorphism of the sym plectic algebra.\nIn a dissipative system by contrast the Hamiltonian is no longer the op erator of time\ntranslation, but, if Tarasov’s construction can be implemented, tim e translation is still\nanautomorphismofthealgebra,andassuchmayberegardedasan outerautomorphism.\nFinally, fromproperty (3)it follows after a brief calculation that the two-form ωmust be\ntime idependent in the full dissipative system,dω\ndt= 0. In terms of symplectic geometry,\nthis is metric compatibility of the dissipative flow.\nTo proceed with the construction, consider the general flow ˙ xi=χi(/vector x,t). Again,\nthese are not assumed to be Hamiltonian flows. Assuming property ( 1) and using ωij\nto form a bracket {A,B}T=ωij∂iA∂jB, property (2) leads to the condition\nωim∂mωjk+ωjm∂mωki+ωkm∂mωij= 0. (2.4)\nTotal time derivatives and derivatives along phase space directions do not commute in\nthe flow,\n[d\ndt,∂i]A=−∂jA∂iχj. (2.5)\nUsing this and the jacobi identity (2.2), one sees that property (3 ) implies a condition\nrelating the form ωand the flow χi,\n∂ωij(t)\n∂t=∂iχj−∂jχiwhere χj=ωjk(t)χk(2.6)Linear frictional forces cause orbits to neither circulari ze nor precess 6\nGivenχi, we proceed by solving (2.4) for an ωijthat satisfies (2.6). This completes\nTarasov’s construction.\nWebreiflyoffer afewfurther remarkshelpful toorientthereader . First, inthemore\nfamiliar context of Hamiltonian flows, there ˙ xi=χi={xi,H}for a local function Hon\nthe phase space. For this case we can compute in the Darboux fram e and learn that the\nusual symplectic form (automatically satisfying (2.4)) is a solution als o to (2.6) since\nthe RHS in that case is zero. We recognize the RHS of (2.6) as exactly the obstruction\nto the flow, χibeing Hamiltonian.\nConformal transformation of the two-form, ˜ ω= Ωω, where Ω is a a scalar function,\ncan only relate two solutions of (2.4) and (2.6) IFF the Ω is a constant of the motion\ndΩ\ndt= 0. For in that case (2.6) indicates that\n∂Ω\n∂tωij=χj∂iΩ−χi∂jΩ (2.7)\nwhereas (2.4) yields\nωjk∂jΩ+ωkl∂jΩ+ωlj∂kΩ = 0 (2.8)\nso, contracting by χkand comparing with (2.7), we learn that Ω must be a constant of\nthe motion. Thus, each solution is conformally unique.\nWedonotknowwhatconditionson χileadtotheexistence ofevenonenon-singular\nsimultaneous solutionωof (2.4) and (2.6). Tarasov[1] provides an explicit solution for a\ngeneral Hamiltonian system ammended by a general linear frictional force. The general\nquestion of the existence of ω(t) for a more general χiis at this point unclear, but\nbeyond the scope of this present effort.\n3. Dynamical Symmetry in a Damped System\nConsider damped orbital motion in a central field;\n˙/vector x=/vector p (3.1)\n˙/vector p=−∂rV/vector x\nr−β(p)/vector p (3.2)\nwithr=|/vector x|andV(r) the interparticle potential (throughout we take the reduced ma ss\nto be normalized to 1). The function β(p) is some general function parameterizing the\nspeed dependence of the damping, and this form of the damping fun ction is the most\ngeneral consistent with isotropy and homogeniety of the damping f orces. Note that we\ncan understand this set as descending from a limit in which the centra l mass is very\nmuch larger than the orbital mass though, as in general, damping do es inextricably mix\nthe center of mass motion and the relative motion. We call linear damp ing the choice\nofβconstant.\nThe Equation (2.6) takes the form,\n∂ωxp(t)\n∂t=∂x(ωpxχx)−∂p(ωxpχp) =∂pωxp′(β(p)p′) (3.3)Linear frictional forces cause orbits to neither circulari ze nor precess 7\nAgain, we do not know if solutions to Equation (3.3) exist and satisfy J acobi for every\nchoice of β(p). However, for β(p) =const.there is a simple solution to Equation (3.3)\nthat satisfies Jacobi[1],\nωij(t) =eβtˆωij (3.4)\nwhere ˆωis the usual symplectic form of the undamped Kepler problem. Physic ally this\ncorresponds to the uniform shrinkage of phase space volumes und er linear damping.\nClearly, in going from {,}(Poisson bracket) to the new bracket {,}Tthe relations\nin Equation (2.1) gain a factor of e−βt. The algebra in the new bracket resulting from\nthis simple rescaling is still so(4). The utility of this simple change to the algebra of\nEquation (2.1) (which was for the undamped system) is that it is now c ompatible with\nthe evolution under Equation (3.1) and Equation (3.2) of the damped system. To see\nthis in an example, take the first relation in Equation (2.1) and take th e (total) time\nderivative of both sides. Then noted{Li,Lj}\ndt=−2β(2ǫijkLk)/ne}ationslash= 2ǫijk˙Lk;i.e.the usual\nPoisson bracket is no longer compatible with time evolution. Duplicating the previous\nlinefor{Li,Lj}T= 2e−βtǫijkLkonelearnsthat thisiscompatiblewiththeflowEquation\n(3.1)andEquation(3.2). Similarly, onemaycheckthatallthebracke tsinEquation(2.1)\n(after replacing {,}with{,}T) are as well. Also note that {Li,H}T= 0 ={Si,H}T,\nthough since brackets with Hno longer delineate time evolution, these equations do not\nimply that /vectorLand/vectorSare constants of the motion in the dissipative system (also clear\nfrom Equation (3.10) below).\nThe critique here is familiar to any attempt to reconcile symplectic str ucture\nand dissipation; fundamentally, Equation (3.1) and Equation (3.2) st ill treat xand\npdifferently so that time evolution is no longer an element in the dynamica l algebra of\n{,}or{,}T.\nTorelaxthecategoryof’constantsofthemotion’sufficientlyford issipative systems,\nconsider to what extent dynamical quantities averaged over some number of orbits\nchange on a longer time scale, i.e.on a timescale relevant to the dissipation (note\n1/β(p) is essentially that timescale). Let <>denote time averages over many orbits, O\na classical observable, and suppose that ωis a solution to Equation (2.6) and the Jacobi\nidentity for the system as in Equation (3.1) and Equation (3.2). In ge neral,\n<{O,H}T>=< ωxp(t)(˙x∂xO −(−˙p−β(p)p)∂pO)> (3.5)\n=< ω(t)/bracketleftBigdO\ndt−∂O\n∂t/bracketrightBig\n+ωxp(t)β(p)p∂pO> (3.6)\nNote that sums are implied in the x,pindices of the ωxp(t), the new symplectic form.\nAbove we have used isotropy to rewrite the sum in the first term in te rms of the\n(normalized) symplectic trace of ωxp(t) which we denote simply as ω(t). To show one\nintermediate step, integrating by parts and using Equation (3.3) we arrive at\n<{O,H}T>=1\nT∆(ωTO)+< ωxαωpβ(∂αχβ−∂βχα+χl∂lωαβ)O\n+ωxpβ(p)p∂pO−ω∂O\n∂t> (3.7)Linear frictional forces cause orbits to neither circulari ze nor precess 8\nWhere ∆( G) refers simply to the overall change of the quantity Gover time T. Finally,\nusing the Equation (2.5) and the fact that ωxpsatisfies the Jacobi identity we reduce\nthe above to\n<{O,H}T>=1\nT∆(ωO)+<(ωxp′∂p′χp−ωpx′∂x′χx)O+ωxpβ(p)p∂pO−ω∂O\n∂t>(3.8)\nWe now specialize to vector fields of the general form Equation (3.1) and Equation (3.2)\nto find,\n1\nT∆(ωO) =<{O,H}T+ω∂O\n∂t−ωxp(∂p(β(p))pO−β(p)p∂pO)> (3.9)\nand so making the RHS zero indicates conserved quantities in the non -Hamiltonian\nsystem. Again, this last result was derived for general β(p), which assumes only that\nthe friction is isotropic and homogeneous. In the linear friction case β(p) =β=const.\nFor that case, using O=L/ω2in the above equation implies that L/ωare constants of\nthe motion in this system. Similarly, taking O=S/ωindicates that ∆ Sis proportional\nto (2β/vectorL/ω)×< ω/vector p >which, again, is zero to first order in β. This result then applied\nto the case of bounded Kepler orbits with linear damping indicates tha t the (orbit-\naveraged) Runge-Lenz vector, and thus the dynamical algebra o f the Kepler problem,\nis conserved to leading order in the linear friction coefficient.\nIn elementary terms, although angular momentum /vectorLand/vectorSare constants in the\nHamiltonian system for V(r)∼1\nrthey evolve under linear damping of (3.1), (3.2) as,\n˙/vectorL=−β/vectorL˙/vectorS=−2β/vectorL×/vector p. (3.10)\nNote that in the weak damping limit, since /vectorLis conserved to O(β0), the second equation\ntime averages to −2< β/vector p >×</vectorL >. Thus, again we learn that if the damping were\nstrictly linear ( βconstant) then since < /vector p >= 0, the time average of˙/vectorSis 0, again\nindicating that the eccentricity vector would be conserved to leadin g order. Note also\nthat it is straightforward to integrate the /vectorLequation explicitely, finding /vectorL=/vectorL0e−βtthe\ninitial condition /vectorL0being identified now a conserved quantity of the dissipative system.\nWe use these results in the next section of this paper to ammend the ’textbook’ orbital\nsecular evolution equations.\n4. The Damped Kepler Problem\nThe previous section suggests that (linear-) damped bounded Kep ler orbits shrink but\nretain their aspect ratio and do not precess to leading order in the d amping. It is well\nknown that superlinear damping does lead to circularization whereas sublinear damping\nleads to infall orbits in the Kepler case. So far this begs the question s of whether this\ngeneralizes to other central field problems, and, if so, then at wha t order in the linear\ndamping coefficient do orbits undergo shape and precessional chan ge. In this section\nwe address both questions, first describing a problem that arises u sing a time-honored\npertubative method for treating general perturbing forces in th e Kepler problem, and\nsecond, generalize the result of the preceeding section to a broad class of central fieldLinear frictional forces cause orbits to neither circulari ze nor precess 9\npotentials. We then establish in precise terms the fate of Kepler orb its under linear\ndamping.\nConsider the usual secular orbital evolution method (called “the va riations of\nconstants”) most common in literature on cellestial mechanics, for example, in [48]\n(Chapter 11 Section 5, pg. 323, though see also the treatments o f non-linear friction\nin [49, 50, 51]). In the “variations of constants’ method, orbital r esponse to an applied\nforce/vectorF=R/vector x+N/vectorL+B/vectorL×/vector x, in the orbit’s tilt Ω, the orbital plane’s axis, i, the\neccentricity ǫ, the angle of the ascending node ωthe semi-major axis aand the period\nT= 2π/n(in their notation) evolve following[48],\ndΩ\ndt=nar√\n1−ǫ2Nsinu\nsini(4.1)\ndi\ndt=nar√\n1−ǫ2Ncosu (4.2)\ndω\ndt=na2√\n1−ǫ2\nǫ[−Rcosθ+B(1+r\nP)sinθ]−cosidΩ\ndt(4.3)\ndǫ\ndt=na2√\n1−ǫ2[Rsinθ+B(cosθ+cosE)] (4.4)\nda\ndt= 2na2[Raǫ√\n1−ǫ2sinθ+Ba2\nr√\n1−ǫ2] (4.5)\nand where\ndn\ndt=−3n\n2ada\ndt(4.6)\nwithu=θ+ωand for the unperturbed Kepler orbit,P\nr= 1 +ǫcosθ,Pis the latus\nrectum, and Eis the anomaly, i.e.r=P(1−ǫcosE). The central angle θis found\nvia the usual definition of angular momentum. When we specialize thes e Kepler orbit\nevolution equations to the case of isotropic and homogeneous frict ion we learn that (see\n[48], Chapter 11, section 7 but using β(p)pforTin that reference)\nda\ndt= 2pa2β(p)p (4.7)\ndω\ndt=2sinθ\nǫβ(p) (4.8)\nand\ndǫ\ndt= 2(cosθ+ǫ)β(p) (4.9)\nWe can now specialize further to the marginal case, linear friction β(p) =β=const. To\nintegratetheseequations, note r2dθ\ndt=L=L0e−βtand, intermsoftheforcecomponents,\nN= 0, and R=β(p)pcosυandB=β(p)psinυwhereυis the angle between the radius\nvector and the tangent to the orbit. That angle can be written usin g the parametericLinear frictional forces cause orbits to neither circulari ze nor precess 10\n20 40 60 80Time\n/Minus0.02/Minus0.010.010.020.030.04Eccentricity\nFigure 1. The eccentricity in the actual damped Kepler problem (solid curve)\ncompared with the eccentricity from (4.10) and (4.11) (dashed cur ve) versus time.\nNote that for the later the eccentricity can oscillates through zer o and can even, as in\nthis case, asymptote to a negative value.\nform ofrin terms of the constants of the orbit and the angle θ, (sinυ=L/rpand\ncosυ=ǫsinθ/Lp) resulting in a self-contained pair of ODE’s in ǫ,θandt,\ndθ\ndt=e+3βt\nL3\n0(1+ǫcosθ)2(4.10)\ndǫ\ndt=−2β(cosθ+ǫ) (4.11)\nIf we integrate these to leading order in βonly (by, for example, using the first equation\nto eliminate the time derivative to leading order in β) we do indeed find that the\neccentricity is an orbit-averaged constant of the motion. But diffic ulty arises when we\ntry to understand these equations beyond leading order in the dam ping, as a direct\nnumerical integration of the equation set reveals (Figure 1). For a broad set of initial\nangles and small initial eccentricities, the ǫpasses through zero and goes negative.\nFor comparison, the eccentricity (i.e. the square root of the lengt h of the /vectorSvector)\ncomputed by numerical integration of the original equations of mot ion for precisely the\nsame mechanical parameters and initial conditions is included on that figure.\nEven if one only wanted to assign importance to the asymptotic chan ge in the\neccentricity, that asymptotic change from integrating the equat ion pair (4.10) , (4.11)\ndoes not scale correctly with the damping coefficient, as may be chec ked numerically\n(see [48] for further admonisions against using the “variations of c onstants” methodLinear frictional forces cause orbits to neither circulari ze nor precess 11\nover long timescales). Clearly the “variations of constants” metho d at higher orders\nin the evolution leads to unphysical results at short and long timesca les. The fault is\ntraceabletothefactthatinhigherorderthereare β-(thedampingcoefficient)dependent\nterms in the orbit shape whose contributions are ignored substitut ing forrusing the\nundamped Kepler shape of the ellipse. This substitution is however ine ctricably part of\nthe“variationofconstants” method. Tofurtherclarifythisprob lemwiththe“variations\nof constants” method, it is not due to some ambiguity in the eccentr icity of a non-\nclosed orbit, since eccentricity itself, rendered as the length of th e/vectorSvector, has a local\ndefinition. Algebraically, withthisdefinitionoftheeccentricity, note thatǫ2−2L2U=k2\nin the 1/rpotential even under arbitrary damping . A more useful algebraically identical\nform isǫ2= 4V2r2−2R2H, from which, since His negative for any damping function\non a bounded orbit, we see immediately that ǫ2is bounded away from zero.\nWe now, in two parts, describe an approach emphasising the secular evolution of\nthe dynamical symmetry, that addresses this mismatch with the us ual “variation of\nconstants” method. For simplicity we focus in the main on potentials w ith fixed scaling\nwieghtα, deined through V(r) =krα. Orbits in any central potential are characterized\nby a fixed orbital plane and a single dimensionless parameter, the rat iod/cof the\nperihelion distance dto the aphelion distance c. LetLdenote the angular momentum\nso thatVeff(r) =L2\n2r2+V(r) is the effective potential. Then from Veff(c) =U=Veff(d)\nwhereUis the total energy for a V(r) of a fixed α, we have,\nU\nk=cα+2−dα+2\nc2−d2L2\n2k=cα−dα\nc2−d2c2d2(4.12)\nthat then can be reduced to a ’dispersion relation’ between UandL,\nUα+2\nk2L2α=f(d/c) (4.13)\nwherefin this case is a monotonic function on [0 ,1]. Note also that f(x) =f(1/x).\nWe calld/cthe aspect ratio of the orbit (related to the eccentricity in the α=−1 case).\nThus in leading order (only) in the damping we think of the RHS as a func tion of the\norbital eccentricity only. In applications, the differential form of ( 4.13) is particularly\nuseful,\n/bracketleftBig\n(α+2)δU\nδL−2αU\nL/bracketrightBigδL\nU=f′\nfδ(d/c) (4.14)\nEquations of this sort are often written down when referring to th e secular evolution of\norbital system (see for example [52] and references therein). As before consider further\nonlydamping forcesthatareisotropicandhomogeneous; theycan bewrittenintheform\nfrom the previous section, /vectorFdrag=−β(p)/vector p. (to simplify notation we henceforth drop\nthe vector symbol over the pdenoting by pboth|p|and/vector p, unambigious by context).\nIn the limit of weak damping we expect Lto be approximately constant so that, time\naveraging, we arrive at < δL >=−< β(p)> Lto leading order in β(p). Note also that\nδU=−β(p)p2to leading order in β.\nNotethatthetimederivativeof RisthesumofaPoissonbracket withHplusaterm\nprobprtional to β(see Equation (4.16)). This is,dR\ndt= 2V+O(β) which, averaged overLinear frictional forces cause orbits to neither circulari ze nor precess 12\nbounded orbits, indicates (the virial theorem) that <V>=O(β). Thus for V(r) =krα\nthis implies that < p2>=α < V > +O(β) so that < U >=α+2\n2< V >+O(β), which\nto leading order in βin Equation (4.14) indicates\n−2α\n< p2>/parenleftBig\n< β(p)p2>−< β(p)>< p2>/parenrightBig\n=f′\nfδ(d/c) (4.15)\nThus restricted to linear damping ( βconstant), but for any α, the averages in (4.15)\nfactorize trivially and the aspect ratio is unchanged to leading order under linear\ndamping. The Equation (4.15) also indicates that this will, in general, no t be the\ncase for a velocity dependent damping coefficient. Although for pot entials with a fixed\nscaling exponent αthere is but one dimensionless parameter (See LHS Equation (4.13)) ,\nthe introduction of the damping coefficient βintroduces new length and time scales,\nindicating that the orbital aspect ratio d/cmay be a function of βand time. The fact\nthatβis time odd does apparently not preclude its inclusion to linear order in t he\norbital aspect ratio in general. Thus, we repeat, the conclusion th at for any monomial\npotentialslineardampingpreserves theorbitalshapeisnotaconse quence ofdimensional\nanalysis and discrete symmetries.\nAs a final check, note that relation Equation (4.15) and Equation (3 .9) are both\nconsistent with the attractors of the secular flow in the orbital sh ape. For circular orbits\np2isaconstant ofthemotion(againtoleading order in β)andthustheLHSofEquation\n(4.15) is zero, as expected by symmetry. Note that in contrast to (4.15) in Equation\n(3.9) the change in the eccentricity is proportional to the eccentr icity for any β(p), and\nsince the eccentricity vanishes in this limit its orbit-averaged change by (3.9) does as\nwell. Also the strictly radial infall orbit limit is one in which the inner radius ,d→0,\nand so the LHS of Equation (4.15) being non-zero in this limit looks incon clusive. But,\nby the definition of fvia Equation (4.13) we see that in this limit f→0 orf→ ∞\ndepending on the sign of α. Thus, by (4.12), L= 0 and remain zero for any β(p). In\nTarasov’s formulation, since (3.9) isfully vector covariant forisotr opic andhomogeneous\n(but otherwise arbitrary β(p)) the change in the /vectorSmust be along the vector itself for the\nradial infall case. Furthermore, as indicated in the discussion follow ing (3.9), the change\n∆/vectorSis linear in /vectorL(for any β(p)) which vanishes in the radial infall case. In summary,\nboth prescriptions indicate that circular orbits and radial infall mus t satisfy <˙/vectorS >= 0\nfor any damping function as expected on the grounds by symmetry.\nFurthermore, it is straightforward to go further and perturbat ively show using\n(4.15) andthe equations of motionthat β=const.istheonlyshape-preserving damping\nfunctionfortheKeplerpotential( α=−1). Theresultisclearlycommontoallmonomial\ncentral potentials only, as it is straightforward to demonstrate a counterexample in a\nmore complicated potential. This is due to the fact that there are no additional length\nscales in the potential and is not the case with other potentials, suc h as the effective\npotential in General Relativity (where the Schwarzschild radius aris es asa second length\nscale in the potential).\nReturning to the rather general statement (3.9), in the Tarasov formulation, the\nexplicit time dependence of a candidate constant of motion Ogives a second term whichLinear frictional forces cause orbits to neither circulari ze nor precess 13\ncancels the last two terms. If the operator has a fixed momentum s caling weight (for\nexample, Lis weight 1 and Sis essentially wieght 2), the last two terms will be of\nthat same scaling weight only for the case of linear friction, β(p) =const.Note that\nthis argument does not rule out the existence of additional consta nts of the motion\nin the dissipative system that scale to zero as one goes to the Hamilto nian limit.\nThe argument does, however, certify that in the case of linear fric tion the original\nHamiltonian symmetries do survive to leading order in that friction.\nHavingshownthatlinearfrictionpreserves theeccentricity tolead ingorderbegsthe\nquestion of what happens in higher order in the damping. In the spirit of the discussion\nafter (4.13) where the Virial played a key role, consider the time evo lution of that part\nof the dynamical algebra\n˙R={R,H}−β(p)R= 2V −β(p)R (4.16)\n˙V={V,H}−2β(p)(2V −H) =−1\nr(∂rV+1\n2∂r(r∂rV))R−2β(p)(2V −H) (4.17)\n˙H=−2β(p)(2V −H) (4.18)\nand for completeness, we have\n{R,V}=H+V −V+r\n2∂rV+r\n2∂r(r∂rV) (4.19)\nTo orient the reader to the content of these, first note the Hamilt onian limit ( i.e.\nβ(p)→0 limit) for the Kepler case ( α=−1), both <V>→0 and<R/r3>→0\nas expected. We thus expect both of these time averages to be at least proportional to\nsome positive power of β. Now, in the abscence of damping Vis time even and Ris time\nodd. Formally, taking βto be time odd preserves this discrete symmetry of the above\nevolution equations. Since we expect the <V>and<R>to be analytic functions of\nβ, it must thus be that <V>vanishes quadratically as β→0.\nAn elementary argument now certifies that the <V>must be nonpositive in the\ndamped system. Take β(p) =βa constant. Consider the radial component of the\nvelocity, R/r. It must average to zero in the β→0 limit. Since the damped orbit must\nshrink, we thus expect <R/r >∼<−Cβ >for some positive quantity C(a function of\nthe other orbital parameters, etc.). But now take the evolution e quation (4.16) divide\nbyrand time average. Clearly, integrating by parts, <˙R/r >=−<R/r2˙r >=\n−<R2/r3>implying that the time average <2V/r−βR/r >must also be strictly\nnegative. But since <R/r >must already be negative, the <V>must also be strictly\nnegative in the damped system.\nSpecializing to Kepler ( α=−1), differentiating ǫ2=|/vectorS|2in time and applying the\nequations of motion of the system with friction, we learn thatd\ndtǫ2=−8βL2V. Using\nthe fact that <V>is negative and order β2and integrating both sides, we learn that\nthe asymptotic change in the eccentricity to leading order is positive and also of order\nβ2(note the integral itself scales as 1 /β). Furthermore, since these are exact evolution\nequations, we have shown that the integration is well behaved thro ughout. Thus linear\nfriction causes Kepler orbits to become more eccentric by a fixed am ount that scales\nwith the square of the linear damping coefficient.Linear frictional forces cause orbits to neither circulari ze nor precess 14\n5. Conclusion\nTypicallyHamiltoniansymmetrieslosethierrelevancetothegeometry ofthetrajectories\nwhen damping forces are added to the Hamiltonian system. If the da mping is weak,\nhomogeneous and isotropic, then for linear damping in monomial pote ntials, we have\nshown that orbit-averaged shape is stationary. This can underst ood most easily through\nTarasov’s generalization of conserved quantities from the Hamilton ian context to the\nnon-Hamiltonian setting. This approach also quantifies in precise ana lytic terms the\nfate and subsequent utility of the dynamical symmetry algebra in th e associated non-\nHamiltonian system.\nThere are three main frameworks for understanding orbital motio n in a perturbed\ncentral field. The first is directly from the equations of motion; this admits\nstraightforward generalization to the non-Hamiltonian case but so mewhat obscures the\nstructure and fate of the dynamical symmetry group. The secon d, namely the KS\nconstruction, embedstheKeplerorbitprobleminthehigherdimensio nal setofharmonic\noscillators with constraints; this illuminates the dynamical symmetry group but does\nnot seem to readily admit a generalization to the non-Hamiltonian syst em. Lastly,\nthe geometrical approach, namely that which associates the Keple r Hamilton equations\nto geodesic flow on manifolds of constant curvature, also illuminates the dynamical\nsymmetry groupwhile making thegeneralizationtothenon-Hamiltonia ncasesomewhat\nunclear.\nIn light of these difficulties, we used Tarasov’s framework (and applie d to the\ndamped central field problem here) for extending Poisson symmetr ies to dissipative\nsystems, emphasising its utility in making crisp connections between d ynamics, algebra\nand the geometric character of the solutions. Finally, the dynamica l algebra remains\nwhole in first order in the linear dissipative system, but flow at higher o rder is not\ntrivial. The secular perturbative method “variation of constants” is not adequate to\nexplain this, however an elementary method based on the Virial suba lgebra explains the\nchange in the shape of kepler orbits in higher order in linear damping.\nAcknowledgments\nThis work was supported in part by the National Science Foundation through a grant\nto the Institute of Theoretical Atomic and Molecular Physics at Har vard University\nand the Smithsonian Astrophysical Observatory, where this work was begun and by\na fellowship from the Radcliffe Institute for Advanced Studies where this work was\nconcluded. It is a pleasure to acknowledge useful discussions with H arvard-Smithsonian\nCenter for Astrophysics personell Mike Lecar, Hosein Sadgehpou r, Thomas Pohl and\nMatt Holman.Linear frictional forces cause orbits to neither circulari ze nor precess 15\nBibliography\n[1] Tarasov, V E 2005 J. Phys. A 38#10/11 2145\n[2] Ermanno, G J 1710 G. Lett. Ital. 2447\n[3] Hermann, J 1710 Hist. Acad. R. 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(Richmond, Va. , Willman-Bell\nInc.)\n[49] Watson J. C. 1964 Theoretical Astronomy (NY, Dover Press)\n[50] Murray, C. D. and Dermott S. F. 2000 Solar System Dynamics (Cambridge, Cambridge Univ.\nPress)\n[51] Brouwer D and Clemence G M 1961 Methods of Celestial Mechanics (NY, Academic Press)\n[52] Kalimeris A Livaniou-Rovithis H 2006 Astrophys. Space Sci. 304113"    },    {        "title": "0708.4164v1.Asymptotic_improvement_of_the_Gilbert_Varshamov_bound_for_linear_codes.pdf",        "content": "arXiv:0708.4164v1  [cs.IT]  30 Aug 2007Asymptotic improvement of the Gilbert-Varshamov\nbound for linear codes\nPhilippe Gaborit∗Gilles Z´ emor†\nAugust 29, 2007\nAbstract\nThe Gilbert-Varshamov bound states that the maximum size A2(n,d)\nof a binary code of length nand minimum distance dsatisfiesA2(n,d)≥\n2n/V(n,d−1) where V(n,d) =/summationtextd\ni=0/parenleftbign\ni/parenrightbig\nstands for the volume of a Ham-\nming ball of radius d. Recently Jiang and Vardy showed that for binary\nnon-linear codes this bound can be improved to\nA2(n,d)≥cn2n\nV(n,d−1)\nforca constant and d/n≤0.499. In this paper we show that certain\nasymptotic families of linearbinary [ n,n/2]randomdouble circulantcodes\nsatisfy the same improved Gilbert-Varshamov bound. These result s were\npartially presented at ISIT 2006 [3].\nIndex terms: Double circulant codes, Gilbert-Varshamov bound, linear\ncodes, random coding.\n1 Introduction\nThe Gilbert-Varshamov bound asserts that the maximum size Aq(n,d) of a\nq-ary code of length nand minimum Hamming distance dsatisfies\nAq(n,d)≥qn\n/summationtextd−1\ni=0/parenleftbign\ni/parenrightbig\n(q−1)i. (1)\nThis result is certainly one of the most well-known in coding theory, it was\noriginally stated in 1952 by Gilbert [5] and improved by Vars hamov in [15]. In\n1982 Tsfasman, Vladuts and Zink [14] improved the GV bound on the number\nof codewords by an exponential factor in the block length, bu t this spectacular\nresult only holds for some classes of non-binary codes. Rece ntly Jiang and\n∗XLIM, Universit´ e de Limoges, 123, Av. Albert Thomas, 87000 Limoges, France.\ngaborit@unilim.fr\n†Universit´ e de Bordeaux 1, Institut de Math´ ematiques de Bo rdeaux, 351 cours de la\nLib´ eration, 33405 Talence. zemor@math.u-bordeaux1.fr\n1Vardy [6] improved the GV bound for non-linear binary codes b y a linear factor\nin the block length nto\nA2(n,d)≥cn2n\nV(n,d−1), (2)\nford/n≤0.499, for a constant cthat depends only on the ratio d/nand\nwhereV(n,d) =/summationtextd\ni=0/parenleftbign\ni/parenrightbig\nstands for the volume of a Hamming ball of radius d.\nThis new bound asymptotically surpasses previous improvem ents of the binary\nGilbert-Varshamov bound which only managed to multiply the right hand side\nin (1) by a constant (see [6] for references). The method used by Jiang and\nVardy relies on a graph-theoretic framework and more specifi cally on locally\nsparse graphs which are used to yield families of non-linear codes (their result\nwas later slighlty improved in [16]). In this paper we also im prove on the the\nGilbert-Varshamov bound by a linear factor in the block leng th but for linear\ncodes, thereby solving one of the open problems of [6]. The me thod we use is\nnot related to graph theory and relies on double circulant ra ndom codes.\nDouble circulant codes are [2 n,n] codes which are stable under the action\nof permutations composed of two circular permutations of or dernacting si-\nmultaneously on two differents halves of the coordinate set. T hese codes can\nalso be seen as quasi-cyclic codes, a natural generalizatio n of cyclic codes [13].\nTheir study started in 1969 in [8] and since they gave some ver y good codes\nit was natural to wonder whether they could be made to satisfy the Gilbert-\nVarshamov bound. A first step in that direction was made by Che n, Peterson\nand Weldon in [1] who prove that when 2 is a primitive root of th e ringZ/pZfor\npa prime, double circulant [2 p,p] random codes satisfy the Gilbert-Varshamov\nbound; unfortunately it is still unknown (this is Artin’s ce lebrated conjecture,\n1927) whether an infinity of such pexists. Later Kasami [9], building on this\nidea, extended the result of [1] to the case of powers of such p, and obtained\na bound which is worse than the Gilbert-Varshamov bound by an exponential\nfactor in the block length (though a very small one). Later Ka sami’s work\nwas generalized to other cases in [7, 11, 12], and, in particu lar in [2], bounds\nwere proven for certain classes of quasi-cyclic codes that a re worse than the\nGilbert-Varshamov bound only by a subexponential factor in the block length.\nIn this paper, building anew on Kasami’s idea we prove, by usi ng a proba-\nbilistic approach, that randomly chosen double circulant c odes not only satisfy\nthe Gilbert-Varshamov bound with high probability, but als o the same linear\nimprovement as that of Jiang and Vardy (2).\nThe paper is organized as follows: in Section 2 we cover the ma in ideas\ninvolved. We start by recalling the probabilistic method fo r deriving lower\nboundsontheminimumdistanceoflinearcodes(section 2.0) , thenweintroduce\ndouble circulant codes in section 2.1 and derive (5) an upper bound on the\nprobability that a random double circulant code contains a n on-zero vector of\nweight not more than a given w. In section 2.2 we study the probability that\na given vector belongs to a randomly chosen double circulant code. Finally in\nsection 2.3 we derive our improved lower bound on the minimum distance in\nthe simple case when the codelength is 2 pand 2 is a primitive root of Z/pZ:\nthe result is given in Theorem 4.\n2in Section 3, we develop our method in the more complicated ca se of block-\nlengths 2 pm,pa “Kasami” prime, in order to obtain an infinite family of doub le\ncirculant codes with an improved minimum distance. Section 3.1 starts by giv-\ningan informal sketch of the content of section 3, which is in tended to give some\nguidance to the reader and discuss the technical issues invo lved. Section 3.2\nshows how to derive our main result, which is Theorem 8, from a proposition\non the weight distribution of a certain class of cyclic codes . Finally section 3.3\nis devoted to a proof of this last proposition.\nSection 4 concludes by some comments and side results.\n2 Overview of the method, the simple cases\n2.0 The Gilbert Varshamov bound for linear codes and its im-\nprovement\nTo put the rest of the paper into perspective and introduce no tation, let us\nrecall how the probabilistic method derives the Gilbert Var shamov bound for\nlinear codes. Rather than bounding the code size from below b y a function of\nthe minimum distance, as in (2), we fix a lower bound on the code rate and\nfind a lower bound on the minimum distance. We limit ourselves to the rate\n1/2 case because it will be our main object of study.\nLetCrandbethe randomcode of length 2 nand dimension k≥nobtained by\nchoosing randomly and uniformly a n×2nparity-check matrix in {0,1}n×2n.\nThe probability that a given nonzero vector x= (x1...x2n) is a codeword\nis clearly 1 /2n. Letwbe a positive number, not necessarily an integer. We\nare interested in the random variable X(w) equal to the number of nonzero\ncodewords of Crandof weight not more than w. In other words\nX(w) =/summationdisplay\nx∈B2n(w)Xx (3)\nwhereB2n(w) denotes the set of nonzero vectors xofV2n={0,1}2nof weight\nat most w, andXxis the Bernoulli random variable equal to 1 if x∈Crand\nand equal to zero otherwise. Now whenever we prove that the pr obability\nP(X(w)>0) is less than 1, we prove the existence of a [2 n,k,d] code with\nk≥nandd > w. Since the variable X(w) is integer valued we have\nP(X(w)>0)≤E[X(w)] =/summationdisplay\nx∈B2n(w)E[Xx] =|B2n(w)|P(x∈Crand)\n=|B2n(w)|1\n2n.\nHence, for every positive integers nandwsatisfying |B2n(w)|<2nthere exists\na linear code of parameters [2 n,n,d > w ]. Reworded, we have the following\nlower bound on d, essentially equivalent to (1).\nTheorem 1 (GV bound) For every positive integer nthere exists a linear\ncode of parameters [2n,n,d]satisfying\n|B2n(d)| ≥2n.\n3In the present paper we shall prove :\nTheorem 2 There exists a positive constant band an infinite sequence of in-\ntegersnand[2n,n,d]linear codes satisfying\n|B2n(d)| ≥bn2n.\nThis result, equivalent to (2) for rate 1 /2, will be obtained by again choosing\nrandom matrices, but from a restricted class, namely the set of parity-check\nmatrices of double circulant codes.\n2.1 Double circulant codes\nA binary double circulant code is a [2n,n] linear code Cwith a parity-check\nmatrix of the form H= [In|A] whereInis then×nidentity matrix and\nA=\na0an−1... a 1\na1a0... a 2\na2a1... a 3\n....................\nan−1an−2... a 0\n.\nThere is a natural action of the group Z/nZon the space V2n={0,1}2nof\nvectorsx= (x1...xn,xn+1...x2n) namely,\nZ/nZ×V2n→V2n\n(j,x)/ma√sto→j·x\nwhere\n1·x= (xn,x1...xn−1,x2n,xn+1,...x2n−1)\nandj·x= (j−1)·(1·x). The double circulant code Cis clearly invariant\nunder this group action. Consider now Cto be the random code Crandobtained\nby choosing the vector a= (a0...an−1) randomly and uniformly in {0,1}n.\nAs before, we are interested in the random variable X(w) defined by (3) and\nequal to the number of nonzero codewords of Crandof weight not more than\nw. We are interested in the maximum value of wfor which we can claim\nthat P(X(w)>0)<1, for this will prove the existence of codes of parameters\n[2n,n,d > w ]. The core remark is now that, if y=j·x, then\nXy=Xx\nwhereXx(Xx) is the Bernoulli random variable equal to 1 if x∈Crand(y∈\nCrand) and equal to zero otherwise. Let now B′\n2n(w) be a set of representatives\nof the orbits of the elements of B2n(w), i.e. for any x∈B2n(w),|{j·x,j∈\nZ/nZ} ∩B′\n2n(w)|= 1. We clearly have X(w)>0 if and only if X′(w)>0\nwhere\nX′(w) =/summationdisplay\nx∈B′\n2n(w)Xx.\n4Denote by ℓ(x) the length (size) of the orbit of x, i.e.ℓ(x) = #{j·x,j∈Z/nZ}.\nWe have\nX′(w) =/summationdisplay\nx∈B2n(w)Xx\nℓ(x)(4)\nBy writing P( X(w)>0) = P(X′(w)>0)≤E[X′(w)], together with (4) we\nobtain\nP(X(w)>0)≤/summationdisplay\nd|n/summationdisplay\nwt(x)≤w\nℓ(x)=dE[Xx]\nd. (5)\nSuppose in particular that nis a prime, in that case orbits are of size 1 or n,\nand ifw < nthen clearly the orbit of xhas sizenfor anyx∈B2n(w), so that\n(5) becomes\nP(X(w)>0)≤E[X(w)]/n.\nIf we can manage to prove that\nE[X(w)]≤ |B2n(w)|c\n2n(6)\nfor constant c, then we will have proved the existence of double circulant c odes\nof parameters [2 n,n,d > w ], for any wsuch that |B2n(w)|<1\ncn2n.\n2.2 The behaviour of P(x∈Crand)\nTo prove equality (6) we need to study carefully the quantiti es E[Xx], for\nx∈B2n(w), since\nE[X(w)] =/summationdisplay\nx∈B2n(w)E[Xx].\nForx∈V2n, let us write x= (xL,xR) withxL,xR∈ {0,1}n. Consider the\nsyndrome function σ\nσ:V2n→Vn\nx/ma√sto→σ(x) =xtH=σL(x)+σR(x)\nwhereσL(x) =xLandσR(x) =xRtA.\nForanybinaryvectoroflength n,u= (u0,...,u n−1), denoteby u(Z) =u0+\nu1Z+···+un−1Zn−1its polynomial representation in the ring F2[Z]/(Zn+1).\nFor any u∈Vn, letC(u) denote the cyclic code of length ngenerated by the\npolynomial representation u(Z) ofu. SinceσR(x) has polynomial representa-\ntion equal to xR(Z)a(Z), we obtain easily\nLemma 3 The right syndrome σR(x)of any given x∈V2nis uniformly dis-\ntributed in the cyclic code C(xR). Therefore, the probability P(x∈Crand)that\nxis a codeword of the random code Crandis\n•P(x∈Crand) = 1/|C(xR)|ifxL∈C(xR),\n•P(x∈Crand) = 0 ifxL/ne}ationslash∈C(xR).\n52.3 The case nprime and 2primitive modulo n\nIfnis prime and 2 is primitive modulo nthen, over F2[Z], the factorization of\nZn+1 into irreducible polynomials is\nZn+1 = (1+ Z)(1+Z+Z2+···+Zn−1)\nand there is only one non-trivial cyclic code of length n, namely the [ n,n−\n1,2] even-weight code. Therefore P( X(w)>0) = P(X′(w)>0)≤E[X′(w)]\ntogether with (4) and Lemma 3 give\nP(X(w)>0)≤/summationdisplay\nwt(xL)+wt(xR)≤w\nwt(xR) odd1\nn2n+/summationdisplay\nwt(xL)+wt(xR)≤w\nwt(xR) even\nwt(xL) even1\nn2n−1(7)\nP(X(w)>0)≤2|B2n(w)|1\nn2n.\nWe therefore have the following result:\nTheorem 4 Ifpis prime and 2is primitive modulo p, then there exist double\ncirculant codes of parameters [2p,p,d > w ]for any positive number wsuch that\n2|B2p(w)|< p2p.\nUnfortunately, it is not known (though it is conjectured) wh ether there\nexistsaninfinitefamilyofprimes pforwhich2isprimitivemodulo p. Therefore,\nto obtain Theorem 2 we will envisage cases when nis non-prime. This will\ninvolve two technical difficulties, namely dealing with non- trivial divisors dof\nnin (5), and non-trivial cyclic codes C(xR) of length nin Lemma 3.\n3 An infinite family of double circulant codes\n3.1 Preview\nIn this section we will study the behaviour of the minimum dis tance of random\ndouble circulant codes for the infinite sequences of blockle ngths 2nintroduced\nby Kasami : we will have n=pmfor suitably chosen p. We will first specialise\ninequality (5) to this case, for which all the possible orbit sizesℓare powers\nofp,ps,s≤m. Applying Lemma 3 will lead us to an upper bound (13) on\nP(X(w)>0) that involves the weight distributions of the cyclic code s of length\nn. This upper bound can be essentially thought of as the same as (7), plus a\nnumber of parasite terms involving all vectors x= (xL,xR) ofB2n(w) for which\nbothxLandxRare codewords of some cyclic code of length nthat is neither\nthe whole space {0,1}nnor the [ n,n−1,2] even-weight subcode. The problem\nat hand is to control the parasite terms so that they do not pol lute too much\nthe main term i.e. the right hand side of (7). To do this, the cr ucial part will\nbe to bound from above with enough precision terms of the form\n/summationdisplay\ni+j≤wAi(C)Aj(C)1\n|C|(8)\n6whereCis a cyclic code of length nandAi(C) is the number of codewords of\nweighti. In section 3.2 we shall state such an upper bound, namely Pro posi-\ntion 5, and show how it leads to the desired result which will b e embodied by\nTheorem 8.\nSection 3.3 will then be devoted to proving Proposition 5. It is not easy\nin general to estimate the weight distribution of cyclic cod es that don’t have\nextra properties, but it turns out that for these particular code lengths of the\nformn=pm, all cyclic codes Chave a special degenerate structure. Either C\nconsists of a collection of vectors of the form ( x,x,...,x ) wherexis a subvector\noflength n/pandisrepeated ptimes, or Cisthedualofsuchacode. Section 3.2\nwill have reduced the problem to the latter class of cyclic co des only. Ideally,\nwe would like to claim that the cyclic codes Chave a binomial distribution\nof weights, i.e. Ai(C)≈|C|\n2n/parenleftbign\ni/parenrightbig\n, however this is not true, the cyclic codes C\nhave many more low-weight codewords than would be dictated b y the binomial\ndistribution. The problem of the unbalanced couples ( i,j), (ismall and jlarge\nor vice versa) in the sum (8) is therefore dealt with by the tri vial upper bound\nAi(C)≤/parenleftbign\ni/parenrightbig\n: Lemma 11 will show that these terms account for a sufficiently\nsmall fraction of |B2n(w)|/2n. Lemma 10 is the central result of section 3.3\nwhich gives a more refined upper bound on Ai(C) foriwell enough separated\nfrom 0, i.e. i≥κnfor constant positive κ. Fortunately, we do not need Ai(C)\nto be too close to the binomial distribution, and the cruder u pper bound of\nLemma 11 will suffice to derive Proposition 5.\n3.2 Reducing the problem to the study of the weight distribu-\ntion of certain cyclic codes\nFollowing Kasami [9], let us consider nof the form n=pmwhere 2 is primi-\ntive modulo pand 2p−1/ne}ationslash= 1 mod p2. It will be implicit that all the primes p\nconsidered in the remainder of section 3 will satisfy this pr operty. Let us also\nsuppose m≥2, since the case m= 1 is covered by Theorem 4.\nIt is known [9] that the irreducible factors of Zn+ 1 inF2[Z] are 1 + Z\ntogether with all the polynomials of the form\n1+Q(Z)+Q(Z)2+···Q(Z)p−1(9)\nforQ(Z) =Z,=Zp,Zp2,...,Zpm−1.\nSincenis a prime power, (5) gets rewritten through Lemma 3 as:\nP(X(w)>0)≤m/summationdisplay\ns=1/summationdisplay\nwt(x)≤w\nℓ(x)=ps\nC(xL)⊂C(xR)1\nps|C(xR)|(10)\nNote that x∈V2nhas orbit length ℓ(x)< nif and only if both xLandxR\nare made up of psuccessive identical subvectors of length n/p. Equivalently xL\nandxReach belong to the cyclic code generated by the polynomial\nPn(Z) = 1+Zn/p+Z2n/p+···+Z(p−1)n/p. (11)\n7LetCndenote the set of those cyclic codes of length nwhose generator poly-\nnomial is nota multiple of Pn(Z). All the other cyclic codes of length nare\nobtained by duplicating ptimes some cyclic code of length n/p. Therefore, for\ns=m, the inner sum in (10) can be bounded from above by:\n/summationdisplay\nC∈Cn/summationdisplay\ni+j≤wAi(C)Aj(C)1\nn|C|(12)\nwhereAi(C) denotes the number of codewords of Cof weight i. Applying (12)\nrecursively, we obtain from (10)\nP(X(w)>0)≤m−1/summationdisplay\ns=0/summationdisplay\nC∈Cn/ps/summationdisplay\ni+j≤w/psAi(C)Aj(C)1\n|C|n/ps.(13)\nWe now proceed to evaluate the righthandside of (13). The mos t technical\npart of our proof of Theorem 2 is contained in the following Pr oposition.\nProposition 5 There exist positive constants q,K,c1andγ <1such that, for\nanyn=pmwithp≥q, we have |B2n(2Kn)| ≤2nand for any positive real\nnumberw,K≤w/2n≤1/4, and for any cyclic code CofCn, we have\n/summationdisplay\ni+j≤wAi(C)Aj(C)1\n|C|≤c1|B2n(w)|\n2nγn−dimC.\nSuitable numerical values of the constants are q= 143,K= 0.1,γ= 1/21/5,\nc1= 26/5.\nBefore proving Proposition 5, let us derive the consequence s on the proba-\nbility P(X(w)>0). That will lead us to our main result, namely Theorem 8,\nthe consequence of which is Theorem 2. We have:\nLemma 6 There exists a constant c2such that, for any n=pm,p > q, and\nfor anyK≤w/2n≤1/4,\n/summationdisplay\nC∈Cn/summationdisplay\ni+j≤wAi(C)Aj(C)1\n|C|≤c2|B2n(w)|\n2n.\nA suitable numerical value for c2isc2= 4.3.\nProof:From Proposition 5 it is enough to show that the sum/summationtext\nC∈Cnγn−dimC\nis upperbounded by a constant for any γ <1. Choosing a code CinCn\nis equivalent to choosing its generator polynomial, and fro m the list (9) of\nirreducible factors of Zn+ 1, we see that if we order all possible generator\npolynomials by increasing degrees, we have 1 and 1+ Z, then 2 polynomials of\ndegree at least p−1, then 4 polynomials of degree at least p(p−1), ... then 2i\n8polynomials of degree at least p(p−1)i−1and so on. Therefore, since n−dimC\nequals the degree of the generator polynomial, we obtain\n/summationdisplay\nC∈Cnγn−dimC≤1+γ+2γp−1+/summationdisplay\ni≥22iγp(p−1)i−1\n≤1+γ+2γp−1+/parenleftbigg2\np−1/parenrightbigg2/summationdisplay\ni≥2(p−1)iγ(p−1)i\n≤1+γ+2γp−1+/parenleftbigg2\np−1/parenrightbigg2/summationdisplay\nj≥1jγj\n≤1+γ+2γp−1+/parenleftbigg2\np−1/parenrightbigg2γ\n(1−γ)2.\nWith the values γ= 21/5,c1= 26/5andp≥143given in Proposition 5 we\nobtain that c2= 4.3 is suitable.\nFrom (13) and Lemma 6 we obtain that\nP(X(w)>0)≤c21\nn|B2n(w)|\n2n+c2m−1/summationdisplay\ns=1ps\nn|B2n/ps(w/ps)|\n2n/ps (14)\nto deal with this last sum we invoke:\nLemma 7 For any prime p >143and for any positive number wsuch that\n|B2n(w)| ≤n2n, we have\nm−1/summationdisplay\ns=1ps\nn|B2n/ps(w/ps)|\n2n/ps≤2\np\nProof:Chooseptimes a vector of length 2 n/pand weight not more than w/p:\nconcatenate the resulting vectors and one obtains a vector o f length 2 nand\nweight not more than w. Therefore |B2n/p(w/p)|p≤ |B2n(w)|and we have\nm−1/summationdisplay\ns=1ps\nn|B2n/ps(w/ps)|\n2n/ps≤m−1/summationdisplay\ns=1ps\nn/parenleftbigg|B2n(w)|\n2n/parenrightbigg1/ps\n≤m−1/summationdisplay\ns=1ps\nnn1/ps.\nThe result follows from routine computations.\nWe see therefore from (14) and Lemma 7 that, if we choose wsuch that\n|B2n(w)| ≤bn2n, forb <1, then, provided the conditions of Proposition 5 are\nsatisfied, we have P( X(w)>0)≤bc2+2c2/p. Forc2= 4.3 and any p >143this\nquantity is less than 1 when b≤0.23. The largest wfor which |B2n(w)| ≤bn2n\nis readily seen to satisfy K≤w\n2n≤1\n4which means that all conditions of\nProposition 5 are satisfied, so that we have proved:\nTheorem 8 There exist positive constants b≤0.23andq, such that for any\nprimep≥qsuch that 2is primitive modulo pand2p−1/ne}ationslash= 1 mod p2, and\nfor any power n=pmofp, there exist double circulant codes of parameters\n[2n,n,d > w ]for anywsuch that |B2n(w)| ≤bn2n. A suitable value of qis\nq= 143and the first suitable prime pisp= 2789.\n93.3 Proof of Proposition 5\nOur remaining task is now to prove Proposition 5. We start by n oting that\nProposition 5 is stated with a positive real number w, because the discussion\nstarting from (13) involves balls of non-integer radius. Ho wever, it clearly is\nenough to prove it only for integer values of w.\nThe crucial part of the proof will be to bound from above the we ight distri-\nbution of C, forC∈Cn. Let us note that, since the polynomial Pn(Z) defined\nin (11) is an irreducible factor of Zn+1, the code Cbelongs to Cnif and only\nifPn(Z) divides the generator polynomial of the dual code C⊥. This means\nthat any codeword of C⊥must be obtained by repeating ptimes a subvector of\nlengthn/p. Equivalently, a generating matrix of C⊥, i.e. a parity-check matrix\nofCis of the form\nHC= [A|A|···|A]\nmeaning that it equals the concatenation of pidentical copies of an r×n/p\nmatrixA.\nWe shall need the following lemma.\nLemma 9 LetHtr= [Ir|Ir|···|Ir]be ther×trmatrix obtained by concate-\nnatingtcopies of the r×ridentity matrix. Let σtrbe the associated syndrome\nfunction:\nσtr:{0,1}tr→ {0,1}r\nx/ma√sto→σtr(x) =xtHtr.\nLetw≤trbe an integer. Then, for any s∈ {0,1}r, the number of vectors of\nlengthtrand of weight wthat map to sbyσtris not more than:\n√\n2rt/parenleftbigg1+|1−2ω|t\n2/parenrightbiggr/parenleftbiggtr\nw/parenrightbigg\nwherew=ωtr.\nProof: LetXbe a random vector of length trobtained by choosing indepen-\ndently each of its coordinates to equal 1 with probability ω. The probabilities\nthat any given coordinate of σtr(X) equals 0 or 1 are those of a sum of tinde-\npendent Bernoulli random variables of parameter ω, namely:\n1+(1−2ω)t\n2and1−(1−2ω)t\n2.\nSince all the coordinates of σtr(X) are clearly independent,\nmax\ns∈{0,1}rP(σtr(X) =s) =/parenleftbigg1+|1−2ω|t\n2/parenrightbiggr\n. (15)\nNow letW= wt(X) be the weight of X. We have\nP(W=w) =/parenleftbiggtr\nw/parenrightbigg\nωw(1−ω)tr−w=/parenleftbiggtr\nωtr/parenrightbigg\n2−trh(ω)\n10wherehdenotes the binary entropy function, h(x) =−xlog2x−(1−x)log2(1−\nx). By a variant of Stirling’s formula [13][Ch. 10, §11,Lemma 7]\n/parenleftbiggn\nw/parenrightbigg\n≥2nh(ω)//radicalbig\n8nω(1−ω), (16)\ntherefore:\nP(W=w)≥1/radicalbig\n8trω(1−ω)≥1√\n2tr.\nFor given s, letNwdenote the number of vectors of length trand weight wthat\nhave syndrome s. Since P( σtr(X) =s|W=w) =Nw//parenleftbigtr\nw/parenrightbig\nwe have\nP(σtr(X) =s)≥P(σtr(X) =s|W=w)P(W=w)≥Nw/parenleftbigtr\nw/parenrightbig1√\n2tr.\nHence, by (15),\nNw≤√\n2tr/parenleftbigg1+|1−2ω|t\n2/parenrightbiggr/parenleftbiggtr\nw/parenrightbigg\nwhich is the claimed result.\nLemma 10 Let0< κ <1/4. There exist q, such that for any p > q,n=pm,\nand for any code C∈Cn, the following holds:\n•eitherC={0,1}norCequals the even-weight code,\n•or, the weight distribution of Csatisfies, for any i,κn≤i≤n/2,\nAi(C)≤1\n23r/5/parenleftbiggn\ni/parenrightbigg\nwherer=n−dimC.\nForκ= 0.07a suitable value of qisq= 143.\nProof: Ifr= 0 orr= 1, i.e. Cequals the whole space {0,1}nor the even-\nweight code, there is nothing to prove. Suppose therefore r >1. From the\nfactorization (9) of Zn+ 1 into irreducible factors we see that we must have\nr≥p−1. From the discussion preceding Lemma 9 we must have\nr≤n−pm−1(p−1) =n/p (17)\nand a parity-check matrix of Cis made up of pidentical copies of some r×n/p\nmatrixA. Therefore, after permuting coordinates, there exists a pa rity-check\nmatrix of Cof the form\nHC= [B|Ir|Ir|···|Ir]\nwhereBissomer×(n−rt)matrixandisfollowed by tcopies ofthe r×ridentity\nmatrix. The integer tcan be chosen to take any value such that 1 ≤t≤p: we\nshall impose the restriction\nt≤p1/3. (18)\n11For anyx∈ {0,1}n, writex= (x1,x2) wherex1is the vector made up of the\nfirstn−trcoordinates of xandx2consists of the remainding trcoordinates\nNow the syndrome function σassociated to HCtakes the vector x∈ {0,1}n\ntoσ(x) =x1tB+σtr(x2) whereσtris the function defined in Lemma 9. The\ncodeCis the set of vectors xsuch that σ(x) = 0, therefore by partitioning the\nset of vectors of weight iinto all possible values of x1we have, from Lemma 9:\nAi(C)≤√\n2trtr/summationdisplay\nj=0/parenleftBigg\n1+|1−2j\ntr|t\n2/parenrightBiggr/parenleftbiggtr\nj/parenrightbigg/parenleftbiggn−tr\ni−j/parenrightbigg\n(19)\nfor anyisuch that\ni≥tr. (20)\nNotice that:/parenleftbiggtr\nj/parenrightbigg/parenleftbiggn−tr\ni−j/parenrightbigg\n=/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig/parenleftbiggn\ni/parenrightbigg\nso that (19) becomes\nAi(C)≤√\n2trtr/summationdisplay\nj=0/parenleftBigg\n1+|1−2j\ntr|t\n2/parenrightBiggr/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig/parenleftbiggn\ni/parenrightbigg\n≤√\n2tr(tr+1)/parenleftbiggn\ni/parenrightbigg\nmax\n0≤j≤tr/parenleftBigg\n1+|1−2j\ntr|t\n2/parenrightBiggr/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig.(21)\nSeti=ιnandj=αtr, we have:\n/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig≤ij(n−i)tr−j\n/parenleftbign\ntr/parenrightbig\nj!(tr−j)!\n≤ιj(1−ι)tr−jntr\n/parenleftbign\ntr/parenrightbig\nj!(tr−j)!\n≤ιj(1−ι)tr−jntr\n(n−tr)tr/parenleftbigtr\nj/parenrightbig−1since/parenleftbign\ntr/parenrightbig\n≥(n−tr)tr/(tr)!\n≤ιj(1−ι)tr−j/parenleftbigtr\nj/parenrightbig\n(1−tr\nn)tr.\nWe have seen (17) that r≤n/pandt≤p1/3(condition (18)), therefore tr/n≤\n1/p2/3≤1/2: by usingthe inequality 1 −x≥2−2x, valid whenever 0 ≤x≤1/2,\nwe therefore have\n/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig≤22t2r2/nιj(1−ι)tr−j/parenleftbiggtr\nj/parenrightbigg\nand by using/parenleftbigtr\nj/parenrightbig\n≤2trh(α)we finally get\n/parenleftbigi\nj/parenrightbig/parenleftbign−i\ntr−j/parenrightbig\n/parenleftbign\ntr/parenrightbig≤2tr(2tr\nn−D(α||ι))\n12whereD(x||y) =xlog2x\ny+(1−x)log21−x\n1−y. Together with (21) we get:\nAi(C)≤2r(β+f(ι))1\n2r/parenleftbiggn\ni/parenrightbigg\nwith\nf(ι) = max\n0≤α≤1g(α,ι) (22)\nwhere g(α,ι) = log2(1+|1−2α|t)−tD(α||ι) (23)\nandβ=1\nrlog2√\n2tr+1\nrlog2(tr+1)+2t2r/n. Write log2(tr+1)≤1+log2tr\nto getβ≤(3\n2+3\n2log2(tr))/r+2t2r/n. By using t < p1/3andp−1≤r≤n/p,\nwe get\n3\n2log2tr\nr<3\n2log2(r+1)4/3\nr= 2log2(r+1)\nr≤2log2p\np−1\nand\nβ≤3\n2(p−1)+2log2p\np−1+2\np1/3.\nWe see that βcan be made arbitrarily small by increasing the value of p. A\nnumerical computation gives us β <0.152 for all p >143.\nSince we have supposed i≤n/2, we have ι≤1/2 so that the definition (22)\nand (23) of fcan be replaced by the equivalent\nf(ι) = max\n0≤α≤ιg(α,ι)\ng(α,ι) = log2(1+(1−2α)t)−tD(α||ι)\nfrom which we easily see that gandfare decreasing functions of ι. We see that\nf(κ) can be made arbitrarily small, for all κ >0, by choosing tbig enough.\nNumerically, by choosing t= 14,κ= 0.07 andp >143, we see that (20) is\nsatisfied and we get, for all 0 .07≤ι,f(ι)≤f(κ)≤0.24. We obtain therefore\nthat, for all κn≤i≤n/2,\nAi(C)≤2−0.608r/parenleftbiggn\ni/parenrightbigg\nwhich proves the lemma.\nTo prove Proposition 5, we need a final technical lemma, of a pu rely enu-\nmerative nature.\nLemma 11 Let0< κ < K < 1/4. There exist an integer n0andε >0such\nthat, for any n≥n0,w= 2ωnwithK≤ω <1/4,\n2/summationdisplay\ni+j≤w\ni<κn/parenleftbiggn\ni/parenrightbigg/parenleftbiggn\nj/parenrightbigg\n≤1\n2εn|B2n(w)|.\nForκ= 0.07,K= 0.1,n0= 143, a suitable value of εisε= 0.004.\n13Proof: Clearly we have:\n2/summationdisplay\ni+j≤w\ni<κn/parenleftbiggn\ni/parenrightbigg/parenleftbiggn\nj/parenrightbigg\n≤κn2/parenleftbiggn\nκn/parenrightbigg/parenleftbiggn\nw−κn/parenrightbigg\n≤κn22n(h(κ)+h(2ω−κ))=κn222nh(ω)\n2n(2h(ω)−h(κ)−h(2ω−κ))\n≤κn2 22nh(ω)\n2n(2h(K)−h(κ)−h(2K−κ))\nsince 2h(ω)−h(κ)−h(2ω−κ) is an increasing function of ω. By (16) we have\n22nh(ω)≤√\n16n|B2n(w)|, so that we obtain, since κ≤1/4,\n2/summationdisplay\ni+j≤w\ni<κn/parenleftbiggn\ni/parenrightbigg/parenleftbiggn\nj/parenrightbigg\n≤n5/2|B2n(w)|\n2n(2h(K)−h(κ)−h(2K−κ))≤|B2n(w)|\n2εn\nfor anyn≥n0withε≤2h(K)−h(κ)−h(2K−κ)−5\n2log2n0\nn0.\nProof of Proposition 5: IfC={0,1}nor ifCis the even-weight subcode,\nthenAi(C)≤/parenleftbign\ni/parenrightbig\n, and/summationtext\ni+j≤wAi(C)Aj(C)≤/summationtext\ni+j≤w/parenleftbign\ni/parenrightbig/parenleftbign\nj/parenrightbig\n=|B2n(w)|. The\nresult clearly holds for any c1≥2/γ.\nLetC∈Cnwithr=n−dimC >1. Let us write:\n1\n|C|/summationdisplay\ni+j≤wAi(C)Aj(C) =S1+S2\nwith\nS1=1\n|C|/summationdisplay\ni+j≤w\nκn≤i,jAi(C)Aj(C) and S2=2\n|C|/summationdisplay\ni+j≤w\ni<κnAi(C)Aj(C).\nBy Lemma 10 we have\nS1≤1\n|C|/summationdisplay\ni+j≤w/parenleftbiggn\ni/parenrightbigg/parenleftbiggn\nj/parenrightbigg1\n26r/5≤|B2n(w)|\n2n1\n2r/5.\nTo upperbound S2we simply write Ai(C)≤/parenleftbign\ni/parenrightbig\n. By Lemma 11, we have\nS2≤|B2n(w)|\n2n2r\n2εn=|B2n(w)|\n2n2r\n(2εp)n/p≤|B2n(w)|\n2n2r\n(2εp)r\nsince we have seen (17) that r≤n/p. By choosing p≥6\n5εwe obtain\nS2≤|B2n(w)|\n2n1\n2r/5.\nThis proves the result with γ= 1/21/5andc1= 26/5.\n144 Comments\nThe probabilistic method we used easily shows that almost al l double circulant\ncodes of the asymptotic family presented here satisfy an imp roved bound of\nthe form (2). Actually we suspect that this is also the case fo r most choices\nofn: this is suggested by computer experiments with randomly ch osen double\ncirculant codes of small blocklengths.\nWe have tried to strike a balance between giving readable pro ofs and deriv-\ning a non-astronomical lower bound on the prime pin Theorem 8. In principle,\nthe numerical values could be refined. In particular, the con stantbof Theorem\n8 could bemade to approach 1 /2 (as in Theorem 4) but at the cost of a larger p.\nIf we convert the formulation of Theorem 8 in the form (2) (whi ch just involves\nswitching from |B2n(d)|in Theorem 2 to |B2n(d−1)|in (2)) we obtain a con-\nstantcwhich is of the same order of magnitude, but somewhat worse, t han the\nimproved constant c≈0.102 of [16] for Jiang and Vardy’s method.\nIn this paper we only consider the binary case with codes of ra te 1/2 but the\nmethod can be straightforwardly generalized to the case of d ifferent alphabets\nand to quasi-cyclic codes of any rational rate (though at the cost of a wors-\nening of the constant b) by considering for parity check matrices vertical and\nhorizontal concatenations of random circulant matrices.\nFinally, a natural question is to wonder whether the ideas de veloped in this\npaper can be extended to Euclidean lattices in a way similar t o the generaliza-\ntion of Jiang and Vardy’s method to sphere-packings of Eucli dean spaces [10].\nA positive answer to this question is given in the paper [4].\nReferences\n[1] C. L. Chen, W. W. Peterson and E. J. Weldon, “Some results o n quasi-\ncyclic codes,” Inform. Control , Vol. 15, no. 5, pp. 407–423, 1969.\n[2] V.V.Chepyzhov, “Newlower boundsforminimumdistanceo flinearquasi-\ncyclicandalmost linearcycliccodes,” Problemy Peredachi Informatsii , Vol.\n28, no 1, pp. 39–51, 1992.\n[3] P. Gaborit and G. Z´ emor, “Asymptotic improvement of the Gilbert-\nVarshamov bound for linear codes”, ISIT 2006, Seattle, p.28 7-291.\n[4] P. Gaborit and G. Z´ emor, “On the construction of dense la ttices with a\ngiven automorphism group,” Annales de l’Institut Fourier , vol. 57 No. 4\n(2007), pp. 1051–1062.\n[5] E. N. Gilbert, ” A comparison of signalling alphabets”, Bell. Sys. Tech. J. ,\n31, pp. 504-522, 1952.\n[6] T. Jiang and A. Vardy, “Asymptotic improvement of the Gil bert-\nVarshamov bound on the size of binary codes,” IEEE Trans. Inf. Theory ,\nVol. 50, no. 8, pp. 1655–1664, 2004.\n15[7] G. A. Kabatiyanskii, ”On the existence of good cyclic alm ost linear codes\nover non prime fields”, Problemy Peredachi Informatsii , Vol. 13, no 3, pp.\n18–21, 1977.\n[8] M. Karlin, ” New binarycoding results by circulant”, IEEE Trans. Inform.\nTheory15, pp. 81–92, 1969.\n[9] T. Kasami, “A Gilbert-Varshamov bound for quasi-cyclic codes of rate\n1/2,”IEEE Trans. Inf. Theory , Vol. 20, no. 5, pp. 679–679, 1974.\n[10] M. Krivelevich, S. Litsyn, A. Vardy, ”A lower bound on th e density of\nsphere packings via graph theory”, Int. Math. Res. Not , no. 43, 2271–2279,\n2004.\n[11] E. Krouk, “On codes with prescribed group of symmetry,” Voprosy Kiber-\nnetiki, Vol. 34, pp. 105–112, 1977.\n[12] E. Krouk and S. Semenov, “On the existence of good quasi- cyclic codes,”\nproc. of 7th joint Swedish-Russian International Workshop on Information\nTheory, St-Petersburg, Russia, june 1995, pp. 164–166.\n[13] F.J. MacWilliams and N.J.A. Sloane, “The Theory of Erro r-Correcting\nCodes,” North-Holland, Amsterdam 1977.\n[14] M. A. Tsfasman, S.G. Vladuts and Zink, ”Modular curves, Shimura curves\nand Goppa codes better than Varshamov-Gilbert bound”, Math. Nach. ,\n104, pp. 13–28, 1982.\n[15] R.R. Varshamov, ”Estimate of the number of signals in er ror-correcting\ncodes”,Dokl. Acad. Nauk ,117, pp. 739–741, 1957 (in Russian).\n[16] V. Vu and L. Wu, ”Improving the Gilbert-Varshamov bound for q-ary\ncodes”,IEEE Trans. Inf. Theo. ,51(9), pp. 3200–3208, 2005\n16"    },    {        "title": "0709.2937v2.Theory_of_current_driven_magnetization_dynamics_in_inhomogeneous_ferromagnets.pdf",        "content": "arXiv:0709.2937v2  [cond-mat.mes-hall]  23 Jan 2008Theory of current-driven magnetization dynamics in\ninhomogeneous ferromagnets\nYaroslav Tserkovnyak,1Arne Brataas,2and Gerrit E. W. Bauer3\n1Department of Physics and Astronomy,\nUniversity of California, Los Angeles, California 90095, U SA\n2Department of Physics, Norwegian University of\nScience and Technology, NO-7491 Trondheim, Norway\n3Kavli Institute of NanoScience, Delft University\nof Technology, 2628 CJ Delft, The Netherlands\n(Dated: November 8, 2018)\nAbstract\nWe give a brief account of recent developments in the theoret ical understanding of the interac-\ntion between electric currents and inhomogeneous ferromag netic order parameters. We start by\ndiscussingthephysicaloriginofthespintorquesresponsi bleforthisinteraction andconstructaphe-\nnomenological description. We then consider the electric c urrent-induced ferromagnetic instability\nand domain-wall motion. Finally, we present a microscopic j ustification of the phenomenological\ndescription of current-driven magnetization dynamics, wi th particular emphasis on the dissipative\nterms, theso-called Gilbertdamping αandtheβcomponent oftheadiabatic current-driventorque.\n1I. INTRODUCTION\nFerromagnetism is a correlated state in which, at sufficiently low temp eratures, the elec-\ntrons align their spins in order to reduce the exchange energy. Belo w the Curie temperature\nTc, the free energy F[M] then attains its minimum at a finite magnetization M∝negationslash= 0 with\nan arbitrary direction, thus spontaneously breaking the spin-rot ational symmetry. Crystal\nanisotropies that are caused by the spin-orbit interaction and sha pe anisotropies governed\nby the magnetostatic dipolar interaction pin the equilibrium magnetiza tion direction to a\ncertain plane or axis. At low temperatures, T≪Tc, fluctuations of the magnitude of the\nmagnetization around the saturation value Ms(the so-called Stoner excitations) become\nenergetically unfavorable. The remaining low-energy long-waveleng th excitations are spin\nwaves (or magnons, which, in technical terms, are viewed as Goldst one modes that restore\nthe broken symmetry). These are slowly varying modulations of the magnetization direction\nin space and time.\nA phenomenological description of the slow collective magnetization d ynamics without\ndissipation proceeds from the free energy F[M(r)] as a functional of the inhomogeneous\n(and instantaneous) magnetic configuration M(r) [1]. The equation of motion\n∂M(r,t)\n∂t=γM(r,t)×δF[M]\nδM(1)\npreserves the total free energy of the system, since the rate o f change of the magneti-\nzation is perpendicular to the “gradient” of the free energy. The f unctional derivative\nof the free energy with respect to the local magnetization is called t heeffective field:\nHeff(r,t) =−δF[M]/δM. [In the following, we use the abbreviations ∂t=∂/∂tfor the\npartial derivative in time and ∂M=δ/δMfor the functional derivative with respect to M.]\nIn the presence of only an externally applied magnetic field H,F[M] =−/integraltext\nd3rM(r)·H(r),\nsoγis identified as the effective gyromagnetic ratio. In general, Heffincludes the crys-\ntal anisotropy due to spin-orbit interactions, modulation of the ex change energy due to\nmagnetization gradients, and demagnetization fields due to dipole-d ipole interactions. The\nLandau-Lifshitz equation (1) qualitatively describes many ferroma gnetic resonance (FMR)\n[2, 3] and Brillouin light scattering [4] experiments. With M=Msm, wheremis the\nmagnetic direction unit vector, we may rewrite Eq. (1) as\n∂tm(r,t) =−γm(r,t)×Heff(r,t). (2)\n2By definition, the effective field on the right-hand side of Eq. (2) is de termined by the\ninstantaneous magnetic configuration. This can be true only if the m otion is so slow that\nall relevant microscopic degrees of freedom manage to immediately r eadjust themselves to\nthe varying magnetization. If this is not the case, the effective field acquires a finite time\nlag that to lowest order in frequency can be schematically expanded as\n˜Heff→ −∂MF[M(r,t−τ)]≈Heff−τ(∂tM·∂M)Heff, (3)\nwhereτisa characteristic delay time. This dynamic correction tothe instant aneous effective\nfield,δHeff, leads to a new term in the equation of motion ∝m×δHeff. Although in general\nnonlocal and anisotropic [5], it makes sense to identify first the simple st, i.e., local and\nisotropic contribution. We can then construct two new terms out o f the vectors mand\n∂tm. The first one, ∝m×∂tm, is dissipative, meaning that it is odd under time reversal\n(i.e., under the transformation t→ −t,H→ −H, andm→ −m), and thus violates the\ntime-reversal symmetry of the Landau-Lifshitz equation (2). Th is argument leads to the\nLandau-Lifshitz-Gilbert (LLG) equation [6, 7]:\n∂tm(r,t) =−γm(r,t)×Heff(r,t)+αm(r,t)×∂tm(r,t), (4)\nintroducing the phenomenological Gilbert damping constant α. The second local and\nisotropic term linear in ∂tmand perpendicular to m, that can be composed out of m\nand∂tmis proportional to ∂tmand can be combined with the left-hand side of the LLG\nequation. In principle, any physical process that contributes to t he Gilbert damping can\nthus also renormalize the gyromagnetic ratio γ. The latter should therefore be interpreted\nas an effective parameter in the equation of motion (4). The second law of thermodynamics\nrequires that αγ≥0, which guarantees that the dissipation of energy P∝Heff·∂tm≥0\n(assuming the magnetization dynamics are slow and isolated from any external sinks of\nentropy). Since the implications of small modifications of the gyroma gnetic ratio are mi-\nnor, we will be mainly concerned with the Gilbert damping constant α. Furthermore, we\nwill focus on dissipative effects due to spin dephasing by magnetic or s pin-orbit impurities\n[8, 9, 10, 11, 12, 13], noting that many other Gilbert damping mechan isms have been pro-\nposed in the past [14, 15, 16, 17, 18, 19, 20, 21]. According to the fl uctuation-dissipation\ntheorem, the dissipation, whatever its microscopic origin is, must be accompanied by a\nstochastic contribution h(r,t) to the effective field. Assuming Gaussian statistics with a\n3white noise correlator [22], which is valid in the classical limit with charac teristic frequen-\ncies that are sufficiently small compared to thermal energies:\n∝angb∇acketlefthi(r,t)hj(r′,t′)∝angb∇acket∇ight= 2kBTα\nγMsδijδ(r−r′)δ(t−t′). (5)\nEqs. (1)-(5) form the standard phenomenological basis for unde rstanding dynamics of fer-\nromagnets [2, 3, 4], in the absence of an applied current or voltage b ias.\nII. CURRENT-DRIVEN MAGNETIZATION DYNAMICS\nIn order to understand recent experiments on current-biased m agnetic multilayers [23,\n24, 25, 26, 27, 28, 29, 30, 31] and nanowires [32, 33, 34, 35, 36, 3 7, 38, 39], Eq. (4) has to be\nmodified [40, 41, 42]. The leading correction has to take into account the finite divergence\nof the spin-current density in conducting ferromagnets, with mag netization texture that has\nto be brought into compliance with the conservation of angular mome ntum. We have to\nintroduce a new term\ns0∂tmi|torque=∇·ji (6)\nin the presence of a current density jifor spin-icomponent, where s0is the total equilibrium\nspin density along −m(the minus sign takes into account that electron spin and magnetic\nmoment point in opposite directions). By adding this term to the right -hand side of Eq. (4)\nas a contribution to ∂tmi, we assume that the angular momentum lost in the spin current\nis fully added to the magnetization. This is called the spin-transfer to rque [42, 43, 44, 45].\nThesimplest approximationforthespin-current density inthebulko fisotropicferromagnets\n[43, 45, 46] is ji=Pjmi, wherePis a material-dependent constant that converts charge-\ncurrent density jinto spin-current density. The underlying assumption here is that s pins are\ncarried by the electric current such that the spin-polarization axis adiabatically follows the\nlocal magnetization direction. This is the case for a large exchange fi eld that varies slowly\nin space. This condition is fulfilled very well in transition-metal ferrom agnets. The spin\nconversion factor\nP=/planckover2pi1\n2eσ↑−σ↓\nσ↑+σ↓(7)\ncharacterizes the polarization of the spin-dependent conductivit yσs(s=↑ors=↓) with↑\nchosen along −m. Hence\n∂tm=−γm×Heff+P\ns0(j·∇)m, (8)\n4where we took into account local charge neutrality by ∇·j= 0.\nThe phenomenological Eq. (8) is “derived” without taking into accou nt spin relaxation\nprocesses. Its inclusion requires some care since both Gilbert damp ing and spin-transfer\ntorque are nontrivially affected [47, 48, 49, 50, 51, 52]. Spin relaxat ion is generated by\nimpurities with potentials that do not commute with the spin density op erator, such as\na quenched random magnetic field or spin-orbit interaction associat ed with randomly dis-\ntributed non-magnetic impurities [13, 47, 49, 50, 51]. In the absenc e of an applied current\nj, imperfections with potentials that mix the spin channels contribute to the Gilbert con-\nstantα[8, 9, 10, 13, 50]. It is instructive to interpret the right-hand side o f the equation\nof motion (8) as an analytic expansion of driving and damping torques in∇and∂t. The\nLLG equation (4) corresponds to the most general (local and isot ropic) expression for the\ndamping to the zeroth order in ∇and first order in ∂t. We will not be concerned with\nhigher order terms in ∂t, since the characteristic frequencies of magnetization dynamics a re\ntypically small on the scale of the relevant microscopic energies, at le ast in metallic systems.\nThe contribution to the effective field due to a finite magnetic stiffnes s [1] is proportional\nto∇2. In the presence of inversion symmetry, the terms proportional to∇cannot appear\nwithout applied electric currents. In the following, we focus on the c urrent-driven terms lin-\near in∇, assuming that the spatial variations in the magnetization direction are sufficiently\nsmooth to rule out higher-order contributions. The dynamics of iso tropic spin-rotationally\ninvariant ferromagnets can then in general be described by the ph enomenological equation\nof motion [47, 49, 53]\n∂tm=−γm×Heff+αm×∂tm+P\ns0(1−βm×)(j·∇)m, (9)\nin which αandβcharacterize those terms that break time-reversal symmetry. Both arise\nnaturally in the presence of spin-dependent impurities [47, 49, 50]. E ven though in practice\nβ≪1,itgivesanimportantcorrectiontothecurrent-driven spin-tra nsfertorque[47, 49, 53],\nasdiscussedinmoredetailbelow. Forthespecialcaseof α=β: Eq.(9)canthenberewritten\n(after multiplying it by 1+ αm×on the left) as\n∂tm=−γ∗m×Heff+αγ∗m×Heff×m+P\ns0(j·∇)m, (10)\nwhereγ∗=γ/(1 +α2). The dissipative term proportional to m×Heff×mis called the\nLandau-Lifshitz damping. Eq. (9) cannot be transformed into equ ation (10) if α∝negationslash=β[in\n5which case Eq. (10) necessarily retains a βterm]. A special feature of Eq. (10) appears when\nHeff(m) is time independent and translationally invariant. A general solution m(r,t) in the\nabsence of an electric current j= 0 (such as a static domain wall or a spin wave) can be\nused to construct a solution\n˜m(r,t) =m(r+Pjt/s0,t) (11)\nof Eq. (10) for an arbitrary uniform j. This unique feature of the solutions of Eq. (9) only\narises for α=β.\nInterestingly, the argument above has been turned around by Ba rnes and Maekawa [54],\nwho find that Galilean invariance of a system would dictate α=β. Galilean invariance\nrequires the existence of solutions of the form ˜m(r−vt), where ˜m(r) is an arbitrary static\nsolution (say, a domain wall) and vis an arbitrary velocity. As explained above, this is\nonly possible when α=β. However, the general validity of the Galilean invariance assump-\ntion for the current-carrying state needs to be discussed in more detail from a microscopic\npoint of view. The Galilean invariance argument [54] implies that the bias -induced electron\ndrift exactly corresponds to the domain-wall velocity, since other wise electron motion would\npersist in the frame that moves with the domain wall. Referring to Eq. (11), we must,\ntherefore, identify v=−Pj/s0with the average electron drift velocity in the presence of the\ncurrentj. We argue in the following that this is indeed true in certain special limits , but\nis not generic, however. In the itinerant Stoner model for ferrom agnets, the spin-dependent\nDrude conductivity reads σs∝nsτs, wherensandτsare the densities and scattering times\nof spins, respectively. When there is no asymmetry between the scatterin g times,τ↑=τ↓,\nvindeed equals the electron drift velocity and Galilean invariance is effec tively fulfilled. In\ngeneral, however, since the spin dependence of wave functions an d densities of states for\nthe electrons at the Fermi energy lead to different scattering cro ss sections in conducting\nferromagnets, the equality between −Pj/s0and the average drift velocity disappears. In\nthe simplest model of perturbative white-noise impurity potentials, for example, 1 /τs∝νs,\nwhereνsis the spin-dependent density of states. Assuming parabolic free- electron bands\nand weak ferromagnets, in which the ferromagnetic exchange split ting is much less than the\nFermi energy, the domain-wall velocity v=−Pj/s0actually becomes 2 /3 of the average\ndrift velocity ¯v,\nv=−Pj\ns0=n↑τ↑−n↓τ↓\nn↑τ↑+n↓τ↓n↑v↑+n↓v↓\nn↑−n↓≈n↑/ν↑−n↓/ν↓\n(n↑−n↓)(ν↑+ν↓)/2¯v≈2\n3¯v.(12)\n6Here,vsis the spin- selectron drift velocity, and we used the relation νs∝n1/3\ns, which\nis valid in three dimensions. Clearly, the potential disorder breaks Ga lilean invariance.\nAn identity of αandβcan therefore not be deduced from general symmetry principles.\nFurthermore, spin-orbit interaction or magnetic disorder that st rongly affect the values of α\nandβ(see below) also break Galilean invariance at the level of the microsco pic Hamiltonian.\nNevertheless, for itinerant ferromagnets we show below that α∼β(where by ∼we mean\n“of the order”), with α≈βin the simplest model of weak and isotropic spin-dephasing\nimpurities [49], which implies that deviations from translational invarian ce are not very\nimportant in metallic ferromagnets, such as transition metals and th eir alloys, in which\nthe Stoner model is applicable. Very recently, two independent gro ups measured α≈βin\npermalloy nanowires [36, 37].\nLet us also consider the s−dmodel of ferromagnetism. When, as is usually done, the d-\norbital lattice is assumed spatially locked, Galilean invariance is broken even in the absence\nof disorder. In this case, the ratio α/βdeviates strongly from unity, although it remains to\nbe relatively insensitive to the strength of spin-dependent impuritie s. In other words, αand\nβscale similarly with the strength of spin-dephasing processes, and t heir ratio appears to\nbe determined mainly by band-structure effects and the nature (r ather than the strength)\nof the disorder [49, 50]. A predictive material-dependent theory of magnetization damping\nand current-induced domain wall motion that transcends the toy m odels mentioned above\nis beyond the scope of our paper.\nThe form of the equation of motion (10) for the special case α=βhas also triggered\nthe suggestion [55] that the Landau-Lifshitz form of damping, ∝m×Heff×m, is more\nnatural than the Gilbert form, ∝m×∂tm. In our opinion, however, such a distinction\nis purely semantic. Both forms are odd under time reversal, and one can easily imagine\nsimple models in which either form arises more naturally than the other : For example, a\nBloch-like T2relaxation added to the Stoner model naturally leads to a Landau-L ifshitz\nform of damping [49], whereas the dynamic interface spin pumping [56, 57] very generally\nobeys the Gilbert damping form. Moreover, mathematically, both eq uations are identical\n(in the absence of any additional torques), since we have shown ab ove that the Landau-\nLifshitz form of damping follows from the Gilbert one simply by multiplying both sides of\nthe LLG form by 1+ αm×from the left (and vice versa by 1 −αm×). Only at the special\npointα=β, the Landau-Lifshitz form (10) does not involve the “ βterm.” In that limit,\n7it may be a more transparent expression for the equation of motion . On the other hand,\nwe noted above that in general α∝negationslash=βand the ratio α/βdepends on material and sample.\nThe current-driven dynamics of domain walls and other spatially nonu niform magnetization\ndistributions turn out to be very sensitive to small deviations of α/βfrom unity, which\nstrongly reduces any advantage a Landau-Lifshitz damping formu lation might have over the\nGilbert phenomenology. In general, we therefore prefer to use th e Gilbert phenomenology.\nUnder time reversal, the electric current as well as the magnetizat ion vector change sign\nand the adiabatic current-induced torque is symmetric, thus nond issipative. The Ohmic\ndissipation generated by this current does not depend on the magn etization texture in this\nlimit and is intentionally disregarded. Saslow [58] prefers to discuss a t orque driven by\nvoltage rather current, which, after inserting Ohm’s law for the cu rrent, becomes odd under\ntime reversal and thus appears dissipative (see Ref. [59] for anot her discussion of this point).\nObviously, the βcorrection torque is odd for the current-biased and even for the voltage-\nbiased configurations. The current-bias picture appears to be mo re natural, since it reflects\nthe absence of additional dissipation by the magnetization texture in the adiabatic limit as\nwell as the close relation between the βcorrection and the Gilbert dissipation.\nIII. CURRENT-DRIVEN INSTABILITY OF FERROMAGNETISM\nLet us now pursue some special aspects of the solutions of the phe nomenological Eq. (9),\nhighlighting the role of various parameters, before we discuss the m icroscopic derivation\nof the magnetization dynamics in Sec. V. It is interesting, for examp le, to investigate\nthe possibility to destabilize a single-domain ferromagnet by sufficient ly large spin torques\n[43, 46]. We consider a homogeneous ferromagnet with an easy-axis anisotropy along the x\naxis, characterized by the anisotropy constant K, and an easy-plane anisotropy in the xy\nplane, with the anisotropy constant K⊥, see Fig. 1. Typically, the anisotropies originate\nfrom the demagnetization fields: For a ferromagnetic wire, for exa mple, the magnetostatic\nenergy is lowest when the magnetization is in the wire direction, so tha t there are no stray\nfield lines outside the ferromagnet. The effective field governing mag netization dynamics is\nthen given by\nHeff= (H+Kmx)x−K⊥mzz+A∇2m, (13)\n8where we also included an applied field Halong the xaxis and the exchange coupling\nparametrized by the stiffness constant A.A,H,K,K ⊥≥0. We then look for spin-wave\nsolutions of the form\nm=x+uei(q·r−ωt), (14)\nplugging it into Eq. (9) with the effective field (13) in the presence of a constant current\ndensityj, and linearizing it with respect to small deviations u. Whenα= 0 and j= 0, we\nrecover the usual spin-wave dispersion (Kittel formula):\nω0(q) =γ/radicalbig\n(H+K+Aq2)(H+K+K⊥+Aq2). (15)\nAfiniteαγ >0results ina negative Im ω(q), asrequired by thestability oftheferromagnetic\nstate. Asufficientlylargeelectriccurrentmay, however, reverse thesignofIm ω(q)forcertain\nwave vectors q, signaling the onset of an instability. The critical value of the curren t for the\ninstability corresponds to the condition Im ω(q) = 0. Straightforward manipulations based\non Eqs. (9) and (13) show that this condition is satisfied when\nP\ns0/parenleftbigg\n1−β\nα/parenrightbigg\n(q·jc) =±ω0(q). (16)\nwhich leads to a critical current density\njc=jc0\n|1−β/α|. (17)\njc0is the lowest current satisfying equation ( P/s0)(q·jc0) =ω0(q) for some q, where the\nleft-hand side can be loosely interpreted as the current-induced D oppler shift to the nat-\nural frequency given by the right-hand side [46]. According to Eq. ( 17), a current-driven\ninstability is absent when α=β. This conclusion is in line with the arguments leading to\nEq. (11): For the special case of α=β, a spin-wave solution in the presence of a finite\ncurrent density jwould acquire a frequency boost proportional to q·j, but with a stable\namplitude. Note that in general the onset of the current-driven f erromagnetic instability\nis significantly modified by the existence of βeven with β≪1, provided that the ratio\nβ/αis appreciable. In fact, αis typically measured to be ∼0.001−0.01, and the existing\nmicroscopic theories [49, 50] predict βto be not too different from α.\n9FIG. 1: Transverse head-to-head (N´ eel) domain wall parall el to the yaxis in the easy xyplane.\nThe uniform magnetization has two stable solutions m=±xalong the easy axis x, which is\ncharacterized by the anisotropy constant K. These are approached far away from the domain wall:\nm→ ±xatx=∓∞, respectively. In equilibrium, the magnetization directi onmis forced into the\nxyplane by the easy-plane anisotropy parametrized by K⊥. A weak magnetic field Hor electric\ncurrentjapplied along the xaxis can induce a slow domain-wall drift along the xaxis, during\nwhich the magnetization close to the domain wall is tilted sl ightly out of the xyplane. At larger H\norj(above the so-called Walker threshold), the magnetization is significantly pushed out of the xy\nplane and undergoes precessional motion during the drift. I n the moving frame, the magnetization\nprofile may remain still close to the equilibrium one.\nIV. CURRENT-DRIVEN DOMAIN-WALL MOTION\nEven more interesting phenomena are associated with the effect of the applied electric\ncurrent on a stationary domain-wall. In particular, we wish to discus s how the spin torques\nmove and distort a domain wall. These questions date back about thr ee decades [40],\nalthough only relatively recently they sparked an intense activity by several groups [49,\n50, 51, 53, 54, 60]. This is motivated by the growing number of intrigu ing experiments\n[32, 33, 34, 35, 36, 38, 39] as well as the promise of practical pote ntial, such as in the so-\ncalled racetrack memory [61] or magnetic logics [62]. Current-induce d domain-wall motion\nis a central topic of the present review.\nFor not too strong driving currents and in the absence of any signifi cant transverse dy-\nnamics, one can make progress analytically by using the one-dimensio nal Walker ansatz,\nwhich was first employed in studies of magnetic-field driven domain-wa ll dynamics [63].\n10This approach has proven useful in the present context as well [6 0, 64, 65]. The key idea\nis to approximately capture the potentially complex domain-wall motio n by few parameters\ndescribing the displacement of its center and a net distortion of the domain-wall structure.\nIn a quasi-one-dimensional set-up, such as a narrow magnetic wire , the domain wall is con-\nstrained to move along a certain axis, whereas the transverse dyn amics are suppressed. This\nregime is relevant for a number of existing experiments, although it s hould be pointed out\nthat the common vortex-type domain walls do not necessarily fall int o this category. Let us\nconsider an idealized situation with an effective field (13) and an equilibr ium domain wall\nmagnetization in the xyplane. The magnetization prefers to be collinear with the xaxis\ndue to the easy-axis anisotropy K. A transverse head-to-head domain wall parallel to the y\naxis corresponds to a magnetization direction that smoothly rotat es in the xyplane between\nxatx→ −∞and−xatx→ ∞, as sketched in Fig. 1.\nThe collective domain-wall dynamics can be described by the center p ositionX(t) and\nan out-of-plane tilting angle Φ( t). [For a more technically-interested reader, we note that\nin the effective treatment of Ref. [60], these variables are canonica lly conjugate.] There is\nalso a width distortion, but that is usually considered less important. WhenH < K, the\ntwo uniform stable states are m=±x. When H= 0, a static transverse head-to-head\ndomain-wall solution centered at x= 0 is given by\nϕ(x)≡0,lntanθ(x)\n2=x\nW, (18)\nwhere position-dependent angles ϕandθparametrize the magnetic configuration:\nm= (mx,my,mz) = (cosθ,sinθcosϕ,sinθsinϕ). (19)\nW=/radicalbig\nA/Kis the wall width, which is governed by the interplay between the stiffn essA\nthat tends to smooth the wall extent and the easy-axis anisotrop yKthat tends to sharpen\nthe wall.\nThe external magnetic field Hor the current density jalong the xaxis disturb the static\nsolution (18), distorting the domain-wall structure and displacing it s position. At weak field\nand current biases, magnetic dynamics can be captured by the Walk er ansatz [63, 64]:\nϕ(r,t)≡Φ(t),lntanθ(r,t)\n2≡x−X(t)\n˜W(t). (20)\nHere, it is assumed that the driving perturbations ( Handj) are not too strong, such that\nthe wall preserves its shape, except for a small change of its width ˜W(t) and a uniform\n11out-of-plane tilt angle Φ( t).X(t) parametrizes the net displacement of the wall along the\nxaxis. Note that although ϕis assumed to be spatially uniform, it has an effect on the\nmagnetization direction only when m∝negationslash=±x, i.e., only near the wall center. A more detailed\ndiscussion concerning the range of validity of this approximation can be found in Ref. [63].\nInserting the ansatz (20) into the equation of motion (9) with j=jx(since the other current\ndirections do not couple to the wall), and using Eq. (13) for the effec tive field, one finds\n[53, 64]\n˙Φ+α˙X\n˜W=γH−βPj\ns0˜W,\n˙X\n˜W−α˙Φ =γK⊥sin2Φ\n2−Pj\ns0˜W,\n˜W=/radicalBigg\nA\nK+K⊥sin2Φ. (21)\nIt iseasyto verify thatthestaticsolution(18)is consistent withth ese equations when H= 0\nandj= 0. Two different dynamic regimes can be distinguished based on Eqs. (21): When\nthe driving forces are weak, a slightly distorted wall moves at a cons tant speed, ˙X= const,\nand constant tilt angle, ˙Φ = 0 (assuming constant Handj). The corresponding Walker\nansatz (20) then actually provides the exact solution, which is appr oached at long times\nafter the constant driving field and/or current are switched on [6 3, 64]. Beyond certain\ncritical values of Horj, called Walker thresholds, however, no solution with constant angle\nΦ and constant velocity ˙Xexist. Both undergo periodic oscillations in time, albeit with a\nfinite average drift velocity ∝angb∇acketleft˙X∝angb∇acket∇ight ∝negationslash= 0. In the spacial case of α=β, Eqs. (21) are exact at\narbitrary dc currents when H= 0: According to Eq. (11), the static domain-wall solution\n(of an arbitrary domain-wall shape) then simply moves with velocity −Pj/s0without any\ndistortions. When β∝negationslash=α, the Walker threshold current diverges when βapproaches α,\nreminiscent of the critical current (17) discussed in the previous s ection.\nFor subthreshold fields and currents with Φ( t)→const as t→ ∞, the steady state\nterminal velocity is given by [47]\nv=˙X(t→ ∞) =γH˜W−βPj/s0\nα. (22)\nIn particular, when j= 0, the wall depicted in Fig. 1 moves along the direction of the\napplied magnetic field Hin order to decrease the free energy [63]. Let us in the following\n12focus on the current-driven dynamics with H= 0. At a finite but small j, the wall is slightly\ncompressed according to\n1−˜W\nW≈(Pj/s0)2\n2γ2AK⊥/parenleftbigg\n1−β\nα/parenrightbigg2\n, (23)\nwhereW=/radicalbig\nA/Kis the equilibrium width. When α=β, the domain-wall velocity\nv→ −Pj/s0. In this case, if we consider the electron spins following the magnetiz ation\ndirection from ±mto∓mon traversing the domain wall with current density j, the entire\nangular momentum change is transferred to the domain-wall displac ement. In this sense,\nthe ratio β/αcan be loosely interpreted as a spin-transfer efficiency from the cu rrent density\nto the domain-wall motion.\nOnly when α=β, the rigidly moving domain-wall solution is exact at arbitrary current\ndensities, leading to an infinite Walker threshold current. The latter becomes finite and\ndecreases with β < α, approaching a finite value jt0atβ= 0 [60], see Fig. 2. In the absence\nof a strong disorder pinning centers, as assumed so far, jt0∝K⊥(which is also the case\nwith the Walker threshold fieldin the absence of an applied current [63]), with an average\nvelocity that slightly above the threshold reads\n∝angb∇acketleft˙X∝angb∇acket∇ight ∝/radicalBig\nj2−j2\nt0. (24)\nSee theβ= 0 curve in Fig.2. At finite β, thedepinning current is determined by the pinning\nfields, which should be included into the effective field (13). The domain -wall velocity at\ncurrents slightly above the depinning current is predicted in Ref. [5 4] to grow linearly with\nj.\nSo far in our discussion, we have completely disregarded the random noise contribution\nto the magnetization dynamics. As noted above, see Eq. (5), ther mal fluctuations are\nubiquitous in dissipative systems. Below the (zero-temperature) d epinning currents, applied\ncurrentscandrivethedomainwallwithfiniteaveragevelocity ∝angb∇acketleftv∝angb∇acket∇ightonlybythermalactivation.\nThe question how ln ∝angb∇acketleftv∝angb∇acket∇ightscales with the current at low temperatures and weak currents is\nof fundamental interest beyond the field of magnetism. Experimen ts on thermally-activated\ndomain-wall motion in magnetic semiconductors [33, 39] reveal a “cr eep” regime [66], in\nwhich the effective thermal-activation barrier diverges at low curre nt density j, so that\nln∝angb∇acketleftv∝angb∇acket∇ightscales as const −j−µ, with an exponent µ∼1/3. This is inconsistent with the\ntheory based on the Walker ansatz for rigid domain-wall motion [67], w hich yields a simple\nlinear scaling of the effective activation barrier and ln ∝angb∇acketleftv∝angb∇acket∇ight ∝const +j. A refinement of the\n13FIG. 2: Average current-driven domain-wall velocity vnumerically calculated using the Walker\nansatz [Eqs. (21)] in Ref. [53]. Here, the domain-wall width has been approximated by its equilib-\nrium value, ˜W≈W, assuming K⊥≪K. The curves are very similar to the full micromagnetic\nsimulations [53]. u=−Pj/s0has the units of velocity (proportional to electron drift ve locity) and\nvw=γK⊥ζ/2 is its value for j=jt0. The length ζ≈W, if we assume K⊥≪K(as was done\nin this calculation), while ζ≈/radicalbig\n2A/K⊥in the opposite limit, K⊥≫K, which is relevant for a\nthin-film with large demagnetization anisotropy K⊥= 4πMs(in which case ζis called exchange\nlength) [64]. α= 0.02, and we refer to Ref. [53] for the remaining details.\nWalker-ansatz treatment [68] cannot explain the experiments eith er. A scaling theory of\ncreep motion close to the critical temperature [39] does offer a qua litative agreement with\nmeasurements by Yamanouchi et al.[39]. However, the intrinsic spin-orbit coupling in p-\ndoped (Ga,Mn)As leads to current-driven effects beyond the stan dard spin-transfer theories,\nsee, e.g., Refs. [69, 70], which needs to be understood better in the present context.\nEven at zero temperature, there are stochastic spin-torque so urces in the presence of\nan applied current, which stem from the discreteness of the angula r momentum carried by\nelectron spins, in analogy with the telegraph-like shot noise of electr ic current carried by\ndiscrete particles. A theoretical study of the combined thermal a nd shot-noise contributions\n14to the stochastic torques for inhomogeneous magnetic configura tions [71] did not yet explore\nconsequences forthedomain-wall dynamics. Forexample, itisnotk nown whether shot noise\nassists thecurrent-driven domain-wall depinning atlowtemperatu res. Questions alongthese\nlines pose challenging problems for future research.\nEffects beyond the theory discussed above are generated by non adiabatic spin torques,\nwhich lead to higher-order in ∂tand∇terms in the equation of motion (9). It is in principle\npossible to extend linear-response diagrammatic Green’s function c alculation [13, 49, 50]\nby systematically calculating higher-order terms as an expansion in t he small parameters,\ni.e., spin-wave frequency and momentum [72]. A dynamic correction to the spin torque in\nEq. (9) has been found in Refs. [49, 72], which comes down to replacin gβ→β+n(/planckover2pi1/∆xc)∂t,\nwhere ∆ xcis the ferromagnetic exchange splitting and n= 1(2) for the Stoner ( s−d)\nmodel [49, 72]. Since this term scales like ∂t∇, it is symmetric under time reversal and\ntherefore nondissipative. Although this dynamic correction is rath er small at the typical\nFMR frequencies /planckover2pi1ω≪∆xc, it can cause significant effects at large currents [72]. Starting\nfromaninhomogeneousequilibriumconfiguration[72, 73,74,75], suc hasamagneticspiralor\na domain wall, one can capture nonadiabatic terms in the equation of m otion that vanish in\nlinear response withrespect totheuniformmagnetizationconsider ed inRefs. [13, 49, 50, 51].\nFor strongly-inhomogeneous magnetic structures, perturbativ e expansions around a uni-\nform magnetic state fail. For example, for sharp domain walls, the eff ective equations (21)\ndescribing wall dynamics and displacement acquire a new term, which c an be understood\nas a force transferred by electrons reflected at the potential b arrier caused by the domain\nwall [60, 76]. Electron reflection at a domain wall increases the resist ance. Adiabaticity\nimplies a vanishing intrinsic domain-wall resistance (see, however, Re f. [69] for a model with\nstrong intrinsic spin-orbit coupling). The force term, becomes impo rtant only for abrupt\nwalls with width W∼λxc≡/planckover2pi1vF/∆xc. Such nonadiabatic effects are not expected to be\nstrong in metallic ferromagnets, where typically W≫λxc∼λF(the Fermi wavelength).\nDilute magnetic semiconductors [such as (Ga,Mn)As] are a different c lass of materials with\nlongerλxcand a strong spin-orbit coupling [70].\nIn metallic systems, effects of the spin-torque in the most relevant regime of slow dy-\nnamics,/planckover2pi1ω≪∆xc, with smooth walls, W≫λxc, and at moderate applied currents is in\nour opinion captured by the adiabatic terms linear in ∂tand∇. We will now discuss the\nmicroscopic basis for Eq. (9) containing such terms.\n15V. MICROSCOPIC THEORY OF MAGNETIZATION DYNAMICS\nOnce the phenomenological equation for current-driven magnetiz ation dynamics is re-\nduced to the form (9), which requires smooth magnetization variat ion, slow dynamics, and\nisotropic ferromagnetism, the remaining key questions concern th e magnitude and relation\nbetween the two dimensionless parameters αandβ. The size of the Gilbert damping con-\nstantαis a long-standing open question in solid-state physics, and even a br ief review of the\nrelevant ideas and literature is beyond the scope of this paper. A re cent model calculation\nhighlighting the multitude of relevant energy scales that control ma gnetic damping can be\nfound in Ref. [12]. Here, we discuss only the ratio β/α, since it is of central importance\nfor macroscopic current-driven phenomena. As noted above, th e ratioβ/αdetermines, for\nexample, the onset of the ferromagnetic current-driven instabilit y [see Eq. (17)] as well as\nthe Walker threshold current (both diverging when β/α→1). The subthreshold current-\ndriven domain-wall velocity is proportional to β/α[see Eq. (22)], while β/α= 1 is a special\npoint, at which the effect of a uniform current density jon the magnetization dynamics is\neliminated in the frame of reference that moves with velocity v=−Pj/s0[see Eq. (11)].\nAlthough the exact ratio β/αis a system-dependent quantity, some qualitative aspects not\ntoo sensitive to the microscopic origin of these parameters have re cently been discussed\n[13, 49, 50].\nIn Ref. [49], we developed a self-consistent mean-field approach, in which itinerant elec-\ntrons are described by a time-dependent single-particle Hamiltonian\nˆH= [H0+U(r,t)]ˆ1+γ/planckover2pi1\n2ˆσ·(H+Hxc)(r,t)+ˆHσ, (25)\nwhere the unit matrix ˆ1 and the vector of the Pauli matrices ˆσ= (ˆσx,ˆσy,ˆσz) form a basis\nfor the Hamiltonian in spin space. H0is the crystal Hamiltonian including kinetic and\npotential energy. Uis the scalar potential consisting of disorder and applied electric-fie ld\ncontributions. The total magnetic field consists of the applied, H, and exchange, Hxc, fields.\nFinally, thelasttermintheHamiltonian, ˆHσ, accountsforspin-dephasing processes, e.g, due\nto quenched magnetic disorder or spin-orbit scattering associate d with impurity potentials.\nThis last term is responsible for low-frequency dissipative processe s affecting αandβin the\ncollective equation of motion (9).\nIn time-dependent spin-density-functional theory [44, 77, 78] o f itinerant ferromagnetism,\n16the exchange field Hxcis a functional of the time-dependent spin-density matrix\nραβ(r,t) =∝angb∇acketleftΨ†\nβ(r)Ψα(r)∝angb∇acket∇ightt (26)\nthat should be computed self-consistently from the Schr¨ odinger equation corresponding to\nˆH. The spin density of conducting electrons is given by\ns(r) =/planckover2pi1\n2Tr[ˆσˆρ(r)]. (27)\nFocusing on low-energy magnetic fluctuations that are long range a nd transverse, we restrict\nour attention to a single parabolic band. Consideration of realistic ba nd structures is pos-\nsible from this starting point. We adopt the adiabatic local-density ap proximation (ALDA,\nessentially the Stoner model) for the exchange field:\nγ/planckover2pi1Hxc[ˆρ](r,t)≈∆xcm(r,t), (28)\nwith direction m=−s/slocked to the time-dependent spin density (27) (assuming γ >0).\nIn another simple model of ferromagnetism, the so-called s-dmodel, conducting selec-\ntrons interact with the exchange field of the delectrons which are assumed to be localized to\nthe crystal lattice sites. The d-orbital electron spins are supposed to account for most of the\nmagnetic moment. Because d-electron shells have large net spins and strong ferromagnetic\ncorrelations, they are usually treated classically. In a mean-field s-ddescription, therefore,\nconducting sorbitals are described by the same Hamiltonian (25) with an exchange field\n(28). The differences between the Stoner and s-dmodels for the magnetization dynamics\nare rather minor and subtle. In the ALDA/Stoner model, the excha nge potential is (on the\nscale of the magnetization dynamics) instantaneously aligned with th e total magnetization.\nIn contrast, the direction unit vector min thes-dmodel corresponds to the dmagnetization,\nwhich is allowed to be misaligned with the smagnetization, transferring torque between the\nsanddmagnetic moments. Since most of the magnetization is carried by the latter, the\nexternal field Hcouples mainly to the dspins, while the sspins respond to and follow the\ntime-dependent exchange field (28). As ∆ xcis usually much larger than the external (includ-\ning demagnetization and anisotropy) fields that drive collective magn etization dynamics, the\ntotal magnetic moment will always be very close to m. A more important difference of the\nphilosophy behind the two models is the presumed shielding of the dorbitals from exter-\nnal disorder. The reduced coupling with dissipative degrees of free dom would imply that\n17their dynamics are much less damped. (Whether this is actually the ca se in real systems\nremains to be proven, however.) Consequently, the magnetization damping has to come\nfrom the disorder experienced by the itinerant selectrons. As in the case of the itinerant\nferromagnets, the susceptibility has to be calculated self-consist ently with the magnetization\ndynamics parametrized by m. For more details on this model, we refer to Refs. [10, 49].\nWith the above differences in mind, the following discussion is applicable t o both models. In\norder to avoid confusion, we remark that the equilibrium spin density s0introduced earlier\nrefers to the total spin density, i.e., d- pluss-electron spin density, while Eq. (27) refers\nonly to the latter. The Stoner model is more appropriate for trans ition-metal ferromagnets\nbecause of the strong hybridization between dands,pelectrons. Magnetic semiconductors\nare characterized by deep magnetic impurity states for which the s-dmodel may be a better\nchoice.\nThe single-particle itinerant electron response to electric and magn etic fields in Hamil-\ntonian (25) is all that is needed to compute the magnetization dynam ics microscopically.\nAs mentioned above, the distinction between the Stoner and s-dmodels will appear only\nat the end of the day, when we self-consistently relate m(r,t) to the itinerant electron spin\nresponse. Before proceeding, we observe that since the consta ntsαandβwhich parametrize\nthe magnetic equation of motion (9) affect the linear response to a s mall transverse applied\nfield with respect to a uniform magnetization, we can obtain them by a linear-response\ncalculation for the single-domain bulk ferromagnet. The large-scale magnetization texture\nassociated with a domain wall does not affect the value of these para meters, in the consid-\nered limit. The linear response to a small magnetic field is complicated by the presence of\nan electrically-driven applied current, however. Since the Kubo for mula based on two-point\nequilibrium Green’s functions is insufficient to calculate the response t o simultaneous mag-\nnetic and electric fields, we chose to pursue a nonequilibrium (Keldysh ) Green’s function\nformalism in Refs. [13, 49]. A technically impressive equilibrium Green’s fu nction calcula-\ntion has been carried out in Ref. [50], which to a large extent confirme d our results, but also\ncontributed some important additions that will be discussed below.\nThe central quantity in the kinetic equation approach [13, 49] is the nonequilibrium\ncomponent of the 2 ×2 distribution function ˆfk(r,t). In the quasiparticle approximation,\nvalidwhen∆ xc≪EF[49], thekineticequationcanbereducedtoasemiclassical Boltzmann -\nlike equation that accounts for electron drift in response to the ele ctric field as well as the\n18spin precession in the magnetic field. The nonequilibrium component of the spin density\nreadss′= (/planckover2pi1/2)/integraltext\nd3kfk/(2π)3, wherefk= Tr[ˆfkˆσ]:\n∂ts′−∆xc\n/planckover2pi1z×s′−∆xcs\n/planckover2pi1z×u=−/planckover2pi1\n2/integraldisplayd3k\n(2π)3(vk·∂r)fk−s′+su\nτσ. (29)\nshere is the equilibrium spin density of itinerant electrons, vk=∂kεks//planckover2pi1is the momentum-\ndependent group velocity, and the magnetization direction m=z+uis assumed to undergo\na small precession urelative to the uniform equilibrium direction z. The first term on the\nright-hand side is the spin-current divergence and the last term is t he spin-dephasing term\nintroduced phenomenologically in Ref. [49] and studied microscopically in Ref. [13]. As\ndetailed in Ref. [49], the spin currents have to be calculated from the full kinetic equation\nand then inserted in Eq. (29). The final result (for the Stoner mod el) is given by Eq. (10)\nor, equivalently, Eq. (9), with α=β. The latter is proportional to the spin-dephasing rate:\nβ=/planckover2pi1\nτσ∆xc. (30)\nThe derivation assumes ω,τ−1\nσ≪∆xc//planckover2pi1, which is typically the case in real materials suffi-\nciently below the Curie temperature. The s-dmodel yields the same result for β, but\nα=ηβ (31)\nis reduced by the η=s/s0ratio, i.e., the fraction of the itinerant to the total angular\nmomentum. [Note that Eq. (31) is also valid for the Stoner model sinc e thens0=s.] For\nthes-dmodel, the equation of motion (9) clearly cannot be reduced to Eq. ( 10), since α∝negationslash=β.\nThe steady-state current-driven velocity (22) for both mean-fi eld models becomes\nv=−βPj\nαs0=−Pj\ns, (32)\nwheresis the itinerant electron spin density. Interestingly, the velocity (3 2) is completely\ndetermined by properties of the conducting electrons, even for t hes−dmodel. In the Drude\nmodel,\nv∝Eτ\nm∗, (33)\nwhereEis the applied electric field, τis the characteristic momentum scattering time, and\nm∗is the effective mass of the itinerant bands at the Fermi energy. We expect the velocity\n(33), which is essentially the conducting electron drift velocity, to b e suppressed for the\n19s-dmodel if the dorbitals are coupled to their own dissipative bath, which has not been\nincluded in the above treatment.\nRef. [50] refines these results by relaxing the assumption that ∆ xc≪EFand by consid-\nering also anisotropic spin-dephasing impurities, which results in α∝negationslash=βfor both Stoner and\ns-dmodels. Ref.[51]laterofferedaKeldysh functional-integral appro achleading tothesame\nresults. (These authors also found stochastic torques express ed in terms of thermal fluctu-\nations (5) in the weak current limit; see, however, Ref. [71] for add itional current-induced\nstochastic terms present in the case of an inhomogeneous magnet ization.) Consider, for\nexample, weak magnetic disorder described by the potential ˆHσ=h(r)·ˆσwith Gaus-\nsian white-noise correlations ∝angb∇acketleftha(r)hb(r′)∝angb∇acket∇ight ∝Uaδabδ(r−r′), where Ua=U⊥(U/bardbl) whenais\nperpendicular (parallel) to the equilibrium magnetization direction. (S pin-orbit interaction\nassociated with scalar disorder gives similar results.) For isotropic dis order,U⊥=U/bardbl, and\n∆xc≪EF,α/β≈ηwithη=s/s0, as was already discussed (reducing to η= 1 for the\nStoner model). Even for larger exchange, the correction to this α/βratio turns out to be\nrather small: For parabolic bands, for example, α/β≈[1−(∆xc/EF)2/48]η. This ratio is\nmore sensitive to anisotropies U/bardbl∝negationslash=U⊥, however, so that in general α/β∝negationslash=ηeven in the\nlimit ∆ xc/EF→0 [50].\nVI. SUMMARY AND OUTLOOK\nOur microscopic understanding is based on a mean-field approximatio n, in which itiner-\nant electrons interact self-consistently with a space- and time-de pendent exchange field. We\npresented results for the local-spin-density approximation and th e mean-field s-dmodel. We\nidentified a relation between dissipative terms parametrized by αandβand spin-dephasing\nscattering potentials. The central result for the collective low-fr equency long-wavelength\ncurrent-driven magnetization dynamics can be formulated as a gen eralization of the phe-\nnomenological Landau-Lifshitz-Gilbert equation, accounting for t he current-driven torques.\nOne should in general also include stochastic terms due to thermal fl uctuations as well as\nnonequilibrium shot-noise contribution in the presence of applied cur rentj[71]. Despite\nsome recent efforts, stochastic effects remain to be relatively une xplored both theoretically\nand experimentally, however.\nThe most important parameter that determines the effect of an ele ctric current on the\n20collective magnetization dynamics in extended systems is the ratio β/α. We find that\nthis ratio is not universal and in general depends on details of the ba nd structure and\nspin-dephasing processes. Nevertheless, simple models give α∼βwith the special limit\nα≈βfor the Stoner model with weak and isotropic spin-dephasing disord er. Solving the\nmagnetization equation of motion for a domain wall is rather straight forward at low dc\ncurrents, when the wall is only slightly compressed. The domain-wall motion can then be\nmodeled within the Walker ansatz, based on parametrizing the magne tic dynamics in terms\nof wall position and spin distortion. Two regimes can then be distinguis hed: At the lowest\ncurrents, the wall moves steadily in the presence of a constant un iform current, while above\nthe so-called Walker threshold, the magnetization close to the wall c enter starts oscillating,\nresulting in a singular dependence of the average velocity on the app lied current.\nThe values of the αandβparameters are not affected by the magnetization textures.\nMicromagnetic simulations can provide better understanding of exp erimental results in the\nregimeswheredomainwallsarenotwell describedbyaone-dimensiona l model. Experiments\ncan contribute to the understanding by studying ferromagnets w ith systematic variations\nof impurity types and concentration, for Py and other different ma terials. Experimental\ninvestigation of creep in metallic ferromagnets at temperatures fa r below the critical ones,\nas compared to studies [33, 39] on magnetic semiconductors close t o the Curie transition,\nare highly desirable in order to advance our understanding.\nBesides realistic microscopic evaluations of the key parameters αandβ, the collective\ncurrent-driven magnetization dynamics pose many theoretical ch allenges, in the spirit of\nclassical nonlinear dynamical systems. Current-driven magnetism displays a rich behavior\nwell beyond what can be achieved by applied magnetic fields only. At su percritical currents,\nferromagnetism becomes unstable, possibly leading to chaotic dyna mics [79], although al-\nternative scenarios have been also suggested [80]. Domain-wall dyn amics in a medium with\ndisordered pinning potentials pose an interesting yet, at weak applie d currents, tractable\nproblem. Spin torques and dynamics in sharp walls and the role of stro ng intrinsic spin-orbit\ncoupling (relevant for dilute magnetic semiconductors) are not yet completely understood.\nOscillatory domain-wall motion under ac currents and in curved geom etries is also starting\nto attract attention both experimentally and theoretically [35, 81]. Another direction of\nrecent activities concern the backaction of a moving domain wall on t he charge degrees of\nfreedom [82, 83, 84, 85, 86].\n21With the exciting recent and forthcoming experimental developmen ts, the questions con-\ncerning interactions of the collective ferromagnetic order with elec tric currents will certainly\nchallenge theoreticians for many years to come. The prospects of using purely electric means\nto efficiently manipulate magnetic dynamics are also promising for prac tical applications.\nVII. ACKNOWLEDGMENTS\nWewould like tothanktheEditors forcarefullyreading themanuscrip t andmaking many\nuseful comments. 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Lett. 94, 076601 (2005).\n[81] B. Kr¨ uger, D. Pfannkuche, M. Bolte, G. Meier, andU. Mer kt, Phys.Rev. B 75, 054421 (2007).\n[82] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007).\n[83] R. A. Duine, cond-mat/0706.3160.\n[84] M. Stamenova, T. N. Todorov, and S. Sanvito, cond-mat/0 708.1167.\n[85] S. A. Yang, D. Xiao, and Q. Niu, cond-mat/0709.1117.\n[86] Y. Tserkovnyak and M. Mecklenburg, cond-mat/0710.519 3.\n25"    },    {        "title": "0710.2826v2.Ferromagnetic_resonance_study_of_polycrystalline_Fe__1_x_V_x_alloy_thin_films.pdf",        "content": "arXiv:0710.2826v2  [cond-mat.mes-hall]  17 Oct 2007Ferromagnetic resonance study of polycrystalline Fe 1−xVxalloy thin films\nJ-M. L. Beaujour, A. D. Kent\nDepartment of Physics, New York University, 4 Washington Pl ace, New York, NY 10003, USA\nJ. Z. Sun\nIBM T. J. Watson Research Center, Yorktown Heights, NY 10598 , USA\n(Dated: November 18, 2018)\nFerromagnetic resonance has been used to study the magnetic properties and magnetization dy-\nnamics of polycrystalline Fe 1−xVxalloy films with 0 ≤x <0.7. Films were produced by co-\nsputtering from separate Fe and V targets, leading to a compo sition gradient across a Si substrate.\nFMR studies were conducted at room temperature with a broadb and coplanar waveguide at fre-\nquencies up to 50 GHz using the flip-chip method. The effective demagnetization field 4 πMeffand\nthe Gilbert damping parameter αhave been determined as a function of V concentration. The\nresults are compared to those of epitaxial FeV films.\nI. INTRODUCTION\nA decade ago, it was predicted that a spin polar-\nized current from a relatively thick ferromagnet (FM)\ncould be used to switch the magnetization of a thin FM\n[1]. Since then, this effect, known as spin-transfer, has\nbeen demonstrated in spin-valves [2] and magnetic tun-\nnel junctions [3]. In a macrospin model with collinear\nlayer magnetizations, there is a threshold current den-\nsityJcfor an instability necessary for current-induced\nmagnetization switching of the thin FM layer [1, 4]:\nJc=2eαMstf(Hk+2πMs)\n/planckover2pi1η, (1)\nwhereαisthedampingconstant. tfandMsarethethick-\nness and the magnetization density of the free layer, re-\nspectively. Hkis the in-plane uniaxial anisotropy field. η\nis the currentspin-polarization. In orderfor spin-transfer\nto be used in high density memory devices Jcmust be re-\nduced. From Eq. 1 it is seen that this can be achieved by\nemploying materials with low Msandαin spin-transfer\ndevicesor, equivalently materialswith lowGilbert damp-\ning coefficients, G = αMs(gµB//planckover2pi1).\nVery recently, an experimental study of epitaxial FeV\nalloy thin films demonstrated a record low Gilbert damp-\ning coefficient [5]. This material is therefore of interest\nfor spin transfer devices. However, such devices are gen-\nerally composed of polycrystalline layers. Therefore it is\nof interest to examine polycrystalline FeV films to assess\ntheir characteristics and device potential.\nIn this paper, we present a FMR study of thin poly-\ncrystalline Fe 1−xVxalloy films with 0 ≤x <0.7 grown\nby co-sputtering. The FeV layers were embedded be-\ntween two Ta |Cu layers, resulting in the layer structure\n||5 Ta|10 Cu|FeV|5 Cu|10 Ta||, where the numbers are\nlayer thickness in nm. FeV polycrystalline films were\nprepared by dc magnetron sputtering at room tempera-\nture from two separate sources, oriented at a 45oangle\n(Fig. 1a). The substrate, cut from a Silicon wafer with\n100 nm thermal oxide, was 64 mm long and about 5 mm\nwide. The Fe and V deposition rates were found to vary/s49 /s50 /s51 /s52/s45/s50/s45/s49/s48/s40/s98/s41\n/s32/s86/s32/s116/s97/s114/s103/s101/s116\n/s52/s53/s111\n/s32\n/s32/s115/s117/s98/s115/s116/s114/s97/s116/s101/s70/s101/s32/s116/s97/s114/s103/s101/s116/s40/s97/s41\n/s72\n/s114/s101/s115/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s108/s105/s110/s101/s32/s32/s40/s97/s117/s41\n/s72\n/s97/s112/s112/s32/s32/s40/s107/s79/s101/s41/s49/s52/s32/s71/s72/s122\n/s32/s32/s32\n/s32/s120 /s61/s48/s46/s51/s55/s120 /s61/s48/s46/s53/s50\n/s120 /s61/s48/s46/s49/s57/s72\n/s115/s117/s98/s115/s116/s114/s97/s116/s101/s32/s104/s111/s108/s100/s101/s114\n/s97/s120/s105/s115\nFIG. 1: a) The co-sputtering setup. b) Typical absorp-\ntion curves at 14 GHz for a selection of ||Ta|Cu|7.5 nm\nFe1−xVx|Cu|Ta||films, with x=0.19, 0.37 and 0.52. The res-\nonance field Hresand the linewidth ∆ Hare indicated.\nlinearly across the wafer. The Fe and V rates were then\nadjusted to produce a film in which xvaries from 0.37\nto 0.66 across the long axis of the wafer. The base pres-\nsure in the UHV chamber was 5 ×10−8Torr and the\nAr pressure was set to 3.5 mTorr. The FeV was 7.5 nm\nin thickness, varying by less than 0.3 % across the sub-\nstrate. An Fe 1−xVxfilm 3 nm thick was also fabricated.\nTo produce films with x <0.30 the rate of the V source\nwas decreased. Finally, pure Fe films with a thickness\ngradient ranging from 7 nm to 13.3 nm were deposited.\nTheFMRmeasurementswerecarriedoutatroomtem-\nperature using a coplanar wave-guide (CPW) and the\nflip-chip method. Details of the experimental setup and\nofthe CPWstructuralcharacteristicscan be found in [6].\nAdc magnetic field, up to 10 kOe, wasapplied in the film\nplane, perpendiculartotheacmagneticfield. Absorption\nlines at frequencies from 2 to 50 GHz were measured by\nmonitoring the relative change in the transmitted signal\nas a function of the applied magnetic field.2\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s49/s50/s51\n/s120\n/s32/s52 /s77\n/s101/s102/s102/s32/s32/s40/s107/s71/s41/s55/s46/s53/s32/s110/s109/s32/s70/s101\n/s49/s45 /s120/s86\n/s120\n/s51/s32/s110/s109/s32/s70/s101\n/s48/s46/s54/s51/s86\n/s48/s46/s51/s55\n/s49/s50/s46/s57/s32/s110/s109/s32/s70/s101\n/s120/s32\n/s32/s103/s45/s102/s97/s99/s116/s111/s114/s40/s98/s41/s40/s99/s41/s32\n/s32\n/s32/s72\n/s114/s101/s115/s32/s32/s40/s107/s79/s101/s41\n/s49/s49/s32/s71/s72/s122/s40/s97/s41\nFIG. 2: a) The resonance field at 11 GHz versus xand b) the\neffective demagnetization field versus x. The solid line is a\nguide to the eye. c) The Land´ e gfactor as a function of x.\nThe dotted line shows the g-factor value of bulk Fe.\nII. RESULTS\nTypical absorption lines at 14 GHz of selected FeV al-\nloyfilmsareshowninFig. 1b. Thelinesarelorentzianfor\nmost frequencies. At a fixed frequency, the FMR absorp-\ntion decreaseswith increasingV content. The FMR peak\nof a film 7.5 nm thick with x= 0.66 is about 100 times\nsmallerthanthatofapureFeofthesamethickness. This\nis accompaniedby a shift of Hrestowardshigher field val-\nues (Fig. 2a). The effective demagnetization field 4 πMeff\nand the Land´ eg-factorgwere determined by fitting the\nfrequency dependence of the resonance field Hresto the\nKittel formula [7]:\nf2=/parenleftBiggµB\nh/parenrightBig2\nHres(Hres+4πMeff),(2)\nwhere the effective demagnetization field is:\n4πMeff= 4πMs−H⊥. (3)\nNote that in the absence of a perpendicular anisotropy\nfieldH⊥, the effective field would be directly related to\nMs. The dependence of 4 πMeffon V concentration is\nshown in Fig. 2c. As xincreases the effective demagneti-\nzation field decreases dramatically, going from about 16\nkG forx= 0 to 1.1 kG for x= 0.66. Note that the effec-\ntive demagnetization field of the 7.5 nm Fe film is about\n25 % lower than that of bulk Fe (21.5 kG). The 12.9\nnm Fe film exhibits a larger 4 πMeff, which is, however,\nstill lower than 4 πMsof the bulk material. Similarly,\nthe 4πMeffof an Fe 0.63V0.37film is thickness dependent:\ndecreasing with decreasing layer thickness.\nThe Land´ eg-factor increases monotonically with in-\ncreasing V concentration (Fig. 2b). The minimum g-\nfactor is measured for the Fe film: g= 2.11±0.01,\nwhich is slightly larger than the value of bulk material\n(g= 2.09). Note that gof a Fe film 12.9 nm thick is\nequal to that of Fe bulk. However, the g-factor of the/s48 /s50/s48 /s52/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s40/s99/s41/s120 /s32/s61/s32/s48/s46/s53/s50/s120/s32 /s61/s32/s48/s46/s52/s51/s120/s32 /s61/s32/s48/s40/s98/s41/s32/s40/s49/s48/s45/s50\n/s41/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s40/s107/s79/s101/s41/s32\n/s32/s72 /s32/s32/s40/s107/s79/s101/s41\n/s72\n/s48/s72\n/s48/s72\n/s49/s52/s32/s32\n/s32/s40/s97/s41\n/s120/s32 /s32/s102/s32/s32/s40/s71/s72/s122/s41\nFIG. 3: a)Frequency dependence of the linewidth for 7.5 nm\nFe1−xVxalloy film with x=0, 0.43 and 0.52. The solid lines\nare the best linear fit of the experimental data. b) ∆ H14, the\nlinewidth at 14 GHz , and ∆ H0are shown as a function of x.\nc) The magnetic damping parameter versus V concentration.\nFe0.63V0.37, does not appear to be thickness dependent:\nthe 3 nm Fe 0.63V0.37layer has about the same gvalue\nthan the 7.5 nm Fe 0.63V0.37layer.\nThe half-power linewidth, ∆ H, was studied as a func-\ntion of the frequency and of the V concentration. Fig.\n3b shows the dependence of the FMR linewidth on xat\n14 GHz. The general trend is that ∆ Hincreases with\nx. However, there are two regimes. For x >0.4, the\nlinewidth depends strongly on x, increasing by a factor\n5 whenxis increased from 0.4 to 0.66. The dependence\nof the linewidth on xis more moderate for the films with\nx <0.4: it increases by about 30 %. For all samples,\nthe linewidth scales linearly with the frequency. A least\nsquare fit of ∆ H(f) gives ∆ H0, the intercept at zero\nfrequency, and the Gilbert damping parameter αwhich\nis proportional to the slope: d∆H/df= (2h/gµB)α[8].\n∆H0is typically associated with an extrinsic contribu-\ntion to the linewidth and related to magnetic and struc-\ntural inhomogeneities in the layer. For two samples with\nthe highest Vanadium concentration, x= 0.60, 0.66, the\nlinewith is dominated by inhomogeneous broadening and\nit wasnot possible to extract α. Asxincreases, ∆ H0and\nαincreases. The damping parameter and ∆ H0remain\npractically unchanged for x≤0.4 and when x >0.4,\nboth the intercept and the slope of ∆ Hversusfincrease\nrapidly.\nIII. DISCUSSION\nSeveral factors can contribute to the dependence of\n4πMeffon the V concentration. First, the decrease of\nthe effective demagnetizationfield canbe associatedwith\nthe reduction of the alloy magnetization density Mssince\nthe Fe content is reduced. In addition, a neutron scat-\ntering study showed that V acquires a magnetic moment3\nantiparallel to the Fe, and that the Fe atom moment de-\ncreases with increasing V concentration [9]. The Curie\ntemperature of Fe 1−xVxdepends on x. In fact, Tcfor\nx=0.65 is near room temperature [10]. It is important to\nmention that a factor that can further decrease 4 πMeff\nis an out-of-plane uniaxial anisotropy field H⊥(Eq. 3).\nIn thin films, the perpendicular anisotropy field is com-\nmonly expressed as H⊥= 2K⊥/(Mst), where K⊥>0\nis the anisotropy constant and tthe ferromagnetic film\nthickness [11]. In this simple picture, it is assumed that\nK⊥is nearly constant over the thickness range of our\nfilms. This anisotropy can be associated with strain\ndue to the lattice mismatch between the FeV alloy and\nthe adjacent Cu layers and/or with an interface contri-\nbution to the magnetic anisotropy. For Fe films with\nt= 7.5 and 12.9 nm, a linear fit of 4 πMeffversus 1/t\ngives 4πMs= 20.2 kG and K⊥= 2.5 erg/cm2. The\nvalue extracted for 4 πMsis in the range of the value of\nFebulk. AsimilaranalysisconductedonFe 0.63V0.37films\nof thickness t=3 and 7.5 nm gives 4 πMs= 12.2 kG and\nK⊥= 0.1 erg/cm2. The result suggests that the surface\nanisotropy constant decreases with increasing x.\nIV. SUMMARY\nThe effective demagnetization field of the polycrys-\ntalline Fe 1−xVxalloy films decreases with increasing xandalmostvanishesfor x≈0.7. AFMRstudyonepitax-\nial films haveshown a similar xdependence of 4 πMeff[5].\nUsing the value of Mscalculated in the analysis above,\nwe estimate the Gilbert damping constant of a 7.5 nm Fe\nlayer and 7.5 nm Fe 0.63V0.37alloy film to be G Fe= 239\nMHz and G FeV= 145 MHz respectively. The decrease of\nthe effective demagnetization field of Fe 1−xVxwith in-\ncreasing xis accompanied by a decrease of the Gilbert\ndamping constant. A similar xdependence of G was\nobserved in epitaxial films [5]. The authors explained\nthe decrease of G by the reduced influence of spin-orbit\ncoupling in lighter ferromagnets. Note that the Gilbert\ndampingofourfilmsislargerthanwhatwasfoundforthe\nepitaxial films (G=57 MHz for epitaxial Fe 8 nm thick).\nWe note that the Fe 0.63V0.37alloy film, which has\n4πMsapproximatly the same as that of Permalloy, has a\nmagnetic damping constant of the same order than that\nof Py layer in a similar layer structure [12]. Hence, with\ntheir low Msandα, polycrystalline FeV alloy films are\npromising materials to be integrated in spin-tranfer de-\nvices.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159(1-2), L1\n(1996) ; L. Berger, Phys. Rev. B 54(12), 9353 (1996).\n[2] see, for example, J. A. Katine et al., Phys. Rev. Lett. 84,\n3149 (2000) ; B. Oezyilmaz et al., Phys. Rev. Lett. 91,\n067203 (2003).\n[3] see, for example, G. D. Fuchs et al., J. Appl. Phys. 85\n(7), 1205 (2004).\n[4] J. Z. Sun, Phys. Rev. B 62(1), 570 (2000).\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] J-M. L. Beaujour et al., Europhys. J. B, DOI:\n10.1140/epjb/e2007-00071-1 (2007).\n[7] C. Kittel in Introduction to Solid State Physics, Ed. 7,\np.505.[8] see, for example, D. L. Mills and S. M. Rezende in Spin\nDynamics in Confined Magnetic Structures II (Eds. B.\nHillebrands andK.Ounadjela), pp.27-58, (Springer, Hei-\ndelberg 2002).\n[9] I. Mirebeau, G. Parette, and J. W. Cable. J. Phys. F:\nMet. Phys. 17, 191 (1987).\n[10] Y. Kakehashi, Phys. Rev. B 32(5), 3035 (1985).\n[11] Y. K. Kim and T. J. Silva, Appl. Phys. Lett. 68, 2885\n(1996).\n[12] S. Mizukami et al., J. Magn. Magn. Mater. 239, 42\n(2002)."    },    {        "title": "0802.1740v1.Temperature_dependent_magnetization_dynamics_of_magnetic_nanoparticles.pdf",        "content": "arXiv:0802.1740v1  [cond-mat.other]  12 Feb 2008Temperature dependent magnetization dynamics of\nmagnetic nanoparticles\nA. Sukhov1,2and J. Berakdar2\n1Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-0 6120 Halle/Saale,\nGermany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenbe rg,\nHeinrich-Damerow-Str. 4, 06120 Halle, Germany\nAbstract. Recent experimental and theoretical studies show that the switc hing\nbehavior of magnetic nanoparticles can be well controlled by extern al time-dependent\nmagnetic fields. In this work, we inspect theoretically the influence o f the temperature\nand the magnetic anisotropy on the spin-dynamics and the switching properties of\nsingle domain magnetic nanoparticles (Stoner-particles). Our theo retical tools are\nthe Landau-Lifshitz-Gilbert equation extended as to deal with finit e temperatures\nwithin a Langevine framework. Physical quantities of interest are t he minimum\nfield amplitudes required for switching and the corresponding rever sal times of the\nnanoparticle’s magnetic moment. In particular, we contrast the ca ses of static and\ntime-dependent external fields and analyze the influence of dampin g for a uniaxial and\na cubic anisotropy.\nPACS numbers: 75.40.Mg, 75.50.Bb, 75.40.Gb, 75.60.Jk, 75.75.+aTemperature dependent magnetization dynamics of magnetic nanoparticles 2\n1. Introduction\nInrecent years, therehasbeenasurgeofresearchactivities fo cusedonthespindynamics\nand the switching behavior of magnetic nanoparticles [1]. These stud ies are driven\nby potential applications in mass-storage media and fast magneto- electronic devices.\nIn principle, various techniques are currently available for controllin g or reversing the\nmagnetization of a nanoparticle. To name but a few, the magnetizat ion can be reversed\nby a short laser pulse [2], a spin-polarized electric current [3, 4] or an alternating\nmagnetic field [5, 6, 7, 8, 9, 10, 11, 12, 13]. Recently [6], it has been sh own for a\nuniaxial anisotropythattheutilizationofaweak time-dependent ma gneticfieldachieves\na magnetization reversal faster than in the case of a static magne tic field. For this case\n[6], however, the influence of the temperature and the different ty pes of anisotropy\non the various dependencies of the reversal process have not be en addressed. These\nissues, which are the topic of this present work, are of great impor tance since, e.g.\nthermal activation affects decisively the stability of the magnetizat ion, in particular\nwhen approaching the superparamagnetic limit, which restricts the density of data\nstorage [14]. Here we study the possibility of fast switching at finite t emperature with\nweak external fields. We consider magnetic nanoparticles with an ap propriate size as to\ndisplayalong-rangemagneticorderandtobeinasingledomainremane ntstate(Stoner-\nparticles). Uniaxial and cubic anisotropies are considered and show n to decisively\ninfluence the switching dynamics. Numerical results are presented and analyzed for\niron-platinum nanoparticles. In principle, the inclusion of finite tempe ratures in spin-\ndynamics studies is well-established (cf. [19, 20, 23, 15, 16, 1] and references therein)\nand will be followed here by treating finite temperatures on the level of Langevine\ndynamics. For the analysis of switching behaviour the Stoner and Wo hlfarth model\n(SW) [17] is often employed. SW investigated the energetically metas table and stable\nposition of the magnetization of a single domain particle with uniaxial an isotropy in\nthe presence of an external magnetic field. They showed that the minimum static\nmagnetic field (generally referred to as the Stoner-Wohlfarth (SW ) field or limit) needed\nto coherently reverse the magnetization is dependent on the direc tion of the applied\nfield with respect to the easy axis. This dependence is described by t he so-called\nStoner-Wohlfarth astroid. The SW findings rely, however, on a sta tic model at zero\ntemperature. Application of a time-dependent magnetic field reduc es the required\nminimum switching field amplitude below the SW limit [6]. It was, however, no t yet\nclear how finite temperatures will affect these findings. To clarify th is point, we utilize\nan extension of the Landau-Lifshitz-Gilbert equation [18] including fi nite temperatures\non the level of Langevine dynamics [19, 20, 23]. Our analysis shows t he reversal time\nto be strongly dependent on the damping, the temperature and th e type of anisotropy.\nThese dependencies are also exhibited to a lesser extent by the crit ical reversal fields.\nThepaperisorganizedasfollows: nextsection2presents detailsof thenumerical scheme\nand the notations whereas section 3 shows numerical results and a nalysis for Fe 50Pt50\nand Fe 70Pt30nanoparticles. We then conclude with a brief summary.Temperature dependent magnetization dynamics of magnetic nanoparticles 3\n2. Theoretical model\nIn what follows we focus on systems with large spins such that their m agnetic dynamics\ncan be described by the classical motion of a unit vector Sdirected along the particle’s\nmagnetization µ, i.e.S=µ/µSandµSisthe particle’s magnetic moment at saturation.\nThe energetics of the system is given by\nH=HA+HF. (1)\nwhereHA(HF) stands for the anisotropy (Zeeman energy) contribution. Furt hermore,\nthe anisotropy contribution is expressed as HA=−Df(S) withDbeing the anisotropy\nconstant. Explicit formof f(S)isprovidedbelow. Themagnetizationdynamics, i.e. the\nequation of motion for S, is governed by the Landau-Lifshitz-Gilbert (LLG) equation\n[18]\n∂S\n∂t=−γ\n(1+α2)S×/bracketleftBig\nBe(t) +α(S×Be(t))/bracketrightBig\n. (2)\nHere we introduced the effective field Be(t) =−1/(µS)∂H/∂Swhich contains the\nexternal magnetic field and the maximum anisotropy field for the unia xial anisotropy\nBA= 2D/µS.γis the gyromagnetic ratio and αis the Gilbert damping parameter. The\ntemperature fluctuations will be described on the level of the Lang evine dynamics [19].\nThis means, a time-dependent thermal noise ζ(t) adds to the effective field Be(t) [19].\nζ(t) is a Gaussian distributed white noise with zero mean and vanishing time correlator\n/angbracketleftζi(t′)ζj(t)/angbracketright=2αkBT\nµsγδi,jδ(t−t′). (3)\ni,jare Cartesian components, Tis the temperature and kBis the Boltzmann constant.\nIt is convenient to express the LLG in the reduced units\nb=Be\nBA, τ=ωat, ωa=γBA. (4)\nThe LLG equation reads then\n∂S\n∂τ=−1\n(1+α2)S×/bracketleftBig\nb(τ) +α(S×b(τ))/bracketrightBig\n, (5)\nwhere the effective field is now given explicitly by\nb(τ) =−1\nµSBA∂H\n∂S+Θ(τ) (6)\nwith\n/angbracketleftΘi(τ′)Θj(τ)/angbracketright=ǫδi,jδ(τ−τ′);ǫ=2αkBT\nµsBA. (7)\nThereducedunitsareindependent ofthedampingparameter α. Inthefollowingsections\nwe use extensively the parameter\nq=kBT\nD. (8)\nqis a measure for the thermal energy in terms of the anisotropy ene rgy. And\nd=D/(µSBA) expresses the anisotropy constant in units of a maximum anisotro pyTemperature dependent magnetization dynamics of magnetic nanoparticles 4\nenergy for the uniaxial anisotropy and is always 1 /2. The stochastic LLG equation (5)\nin reduced units (4) is solved numerically using the Heun method which c onverges in\nquadratic mean to the solution of the LLG equation when interprete d in the sense of\nStratonovich [20]. For each type of anisotropy we choose the time s tep ∆τto be one\nthousandth part of the corresponding period of oscillations. The v alues of the time\ninterval in not reduced units for uniaxial and cubic anisotropies are ∆tua= 4.61·10−15s\nand ∆tca= 64.90·10−15s, respectively, providing us thus with correlation times on the\nfemtosecond time scale. The reason for the choice of such small tim e intervals is given in\n[19], where it is argued that the spectrum of thermal-agitation forc es may be considered\nas white up to a frequency of order kBT/hwithhbeing the Planck constant. This value\ncorresponds to 10−13sfor room temperature. The total scale of time is limited by a\nthousand of such periods. Hence, we deal with around one million iter ation steps for\na switching process. Details of realization of this numerical scheme c ould be found in\nreferences [21, 22, 20]. We note by passing, that attempts have b een made to obtain,\nunder certain limitations, analytical results for finite-temperatur e spin dynamics using\nthe Fokker-Planck equation (cf. [15, 16] and references therein ). For the general case\ndiscussed here one has however to resort to fully numerical appro aches.\n3. Results and interpretations\nWeconsider a magneticnanoparticleina singledomainremanent state (Stoner-particle)\nwith aneffective anisotropy whose origin can be magnetocrystalline, magnetoelastic and\nsurface anisotropy. We assume the nanoparticle to have a spheric al form, neglecting\nthus the shape anisotropy contributions. In the absence of exte rnal fields, thermal\nfluctuations may still drive the system out of equilibrium. Hence, the stability of\nthe system as the temperature increases becomes an important is sue. The time tat\nwhich the magnetization of the system overcomes the energy barr ier due to the thermal\nactivation, also called the escape time , is given by the Arrhenius law\nt=t0eD\nkBT, (9)\nwhere the exponent is the ratio of the anisotropy to the thermal e nergy. The coefficient\nt0may be inferred when D≫kBTand for high damping [19] (see [25] for a critical\ndiscussion)\nt0=1+α\nαγπµS\n2D/radicalBigg\nkBT\nD. (10)\nHere we focus on two different types of iron-platinum-nanoparticle s: The compound\nFe50Pt50which has a uniaxial anisotropy [26, 27], whereas the system Fe 70Pt30possesses\na cubic anisotropy [24]. Furthermore, the temperature dependen ce will be studied by\nvaryingq(cf. eq.(8)).\nFor Fe 50Pt50the important parameters for simulations are the diameter of the\nnanoparticles 6 .3nm, the strength of the anisotropy Ku= 6·106J/m3, the magnetic\nmoment per particle µp= 21518 ·µBand the Curie-temperature Tc= 710K[26, 27].Temperature dependent magnetization dynamics of magnetic nanoparticles 5\nThe relation between KuandDuisDu=KuVu, whereVuis the volume of Fe 50Pt50\nnanoparticles. In the calculations for Fe 50Pt50nanoparticles the following qvalues\nwere chosen: q1= 0.001,q2= 0.005 orq3= 0.01 which correspond to the real\ntemperatures 56 K, 280Kor 560K, respectively (these temperatures are below the\nblocking temperature). The corresponding escape times are tq1≈2·10217s,tq2≈1075s\nandtq3≈7·1031s, respectively. In some cases we also show the results for an additio nal\ntemperature q01= 0.0001 with the corresponding real temperature to be equal to 5 K.\nThecorrespondingescapetimeforthisis tq01≈104300s. Thesetimesshouldbecompared\nwith the measurement period which is about tm≈5ns, endorsing thus the stability of\nthe system during the measurements.\nFor Fe 70Pt30the parameters are as follows: The diameter of the nanoparticles 2 .3nm,\nthe strength of the anisotropy Kc= 8·105J/m3, the magnetic moment per particle\nµp= 2000·µB, the Curie-temperature is Tc= 420K[24], and Dc=KcVc(Vcis the\nvolume.) For Fe 70Pt30nanoparticles the values of qwe choose in the simulations are\nq4= 0.01,q5= 0.03 orq6= 0.06 which means that the temperature is respectively\n0.3K, 0.9Kor 1.9K. The escape times are tq4≈1034s,tq5≈2·105sandtq6≈2·10−2s,\nrespectively. Here we also choose an intermediate value q04= 0.001 and the real\ntemperature 0 .03Kwith the corresponding escape time to be equal to tq04≈10430s.\nThe measurement period is the same, namely about 5 ns. All values of the escape times\nwere given for α= 0.1.\nCentral to this study are two issues: The critical magnetic field and the corresponding\nreversal time . The critical magnetic field we define as the minimum field amplitude\nneeded to completely reverse the magnetization. The reversal tim e is the corresponding\ntime for this process. In contrast, in other studies [6] the rever sal time is defined as the\ntime needed for the magnetization to switch from the initial position t o the position\nSz= 0, our reversal time is the time at which the magnetization reaches the very\nproximity of the antiparallel state (Fig. 1). The difference in the defi nition is in so far\nimportant as the magnetization position Sz= 0 at finite temperatures is not stabile so\nit may switch back to the initial state due to thermal fluctuations an d hence the target\nstate is never reached.\n3.1. Nanoparticles having uniaxial anisotropy: Fe 50Pt50\nA Fe50Pt50magnetic nanoparticle has a uniaxial anisotropy whose direction defi nes the\nzdirection. The magnetization direction Sis specified by the azimuthal angle φand\nthe polar angle θwith respect to z. In the presence of an external field bapplied at\nan arbitrarily chosen direction, the energy of the system in dimensio nless units derives\nfrom\n˜H=−dcos2θ−S·b. (11)\nThe initial state of the magnetization is chosen to be close to Sz= +1 and we aim at\nthe target state Sz=−1.Temperature dependent magnetization dynamics of magnetic nanoparticles 6\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization SzT0=0 K T3=560 K\nFigure 1. (Color online) Magnetization reversal of a nanoparticle when a stat ic field\nis applied at zero Kelvin ( q0= 0, black) and at reduced temperature q3= 0.01≡560K\n(blue). The strengths of the fields in the dimensionless units (4) and (8) areb= 1.01\nandb= 0.74, respectively. The damping parameter is α= 0.1. The start position of\nthe magnetization is given by the initial angle θ0=π/360 between the easy axis and\nthe magnetization vector.\n3.1.1. Static field For an external static magnetic field applied antiparallel to the z\ndirection ( b=−bez) eq.(11) becomes\n˜H=−dcos2θ+bcosθ. (12)\nTo determine the critical field magnitude needed for the magnetizat ion reversal we\nproceed as follows (cf. Fig. 1): At first, the external field is increa sed in small steps.\nWhen the magnetization reversal is achieved the corresponding va lues of the critical\nfield versus the damping parameter αare plotted as shown in the inset of Fig. 3. The\nreversal times corresponding to the critical static field amplitudes of Fig. 3 are plotted\nversus damping in Fig. 4.\nIn the Stoner-Wohlfarth (static) model the mechanism of magnet ization reversal is not\ndue to damping. It is rather caused by a change of the energy profi le in the presence of\nthe field. The curves displayed on the energy surface in Fig. 2 mark t he magnetization\nmotion in the E(θ,φ) landscape. The magnetization initiates from φ0= 0 and θ0and\nends up at θ=π. As clearly can be seen from the figure, reversal is only possible if th e\ninitial state is energetically higher than the target state. This ”low d amping” reversal\nis, however, quite slow, which will be quantified more below. For the re versal at T= 0,\nthe SW-model predicts a minimum static field strength, namely bcr=B/BA= 1 (the\ndashed line in Fig. 3 ).\nThis minimum field measured with respect to the anisotropy field stren gth does not\ndepend on the damping parameter α, provided the measuring time is infinite. For T >0\nthe simulations were averaged over 500 cycles with the result shown in Fig. 3. The one-\ncycle data are shown in the inset. Fig. 3 evidences that with increasin g temperature\nthermal fluctuations assist a weak magnetic field as to reverse the magnetization.\nFurthermore, the required critical field is increased slightly at very large and strongly\nat very small damping with the minimum critical field being at α≈1.0. The reason forTemperature dependent magnetization dynamics of magnetic nanoparticles 7\nFigure 2. (Color online) The trajectories of the magnetization unit vector\nparameterized by the angles θandφat zero temperature. Other parameters are as in\nFig. 1 for q0.\n00.511.522.533.544.5 5 5.5 6\nDamping α00.20.40.60.81Critical DC field T0=0 K (SW)\nT01=5 K\nT1=56 K\nT2=280 K\nT3=560 K0 1 2 3 4 5 6\nDamping α00.20.40.60.81Critical DC field T3=560 K\nFigure 3. (Color online) Critical static field amplitudes vs. the damping paramet ers\nfor different temperatures averaged over 500 times. Inset show s not averaged data for\nq3= 0.01≡560K.\nthis behavior is that for low damping the second term of equation (2) is much smaller\nthan the first one, meaning that the system exhibits a weak relaxat ion. In the absence\nof damping, higher fields are necessary to switch the magnetization . For high α, both\nterms in equation (2) become small (compared to a low-damping case ) leading to a\nstiff magnetization and hence higher fields are needed to drive the ma gnetization. For\nmoderate damping, we observe a minimum of switching fields which is due to an optimal\ninterplay between precessional and damping terms. Obviously, finit e temperatures do\nnot influence this general trend.\nFor the case of q0= 0, the Landau-Lifshitz-Gilbert equation of motion can be solved\nanalytically in spherical coordinates. The details of the solution can b e found in Ref.\n[20] (eq. (A1)-(A8)). The final result of the solution in this refere nce differs, however,\nfrom the one given here due to to different geometries in these syst ems. In contrast to\nour alignment of the magnetization and the external field, the stat ic field in Ref. [20] is\napplied parallel to the initial position of the magnetization. For the so lution, we assume\nthat the magnetization starts at θ=θ0=π/360 and arrives at θ=π. Note, that the\nexpression θ/negationslash= 0 is important only for zero Kelvin since the switching is not possible ifTemperature dependent magnetization dynamics of magnetic nanoparticles 8\nthe magnetization starts at θ0= 0 (the vector product in equation (2) vanishes). The\nreversal time in the SW-limit is then given by\ntrev=g(θ0,b)1+α2\nα, (13)\nwheregis defined as\ng(θ0,b) =µS\n2γD1\nb2−1ln/parenleftBiggtg(θ/2)bsinθ\nb−cosθ/parenrightBigg/vextendsingle/vextendsingle/vextendsingleπ\nθ0. (14)\nFromthisrelationweinferthatswitchingispossibleonlyiftheappliedfie ldislargerthan\nthe anisotropy field and the reversal time decreases with increasin gb. This conclusion is\nindependent of the Stoner-Wohlfarth model and follows directly fr om the solution of the\nLLG equation. An illustration is shown by the dashed curve in Fig. 4, wh ich was a test\nto compare the appropriate numerical results with the analytical o ne. As our aim is the\nstudy of the reversal-time dependence on the magnetic moment an d on the anisotropy\nconstant, we deem the logarithmic dependence in Eq.(14) to be weak and write\ng(b,µS,D)≈µS\nγ2D\nB2µ2\nS−4D2. (15)\nThis relation indicates that an increase in the magnetic moment result s in a decrease\nof the reversal time. The magnetic moment enters in the Zeeman en ergy and therefore\nthe increase in magnetic moment is very similar to an increase in the mag netic field.\nAn increase of the reversal time with the increasing anisotropy orig inates from the fact\nthat the anisotropy constant determines the height of the poten tial barrier. Hence, the\nhigher the barrier, the longer it takes for the magnetization to ove rcome it.\nFor the other temperatures the corresponding reversal times ( also averaged over 500\ncycles) are shown in Fig. 4. In contrast to the case T= 0, where an appreciable\ndependence on damping is observed, the reversal times for finite t emperatures show\na weaker dependence on damping. If α→0 only the precessional motion of the\nmagnetization is possible and therefore trev→ ∞. At high damping the system relaxes\non a time scale that is much shorter than the precession time, giving t hus rise to an\nincrease in switching times. Additionally, one can clearly observe the in crease of the\nreversal times with increasing temperatures, even though these time remain on the\nnanoseconds time scale.\n3.1.2. Alternating field As was shown in Ref. [6, 7, 15] theoretically and in Ref. [5]\nexperimentally, a rotating alternating field with no static field being ap plied can also\nbe used for the magnetization reversal. A circular polarized microwa ve field is applied\nperpendicularly to the anisotropy axis. Thus, the Hamiltonian might b e written in form\nof equation (11) and the applied field is\nb(t) =b0cosωtex+b0sinωtey, (16)\nwhereb0is the alternating field amplitude and ωis its frequency. For a switching of\nthe magnetization the appropriate frequency of the applied altern ating field should beTemperature dependent magnetization dynamics of magnetic nanoparticles 9\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]    Theory T0=0 K\nT01=5 K\nT1=56 K\nT2=280 K\nT3=560 K 0 1 2 3 4 5 6\nDamping α012345Reversal time, [ns]    T3=560 K\nFigure 4. (Color online) Reversal times corresponding to the critical static fi elds in\nFig. 3 vs. damping averaged over 500 cycles. Inset shows the as-c alculated numerical\nresults for q3= 0.01≡560K(one cycle).\n00.050.10.150.20.25\nTime, [ns]-1-0.500.51Magnetization SzT0=0 K\n0 1 2 3 4 5\nTime, [ns]-1-0.500.51Magnetization Sz T3=560 K\nFigure 5. (Color online) Magnetization reversal in a nanoparticle using a time\ndependentfieldfor α= 0.1andatazerotemperature. Thefieldstrengthandfrequency\nin the units (4) are respectively b0= 0.18 andω=ωa/1.93. Inset shows for this case\nthe magnetization reversal for the temperature q3= 0.01≡560Kwithb0= 0.17 and\nthe same frequency.\nchosen. In Ref. [15] analytically and in [6] numerically a detailed analysis of the optimal\nfrequency is given which is close to the precessional frequency of t he system. The role\nof temperature and different types of anisotropy have not yet be en addressed, to our\nknowledge.\nFig. 5 shows our calculations for the reversal process at two differ ent temperatures.\nIn contrast to the static case, the reversal proceeds through many oscillations on a\ntime scale of approximately ten picoseconds. Increasing the tempe rature results in an\nincrease of the reversal time.\nFig. 6 shows the trajectory of the magnetization in the E( θ,φ) space related to the\ncase of the alternating field application. Compared with the situation depicted in Fig.\n2, the trajectory reveals a quite delicate motion of the magnetizat ion. It is furthermore,\nnoteworthy that the alternating field amplitudes needed for the re versal (cf. Fig. 7) are\nsubstantially lower than their static counterpart, meaning that th e energy profile of theTemperature dependent magnetization dynamics of magnetic nanoparticles 10\nFigure 6. (Color online) Trajectories followed by magnetization as specified by θand\nφforq0= 0. Other parametersare b0= 0.18,α= 0.1 andω=ωa/1.93. Energy-profile\nvariations due to the oscillating external field are not visible on this sc ale.\nsystem is not completely altered by the external field.\nFig. 7 inspects the dependence of the minimum switching field amplitude on\ndamping. The critical fields are obtained upon averaging over 500 cy cles. The SW-\nlimit lies by 1 on this scale. In contrast to the static case, the critical fields increase\nwith increasing α. In the low damping regime the critical field is smaller than in the\ncase of a static field. This behavior can be explained qualitatively by a r esonant energy-\nabsorptionmechanism when thefrequencies oftheappliedfieldmatc hes thefrequency of\nthe system. Obviously, at very low frequencies (compared to the p recessional frequency)\nthe dynamics resembles the static case.\nThe influence of the temperature on the minimum alternating field amp litudes is\ndepicted in Fig. 7. With increasing temperatures, the minimum amplitud es become\nsmaller due to an additional thermal energy pumped from the enviro nment. The curves\nin this figure can be approached with two linear dependencies with diffe rent slopes for\napproximately α <1 and for α >1; for high damping it is linearly dependent on α,\nmore specifically it can be shown that for high damping the critical field s behave as\nbcr≈1+α2\nα. (17)\nThe proportionality coefficient contains the frequency of the alter nating field and the\ncritical angle θ. The solution (17) follows from the LLG equation solved for the case\nwhen the phase of the external field follows temporally that of the m agnetization, which\nwe checked numerically to be valid.\nThe reversal times associated with the critical switching fields are s hown in (Fig.\n8). Qualitatively, we observe the same behavior as for the case of a static field. The\nvalues of the reversal times for T= 0 are, however, significantly smaller than for the\nstatic case. For the same reason as in the static field case, an incre ased temperature\nresults in an increase of the switching times.Temperature dependent magnetization dynamics of magnetic nanoparticles 11\n00.511.522.533.544.5 5 5.5 6\nDamping α00.511.522.5 Critical AC fieldT0=0 K\nT1=56 K\nT2=280 K\nT3=560 K0 1 2 3 4 5 6\nDamping α00.511.522.5 Critical AC fieldT3=560 K\nFigure 7. (Color online) Critical alternating field amplitudes vs. damping for\ndifferent temperatures averaged over 500 times. Inset shows no t averaged data for\nq3= 0.01≡560K.\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]T1=56 K\nT2=280 K\nT3=560 K\n0 1 2 3 4 5 6\nDamping α00.10.20.30.40.5Reversal time, [ns]T0=0 K\nFigure 8. (Coloronline) The damping dependence ofthe reversaltimes corre sponding\nto the critical field amplitudes of Fig. 7 for different temperatures. Inset shows the\ncase of zero Kelvin.\n3.2. Nanoparticles with cubic anisotropy: Fe 70Pt30\nNow we focus on another type of the anisotropy, namely a cubic anis otropy which is\nsupposed to be present for Fe 70Pt30nanoparticles [24]. The energetics of the system is\nthen described by the functional form\n˜H=−d(S2\nxS2\ny+S2\nyS2\nz+S2\nxS2\nz)−S·b, (18)\nor in spherical coordinates\n˜H=−d(cos2φsin2φsin4θ+cos2θsin2θ)−S·b. (19)\nIn contrast to the previous section, there are more local minima or in other words more\nstable states of the magnetization in the energy profile for the Fe 70Pt30nanoparticles.\nIt can be shown that the minimum barrier that has to be overcome is d/12 which is\ntwelve times smaller than that in the case of a uniaxial anisotropy. Th e maximal one is\nonlyd/3.\nThe magnetization of these nanoparticles is first relaxed to the initia l state close toTemperature dependent magnetization dynamics of magnetic nanoparticles 12\nFigure 9. (Color online) Trajectories of the magnetization in the θ(φ) space (q0= 0).\nIn the units (4) we choose b= 0.82 andα= 0.1.\nφ0=π/4 andθ0= arccos(1 /√\n3), whereas in the target state it is aligned antiparallel to\nthe initial one, i. e. φe= 3π/4 andθe=π−arccos(1/√\n3). In order to be close to the\nstarting state for the uniaxial anisotropy case we choose φ0= 0.2499·π,θ0= 0.3042·π.\n3.2.1. Static driving field A static field is applied antiparallel to the initial state of the\nmagnetization, i.e.\nb=−b/√\n3(ex+ey+ez). (20)\nIn Fig. 9 the trajectory of the magnetization in case of an applied st atic field is shown.\nSimilar to the previous section the energy of the initial state lies highe r than that of\nthe target state. The magnetization rolls down the energy landsca pe to eventually\nend up by the target state. The trajectory the magnetization fo llows is completely\ndifferent from the one for the uniaxial anisotropy. Fig. 10 suppleme nts this scenario\nof the magnetization reversal by showing the time evolution of the Szvector. Because\nof the different anisotropy type, the trajectory is markedly differ ent from the case of\nthe uniaxial anisotropy and a static field. Here we show only the Szmagnetization\ncomponent even though the other components also have to be tak en into account in\norder to avoid a wrong target state.\nThe procedure to determine the critical field amplitudes is similar to th at described in\nthe previous section. In Fig. 11 the critical fields versus the dampin g parameter for\ndifferent temperatures are shown. For q0, the critical field strength is smaller than 1.\nThis is consistent insofar as the maximum effective field for a cubic anis otropy is2\n3BA.\nIn principle, the critical field turns out to be constant for all αbut for an infinitely large\nmeasuring time. Since we set this time to be about 5 nanoseconds, th e critical fields\nincrease for small and high damping. On the other hand, at lower tem peratures smaller\ncritical fields are sufficient for the (thermal activation-assisted) reversal process.\nThe behaviour of the corresponding switching times presented in Fig . 12 only\nsupplements the fact of too low measuring time, which is chosen as 5 nsfor a better\ncomparison of these results with ones for uniaxial anisotropy. Ind eed, constant jumps\nin the reversal times for T= 0Kas a function of damping can be observed. The reasonTemperature dependent magnetization dynamics of magnetic nanoparticles 13\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization Sz\nT0=0 K\nT6=1.9 K\nFigure 10. (Color online) Magnetization reversal of a nanoparticle when a stat ic field\nb= 0.82 is applied and for α= 0.1 at zero temperature (black). The magnetization\nreversal for α= 0.1,b= 0.22 andq6= 0.06≡1.9Kis shown with blue color.\n0 1 2 3 4 56\nDamping α00.20.40.60.811.2 Critical DC field T0=0 K\nT04=0.03 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K0 1 2 3 4 5 6\nDamping α00.51Critical DC field T6=1.9 K\nFigure 11. (Color online) Critical static field amplitudes vs. the damping paramet ers\nfor different temperatures averaged over 500 times. Inset show s not averaged data for\nq6= 0.06≡1.9K.\n0 1 2 3 4 56\nDamping α012345Reversal time, [ns]   T0=0 K\nT04=0.03 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K\nFigure 12. (Color online) Reversal times corresponding to the critical static fi elds of\nFig. 11 vs. damping averaged over 500 times.\nwhy the reversal times for finite temperatures are lower is as follow s: The initial state\nforT= 0Kis chosen to be very close to equilibrium. This does not happen for finit eTemperature dependent magnetization dynamics of magnetic nanoparticles 14\nFigure 13. (Color online) Trajectories of the magnetization vector specified b y the angles θandφat\nzero temperature. The chosen parameters are b0= 0.055 and ω= ˜ωa/1.93, where ˜ ωa= 2/3ωa.\ntemperatures, where the system due to thermal activation jump s out of equilibrium (cf.\nsee Fig. 10).\n3.2.2. Time-dependent external field Here we consider the case of an alternating field\nthat rotates in the plain perpendicularly to the initial state of the ma gnetization. It is\npossible to switch the magnetization with a field rotating in the xy−plane but the field\namplitudes turn out to be larger than those when the field rotates p erpendicular to the\ninitial state. For the energy this means that the field entering equa tion (19) reads\nb(t) = (b0cosω1tcosφ0+b0sinω1tsinφ0cosθ0)ex\n+(−b0cosω1tsinφ0+b0sinω1tcosφ0cosθ0)ey+(−b0sinθ0sinω1t)ez,(21)\nwhereb0is the alternating field amplitude and ω1is the frequency associated with the\nfield. This expression is derived upona rotationof thefield planeby th e anglesφ0=π/4\nandθ0= arccos(1 /√\n3).\nThe magnetization trajectories depicted in Fig. 13 reveal two inter esting features:\nFirstly, particularly for small damping, the energy profile changes v ery slightly (due\nto the smallness of b0) while energy is pumped into the system during many cycles.\nSecondly, thesystemswitchesmostlyinthevicinityoflocalminimatoa cquireeventually\nthe target state. Fig. 14 hints on the complex character of the ma gnetization dynamics\nin this case. As in the static field case with a cubic anisotropy the critic al field\namplitudes shown in Fig. 15 are smaller than those for a uniaxial anisot ropy. Obviously,\nthe reason is that the potential barrier associated with this anisot ropy is smaller in this\ncase, giving rise to smaller amplitudes. As before an increase in tempe rature leads to a\ndecrease in the critical fields.\nThe reversal times shown inFig. 16 exhibit the same feature asin the cases for uniaxial\nanisotropy: With increasing temperatures the corresponding rev ersal times increase. A\nphysically convincing explanation of the (numerically stable) oscillation s for the reversal\ntimes is still outstanding.Temperature dependent magnetization dynamics of magnetic nanoparticles 15\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization Sz\nT0=0 K\nT6=1.9 K\nFigure 14. (Color online) Magnetization reversal in a nanoparticle using a time de pendent field for\nα= 0.1 andq0(black) and for q6= 0.06≡1.9K(blue). Other parameters are as in Fig. 13.\n0 1 2 3 4 5 6\nDamping α00.51Critical AC fieldT6=1.9 K\n00.511.522.533.544.5 5 5.5 6\nDamping α00.511.52Critical AC fieldT0=0 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K\nFigure 15. (Color online) Critical alternating field amplitudes vs. damping for diffe rent temperatures\naveraged over 500 cycles. Inset shows the single cycle data at q6= 0.06≡1.9K.\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]    T4=0.3 K\nT5=0.9 K\nT6=1.9 K\n0 1 2 3 4 5 6\nDamping α012345Reversal time, [ns]T0=0 K\nFigure 16. (Color online) The damping dependence of the reversal times corre sponding to the critical\nfields of the Fig. 15 for different temperatures averaged over 500 runs. Inset shows the T= 0 case.\n4. Summary\nIn this work we studied the critical field amplitudes required for the m agnetization\nswitching of Stoner nanoparticles and derived the corresponding r eversal times forTemperature dependent magnetization dynamics of magnetic nanoparticles 16\nstatic and alternating fields for two different types of anisotropies . The general trends\nfor all examples discussed here can be summarized as follows: Firstly , increasing the\ntemperature results in a decrease of all critical fields regardless o f the anisotropy type.\nAnisotropy effects decline with increasing temperatures making it ea sier to switch the\nmagnetization. Secondly, elevating thetemperature increases th e corresponding reversal\ntimes. Thirdly, thesametrendsareobservedfordifferenttemper atures: Thecriticalfield\namplitudes for a static field depend only slightly on α, whereas the critical alternating\nfield amplitudes exhibit a pronounced dependence on damping. In the case of a uniaxial\nanisotropy we find the critical alternating field amplitudes to be smalle r than those for a\nstatic field, especially in the low damping regime and for finite temperat ures. Compared\nwithastaticfield, alternating fieldsleadtosmaller switching times( T= 0K). However,\nthis is not the case for the cubic anisotropy. The markedly different trajectories for the\ntwo kinds of anisotropies endorse the qualitatively different magnet ization dynamics.\nIn particular, one may see that for a cubic anisotropy and for an alt ernating field\nthe magnetization reversal takes place through the local minima lea ding to smaller\namplitudesoftheappliedfield. Generally, acubicanisotropyissmallert hantheuniaxial\none giving rise to smaller slope of critical fields, i.e. smaller alternating fi eld amplitudes.\nIt is useful to contrast our results with those of Ref. [15]. Our re versal times for\nAC-fields increase with increasing temperatures. This is not in contr adiction with the\nfindings of [15] insofar as we calculate the switching fields at first, an d then deduce the\ncorresponding reversal times. If the switching fields are kept con stant while increasing\nthe temperature [15] the corresponding reversal times decreas e. We note here that\nexperimentally known values of the damping parameter are, to our k nowledge, not\nlarger than 0 .2. The reason why we go beyond this value is twofold. Firstly, the valu es\nof damping are only well known for thin ferromagnetic films and it is not clear how to\nextend them to magnetic nanoparticles. For instance, in FMR exper iments damping\nvalues are obtained from the widths of the corresponding curves o f absorption. The\ncurves for nanoparticles can be broader due to randomly oriented easy anisotropy axes\nand, hence, the values of damping could be larger than they actually are. Secondly, due\ntoaverystrongdependenceofthecriticalAC-fields(Fig. 7, e.g.) t heycanevenbelarger\nthan static field amplitudes. This makes the time-dependent field disa dvantageous for\nswitching in an extreme high damping regime.\nFinally, as can be seen from all simulations, the corresponding rever sal times are much\nmore sensitive a quantity thantheir critical fields. This follows from t he expression (13),\nwhere a slight change in the magnetic field bleads to a sizable difference in the reversal\ntime. This circumstance is the basis for our choice to average all the reversal times and\nfields over many times. This is also desirable in view of an experimental r ealization, for\nexample, in FMR experiments or using a SQUID technique quantities like critical fields\nand their reversal times are averaged over thousands of times. T he results presented\nin this paper are of relevance to the heat-assisted magnetic recor ding, e.g. using a\nlaser source. Our calculations do not specify the source of therma l excitations but\nthey capture the spin dynamics and switching behaviour of the syst em upon thermalTemperature dependent magnetization dynamics of magnetic nanoparticles 17\nexcitations.\nAcknowledgments\nThis work is supported by the International Max-Planck Research School for Science\nand Technology of Nanostructures.\nReferences\n[1]Spindynamics in confined magnetic structures III B. Hillebrands, A. Thiaville (Eds.) (Springer,\nBerlin, 2006); Spin Dynamics in Confined Magnetic Structures II B. Hillebrands, K. Ounadjela\n(Eds.) (Springer, Berlin, 2003); Spin dynamics in confined magnetic structures B. Hillebrands,\nK. Ounadjela (Eds.) (Springer, Berlin, 2001); Magnetic Nanostructures B. Aktas, L. Tagirov, F.\nMikailov (Eds.), (Springer Series in Materials Science, Vol. 94) (Spring er, 2007) and references\ntherein.\n[2] M. Vomir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Phys. Rev. Lett. 94,\n237601 (2005).\n[3] J. Slonczewski, J. Magn. Magn. Mater., 159, L1, (1996).\n[4] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[5] C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2, 524 (2003).\n[6] Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 (2006).\n[7] Z. Z. Sun and X. R. Wang, Phys. Rev. Lett. 97, 077205 (2006).\n[8] L. F. Zhang, C. Xu, Physics Letters A 349, 82-86 (2006).\n[9] C. Xu, P. M. Hui , Y. Q. Ma, et al., Solid State Communications 134, 625-629 (2005).\n[10] T. Moriyama, R. Cao, J. Q. Xiao, et al., Applied Physics Letters 90, 152503 (2007).\n[11] H. K. Lee, Z. M. Yuan, Journal of Applied Physics 101, 033903 (2007).\n[12] H. T. Nembach, P. M. Pimentel, S. J. Hermsdoerfer, et al., Physics Letters 90, 062503 (2007).\n[13] K. Rivkin, J. B. Ketterson, Applied Physics Letters 89, 252507 (2006).\n[14] R. W. Chantrell and K. O’Grady The Magnetic Properties of fine Particles in R. Gerber, C. D.\nWright and G. Asti (Eds.), Applied Magnetism (Kluwer, Academic Pub., Dordrecht, 1994).\n[15] S. I. Denisov, T. V. Lyutyy, P. H¨ anggi, and K. N. Trohidou, Ph ys. Rev. B 74, 104406 (2006).\n[16] S. I. Denisov, T. V. Lyutyy, and P. H¨ anggi, Phys. Rev. Lett. 97, 227202 (2006).\n[17] E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. Londo n, Ser A 240, 599 (1948).\n[18] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935).\n[19] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[20] J. L. Garcia-Palacios and F. Lazaro, Phys. Rev. B 58, 14937 (1998).\n[21]Algorithmen in der Quantentheorie und Statistischen Physi kJ. Schnakenberg (Zimmermann-\nNeufang, 1995).\n[22] U. Nowak, Ann. Rev. Comp. Phys. 9, 105 (2001).\n[23] K. D. Usadel, Phys. Rev. B 73, 212405 (2006).\n[24] C. Antoniak, J. Lindner, and M. Farle, Europhys. Lett. 70, 250 (2005).\n[25] I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990).\n[26] C. Antoniak, J. Lindner, M. Spasova, D. Sudfeld, M. Acet, and M. Farle, Phys. Rev. Lett. 97,\n117201 (2006).\n[27] S. Ostanin, S. S. A. Razee, J. B. Staunton, B. Ginatempo and E . Bruno, J. Appl. Phys. 93, 453\n(2003)."    },    {        "title": "0802.2043v2.Light_induced_magnetization_precession_in_GaMnAs.pdf",        "content": "Light-induced magnetization precession in GaMnAs \n \nE. Rozkotová, P. N ěmeca), P. Horodyská, D. Sprinzl, F. Trojánek, and P. Malý \nFaculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3,  \n121 16 Prague 2, Czech Republic \n \nV. Novák, K. Olejník, M. Cukr, and T. Jungwirth \nInstitute of Physics ASCR  v.v.i., Cukrovarnická 10, 162 53 Prague, Czech Republic \n \n \nWe report dynamics of the transient polar Kerr rotation (KR) and of the transient \nreflectivity induced by femtosecond laser pulses in ferromagnetic (Ga,Mn)As with no \nexternal magnetic field applied. It is s hown that the measured KR signal consist of \nseveral different contributions, among which only the oscillatory signal is directly \nconnected with the ferromagnetic order in (Ga,Mn)As. The origin of the light-induced magnetization precession is discussed and the magnetization precession damping (Gilbert damping) is found to be strongly influenced by annealing of the sample. \n \n \n (Ga,Mn)As is the most intensively studied member of the family of diluted magnetic \nsemiconductors with carrier-mediated ferromagn etism [1]. The sensitiv ity of ferromagnetism \nto concentration of charge carriers opens up th e possibility of magneti zation manipulation on \nthe picosecond time scale using light pulses from ultrafast lasers [2]. Photoexcitation of a \nmagnetic system can strongly disturb the equili brium between the mobile  carriers (holes), \nlocalized spins (Mn ions), and the lattice. This in turn tr iggers a variety of dynamical \nprocesses whose characteristic time scales and st rengths can be investigated by the methods of \ntime-resolved laser spectroscopy [2]. In part icular, the magnetization reversal dynamics in \nvarious magnetic materials attracts a significant attention because it is directly related to the \nspeed of data storage in the magnetic reco rding [3]. The laser-induced precession of \nmagnetization in ferromagnetic (Ga,Mn)As has been recently re ported by two research groups \n[4-6] but the physical processes responsible fo r it are still not well unde rstood. In this paper \nwe report on simultaneous measurements of  the light-induced magnetization precession \ndynamics and of the dynamics of  photoinjected carriers.  \nThe experiments were performed on a 500 nm thick ferromagnetic Ga 1-xMn xAs film \nwith x = 0.06 grown by the low temperature molecular beam epitaxy (LT-MBE) on a \nGaAs(001) substrate. We studied both the as-grown sample, with the Curie temperature TC  ≈ \n60 K and the conductivity of 120 Ω-1cm-1, and the sample annealed at 200°C for 30 hours, \nwith TC  ≈ 90 K and the conductivity of 190 Ω-1cm-1; using the mobility vs. hole density \ndependence typical for GaMnAs [7] we can r oughly estimate their hole densities as 1.5 x 1020 \ncm-3 and 3.4 x 1020 cm-3, respectively. Magnetic properties of the samples were measured \nusing a superconducting quantum interference de vice (SQUID) with magnetic field of 20 Oe \napplied along different crysta llographic directions. The photo induced magnetization dynamics \nwas studied by the time-resolved Kerr rotation  (KR) technique [2] using a femtosecond \ntitanium sapphire laser (Tsunami, Spectra Physics) . Laser pulses, with th e time width of 80 fs \nand the repetition rate of 82 MHz, were tune d to 1.54 eV. The energy  fluence of the pump \npulses was typically 15 μJ.cm-2 and the probe pulses were always at least 10 times weaker. \nThe polarization of the pump pulses was either circular or linear, while the probe pulses were \n                                                 \na) Electronic mail: nemec@karlov.mff.cuni.cz \n 1linearly polarized (typically along the [010] crystallographic direction in the sample, but \nsimilar results were obtained also for other orientations). The rotation angle of the \npolarization plane of the reflected probe pulses was obtained by taking the difference  of \nsignals measured by detectors in an optical br idge detection system [2]. Simultaneously, we \nmeasured also the sum of signals from the detectors, which corresponded to a probe intensity \nchange due to the pump induced modification of  the sample reflectivity. The experiment was \nperformed with no external magnetic field applie d. However, the sample was cooled in some \ncases with no external magnetic field applied or  alternatively with a ma gnetic field of 170 Oe \napplied along the [-110] direction. \n \n \n \nFig. 1. Dynamics of photoinduced Kerr rotation angle (KR) measured for the as-grown sample at 10 K. (a) KR \nmeasured for σ + and σ - circularly polarized (CP) pump  pulses; (b) KR measured for p and s linearly polarized \n(LP) pump pulses. Polarization-independent part (c) (polar ization-dependent part (d)) of KR signal, which was \ncomputed from the measured traces as an average of the signals (a half of  the difference between the signals) \ndetected for pump pulses with the opposite CP (LP). Inset: Fourier transform of the oscillations. No external \nmagnetic field was applied during the sample cooling. \n \n In Fig. 1 we show typical te mporal traces of the transien t angles of KR measured for \nthe as-grown sample at 10 K. The KR signal wa s dependent on the light polarization but there \nwere certain features presen t for both the circular (Fig. 1 (a)) and linear (Fig. 1 (b)) \npolarizations. In Fig. 1 (c) we show the polari zation-independent part of the measured KR \nsignal, which was the same for circular and line ar polarization of pump pulses. On the other \nhand, the amplitude of the polarization-dependent part of the signal (Fig. 1 (d)) was larger for \nthe circular polarization. The interpretation of the polarization-dependent part of the signal is \nsignificantly complicated by the fact that the circularly polarized light generates spin-\npolarized carriers (electrons in particular), whose contribution to the measured KR signal can \neven exceed that of ferromagnetic ally coupled Mn spins [8]. In  the following we concentrate \non the polarization-independent part of the KR signal (Fig. 1 (c)). This signal can be fitted \nwell (see Fig. 2) by an exponentially damped sine harmonic oscillation superimposed on a \npulse-like function: \n \n() ( ) ( ) () [ ]()2 1 / exp / exp 1 sin / exp τ τ ϕωτ t t B t t A t KRD − −−+ + − = .    (1) \n \n 2The oscillatory part of the KR signa l is characterized by the amplitude (A ), damping time ( τD), \nangular frequency ( fπω2= ), and phase ( ϕ). The pulse-like part of the KR signal is \ndescribed by the amplitude ( B), rise time ( τ1), and decay time ( τ2). In the inset of Fig. 2 we \nshow the dynamics of the sample reflectivity change ΔR/R. This signal monitored the change \nof the complex index of refraction of the sa mple due to carriers photoinjected by the pump \npulse. From the dynamics of ΔR/R we can conclude that the population of photogenerated free \ncarriers (electrons in par ticular [9]) decays within ≈ 50 ps after the photoinjection. This rather \nshort lifetime of free electrons is similar to th at reported for the low temperature grown GaAs \n(LT-GaAs), which is generally interpreted as  a consequence of a high concentration of \nnonradiative recombination centers induced by th e low temperature growth mode of the MBE \n[9]. It is also clearly apparent from the inset of Fig. 2 that the KR data can be fitted well by \nEq. (1) only for time delays larger than ≈ 50 ps (i.e., just after th e population of photoinjected \nfree electrons nonradiatively decayed). We will come back to this point later. \n \n \n \nFig. 2. The fitting procedure applied to the polarization-independent part of KR signal. (a) The measured data \nfrom Fig. 1 (c) (points) are fitted (solid line) by a sum of the exponentially damped sine harmonic oscillation \n(solid line in part (b)) and the pulse-like KR signal (dashe d line in part (b)). Inset: Dynamics of the reflectivity \nchange (thick solid line) and the detail of the fitted KR signal. \n \n In Fig. 3(a) we show the intensity dependence of A and B, and in Fig. 3(b) of ω and τD  \nmeasured at 10 K. For the increasing inte nsity of pump pulses  the magnitudes of A and B \nwere increasing, ω was decreasing and the values of τD were not changing significantly. The \napplication of magnetic field applied along the [-110] directi on during the sample cooling \nmodified the value of ω. For 10 K (and pump intensity I0) the frequency decreased from 24.5 \nto 20 GHz (open and solid point in Fig. 3 (d ), respectively). The measured temperature \ndependence of A and B (Fig. 3 (c)) revealed that the oscillatory signal vanished above TC, \nwhile a certain fraction of  the pulse-like KR signal persisted even above TC. This shows that \nonly the oscillatory part of the KR si gnal was directly c onnected with the ferromagnetic  order \nin (Ga,Mn)As. (It is worth noti ng that also the polarization-de pendent part of the KR signal \nwas non-zero even above TC.) The frequency of oscillations  was decreasing with the sample \ntemperature (Fig. 3 (d)), but the values of τD were not changing sign ificantly (not shown \nhere). \n 3 \n \nFig. 3. Intensity dependence of ⎪A⎪and ⎪B⎪ (a), ω and τD (b) measured at 10 K; I0 = 15 μJ.cm-2, no external \nmagnetic field was applied during the sample cooling. (c), (d) Temperature dependence of ⎪A⎪, ⎪B⎪ and ω \n(points) measured at pump intensity I0. The open point in (d) was obtained for the sample cooled with no \nexternal magnetic field applied and the data in (c) and the solid points in (d) were obtained for the sample cooled \nwith magnetic field applied along the [-110]. The lines in (d) are the temperature dependence of the sample \nmagnetization projections to different crystallographic directions measured by SQUID. \n \n The photoinduced magnetization precession was reported by A. Oiwa  et al. , who \nattributed it to the precession of ferromagnetically coupled Mn spins induced by a change in \nmagnetic anisotropy initiated by an increase in hole concentration [4]. It was also shown that \nthe photoinduced magnetization precession a nd the ferromagnetic resonance (FMR) can \nprovide similar information [4]. Magnetic an isotropy in (Ga,Mn)As is influenced by the \nintrinsic cubic anisotropy, which is arising from its zinc-blende symmetry, and by the uniaxial anisotropy, which is a result of a strain induced  by different lattice constants of GaMnAs and \nthe substrate. For the standard stressed GaMnAs  films with  Mn content above 2% grown on \nGaAs substrates the magnetic easy axes are in -plain. Consequently, the measured polar Kerr \nrotation is not sensitive to the steady state magn etization of the sample, but only to the light-\ninduced transient out-of-plane magnetization due to the polar Kerr effect [2]. In our \nexperiment, the pump pulses with a fluence I\n0 = 15 μJ.cm-2 photoinjected electron-hole pairs \nwith an estimated concentration Δp = Δn ≈ 8 x 1017 cm-3. This corresponded to Δp/p ≈ 0.5% \nand such a small increase in the hole concentrati on is highly improbable to lead to any sizable \nchange of the sample anisotropy [1]. Another hypothesis about the origin  of the light-induced \nmagnetization precession was reported recently by J. Qi et al. [6]. The authors suggested that \nnot only the transient increase in local hole concentration Δp but also the local temperature \nincrease ΔT contributes to the change of anisotr opy constants.  This modification of the \nsample anisotropy changes in turn the direction of the in-plane magnetic easy axis and, \nconsequently, triggers a precessional motion of  the magnetization around the altered magnetic \nanisotropy field . The magnitude of decreases as T (the sample temperature) or ΔT \nincreases, primarily due to the decrease in the cubic anisotropy constant KMn\nanisHMn\nanisH\n1c [6]. Our samples \nexhibit in-plane easy axis behavior typica l for stressed GaMnAs layers grown on GaAs \nsubstrates. To characterize their in-plane anis otropy we measured the temperature dependent \nmagnetization projections to [110] , [010], and [-110] crystallographic directions – the results \nare shown in Fig. 3 (d) and in inset of Fig. 4 for the as-grown and the annealed sample, \nrespectively. At low temperatures the cubi c anisotropy dominates (as indicated by the \n 4maximal projection measured along the [010] di rection) but the uniaxial in-plane component \nis not negligible and the sample magnetization is slightly tilted from the [010] direction \ntowards the [-110] direction. Both samples exhibit rotation of magnetizat ion direction in the \ntemperature region 10-25K, which is in agreement with the expected fast weakening of the \ncubic component with an increasing temperatur e. In our experiment, the excitation fluence I0 \nled to ΔT ≈ 10 K (as estimated from the GaAs specifi c heat of 1 mJ/g/K [6]) that can be \nsufficient for a change of the easy axis positi on. This temperature-ba sed hypothesis about the \norigin of magnetization precession is supported also by our observation that the oscillations \nwere not fully developed immediately after the photoinj ection of carriers but only after ≈ 50 \nps when phonons were emitted by the nonradiative  decay of the population of free electrons \n(see inset in Fig. 2). We also point out that the measured precession frequency ω and the \nsample magnetization M (measured by SQUID) had very similar temperature dependence (see \nFig. 3(d)).  \n \n \n \nFig. 4. Polarization-independent part of KR signal measured for the annealed sample at 10 K; I0 = 15 μJ.cm-2, the \nsample was cooled with magnetic field applied along the [-110]. Inset: Temperature dependence of the sample \nmagnetization projections to different crystallographic directions measured by SQUID. \n \n An example of the results measured for the annealed sample is shown in Fig. 4. The \nanalysis of the data revealed that at simila r conditions the precession frequency was slightly \nhigher in the annealed sample (20 GHz and 24 GHz for the as-grown and the annealed \nsample, respectively). However, a major effect of the sample annealing was on the oscillation \ndamping time τD, which increased from 0.4 ns to  1.1 ns. This prolongation of τD can be \nattributed to the improved quality of the ann ealed sample, which is indicated by the higher \nvalue of TC and by the more Brillouin-like temperature dependence of the magnetization (cf. \nFig. 3 (d) and inset in Fig. 4). The damping of  oscillations is connected with the precession \ndamping in the Landau-Lifshitz-Gilbert equation [1]. The exact determination of the intrinsic \nGilbert damping coefficient α from the measured data is not straightforward because it is \ndifficult to decouple the contribution due to the inhomogeneous broadening [10]. In Ref. 6 the \nvalues of α from 0.12 to 0.21 were deduced for the as -grown sample from the analysis of the \noscillatory KR signal. The time-domain KR should provide similar information as the \nfrequency-domain based FMR, where the relaxa tion rate of the magnetization is connected \nwith the peak-to-peak ferromagnetic resonance linewidth ΔHpp [10]. Indeed both methods \nshowed that the relaxation rate of the magnetization is consider ably slower in the annealed \nsamples (as indicated by the prolongation of τD in our experiment and by the reduction of \nΔHpp in FMR [10]).  \n 5 In conclusion, we studied the transient Ke rr rotation (KR) and the reflectivity change \ninduced by laser pulses in (Ga ,Mn)As with no external magnetic field applied. We revealed \nthat the measured KR signals consisted of seve ral different contributions  and we showed that \nonly the oscillatory KR signal was directly connected with the fe rromagnetic order in \n(Ga,Mn)As. Our data indicated that the phonons  emitted by photoinjected  carriers during their \nnonradiative recombination in (Ga,Mn)As can be re sponsible for the magnetic anisotropy \nchange that was triggering the magnetization precession. We also observed that the precession \ndamping was strongly suppressed in the ann ealed sample, which reflected its improved \nmagnetic properties. This work was supported by Ministry of Education of the Czech Republic in the framework of the rese arch centre LC510, the research plans \nMSM0021620834 and AV0Z1010052, by the Grant Agen cy of the Charles University in \nPrague under Grant No. 252445, and by the Grant Agency of Academy of Sciences of the Czech Republic Grants FON/06/E 001, FON/06/E002, and KAN400100652. \n \n \nReferences \n \n[1] T. Jungwirth, J. Sinova, J. Maše k, A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). \n[2] J. Wang, Ch. Sun, Y. Hashimoto, J. Kono, G.A. Khodaparast, L. Cywinski, L.J. Sham, G.D. Sanders, Ch.J. Stanton, H. Munekata, J. Phys.: Condens. Matter \n18, R501 (2006). \n[3] A.V. Kimel, A. Kirilyuk, F.  Hansteen, R.V. Pisarev, T. Ra sing, J. Phys.: Condens. Matter \n19, 043201 (2007). \n[4] A. Oiwa, H. Takechi, H.  Munekata, J. Supercond. 18, 9 (2005). \n[5] H. Takechi, A. Oiwa, K. Nomura, T. Kondo, H. Munekata, phys. stat. sol. (c) 3, 4267 \n(2006). \n[6] J. Qi, Y. Xu, N.H. Tolk, X. Liu, J.K.  Furdyna, I.E. Perakis, Appl. Phys. Lett. 91, 112506 \n(2007). \n[7] T. Jungwirth et al., Phys. Rev. B 76, 125206 (2007). \n[8] A.V. Kimel, G.V. Astakhov,  G.M. Schott, A. Kirilyuk, D. R. Yakovlev, G. Karczewski, \nW. Ossau, G. Schmidt, L.W. Molenkamp, Th. Rasing, Phys. Rev. Lett. 92, 237203 (2004). \n[9] M. Stellmacher, J. Nagle, J.F. Lampin, P. Santoro, J. Van eecloo, A. Alexandrou, J. Appl. \nPhys. 88, 6026 (2000). \n[10] X. Liu, J.K. Furdyna, J. Phys.: Condens. Matter 18, R245 (2006). \n 6"    },    {        "title": "0804.0820v2.Inhomogeneous_Gilbert_damping_from_impurities_and_electron_electron_interactions.pdf",        "content": "arXiv:0804.0820v2  [cond-mat.mes-hall]  9 Aug 2008Inhomogeneous Gilbert damping from impurities and electro n-electron interactions\nE. M. Hankiewicz,1,2,∗G. Vignale,2and Y. Tserkovnyak3\n1Department of Physics, Fordham University, Bronx, New York 10458, USA\n2Department of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA\n3Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: October 30, 2018)\nWe present a unified theory of magnetic damping in itinerant e lectron ferromagnets at order q2\nincluding electron-electron interactions and disorder sc attering. We show that the Gilbert damping\ncoefficient can be expressed in terms of the spin conductivity , leading to a Matthiessen-type formula\nin which disorder and interaction contributions are additi ve. Inaweak ferromagnet regime, electron-\nelectron interactions lead to a strong enhancement of the Gi lbert damping.\nPACS numbers: 76.50.+g,75.45.+j,75.30.Ds\nIntroduction – In spite of much effort, a complete\ntheoretical description of the damping of ferromagnetic\nspin waves in itinerant electron ferromagnets is not yet\navailable.1Recent measurements of the dispersion and\ndamping of spin-wave excitations driven by a direct spin-\npolarized current prove that the theoretical picture is in-\ncomplete, particularly when it comes to calculating the\nlinewidth of these excitations.2One of the most impor-\ntant parameters of the theory is the so-called Gilbert\ndamping parameter α,3which controls the damping rate\nand thermal noise and is often assumed to be indepen-\ndent of the wave vector of the excitations. This assump-\ntion is justified for excitations of very long wavelength\n(e.g., a homogeneous precession of the magnetization),\nwhereαcanoriginateinarelativelyweakspin-orbit(SO)\ninteraction4. But it becomes dubious as the wave vector\nqof the excitations grows. Indeed, both electron-electron\n(e-e) and electron-impurity interactions can cause an in-\nhomogeneous magnetization to decay into spin-flipped\nelectron-hole pairs, giving rise to a q2contribution to the\nGilbert damping. In practice, the presence of this contri-\nbution means that the Landau-Lifshitz-Gilbert equation\ncontains a term proportional to −m×∇2∂tm(wherem\nis the magnetization) and requires neither spin-orbit nor\nmagnetic disorder scattering. By contrast, the homoge-\nneous damping term is of the form m×∂tmand vanishes\nin the absence of SO or magnetic disorder scattering.\nThe influence of disorder on the linewidth of spin\nwaves in itinerant electron ferromagnets was discussed in\nRefs. 5,6,7, and the role of e-e interactions in spin-wave\ndamping was studied in Refs. 8,9 for spin-polarized liq-\nuid He3and in Refs. 10,11fortwo-and three-dimensional\nelectron liquids, respectively. In this paper, we present\na unified semiphenomenological approach, which enables\nus to calculate on equal footing the contributions of dis-\norder and e-e interactions to the Gilbert damping pa-\nrameter to order q2. The main idea is to apply to the\ntransverse spin fluctuations of a ferromagnet the method\nfirst introduced by Mermin12for treating the effect of\ndisorder on the dynamics of charge density fluctuations\nin metals.13Following this approach, we will show that\ntheq2contribution to the damping in itinerant electron\nferromagnets can be expressed in terms of the transversespin conductivity, which in turn separates into a sum of\ndisorder and e-e terms.\nA major technical advantage of this approach is that\nthe ladder vertex corrections to the transverse spin-\nconductivity vanish in the absence of SO interactions,\nmaking the diagrammatic calculation of this quantity a\nstraightforwardtask. Thusweareabletoprovideexplicit\nanalytic expressions for the disorder and interaction con-\ntribution to the q2Gilbert damping to the lowest order\nin the strength of the interactions. Our paper connects\nand unifies different approaches and gives a rather com-\nplete and simple theory of q2damping. In particular, we\nfind that for weak metallic ferromagnets the q2damping\ncan be strongly enhanced by e-e interactions, resulting in\na value comparable to or larger than typical in the case\nof homogeneous damping. Therefore, we believe that the\ninclusionofadampingtermproportionalto q2inthephe-\nnomenologicalLandau-Lifshitzequationofmotionforthe\nmagnetization14is a potentially important modification\nof the theory in strongly inhomogeneous situations, such\nas current-driven nanomagnets2and the ferromagnetic\ndomain-wall motion15.17\nPhenomenological approach – In Ref. 12, Mermin con-\nstructed the density-density response function of an elec-\ntron gas in the presence of impurities through the use\nof a local drift-diffusion equation, whereby the gradient\nof the external potential is cancelled, in equilibrium, by\nan opposite gradient of the local chemical potential. In\ndiagrammatic language, the effect of the local chemical\npotential corresponds to the inclusion of the vertex cor-\nrection in the calculation of the density-density response\nfunction. Here, we use a similar approach to obtain the\ntransverse spin susceptibility of an itinerant electron fer-\nromagnet, modeled as an electron gas whose equilibrium\nmagnetization is along the zaxis.\nBefore proceeding we need to clarify a delicate point.\nThe homogeneous electron gas is not spontaneously fer-\nromagnetic at the densities that are relevant for ordinary\nmagneticsystems.13Inordertoproducethe desired equi-\nlibrium magnetization, we must therefore impose a static\nfictitious field B0. Physically, B0is the “exchange” field\nBexplus any external/applied magnetic field Bapp\n0which\nmaybeadditionallypresent. Therefore,inordertocalcu-2\nlate the transverse spin susceptibility we must take into\naccount the fact that the exchange field associated with\na uniform magnetization is parallel to the magnetization\nand changes direction when the latter does. As a result,\nthe actual susceptibility χab(q,ω) differs from the sus-\nceptibility calculated at constant B0, which we denote\nby ˜χab(q,ω), according to the well-known relation:11\nχ−1\nab(q,ω) = ˜χ−1\nab(q,ω)−ωex\nM0δab. (1)\nHere,M0is the equilibrium magnetization (assumed to\npoint along the zaxis) and ωex=γBex(whereγis the\ngyromagnetic ratio) is the precession frequency associ-\nated with the exchange field. δabis the Kronecker delta.\nThe indices aandbdenote directions ( xory) perpen-\ndicular to the equilibrium magnetization and qandω\nare the wave vector and the frequency of the external\nperturbation. Here we focus solely on the calculation of\nthe response function ˜ χbecause term ωexδab/M0does\nnot contribute to Gilbert damping. We do not include\nthe effects of exchange and external fields on the orbital\nmotion of the electrons.\nThe generalized continuity equation for the Fourier\ncomponent of the transverse spin density Main the di-\nrectiona(xory) at wave vector qand frequency ωis\n−iωMa(q,ω) =−iγq·ja(q,ω)−ω0ǫabMb(q,ω)\n+γM0ǫabBapp\nb(q,ω), (2)\nwhereBapp\na(q,ω)isthetransverseexternalmagneticfield\ndriving the magnetization and ω0is the precessional fre-\nquency associated with a static magnetic field B0(in-\ncluding exchange contribution) in the zdirection. jais\ntheath component of the transverse spin-current density\ntensor and we put /planckover2pi1= 1 throughout. The transverse\nLevi-Civita tensor ǫabhas components ǫxx=ǫyy= 0,\nǫxy=−ǫyx= 1, and the summation over repeated in-\ndices is always implied.\nThe transverse spin current is proportional to the gra-\ndient of the effective magnetic field, which plays the role\nanalogousto the electrochemicalpotential, and the equa-\ntion that expressesthis proportionalityis the analogueof\nthe drift-diffusion equation of the ordinary charge trans-\nport theory:\nja(q,ω) =iqσ⊥/bracketleftbigg\nγBapp\na(q,ω)−Ma(q,ω)\n˜χ⊥/bracketrightbigg\n,(3)\nwhereσ⊥(=σxxorσyy) is the transverse dc (i.e., ω= 0)\nspin-conductivity and ˜ χ⊥=M0/ω0is the static trans-\nverse spin susceptibility in the q→0 limit.18Just as in\nthe ordinary drift-diffusion theory, the first term on the\nright-hand side of Eq. (3) is a “drift current,” and the\nsecond is a “diffusion current,” with the two canceling\nout exactly in the static limit (for q→0), due to the\nrelationMa(0,0) =γ˜χ⊥Bapp\na(0,0). Combining Eqs. (2)\nand (3) gives the following equation for the transversemagnetization dynamics:\n/parenleftbigg\n−iωδab+γσ⊥q2\n˜χ⊥δab+ω0ǫab/parenrightbigg\nMb=\n/parenleftbig\nM0ǫab+γσ⊥q2δab/parenrightbig\nγBapp\nb,(4)\nwhich is most easily solved by transforming to the\ncircularly-polarized components M±=Mx±iMy, in\nwhich the Levi-Civita tensor becomes diagonal, with\neigenvalues ±i. Solving in the “+” channel, we get\nM+=γ˜χ+−Bapp\n+=M0−iγσ⊥q2\nω0−ω−iγσ⊥q2ω0/M0γBapp\n+,\n(5)\nfrom which we obtain to the leading order in ωandq2\n˜χ+−(q,ω)≃M0\nω0/parenleftbigg\n1+ω\nω0/parenrightbigg\n+iωγσ⊥q2\nω2\n0.(6)\nThe higher-orderterms in this expansion cannot be legit-\nimately retained within the accuracy of the present ap-\nproximation. We also disregard the q2correction to the\nstatic susceptibility, since in making the Mermin ansatz\n(3) we are omitting the equilibrium spin currents respon-\nsible for the latter. Eq. (6), however, is perfectly ade-\nquate for our purpose, since it allows us to identify the\nq2contribution to the Gilbert damping:\nα=ω2\n0\nM0lim\nω→0ℑm˜χ+−(q,ω)\nω=γσ⊥q2\nM0.(7)\nTherefore, the Gilbert damping can be calculated from\nthe dc transverse spin conductivity σ⊥, which in turn\ncan be computed from the zero-frequency limit of the\ntransverse spin-current—spin-current response function:\nσ⊥=−1\nm2∗Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtextN\ni=1ˆSiaˆpia;/summationtextN\ni=1ˆSiaˆpia/angb∇acket∇ight/angb∇acket∇ightω\nω,(8)\nwhereˆSiaisthexorycomponentofspinoperatorforthe\nith electron, ˆ piais the corresponding component of the\nmomentum operator, m∗is the effective electron mass, V\nisthe systemvolume, Nisthe totalelectronnumber, and\n/angb∇acketleft/angb∇acketleftˆA;ˆB/angb∇acket∇ight/angb∇acket∇ightωrepresents the retarded linear response func-\ntion for the expectation value of an observable ˆAunder\nthe action of a field that couples linearly to an observable\nˆB. Both disorder and e-e interaction contributions can\nbe systematically included in the calculation of the spin-\ncurrent—spin-current response function. In the absence\nof spin-orbit and e-e interactions, the ladder vertex cor-\nrections to the conductivity are absent and calculation\nofσ⊥reduces to the calculation of a single bubble with\nGreen’s functions\nG↑,↓(p,ω) =1\nω−εp+εF±ω0/2+i/2τ↑,↓,(9)\nwhere the scattering time τsin general depends on the\nspin band index s=↑,↓. In the Born approximation,3\nthe scattering rate is proportional to the electron den-\nsity of states, and we can write τ↑,↓=τν/ν↑,↓, whereνs\nis the spin- sdensity of states and ν= (ν↑+ν↓)/2.τ\nparametrizes the strength of the disorder scattering. A\nstandard calculation then leads to the following result:\nσdis\n⊥=υ2\nF↑+υ2\nF↓\n6(ν−1\n↓+ν−1\n↑)1\nω2\n0τ. (10)\nThis, inserted in Eq. (7), gives a Gilbert damping pa-\nrameter in full agreement with what we have also calcu-\nlated from a direct diagrammatic evaluation of the trans-\nverse spin susceptibility, i.e., spin-density—spin-density\ncorrelation function. From now on, we shall simplify the\nnotation by introducing a transversespin relaxation time\n1\nτdis\n⊥=4(EF↑+EF↓)\n3n(ν−1\n↓+ν−1\n↑)1\nτ, (11)\nwhereEFs=m∗υ2\nFs/2istheFermienergyforspin- selec-\ntrons and nis the total electron density. In this notation,\nthe dc transverse spin-conductivity takes the form\nσdis\n⊥=n\n4m∗ω2\n01\nτdis\n⊥. (12)\nElectron-electron interactions – One of the attractive fea-\ntures of the approach based on Eq. (8) is the ease with\nwhich e-e interactions can be included. In the weak cou-\npling limit, the contributions of disorder and e-e inter-\nactions to the transverse spin conductivity are simply\nadditive. We can see this by using twice the equation of\nmotion for the spin-current—spin-current response func-\ntion. This leads to an expression for the transverse\nspin-conductivity (8) in terms of the low-frequency spin-\nforce—spin-force response function:\nσ⊥=−1\nm2∗ω2\n0Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext\niˆSiaˆFia;/summationtext\niˆSiaˆFia/angb∇acket∇ight/angb∇acket∇ightω\nω.(13)\nHere,ˆFia=˙ˆpiais the time derivative of the momentum\noperator, i.e., the operator of the force on the ith elec-\ntron. The total force is the sum of electron-impurity and\ne-e interaction forces. Each of them, separately, gives a\ncontribution of order |vei|2and|vee|2, whereveiandvee\nare matrix elements of the electron-impurity and e-e in-\nteractions, respectively, while cross terms are of higher\norder, e.g., vee|vei|2. Thus, the two interactions give ad-\nditive contributions to the conductivity. In Ref.16, a phe-\nnomenological equation of motion was used to find the\nspin current in a system with disorder and longitudinal\nspin-Coulomb drag coefficient. We can use a similar ap-\nproach to obtain transversespin currents with transverse\nspin-Coulomb drag coefficient 1 /τee\n⊥. In the circularly-\npolarized basis,\ni(ω∓ω0)j±=−nE\n4m∗+j±\nτdis\n⊥+j±\nτee\n⊥,(14)and correspondingly the spin-conductivities are\nσ±=n\n4m∗1\n−(ω∓ω0)i+1/τdis\n⊥+1/τee\n⊥.(15)\nIn the dc limit, this gives\nσ⊥(0) =σ++σ−\n2=n\n4m∗1/τdis\n⊥+1/τee\n⊥\nω2\n0+/parenleftbig\n1/τdis\n⊥+1/τee\n⊥/parenrightbig2.(16)\nUsing Eq. (16), an identification of the e-e contribution is\npossible in a perturbative regime where 1 /τee\n⊥,1/τdis\n⊥≪\nω0, leading to the following formula:\nσ⊥=n\n4m∗ω2\n0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg\n. (17)\nComparison with Eq. (13) enables us to immediately\nidentify the microscopic expressions for the two scatter-\ning rates. For the disorder contribution, we recover what\nwe already knew, i.e., Eq. (11). For the e-e interaction\ncontribution, we obtain\n1\nτee\n⊥=−4\nnm∗Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext\niˆSiaˆFC\nia;/summationtext\niˆSiaˆFC\nia/angb∇acket∇ight/angb∇acket∇ightω\nω,(18)\nwhereFCis just the Coulomb force, and the force-force\ncorrelation function is evaluated in the absence of disor-\nder. The correlation function in Eq. (18) is proportional\nto the function F+−(ω) which appeared in Ref. 11 [Eqs.\n(18) and (19)] in a direct calculation of the transverse\nspin susceptibility. Making use of the analytic result for\nℑmF+−(ω)presentedinEq. (21)ofthatpaperweobtain\n1\nτee\n⊥= Γ(p)8α0\n27T2r4\nsm∗a2\n∗k2\nB\n(1+p)1/3, (19)\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s49/s48/s45/s54/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s112/s61/s48/s46/s57/s57/s40/s110/s111/s32/s101/s45/s101/s32/s105/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s41\n/s112/s61/s48/s46/s53/s112/s61/s48/s46/s49/s112/s61/s48/s46/s49\n/s32/s32\n/s49/s47 /s32/s91/s49/s47/s110/s115/s93\nFIG. 1: (Color online) The Gilbert damping αas a function\nof the disorder scattering rate 1 /τ. Red (solid) line shows the\nGilbertdampingfor polarization p= 0.1inthepresenceofthe\ne-e and disorder scattering, while dashed line does not incl ude\nthee-escattering. Blue(dotted)andblack(dash-dotted)l ines\nshow Gilbert damping for p= 0.5 andp= 0.99, respectively.\nWe took q= 0.1kF,T= 54K,ω0=EF[(1+p)2/3−(1−p)2/3],\nM0=γpn/2,m∗=me,n= 1.4×1021cm−3,rs= 5,a∗= 2a04\nwhereTis the temperature, p= (n↑−n↑)/nis the degree\nof spin polarization, a∗is the effective Bohr radius, rsis\nthe dimensionless Wigner-Seitz radius, α0= (4/9π)1/3\nand Γ(p) – a dimensionless function of the polarization\np– is defined by Eq. (23) of Ref. 11. This result is valid\nto second order in the Coulomb interaction. Collecting\nour results, we finally obtain a full expression for the q2\nGilbert damping parameter:\nα=γnq2\n4m∗M01/τdis\n⊥+1/τee\n⊥\nω2\n0+/parenleftbig\n1/τdis\n⊥+1/τee\n⊥/parenrightbig2.(20)\nOne of the salient features of Eq. (20) is that it scales\nas the total scattering ratein the weak disorder and\ne-e interactions limit, while it scales as the scattering\ntimein the opposite limit. The approximate formula\nfor the Gilbert damping in the more interesting weak-\nscattering/strong-ferromagnet regime is\nα=γnq2\n4m∗ω2\n0M0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg\n, (21)\nwhile in the opposite limit, i.e. for ω0≪1/τdis\n⊥,1/τee\n⊥:\nα=γnq2\n4m∗M0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg−1\n. (22)\nOur Eq. (20) agrees with the result of Singh and\nTeˇ sanovi´ c6on the spin-wave linewidth as a function of\nthe disorder strength and ω0. However, Eq. (20) also\ndescribes the influence of e-e correlations on the Gilbert\ndamping. A comparison of the scattering rates originat-\ning from disorder and e-e interactions shows that the lat-\nter is important and can be comparable or even greater\nthan the disorder contribution for high-mobility and/or\nlow density 3D metallic samples. Fig. 1 shows the be-\nhavior of the Gilbert damping as a function of the dis-\norder scattering rate. One can see that the e-e scatter-\ning strongly enhances the Gilbert damping for small po-\nlarizations/weak ferromagnets, see the red (solid) line.\nThis stems from the fact that 1 /τdis\n⊥is proportional to\n1/τand independent of polarization for small polar-\nizations, while 1 /τee\n⊥is enhanced by a large prefactorΓ(p) = 2λ/(1−λ2) + (1/2)ln[(1 + λ)/(1−λ)], where\nλ= (1−p)1/3/(1+p)1/3. On the other hand, for strong\npolarizations(dotted anddash-dottedlinesinFig.1), the\ndisorder dominates in a broad range of 1 /τand the inho-\nmogenous contribution to the Gilbert damping is rather\nsmall. Finally, we note that our calculation of the e-e in-\nteractioncontributiontothe Gilbertdampingisvalidun-\nder the assumption of /planckover2pi1ω≪kBT(which is certainly the\ncase ifω= 0). More generally, as follows from Eqs. (21)\nand (22) of Ref. 11, a finite frequency ωcan be included\nthrough the replacement (2 πkBT)2→(2πkBT)2+(/planckover2pi1ω)2\nin Eq. (19). Thus 1 /τee\n⊥is proportional to the scattering\nrateofquasiparticlesnearthe Fermi level, andour damp-\ning constant in the clean limit becomes qualitatively sim-\nilar to the damping parameter obtained by Mineev9for\nωcorresponding to the spin-wave resonance condition in\nsome external magnetic field (which in practice is much\nsmaller than the ferromagnetic exchange splitting ω0).\nSummary – We have presented a unified theory of the\nGilbert damping in itinerant electron ferromagnets at\nthe order q2, including e-e interactions and disorder on\nequal footing. For the inhomogeneous dynamics ( q/negationslash= 0),\nthese processes add to a q= 0 damping contribution\nthat is governed by magnetic disorder and/or spin-orbit\ninteractions. We have shown that the calculation of the\nGilbertdampingcanbe formulatedinthe languageofthe\nspin conductivity, which takes an intuitive Matthiessen\nform with the disorder and interaction contributions be-\ning simply additive. It is still a common practice, e.g., in\nthe micromagnetic calculations of spin-wave dispersions\nand linewidths, to use a Gilbert damping parameter in-\ndependent of q. However, such calculations are often at\nodds with experiments on the quantitative side, particu-\nlarly where the linewidth is concerned.2We suggest that\nthe inclusion of the q2damping (as well as the associ-\nated magnetic noise) may help in reconciling theoretical\ncalculations with experiments.\nAcknowledgements – This work was supported in part\nby NSF Grants Nos. DMR-0313681 and DMR-0705460\nas well as Fordham Research Grant. Y. T. thanks A.\nBrataas and G. E. W. Bauer for useful discussions.\n∗Electronic address: hankiewicz@fordham.edu\n1Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n2I. N. Krivorotov et al., Phys. Rev. B 76, 024418 (2007).\n3T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B75, 174434 (2007).\n5A. Singh, Phys. Rev. B 39, 505 (1989).\n6A. Singh and Z. Tesanovic, Phys. Rev. B 39, 7284 (1989).\n7V.L.SafonovandH.N.Bertram, Phys.Rev.B 61, R14893\n(2000).\n8V. P. Silin, Sov. Phys. JETP 6, 945 (1958).9V. P. Mineev, Phys. Rev. B 69, 144429 (2004).\n10Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev.\nB60, 4856 (1999).\n11Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 56404 (2002).\n12N. D. Mermin, Phys. Rev. B 1, 2362 (1970).\n13G. F. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid (Cambridge University Press, UK, 2005).\n14E.M.Lifshitz andL.P.Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n15Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn.\nMagn. Mater. 320, 1282 (2008), and reference therein.5\n16I. D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000).\n17In ferromagnets whose nonuniformities are beyond the\nlinearized spin waves, there is a nonlinear q2contribu-\ntion to damping, (see J. Foros and A. Brataas and Y.\nTserkovnyak, and G. E. W. Bauer, arXiv:0803.2175) which\nhas a different physical origin, related to the longitudinalspin-current fluctuations.\n18Although both σ⊥and ˜χ⊥are in principle tensors in trans-\nverse spin space, they are proportional to δabin axially-\nsymmetric systems—hence we use scalar notation."    },    {        "title": "0805.0147v1.Chaotic_Spin_Dynamics_of_a_Long_Nanomagnet_Driven_by_a_Current.pdf",        "content": "arXiv:0805.0147v1  [nlin.CD]  1 May 2008Chaotic Spin Dynamics of a Long Nanomagnet Driven by a\nCurrent\nYueheng Lan and Y. Charles Li\nAbstract. We study the spin dynamics of a long nanomagnet driven by an\nelectrical current. In the case of only DC current, the spin d ynamics has a\nsophisticated bifurcation diagram of attractors. One type of attractors is a\nweak chaos. On the other hand, in the case of only AC current, t he spin\ndynamics has a rather simple bifurcation diagram of attract ors. That is, for\nsmall Gilbert damping, when the AC current is below a critica l value, the\nattractor is a limit cycle; above the critical value, the att ractor is chaotic\n(turbulent). For normal Gilbert damping, the attractor is a lways a limit cycle\nin the physically interesting range of the AC current. We als o developed\na Melnikov integral theory for a theoretical prediction on t he occurrence of\nchaos. Our Melnikov prediction seems performing quite well in the DC case.\nIn the AC case, our Melnikov prediction seems predicting tra nsient chaos.\nThe sustained chaotic attractor seems to have extra support from parametric\nresonance leading to a turbulent state.\nContents\n1. Introduction 2\n2. Mathematical Formulation 3\n3. Isospectral Integrable Theory for the Heisenberg Equation 4\n4. A Melnikov Function 14\n5. Numerical Simulation 17\n6. Appendix: The Connection Between the Heisenberg Equation and the\nNLS Equation 23\nReferences 25\n1991Mathematics Subject Classification. Primary 35, 65, 37; Secondary 78.\nKey words and phrases. Magnetization reversal, spin-polarized current, chaos, D arboux\ntransformation, Melnikov function.\nc/circlecopyrt2008 (copyright holder)\n12 YUEHENG LAN AND Y. CHARLES LI\n1. Introduction\nThe greatest potential of the theory of chaos in partial different ial equations\nlies in its abundant applications in science and engineering. The variety of the spe-\ncific problems demands continuing innovation of the theory [ 17] [16] [18] [23] [24]\n[15] [20] [21]. In these representative publications, two theories were develop ed.\nThe theory developed in [ 17] [16] [18] involves transversal homoclinic orbits, and\nshadowing technique is used to prove the existence of chaos. This t heory is very\ncomplete. The theory in [ 23] [24] [15] [20] [21] deals with Silnikov homoclinic\norbits, and geometric construction of Smale horseshoes is employe d. This theory\nis not very complete. The main machineries for locating homoclinic orbit s are (1).\nDarbouxtransformations,(2). Isospectraltheory,(3). Per sistenceofinvariantman-\nifolds and Fenichel fibers, (4). Melnikov analysis and shooting techn ique. Overall,\nthe two theories on chaos in partial differential equations are resu lts of combining\nIntegrable Theory, Dynamical System Theory, and Partial Differe ntial Equations\n[19].\nIn this article, we are interested in the chaotic spin dynamics in a long n ano-\nmagnet diven by an electrical current. We hope that the abundant spin dynamics\nrevealed by this study can generate experimental studies on long n anomagets. To\nillustrate the general significance of the spin dynamics, in particular the magneti-\nzation reversal issue, we use a daily example: The memory of the har d drive of a\ncomputer. The magnetization is polarized along the direction of the e xternal mag-\nnetic field. By reversing the external magnetic field, magnetization reversal can\nbe accomplished; thereby, generating 0 and 1 binary sequence and accomplishing\nmemory purpose. Memory capacity and speed via such a technique h ave reached\ntheir limits. The “bit” writing scheme based on such Oersted-Maxwell magnetic\nfield(generatedbyanelectricalcurrent)encountersfundamen talproblemfromclas-\nsical electromagnetism: the long range magnetic field leads to unwan ted writing or\nerasing of closely packed neighboring magnetic elements in the extre mely high den-\nsity memory device and the induction laws place an upper limit on the mem ory\nspeed due to slow rise-and-decay-time imposed by the law of inductio n. Discovered\nby Slonczewski [ 35] and Berger [ 1], electrical current can directly apply a large\ntorque to a ferromagnet. If electrical current can be directly ap plied to achieve\nmagnetization reversal, such a technique will dramatically increase t he memory\ncapacity and speed of a hard drive. The magnetization can then be s witched\non the scale of nanoseconds and nanometers [ 39]. The industrial value will be\ntremendous. Nanomagnets driven by currents has been intensive ly studied recently\n[13, 10, 34, 7, 14, 12, 33, 8, 39, 11, 32, 37, 38, 4, 3, 26, 27, 29, 4 2, 31]. The\nresearches have gone beyond the original spin valve system [ 35] [1]. For instance,\ncurrent driven torques have been applied to magnetic tunnel junc tions [36] [6], di-\nlute magnetic semiconductors [ 40], multi-magnet couplings [ 10] [14]. AC currents\nwere also applied to generate spin torque [ 34] [7]. Such AC current can be used to\ngenerate the external magnetic field [ 34] or applied directly to generate spin torque\n[7].\nMathematically, the electrical current introduces a spin torque fo rcing term\nin the conventional Landau-Lifshitz-Gilbert (LLG) equation. The A C current can\ninduce novel dynamics of the LLG equation, like synchronization [ 34] [7] and chaos\n[25] [41]. Both synchronization and chaos are important phenomena to und erstand\nbefore implementing the memory technology. In [ 25] [41], we studied the dynamicsCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 3\nof synchronization and chaos for the LLG equation by ignoring the e xchange field\n(i.e. LLG ordinary differential equations). When the nanomagnetic d evice has the\nsame order of length along every direction, exchange field is not impo rtant, and we\nhave a so-called single domain situation where the spin dynamics is gove rned by\nthe LLG ordinary differential equations. In this article, we will study what we call\n“long nanomagnet” which is much longer along one direction than othe r directions.\nIn such a situation, the exchange field will be important. This leads to a LLG\npartial differential equation. In fact, we will study the case where the exchange\nfield plays a dominant role.\nThe article is organized as follows: Section 2 presents the mathemat ical formu-\nlation of the problem. Section 3 is an integrable study on the Heisenbe rg equation.\nBased upon Section 3, Section 4 builds the Melnikov integral theory f or predicting\nchaos. Section 5 presents the numerical simulations. Section 6 is an appendix to\nSection 3.\n2. Mathematical Formulation\nTo simplify the study, we will investigate the case that the magnetiza tion de-\npends on only one spatial variable, and has periodic boundary condit ion in this\nspatial variable. The application of this situation will be a large ring sha pe nano-\nmagnet. Thus, we shall study the following forced Landau-Lifshitz -Gilbert (LLG)\nequation in the dimensionless form,\n(2.1)∂tm=−m×H−ǫαm×(m×H)+ǫ(β1+β2cosω0t)m×(m×ex),\nsubject to the periodic boundary condition\n(2.2) m(t,x+2π) =m(t,x),\nwheremis a unit magnetization vector m= (m1,m2,m3) in which the three\ncomponents are along( x,y,z) directions with unit vectors( ex,ey,ez),|m|(t,x) = 1,\nthe effective magnetic field Hhas several terms\nH=Hexch+Hext+Hdem+Hani\n=∂2\nxm+ǫaex−ǫm3ez+ǫbm1ex, (2.3)\nwhereHexch=∂2\nxmis the exchange field, Hext=ǫaexis the external field,\nHdem=−ǫm3ezisthe demagnetizationfield, and Hani=ǫbm1existhe anisotropy\nfield. For the materials of the experimental interest, the dimension less parameters\nare in the ranges\na≈0.05, b≈0.025, α≈0.02,\nβ1∈[0.01,0.3], β2∈[0.01,0.3] ; (2.4)\nandǫis a small parameter measuring the length scale of the exchange field . One\ncan also add an AC current effect in the external field Hext, but the results on the\ndynamics are similar.\nOur goal is to build a Melnikov function for the LLG equation around do main\nwalls. The roots of such a Melnikov function provide a good indication o f chaos.\nFor the rest of this section, we will introduce a few interesting nota tions. The\nPauli matrices are:\n(2.5)σ1=/parenleftbigg0 1\n1 0/parenrightbigg\n, σ2=/parenleftbigg0−i\ni0/parenrightbigg\n, σ3=/parenleftbigg1 0\n0−1/parenrightbigg\n.4 YUEHENG LAN AND Y. CHARLES LI\nLet\n(2.6) m+=m1+im2, m−=m1−im2,\ni.e.m+=m−. Let\n(2.7) Γ = mjσj=/parenleftbiggm3m−\nm+−m3/parenrightbigg\n.\nThus, Γ2=I(the identity matrix). Let\nˆH=−H−αm×H+βm×ex,Π =/parenleftbiggˆH3ˆH1−iˆH2\nˆH1+iˆH2−ˆH3/parenrightbigg\n.\nThen the LLG can be written in the form\n(2.8) i∂tΓ =1\n2[Γ,Π],\nwhere [Γ,Π] = ΓΠ −ΠΓ.\n3. Isospectral Integrable Theory for the Heisenberg Equati on\nSettingǫtozero,theLLG(2.1)reducestotheHeisenbergferromagneteq uation,\n(3.1) ∂tm=−m×mxx.\nUsing the matrix Γ introduced in (2.7), the Heisenberg equation (3.1) has the form\n(3.2) i∂tΓ =−1\n2[Γ,Γxx],\nwhere the bracket [ ,] is defined in (2.8). Obvious constants of motion of the\nHeisenberg equation (3.1) are the Hamiltonian,\n1\n2/integraldisplay2π\n0|mx|2dx ,\nthe momentum,/integraldisplay2π\n0m1m2x−m2m1x\n1+m3dx ,\nand the total spin,/integraldisplay2π\n0mdx .\nThe Heisenberg equation (3.1) is an integrable system with the followin g Lax pair,\n∂xψ=iλΓψ , (3.3)\n∂tψ=−λ\n2(4iλΓ+[Γ,Γx])ψ , (3.4)\nwhereψ= (ψ1,ψ2)Tis complex-valued, λis a complex parameter, Γ is the matrix\ndefined in (2.7), and [Γ ,Γx] = ΓΓ x−ΓxΓ. In fact, there is a connection be-\ntween the Heisenberg equation (3.1) and the 1D integrable focusing cubic nonlinear\nSchr¨ odinger (NLS) equation via a nontrivial gauge transformatio n. The details of\nthis connection are given in the Appendix.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 5\n3.1. A Simple Linear Stability Calculation. As shown in the Appendix,\nthe temporally periodic solutions of the NLS equation correspond to the domain\nwalls of the Heisenberg equation. Consider the domain wall\nΓ0=/parenleftbigg0e−iξx\neiξx0/parenrightbigg\n, ξ∈Z; i.e.m1= cosξx, m 2= sinξx, m 3= 0,\nwhich is a fixed point of the Heisenberg equation. Linearizing the Heise nberg equa-\ntion at this fixed point, one gets\ni∂tΓ =−1\n2[Γ0,Γxx]−1\n2[Γ,Γ0xx].\nLet\nΓ =/parenleftbigg\nm3 e−iξx(m1−im2)\neiξx(m1+im2) −m3/parenrightbigg\n,\nwe get\n∂tm1= 0, ∂tm2=m3xx+ξ2m3, ∂tm3=−m2xx−2ξm1x.\nLet\nmj=∞/summationdisplay\nk=0(m+\njk(t)coskx+m−\njk(t)sinkx),\nwherem±\njk(t) =c±\njkeΩt,c±\njkand Ω are constants. We obtain that\n(3.5) Ω =/radicalbig\nk2(ξ2−k2)\nwhich shows that only the modes 0 <|k|<|ξ|are unstable. Such instability is\ncalled a modulational instability, also called a side-band instability. Comp aring\nthe Heisenberg ferromagnet equation (3.1) and the Landau-Lifsh itz-Gilbert equa-\ntion (2.1), we see that if we drop the exchange field Hexch=∂2\nxmin the effective\nmagnetic field H(2.3), such a modulational instability will disappear, and the\nLandau-Lifshitz-Gilbert equation (2.1) reduces to a system of thr ee ordinary differ-\nential equations, which has no chaos as verified numerically. Thus th e modulational\ninstability is the source of the chaotic magnetization dynamics.\nIn terms of m±\njk, we have\nd\ndtm±\n1k= 0,d\ndtm±\n2k= (ξ2−k2)m±\n3k,d\ndtm±\n3k=k2m±\n2k∓2ξkm∓\n1k.\nChoosingξ= 2, we have for k= 0,\n\nm∓\n10\nm±\n20\nm±\n30\n=c1\n1\n0\n0\n+c2\n0\n1\n0\n+c3\n0\n4t\n1\n;\nfork= 1,\n\nm∓\n11\nm±\n21\nm±\n31\n=c1\n1\n±4\n0\n+c2\n0√\n3\n1\ne√\n3t+c3\n0\n−√\n3\n1\ne−√\n3t;\nfork= 2,\nm∓\n12\nm±\n22\nm±\n32\n=c1\n0\n0\n1\n+c2\n1\n0\n∓8t\n+c3\n0\n1\n4t\n;6 YUEHENG LAN AND Y. CHARLES LI\nfork>2,\n\nm∓\n1k\nm±\n2k\nm±\n3k\n=c1\n1\n±4/k\n0\n+c2\n0√\nk2−4cos(k√\nk2−4t)\nksin(k√\nk2−4t)\n\n+c3\n0\n−√\nk2−4sin(k√\nk2−4t)\nkcos(k√\nk2−4t)\n;\nwherec1,c2andc3are arbitrary constants.\nThe nonlinear foliation of the above linear modulational instability can b e es-\ntablished via a Darboux transformation.\n3.2. A Darboux Transformation. A Darboux transformation for (3.3)-\n(3.4) can be obtained.\nTheorem 3.1. Letφ= (φ1,φ2)Tbe a solution to the Lax pair (3.3)-(3.4) at ( Γ,ν).\nDefine the matrix\nG=N/parenleftbigg(ν−λ)/ν 0\n0 (¯ν−λ)/¯ν/parenrightbigg\nN−1,\nwhere\nN=/parenleftbigg\nφ1−φ2\nφ2φ1/parenrightbigg\n.\nThen ifψsolves the Lax pair (3.3)-(3.4) at ( Γ,λ),\n(3.6) ˆψ=Gψ\nsolves the Lax pair (3.3)-(3.4) at ( ˆΓ,λ), where ˆΓis given by\n(3.7) ˆΓ =N/parenleftbigg\ne−iθ0\n0eiθ/parenrightbigg\nN−1ΓN/parenleftbigg\neiθ0\n0e−iθ/parenrightbigg\nN−1,\nwhereeiθ=ν/|ν|.\nThe transformation (3.6)-(3.7) is called a Darboux transformation . This theo-\nrem can be proved either through the connection between the Heis enberg equation\nand the NLS equation (with a well-known Darboux transformation) [ 2], or through\na direct calculation.\nNotice also that ˆΓ2=I. Let/parenleftbigg\nΦ1−Φ2\nΦ2Φ1/parenrightbigg\n=N/parenleftbigg\ne−iθ0\n0eiθ/parenrightbigg\nN−1\n=1\n|φ1|2+|φ2|2/parenleftbigge−iθ|φ1|2+eiθ|φ2|2−2isinθ φ1φ2\n−2isinθφ1φ2eiθ|φ1|2+e−iθ|φ2|2/parenrightbigg\n. (3.8)\nThen\n(3.9) ˆΓ =/parenleftbigg\nˆm3 ˆm1−iˆm2\nˆm1+iˆm2−ˆm3/parenrightbigg\n,\nwhere\nˆm+= ˆm1+iˆm2=Φ12(m1+im2)−Φ2\n2(m1−im2)+2Φ1Φ2m3,\nˆm3=/parenleftbig\n|Φ1|2−|Φ2|2/parenrightbig\nm3−Φ1Φ2(m1+im2)−Φ1Φ2(m1−im2).CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 7\nOne can generate the figure eight connecting to the domain wall, as t he nonlinear\nfoliation of the modulational instability, via the above Darboux trans fomation.\n3.3. Figure Eight Connecting to the Domain Wall . LetΓbethedomain\nwall\nΓ =/parenleftbigg\n0e−i2x\nei2x0/parenrightbigg\n,\ni.e.m1= cos2x,m2= sin2x, andm3= 0. Solving the Lax pair (3.3)-(3.4), one\ngets two Bloch eigenfunctions\n(3.10)ψ=eΩt/parenleftbigg\n2λexp{i\n2(k−2)x}\n(k−2)exp{i\n2(k+2)x}/parenrightbigg\n,Ω =−iλk , k =±2/radicalbig\n1+λ2.\nTo apply the Darboux transformation (3.7), we start with the two B loch functions\nwithk=±1,\nφ+=/parenleftbigg√\n3e−ix\nieix/parenrightbigg\nexp/braceleftBigg√\n3\n2t+i1\n2x/bracerightBigg\n,\n(3.11)\nφ−=/parenleftbigg−ie−ix√\n3eix/parenrightbigg\nexp/braceleftBigg\n−√\n3\n2t−i1\n2x/bracerightBigg\n.\nThe wise choice for φused in (3.7) is:\n(3.12) φ=/radicalbigg\nc+\nc−φ++/radicalbigg\nc−\nc+φ−=/parenleftbigg/parenleftbig√\n3eτ+iχ−ie−τ−iχ/parenrightbig\ne−ix\n/parenleftbig\nieτ+iχ+√\n3e−τ−iχ/parenrightbig\neix/parenrightbigg\n,\nwherec+/c−= exp{σ+iγ},τ=1\n2(√\n3t+σ), andχ=1\n2(x+γ). Then from the\nDarboux transformation (3.7), one gets\nˆm1+iˆm2=−ei2x/braceleftbigg\n1−2 sech2τcos2χ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\nsech2τcos2χ\n+i/parenleftBig√\n3−2 sech2τsin2χ/parenrightBig/bracketrightbigg/bracerightbigg\n, (3.13)\nˆm3=2 sech2τtanh2τcos2χ\n(2−√\n3 sech2τsin2χ)2. (3.14)\nAst→ ±∞,\nˆm1→ −cos2x ,ˆm2→ −sin2x ,ˆm3→0.\nThe expressions (3.13)-(3.14) represent the two dimensional figu re eight separatrix\nconnecting to the domain wall ( m+=−ei2x,m3= 0), parametrized by σand\nγ. See Figure 1 for an illustration. Choosing γ= 0,π, one gets the figure eight\ncurve section of Figure 1. The spatial-temporal profiles correspo nding to the two\nlobes of the figure eight curve are shown in Figure 2. In fact, the tw o profiles\ncorresponding the two lobes are spatial translates of each other byπ. Inside one\nof the lobe, the spatial-temporal profile is shown in Figure 3(a). Out side the figure\neight curve, the spatial-temporal profile is shown in Figure 3(b). He re the inside\nand outside spatial-temporal profiles are calculated by using the int egrable finite\ndifference discretization [ 9] of the Heisenberg equation (3.1),\n(3.15)d\ndtm(j) =−2\nh2m(j)×/parenleftbiggm(j+1)\n1+m(j)·m(j+1)+m(j−1)\n1+m(j−1)·m(j)/parenrightbigg\n,8 YUEHENG LAN AND Y. CHARLES LI\nwherem(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the spatial mesh size. For\nthe computation of Figure 3, we choose N= 128.\nγ\nσ\nFigure 1. The separatrix connecting to the domain wall m+=\n−ei2x,m3= 0.\n0246\n0102030−101\nxtm1\n(a)γ= 00246\n0102030−101\nxtm1\n(b)γ=π\nFigure 2. The spatial-temporal profiles corresponding to the two\nlobes of the figure eight curve.\nBy a translation x→x+θ, one can generate a circle of domain walls:\nm+=−ei2(x+θ), m3= 0,\nwhereθis the phase parameter. The three dimensional figure eight separa trix\nconnecting to the circle of domain walls, parametrized by σ,γandθ; is illustrated\nin Figure 4.\nIn general, the unimodal equilibrium manifold can be sought as follows: Let\nmj=cjcos2x+sjsin2x , j= 1,2,3,\nthen the uni-length condition |m|(x) = 1 leads to\n|c|= 1,|s|= 1, c·s= 0,\nwherecandsare the two vectors with components cjandsj. Thus the unimodal\nequilibrium manifold is three dimensional and can be represented as in F igure 5.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 9\n0246\n10203040−101\nxtm1\n(a) inside0123456\n10152025303540−101\nxtm1\n(b) outside\nFigure 3. The spatial-temporal profiles corresponding to the in-\nside and outside of the figure eight curve.\nγ\nσ\nθ\nFigure 4. The separatrix connecting to the circle of domain walls\nm+=ei2(x+θ),m3= 0.\nUsing the formulae (3.13)-(3.14), we want to build a Melnikov integral. The\nzeros of the Melnikov integral will give a prediction on the existence o f chaos. To\nbuild such a Melnikov integral, we need to first develop a Melnikov vecto r. This\nrequires Floquet theory of (3.3).\n3.4. Floquet Theory. Focusing on the spatial part (3.3) of the Lax pair\n(3.3)-(3.4), let Y(x) be the fundamental matrix solution of (3.3), Y(0) =I(2×2\nidentity matrix), then the Floquet discriminant is defined by\n∆ = trace Y(2π).\nThe Floquet spectrum is given by\nσ={λ∈C| −2≤∆(λ)≤2}.10 YUEHENG LAN AND Y. CHARLES LI\ns c\nFigure 5. A representation of the 3 dimensional unimodal equi-\nlibrium manifold.\nPeriodicandanti-periodicpoints λ±(whichcorrespondtoperiodicandanti-periodic\neigenfunctions respectively) are defined by\n∆(λ±) =±2.\nA critical point λ(c)is defined by\nd∆\ndλ(λ(c)) = 0.\nA multiple point λ(n)is a periodic or anti-periodic point which is also a critical\npoint. The algebraic multiplicity of λ(n)is defined as the order of the zero of\n∆(λ)±2 atλ(n). When the order is 2, we call the multiple point a double point,\nand denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(n)\nis defined as the dimension of the periodic or anti-periodic eigenspace atλ(n), and\nis either 1 or 2.\nCounting lemmas for λ±andλ(c)can be established as in [ 30] [23], which lead\nto the existence of the sequences {λ±\nj}and{λ(c)\nj}and their approximate locations.\nNevertheless, counting lemmas are not necessary here. For any λ∈C, ∆(λ) is a\nconstantofmotionofthe Heisenbergequation(3.1). Thisistheso- calledisospectral\ntheory.\nExample 3.2. For the domain wall m1= cos2x,m2= sin2x, andm3= 0; the\ntwo Bloch eigenfunctions are given in (3.10). The Floquet discriminant is given by\n∆ = 2cos/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.\nThe periodic points are given by\nλ=±/radicalbigg\nn2\n4−1, n∈Z, nis even.\nThe anti-periodic points are given by\nλ=±/radicalbigg\nn2\n4−1, n∈Z, nis odd.\nThe choice of φ+andφ−correspond to n=±1 andλ=ν=i√\n3/2 withk=±1.\n∆′=−4πλ√\n1+λ2sin/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.\n∆′′=−4π(1+λ2)−3/2sin/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n−8π2λ2\n1+λ2cos/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 11\nλ\nFigure 6. The periodic and anti-periodic points corresponding to\nthe potential of domain wall m+=ei2x,m3= 0. The open circles\nare double points, the solid circle at the origin is a multiple point\nof order 4, and the two bars intersect the imaginary axis at two\nperiodic points which are not critical points.\nWhenn= 0, i.e.√\n1+λ2= 0, by L’Hospital’s rule\n∆′→ −8π2λ , λ=±i .\nThat is,λ=±iare periodic points, not critical points. When n=±1, we have two\nimaginary double points\nλ=±i√\n3/2.\nWhenn=±2,λ= 0 is a multiple point of order 4. The rest periodic and anti-\nperiodicpoints areallrealdouble points. Figure 6is anillustrationoft hese spectral\npoints.\n3.5. Melnikov Vectors. Starting from the Floquet theory, one can build\nMelnikov vectors.\nDefinition 3.3. An importantsequenceofinvariants Fjofthe Heisenbergequation\nis defined by\nFj(m) = ∆(λ(c)\nj(m),m).\nLemma 3.4. If{λ(c)\nj}is a simple critical point of ∆, then\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\nProof. We know that\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj+∂∆\n∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj∂λ(c)\nj\n∂m.\nSince\n∂∆\n∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj= 0,\nwe have\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj∂λ(c)\nj\n∂m+∂2∆\n∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj= 0.12 YUEHENG LAN AND Y. CHARLES LI\nSinceλ(c)\njis a simple critical point of ∆,\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj/ne}ationslash= 0.\nThus\n∂λ(c)\nj\n∂m=−/bracketleftBigg\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj/bracketrightBigg−1\n∂2∆\n∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\nNotice that ∆ is an entire function of λandm[23], then we know that∂λ(c)\nj\n∂mis\nbounded, and\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\n/square\nTheorem 3.5. As a function of two variables, ∆ = ∆(λ,m)has the partial deriva-\ntives given by Bloch functions ψ±(i.e.ψ±(x) =e±Λx˜ψ±(x), where˜ψ±are periodic\ninxof period 2π, andΛis a complex constant):\n∂∆\n∂m+=−iλ√\n∆2−4\nW(ψ+,ψ−)ψ+\n1ψ−\n1,\n∂∆\n∂m−=iλ√\n∆2−4\nW(ψ+,ψ−)ψ+\n2ψ−\n2,\n∂∆\n∂m3=iλ√\n∆2−4\nW(ψ+,ψ−)/parenleftbig\nψ+\n1ψ−\n2+ψ+\n2ψ−\n1/parenrightbig\n,\n∂∆\n∂λ=i√\n∆2−4\nW(ψ+,ψ−)/integraldisplay2π\n0/bracketleftbig\nm3/parenleftbig\nψ+\n1ψ−\n2+ψ+\n2ψ−\n1/parenrightbig\n−m+ψ+\n1ψ−\n1+m−ψ+\n2ψ−\n2/bracketrightbig\ndx ,\nwhereW(ψ+,ψ−) =ψ+\n1ψ−\n2−ψ+\n2ψ−\n1is the Wronskian.\nProof. Recall that Yis the fundamental matrix solution of (3.3), we have the\nequation for the differential of Y\n∂xdY=iλΓdY+i(dλΓ+λdΓ)Y , dY (0) = 0.\nUsing the method of variation of parameters, we let\ndY=YQ , Q (0) = 0.\nThus\nQ(x) =i/integraldisplayx\n0Y−1(dλΓ+λdΓ)Ydx ,\nand\ndY(x) =iY/integraldisplayx\n0Y−1(dλΓ+λdΓ)Ydx .\nFinally\nd∆ = trace dY(2π)\n=itrace/braceleftbigg\nY(2π)/integraldisplay2π\n0Y−1(dλΓ+λdΓ)Ydx/bracerightbigg\n. (3.16)\nLet\nZ= (ψ+ψ−)CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 13\nwhereψ±are two linearly independent Bloch functions (For the case that the re is\nonly one linearly independent Bloch function, L’Hospital’s rule has to be used, for\ndetails, see [ 23]), such that\nψ±=e±Λx˜ψ±,\nwhere˜ψ±are periodic in xof period 2πand Λ is a complex constant (The existence\nof such functions is the result of the well known Floquet theorem). Then\nZ(x) =Y(x)Z(0), Y(x) =Z(x)[Z(0)]−1.\nNotice that\nZ(2π) =Z(0)E ,whereE=/parenleftbigg\neΛ2π0\n0e−Λ2π/parenrightbigg\n.\nThen\nY(2π) =Z(0)E[Z(0)]−1.\nThus\n∆ = trace Y(2π) = traceE=eΛ2π+e−Λ2π,\nand\ne±Λ2π=1\n2[∆±/radicalbig\n∆2−4].\nIn terms of Z,d∆ as given in (3.16) takes the form\nd∆ =itrace/braceleftbigg\nZ(0)E[Z(0)]−1/integraldisplay2π\n0Z(0)[Z(x)]−1(dλΓ+λdΓ)Z(x)[Z(0)]−1dx/bracerightbigg\n=itrace/braceleftbigg\nE/integraldisplay2π\n0[Z(x)]−1(dλΓ+λdΓ)Z(x)dx/bracerightbigg\n,\nfrom which one obtains the partial derivatives of ∆ as stated in the t heorem. /square\nIt turns out that the partial derivatives of Fjprovide the perfect Melnikov\nvectors rather than those of the Hamiltonian or other invariants [ 23], in the sense\nthatFjis the invariant whose level sets are the separatrices.\n3.6. An Explicit Expression of the Melnikov Vector Along the Figure\nEight Connecting to the Domain Wall. We continue the calculation in sub-\nsection 3.3 to obtain an explicit expression of the Melnikov vector alon g the figure\neight connecting to the domain wall. Apply the Darboux transformat ion (3.6) to\nφ±(3.11) atλ=ν, we obtain\nˆφ±=±¯ν−ν\n¯νexp{∓1\n2σ∓i1\n2γ}W(φ+,φ−)\n|φ1|2+|φ2|2\nφ2\n−φ1\n.\nIn the formula (3.6), for general λ,\ndetG=(ν−λ)(¯ν−λ)\n|ν|2, W(ˆψ+,ˆψ−) = detG W(ψ+,ψ−).\nIn a neighborhood of λ=ν,\n∆2−4 = ∆(ν)∆′′(ν)(λ−ν)2+ higher order terms in ( λ−ν).\nAsλ→ν, by L’Hospital’s rule\n√\n∆2−4\nW(ˆψ+,ˆψ−)→/radicalbig\n∆(ν)∆′′(ν)\nν−¯ν\n|ν|2W(φ+,φ−).14 YUEHENG LAN AND Y. CHARLES LI\nNotice, by the calculation in Example 3.2, that\nν=i√\n3\n2,∆(ν) =−2,∆′′(ν) =−24π2,\nthen by Theorem 3.5,\n∂∆\n∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3πi\n(|φ1|2+|φ2|2)2φ22,\n∂∆\n∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3π−i\n(|φ1|2+|φ2|2)2φ12,\n∂∆\n∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3π2i\n(|φ1|2+|φ2|2)2φ1φ2,\nwhere ˆmis given in (3.13)-(3.14). With the explicit expression (3.12) of φ, we\nobtain the explicit expressions of the Melnikov vector,\n∂∆\n∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n2isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n(1−2tanh2τ)cos2χ\n+i(2−tanh2τ)sin2χ−i√\n3 sech2τ/bracketrightbigg\ne−i2x, (3.17)\n∂∆\n∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n2−isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n(1+2tanh2 τ)cos2χ\n−i(2+tanh2τ)sin2χ+i√\n3 sech2τ/bracketrightbigg\nei2x, (3.18)\n∂∆\n∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n22isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n2 sech2τ−√\n3sin2χ\n−i√\n3tanh2τcos2χ/bracketrightbigg\n, (3.19)\nwhere again\nm±=m1±im2, τ=√\n3\n2t+σ\n2, χ=1\n2(x+γ),\nandσandγare two real parameters.\n4. A Melnikov Function\nThe forced Landau-Lifshitz-Gilbert (LLG) equation (2.1) can be re written in\nthe form,\n(4.1) ∂tm=−m×mxx+ǫf+ǫ2g\nwherefis the perturbation\nf=−am×ex+m3(m×ez)−bm1(m×ex)\n−αm×(m×mxx)+(β1+β2cosω0t)m×(m×ex),\ng=−αm×[m×(aex−m3ez+bm1ex)].\nThe Melnikov function for the forced LLG (2.1) is given as\nM=/integraldisplay∞\n−∞/integraldisplay2π\n0/bracketleftbigg∂∆\n∂m+(f1+if2)+∂∆\n∂m−(f1−if2)+∂∆\n∂m3f3/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆmdxdt ,CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 15\nwhere ˆmis given in (3.13)-(3.14), and∂∆\n∂w(w=m+,m−,m3) are given in (3.17)-\n(3.19). The Melnikov function depends on several external and int ernal parameters\nM=M(a,b,α,β 1,β2,ω0,σ,γ) whereσandγare internal parameters. We can split\nfas follows:\nf=af(a)+f(0)+bf(b)+αf(α)+β1f(β1)\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)+sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)/bracketrightbigg\n,\nwhere\nf(a)=−m×ex,\nf(0)=m3(m×ez),\nf(b)=−m1(m×ex),\nf(α)=−m×(m×mxx),\nf(β1)=m×(m×ex),\nf(c)= cos/parenleftbigg2√\n3ω0τ/parenrightbigg\nm×(m×ex),\nf(s)= sin/parenleftbigg2√\n3ω0τ/parenrightbigg\nm×(m×ex).\nThusMcan be splitted as\nM=aM(a)+M(0)+bM(b)+αM(α)+β1M(β1)\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nM(c)+sin/parenleftbiggσω0√\n3/parenrightbigg\nM(s)/bracketrightbigg\n, (4.2)\nwhereM(ζ)=M(ζ)(γ),ζ=a,0,b,α,β 1, andM(ζ)=M(ζ)(γ,ω0),ζ=c,s.\nIn general [ 19], the zeros of the Melnikov function indicate the intersection of\ncertain center-unstable and center-stable manifolds. In fact, t he Melnikov function\nis the leading order term of the distance between the center-unst able and center-\nstable manifolds. In some cases, such an intersection can lead to ho moclinic orbits\nand homoclinic chaos. Here in the current problem, we do not have an invariant\nmanifold result. Therefore, our calculation on the Melnikov function is purely from\na physics, rather than rigorous mathematics, point of view.\nIn terms ofthe variables m+andm3, the forced Landau-Lifshitz-Gilbert (LLG)\nequation (2.1) can be rewritten in the form that will be more convenie nt for the\ncalculation of the Melnikov function,\n∂tm+=i(m+m3xx−m3m+xx)+ǫf++ǫ2g+, (4.3)\n∂tm3=1\n2i(m+m+xx−m+m+xx)+ǫf3+ǫ2g3, (4.4)16 YUEHENG LAN AND Y. CHARLES LI\nwhere\nf+=f1+if2=af(a)\n++f(0)\n++bf(b)\n++αf(α)\n++β1f(β1)\n+\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)\n++sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)\n+/bracketrightbigg\n,\nf3=af(a)\n3+f(0)\n3+bf(b)\n3+αf(α)\n3+β1f(β1)\n3\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)\n3+sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)\n3/bracketrightbigg\n,\ng+=g1+ig2=αag(a)\n++αg(0)\n++αbg(b)\n+,\ng3=αag(a)\n3+αg(0)\n3+αbg(b)\n3,\nf(a)\n+=−im3,\nf(0)\n+=−im3m+,\nf(b)\n+=−i1\n2m3(m++m+),\nf(α)\n+=1\n2m+(m+m+xx−m+m+xx)+m3(m3m+xx−m+m3xx),\nf(β1)\n+=1\n2m+(m+−m+)−m2\n3,\nf(c)\n+= cos/parenleftbigg2√\n3ω0τ/parenrightbigg/bracketleftbigg1\n2m+(m+−m+)−m2\n3/bracketrightbigg\n,\nf(s)\n+= sin/parenleftbigg2√\n3ω0τ/parenrightbigg/bracketleftbigg1\n2m+(m+−m+)−m2\n3/bracketrightbigg\n,\nf(a)\n3=1\n2i(m+−m+),\nf(0)\n3= 0,\nf(b)\n3=1\n4i(m2\n+−m+2),\nf(α)\n3=m3xx|m+|2−1\n2m3(m+m+xx+m+m+xx),\nf(β1)\n3=1\n2m3(m++m+),\nf(c)\n3=1\n2cos/parenleftbigg2√\n3ω0τ/parenrightbigg\nm3(m++m+),\nf(s)\n3=1\n2sin/parenleftbigg2√\n3ω0τ/parenrightbigg\nm3(m++m+),CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 17\ng(a)\n+=m2\n3−1\n2m+(m+−m+),\ng(0)\n+=m2\n3m+,\ng(b)\n+=1\n2m2\n3(m++m+)−1\n4m+(m2\n+−m+2),\ng(a)\n3=−1\n2m3(m++m+),\ng(0)\n3=−m3|m+|2,\ng(b)\n3=−1\n4m3(m++m+)2.\nDirect calculation gives that\nM(a)(γ) =M(0)(γ) =M(b)(γ) = 0, M(α)(γ) = 91.3343,\nandM(β1)andM(c)are real, while M(s)is imaginary. The graph of M(β1)is\nshown in Figure 7(a) (Notice that M(β1)is independent of ω0). The graph of M(c)\nis shown in Figure 7(b). The imaginary part of M(s)is shown in Figure 7(c). In\nthe case of only DC current ( β2= 0),M= 0 (4.2) leads to\n(4.5) α=−β1M(β1)/91.3343,\nwhereM(β1)(γ) is a function of the internal parameter γas shown in Figure 7(a).\nIn the general case ( β2/ne}ationslash= 0),M(s)(γ,ω0) = 0 determines curves\n(4.6) γ=γ(ω0) = 0, π/2, π,3π/2,\nandM= 0 (4.2) leads to\n(4.7) |β2|>/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig\n91.3343α+β1M(β1)/parenrightBig/slashbigg\nM(c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhereM(β1)andM(c)are evaluated along the curve (4.6), M(β1)=±43.858 (‘+’\nforγ=π/2,3π/2; ‘−’ forγ= 0, π),M(c)is plotted in Figure 8 (upper curve\ncorresponds to γ=π/2,3π/2; lower curve corresponds to γ= 0, π), and\ncos/parenleftbiggσω0√\n3/parenrightbigg\n=−91.3343α+β1M(β1)\nβ2M(c).\n5. Numerical Simulation\nIn the entire article, we use the finite difference method to numerica lly simulate\nthe LLG (2.1). Due to an integrable discretization [ 9] of the Heisenberg equation\n(3.1), the finite difference performs much better than Galerkin Fou rier mode trun-\ncations. As in (3.15), let m(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the\nspatial mesh size. Without further notice, we always choose N= 128 (which pro-\nvides enough precision). The only tricky part in the finite difference d iscretization\nof (2.1) is the second derivative term in H, for the rest terms, just evaluate mat\nm(j):\n∂2\nxm(j) =2\nh2/parenleftbiggm(j+1)\n1+m(j)·m(j+1)+m(j−1)\n1+m(j−1)·m(j)/parenrightbigg\n.18 YUEHENG LAN AND Y. CHARLES LI\n0246\n024−50050\nγω0Mβ1\n(a)0246\n024−50050\nγω0M(c)\n(b)\n0246\n024−50050\nγω0M(s)\n(c)\nFigure 7. (a). The graph of M(β1)as a function of γ, andM(β1)\nis independent of ω0. (b). The graph of M(c)as a function of γ\nandω0. (c). The graph of the imaginarypart of M(s)as a function\nofγandω0.\n5.1. Only DC Current Case. In this case, β2= 0 in (2.1), and we choose\nβ1as the bifurcation parameter, and the rest parameters as:\n(5.1) a= 0.05,b= 0.025,α= 0.02,ǫ= 0.01.\nThe computation is first run for the time interval [0 ,8120π], then the figures are\nplotted starting from t= 8120π. The bifurcation diagram for the attractors, and\nthe typical spatial profiles on the attractors are shown in Figure 9 . This figure\nindiactes that interesting bifurcations happen over the interval β1∈[0,0.15] which\nis the physically important regime where β1is comparable with values of other\nparameters. There are six bifurcation thresholds c1···c6(Fig. 9). When β1<c1,\nthe attractor is the spatially uniform fixed point m1= 1 (m2=m3= 0). WhenCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 19\n00.511.522.533.54−50−40−30−20−1001020304050\nω0M(c)\nFigure 8. The graphs of M(c)along the curves (4.6).\nβ1c2c3c1c c c4 5 6I II III IV V VI VII\nFigure 9. The bifurcation diagram for the attractors and typ-\nical spatial profiles on the attractors in the case of only DC\ncurrent where β1is the bifurcation parameter, c1···c6are the\nbifurcation thresholds, and c1∈[0,0.01],c2∈[0.0205,0.021],\nc3∈[0.0231,0.0232],c4∈[0.025,0.026],c5∈[0.08,0.1],c6∈\n[0.13,0.15].\nc1≤β1≤c2, the attractoris a spatially non-uniform fixed point as shown in Figur e\n10(a). When c2< β1< c3, the attractor is spatially non-uniform and temporally\nperiodic (a limit cycle) or quasiperiodic (a limit torus) as shown in Figure 1 0(b).\nHere as the value of β1is increased, first there is one basic temporal frequence,\nthen more frequencies enter and the temporal oscillation amplitude becomes bigger\nand bigger. When c3≤β1< c4, the attractor is chaotic, i.e. a strange attractor\nas shown in Figure 10(c). Even though the chaotic nature is not ver y apparent\nin Figure 10(c), due to the smallness of the perturbation paramete r together with\nsmallness of all other parameters, the temporal evolution is chaot ic, and we have\nused Liapunov exponent and power spectrum devices to verify this . Whenc4≤\nβ1< c5, the attractor is spatially non-uniform and temporally periodic (a limit\ncycle) as shown in Figure 11(a). The spatial modulation is small, and it is even not\napparentin Figure11(a). But itisapparentonthe individualtypical spatialprofiles\nas seen in region V in Figure 9. With the increase of β1, the spatial modulation\nbecomes smaller and smaller. When c5≤β1<c6, the attractor is spatially uniform\nand temporally periodic (a limit cycle, the so-called procession) as sho wn in Figure20 YUEHENG LAN AND Y. CHARLES LI\n0123456\n0510152025300.60.81\nxtm1\n(a)β1= 0.0205 spatially non-uniform fixed\npoint0123456\n05101520253000.51\nxtm1\n(b)β1= 0.023 spatially non-uniform and\ntemporally periodic or quasiperiodic attrac-\ntor\n0123456\n051015202530−101\nxtm1\n(c)β1= 0.0235 weak chaotic attractor\nFigure 10. The spatio-temporal profiles of solutions in the at-\ntractors in the case of only DC current.\n11(b). When β1≥c6, the attractor is the spatially uniform fixed point m1=−1\n(m2=m3= 0).\nWhenβ2= 0, the Melnikov function predicts that around β1= 0.041 (4.5),\nthere is probably chaos; while the numerical calculation shows that t here is an\ninterval [0.0231,0.026] forβ1where chaos is the attractor. Since the perturbation\nparameterǫ= 0.01 in the numerical calculation, the Melnikov function prediction\nseems in agreement with the numerical calculation.\nSome of the attractors in Figure 9 are attractors of the corresp onding ordinary\ndifferential equations by setting ∂x= 0 in (2.1), i.e. the single domain case. These\nattractors are the ones in regions I, VI and VII in Figure 9 [ 25] [41]. Because weCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 21\n0123456\n0100200300400500600−101\nxtm1\n(a)β1= 0.026 spatially non-uniform and\ntemporally periodic attractor0123456\n0100200300400500600−0.500.5\nxtm1\n(b)β1= 0.1 spatially uniform and tempo-\nrally periodic attractor\nFigure 11. The spatio-temporal profiles of solutions in the at-\ntractors in the case of only DC current (continued).\nare studying the only DC current case, the ordinary differential eq uations do not\nhave any chaotic attractor [ 25] [41].\nOn the other hand, the partial differential equations (2.1) does ha ve a chaotic\nattractor (region IV in Figure 9).\nIn general, when β1<0, the Gilbert damping dominates the spin torque driven\nby DC current and m1= 1 is the attractor. When β1>0.15, the spin torque driven\nby DC current dominates the Gilbert damping, m1=−1 is the attractor, and we\nhave a magnetization reversal. In some technological applications, β1>0.15 may\ncorrespond too high DC current that can burn the device. On the o ther hand, in\nthe technologically advantageousinterval β1∈[0,0.15], magnetization reversalmay\nbe hard to achieve due to the sophisticated bifurcations in Figure 9.\n5.2. Only AC Current Case. In this case, β1= 0 in (2.1), and we choose\nβ2as the bifurcation parameter, and the rest parameters as:\n(5.2) a= 0.05,b= 0.025,α= 0.0015,ǫ= 0.01,ω0= 0.2.\nUnlike the DC case, here the figures are plotted starting from t= 0. It turns\nout that the types of attractors in the AC case are simpler than th ose of the\nDC case. When β2= 0, the attractor is a spatially non-uniform fixed point as\nshown in Figure 12. In this case, the only perturbation is the Gilbert d amping\nwhich damps the evolution to such a fixed point. When 0 < β2< β∗\n2where\nβ∗\n2∈[0.18,0.19], the attractor is a spatially non-uniform and temporally periodic\nsolution. When β2≥β∗\n2, the attractor is chaotic as shown in Figure 12. Our\nMelnikov prediction (4.7) predicts that when |β2|>0.003, certain center-unstable\nandcenter-stablemanifolds intersect. Ournumericsshowsthat s uch anintersection\nseems leading to transient chaos. Only when |β2|>β∗\n2, the chaos can be sustained\nas an attractor. It seems that such sustained chaotic attracto r gains extra support\nfrom parametric resonance due to the AC current driving [ 22], as can be seen from22 YUEHENG LAN AND Y. CHARLES LI\n0123456\n050001000015000−101\nxtm1\n(a)β2= 0 spatially non-uniform fixed point020004000600080001000012000−1−0.95−0.9−0.85−0.8−0.75−0.7−0.65−0.6−0.55−0.5\ntm1(x1)\n(b)β2= 0 temporal evolution at one spatial\nlocation\n0123456\n050001000015000−101\nxtm1\n(c)β2= 0.21 chaotic attractor020004000600080001000012000−1−0.8−0.6−0.4−0.200.20.40.60.81\ntm1(x1)\n(d)β2= 0.21 temporal evolution at one spa-\ntial location\nFigure 12. The attractors in the case of only AC current.\nthe turbulent spatial structure of the chaotic attractor (Fig. 1 2), which diverges\nquite far away from the initial condition. Another factor that may b e relevant\nis the fact that higher-frequency spatially oscillating domain walls hav e more and\nstronger linearly unstable modes (3.5). By properly choosing initial c onditions, one\ncan find the homotopy deformation from the ODE limit cycle (process ion) [25] [41]\nto the current PDE chaos as shown in Figure 13 at the same paramet er values.\nWe also simulated the case of normal Gilbert damping α= 0.02. For all values\nofβ2∈[0.01,0.3], the attractor is always non-chaotic. That is, the only attracto r\nwe can find is a spatially uniform limit cycle with small temporal oscillation a s\nshown in Figure 14.\nOf course, when neither β1norβ2is zero, the bifurcation diagram is a combi-\nnation of the DC only and AC only diagrams.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 23\nFigure 13. Homotopy deformation of the attractors under differ-\nent initial conditions. See http://www.math.missouri.edu/˜cli\nFigure 14. The attractor when α= 0.02,β2= 0.21 and\nall other parameters’ values are the same with Figure 12. See\nhttp://www.math.missouri.edu/˜cli\n6. Appendix: The Connection Between the Heisenberg Equatio n and\nthe NLS Equation\nIn this appendix, we will show the details on the connection between t he 1D\ncubic focusing nonlinear Schr¨ odinger (NLS) equation and the Heise nberg equation\n(3.1). The nonlinear Schr¨ odinger (NLS) equation\n(6.1) iqt+qxx+2|q|2q= 0,\nis a well-known integrable system with the Lax pair\n∂xφ= (iλσ3+U)φ , (6.2)\n∂tφ=−(2iλ2σ3+2λU+V)φ , (6.3)24 YUEHENG LAN AND Y. CHARLES LI\nwhereλis the complex spectral parameter, σ3is defined in (2.5), and\nU=/parenleftbigg0iq\ni¯q0/parenrightbigg\n, V=−i|q|2σ3+/parenleftbigg0qx\n−¯qx0/parenrightbigg\n.\nLemma 6.1. Ifφ= (φ1,φ2)Tsolves the Lax pair (6.2)-(6.3) at λ, then(−φ2,φ1)T\nsolves the Lax pair (6.2)-(6.3) at ¯λ. Whenqis even, i.e. q(−x) =q(x), then\n(φ2(−x),φ1(−x))Tsolves the Lax pair (6.2)-(6.3) at −¯λ. Whenλis real, and φis\na nonzero solution, then (−φ2,φ1)Tis another linearly independent solution. For\nany two solutions of the Lax pair, their Wronskian is indepen dent ofxandt.\nFor any real λ0, by the well-known Floquet theorem [ 28] and Lemma 6.1, there\nare always two linearly independent Floquet (or Bloch) eigenfunction sφ±to the\nLax pair (6.2)-(6.3) at λ=λ0, such that\nφ+=/parenleftbiggϕ1\nϕ2/parenrightbigg\n, φ−=/parenleftbigg−ϕ2\nϕ1/parenrightbigg\n,\nφ+(x+2π) =ρφ+(x), φ−(x+2π) = ¯ρφ−(x),|ρ|2= 1.\nSincetheWronskian W(φ+,φ−)isindependentof xandt, withoutlossofgenerality,\nwe chooseW(φ+,φ−) = 1. Then\nS=/parenleftbigg\nϕ1−ϕ2\nϕ2ϕ1/parenrightbigg\nis a unitary solution to the Lax pair at λ=λ0:\nS−1=SH=/parenleftbiggϕ1ϕ2\n−ϕ2ϕ1/parenrightbigg\n,|ϕ1|2+|ϕ2|2= 1.\nRecall the definition of Γ (2.7), let\nΓ =S−1σ3S=/parenleftbigg\n|ϕ1|2−|ϕ2|2−2ϕ1ϕ2\n−2ϕ1ϕ2|ϕ2|2−|ϕ1|2/parenrightbigg\n,\ni.e.\nm1+im2=−2ϕ1ϕ2, m3=|ϕ1|2−|ϕ2|2.\nNow for any φsolving the Lax pair (6.2)-(6.3) at λ, defineψas\nψ=S−1φ .\nThenψsolves the pair\nψx=i(λ−λ0)Γψ , (6.4)\nψt=−/braceleftbigg\n2i(λ2−λ2\n0)Γ+1\n2(λ−λ0)[Γ,Γx]/bracerightbigg\nψ . (6.5)\nThe compatibility condition of this pair leads to the equation\n(6.6) Γ t=−/braceleftbigg\n4λ0Γx+1\n2i[Γ,Γxx]/bracerightbigg\n.\nSettingλ0= 0 or performing the translation t=t,ˆx=x−4λ0t, equation (6.6)\nreduces to the Heisenberg equation (3.2). Therefore, the Gauge transformStrans-\nforms NLS equation into the Heisenberg equation. Periodicity in xmay not persist.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 25\nExample 6.2. Consider the temporally periodic solution of the NLS equation\n(6.1),\nq=aeiθ(t), θ(t) = 2a2t+γ .\nThe corresponding Bloch eigenfunction of the Lax pair (6.2)-(6.3) a tλ= 0 is\nϕ=1√\n2/parenleftbigg\neiθ/2\ne−iθ/2/parenrightbigg\neiax.\nThen\nΓ =S−1σ3S=/parenleftbigg\n0 −e−i2ax\n−ei2ax0/parenrightbigg\n,\nwhich is called a domain wall.\nAcknowledgment : The second author Y. Charles Li is grateful to Professor\nShufeng Zhang, Drs. Zhanjie Li and ZhaoyangYang, and Mr. Jiexu an He for many\nhelpful discussions.\nReferences\n[1] L. Berger. Emission of Spin Waves by a Magnetic Multilaye r.Phys. Rev. B , 54:9353–9358,\n1996.\n[2] A. Calini. A Note on a B¨ acklund Transformation for the Co ntinuous Heisenberg Model. Phys.\nLett. A, 203: 333–344, 1995.\n[3] M. Covington et al. Current-Induced Magnetization Dyna mics in Current Perpendicular to\nthe Plane Spin Valves. Phy. Rev. B , 69:184406, 2004.\n[4] A. Fabian et al. 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Chaos, Solitons\nand Fractals , 20, no.4:791–798, 2004.\n[19] Y. Li. Chaos in Partial Differential Equations . International Press, 2004.26 YUEHENG LAN AND Y. CHARLES LI\n[20] Y. Li. Persistent Homoclinic Orbits for Nonlinear Schr ¨ odinger Equation Under Singular Per-\nturbation. Dynamics of PDE , vol.1, no.1, 87-123, 2004.\n[21] Y. Li. Existence of Chaos for Nonlinear Schr¨ odinger Eq uation Under Singular Perturbation.\nDynamics of PDE , vol.1, no.2, 225-237, 2004.\n[22] Y. Li. Chaos and Shadowing Around a Heteroclinically Tu bular Cycle With an Application\nto Sine-Gordon Equation. Studies in Applied Mathematics , vol.116, 145-171, 2006, pp.165.\n[23] Y. Li and D. W. McLaughlin. Morse and Melnikov Functions for NLS Pdes. Comm. Math.\nPhys., 162:175–214, 1994.\n[24] Y. Li et al. Persistent Homoclinic Orbits for a Perturbe d Nonlinear Schr¨ odinger equation.\nComm. Pure Appl. Math. , XLIX:1175–1255, 1996.\n[25] Z. Li, Y. Li, S. Zhang. Synchronization, Modification an d Chaos Induced by Spin-polarized\nCurrents. Phys. Rev. B. , 74:054417, 2006.\n[26] Z. Li and S. Zhang. Magnetization Dynamics with a Spin-T ransfer Torque. Phys. Rev. B ,\n68:024404, 2003.\n[27] Z. Li and S. Zhang. Thermally Assisted Magnetization Re versal in the Presence of a Spin-\nTransfer Torque. Phys. Rev. B , 69:134416, 2004.\n[28] W. Magnus, S. Winkler. Hill’s Equation . Dover, NY, 1979.\n[29] J. Miltat et al. Spin Transfer into an Inhomogeneous Mag netization Distribution. J. Appl.\nPhys., 89:6982, 2001.\n[30] B. Mityagin. Spectral Expansions of One-Dimensional P eriodic Dirac Operators. Dynamics\nof PDE, 1, no.2:125, 2004.\n[31] E. B. Myers et al. Current-Driven Switching of Domains i n Magnetic Multilayer Devices.\nScience, 285:867–870, 1999.\n[32] B. Ozyilmaz et al. Current-Induced Magnetization Reve rsal in High Magnetic Fields in\nCo/Cu/Co Nanopillars. Phys. Rev. Lett. , 93:067203, 2003.\n[33] M. R. Pufall et al. Materials Dependence of the Spin-Mom entum Transfer Efficiency and\nCritical Current in Ferromagnetic Metal/Cu Multilayers. Appl. Phys. Lett. , 83:323, 2003.\n[34] A. Slavin, et al. Nonlinear Self-Phase-Locking Effect i n an Array of Current-Driven Magnetic\nNanocontact. Phys. Rev. B , 72: 092407, 2005.\n[35] J.C.Slonczewski.Current-DrivenExcitation ofMagne tic Multilayers. J. Magn. Magn. Mater ,\n159:L1–L7, 1996.\n[36] J. Z. Sun. Current-Driven Magnetic Switching in Mangan ite TrilayerJunctions. J. Mag. Mag.\nMater., 202:157, 1999.\n[37] S. Urazhdin et al. Current-Driven Magnetic Excitation s in Permalloy-Based Multilayer\nNanopillars. Phys. Rev. Lett. , 91:146803, 2003.\n[38] J. E. Wegrowe. Magnetization Reversal and Two-Level Fl uctuations by Spin Injection in a\nFerromagnetic Metallic Layer. Phy. Rev. B , 68:2144xx, 2003.\n[39] S. A. Wolf et al. Spintronics: a Spin-Based Electronics Vision for the Future. Science,\n294:1488, 2001.\n[40] M. Yamanouchi et al. Current-Induced Domain-Wall Swit ching in a Ferromagnetic Semicon-\nductor Structure. Nature, 428:539, 2004.\n[41] Z. Yang, S. Zhang, Y. Li. Chaotic Dynamics of Spin Valve O scillators. Phys. Rev. Lett. , 99:\n134101, 2007.\n[42] J. G. Zhu et al. Spin Transfer Induced Noise in CPP Read He ads.IEEE Trans. on Magn. ,\n40:182, 2004.\nDepartment of Mechanical Engineering, University of Califo rnia, Santa Barbara,\nCA 93106\nE-mail address :yueheng lan@yahoo.com\nDepartment of Mathematics, University of Missouri, Columbi a, MO 65211\nE-mail address :cli@math.missouri.edu"    },    {        "title": "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": "0805.3306v1.Non_equilibrium_thermodynamic_study_of_magnetization_dynamics_in_the_presence_of_spin_transfer_torque.pdf",        "content": "arXiv:0805.3306v1  [cond-mat.mes-hall]  21 May 2008Non-equilibrium thermodynamic study of magnetization dyn amics in the presence of\nspin-transfer torque\nKazuhiko Seki and Hiroshi Imamura\nNanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\nThe dynamics of magnetization in the presence of spin-trans fer torque was studied. We derived\nthe equation for the motion of magnetization in the presence of a spin current by using the local\nequilibrium assumption in non-equilibrium thermodynamic s. We show that, in the resultant equa-\ntion, the ratio of the Gilbert damping constant, α, and the coefficient, β, of the current-induced\ntorque, called non-adiabatic torque, depends on the relaxa tion time of the fluctuating field τc. The\nequality α=βholds when τcis very short compared to the time scale of magnetization dyn amics.\nWe apply our theory to current-induced magnetization rever sal in magnetic multilayers and show\nthat the switching time is a decreasing function of τc.\nSpin-transfer torque-induced magnetization dynamics\nsuch as current-induced magnetization reversal [1, 2, 3],\ndomain wall motion [4], and microwave generation [5]\nhave attracted a great deal of attention because of their\npotential applications to future nano-spinelectronic de-\nvices. In the absence of spin-transfer torque, magnetiza-\ntion dynamics is described by either the Landau-Lifshitz\n(LL) equation [6] or the Landau-Lifshitz-Gilbert (LLG)\nequation [7]. It is known that the LL and LLG equations\nbecome equivalent through rescaling of the gyromagnetic\nratio.\nHowever, this is not the case in the presence of spin-\ntransfer torque. For domain wall dynamics, the following\nLLG-type equation has been studied by several groups\n[8, 9, 10]:\n∂t/angbracketleftM/angbracketright+v·∇/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+α\nM/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+β\nM/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright],(1)\nwhereMrepresents the magnetization, vis the velocity,\nγis the gyromagnetic ratio and αis the Gilbert damping\nconstant. The second term on the left-hand side repre-\nsents the adiabatic contribution of spin-transfer torque.\nThe first and the second terms on the right-hand side are\nthe torque due to the effective magnetic field Hand the\nGilbert damping. The last term on the right-hand side\nof Eq. (1) represents the current-induced torque, called\n“non-adiabatic torque” or simply the βterm. The direc-\ntions of the adiabaticcontribution of spin-transfertorque\nand non-adiabatic torque are shown in Fig. 1 (a).\nAs shownbyThiaville et al., the value ofthe coefficient\nβstrongly influences the motion of the domain wall [8].\nHowever, the value of the coefficient βis still controver-\nsial, and different conclusions have been drawn from dif-\nferent approaches[9, 10, 11, 12, 13, 14, 15]. For example,\nBarnes and Maekawa showed that the value of βshould\nbeequaltothatoftheGilbertdampingconstant αtosat-\nisfy the requirement that the relaxation should cease at\nthe minimum of electrostatic energy, even under particle\nflow. Kohno et al.performed microscopic calculationsFIG. 1: (a) The direction of the magnetization M, the adia-\nbatic contribution of spin-transfer torque, ( v·∇)M, and the\nβterm,M×[(v·∇)M], are shown. The direction of the ve-\nlocityvis indicated by the dotted arrow. (b) The magnetic\nmultilayers, in which the pinned and the free layers are sepa -\nrated by a nonmagnetic spacer layer are schematically shown .\nThe magnetization vectors of the pinned and free layers are\nrepresented by S1andSs, respectively. The effective mag-\nnetic field to which S2is subject is represented by H. (c)\nThe direction of the magnetization of the free layer, S2, the\nspin-transfer torque ( S2×S1)×S2, and the non-adiabatic\ntorque,S2×S1, are shown. The direction of S1is indicated\nby the dotted arrow.\nof spin torques in disordered ferromagnets and showed\nthat theαandβterms arise from the spin relaxation\nprocesses and that α/negationslash=βin general [10]. Tserkovnyak et\nal.[11] derived the βterm using a quasiparticle approx-\nimation and showed that α=βwithin a self-consistent\npicture based on the local density approximation.\nIn the current-induced magnetization dynamics in the\nmagnetic multilayers shown in Fig. 1 (b) [16, 17, 18], the\nnon-adiabatic torque exerts a strong effect, and therefore\naffectsthedirect-currentvoltageofthe spintorquediode,\nas shown in Refs. [17, 18]. The magnetization dynam-\nics of the free layer, S2, has been studied by using the\nfollowing LLG-type equation,\n∂tS2−I\neg/planckover2pi1(S2×S1)×S2=γH×S2+α\nS2S2×∂tS2\n+ηIS2×S1, (2)\nwhereIis the charge current density, gis the amplitude\nof the spin torque introduced by Slonczewski [1], /planckover2pi1is2\nthe Dirac constant and ηrepresents the magnitude of\nthe “non-adiabatictorque” which is sometimes called the\nfield-like torque [17, 18].\nIn this paper, we study the magnetization dynamics\ninduced by spin-transfer torque in the framework of non-\nequilibrium thermodynamics. We derive the equation of\nmotion of the magnetization in the presence of a spin\ncurrentby usingthe local equilibrium assumption. In the\nresultant equation, the Gilbert damping term and the β\nterm are expressed as memory terms with the relaxation\ntime of the fluctuating field τc. We show that the value\nof the coefficient βis not equal to that of the Gilbert\ndamping constant αin general. However, we also show\nthat the equality α=βholds ifτc≪1/(γH). We apply\nour theory to the current-induced magnetization reversal\nin magnetic multilayersand showthat the switching time\nis a decreasing function of τc.\nLet us first briefly introduce the non-equilibrium sta-\ntistical theory of magnetization dynamics in the absence\nof spin current [19]. The LLG equation describing the\nmotion of magnetization Munder an effective magnetic\nfieldHis given by\n∂tM=γH×M+α\nMM×∂tM.(3)\nThe equivalent LL equation is expressed as\n∂tM=γ\n1+α2H×M−αγ\nM(1+α2)M×(M×H).(4)\nThe Langevin equations leading to Eqs. (3) and (4) by\ntaking the ensemble average of magnetization m, are\n∂tm=γHtot×m (5)\n∂tδH=−1\nτc(δH−χsm)+R(t), (6)\nwhere the total magnetic field Htotis the sum of the\neffective magnetic field Hand the fluctuating magnetic\nfieldδHandχsis the susceptibility ofthe local magnetic\nfieldinducedatthepositionofthespin. AccordingtoEq.\n(6) the fluctuating magnetic field δHrelaxes toward the\nreaction field χsmwith the relaxation time τc. The ran-\ndom field R(t) satisfies /angbracketleftR(t)/angbracketright= 0 and the fluctuation-\ndissipation relation, /angbracketleftRi(t)Rj(t′)/angbracketright=2\nτcχskBTδi,jδ(t−t′),\nwherekBis the Boltzmann constant, Tis the tempera-\nture,/angbracketleft···/angbracketrightdenotestheensembleaverage,and i,j= 1,2,3\nrepresents the Cartesian components. It was shown that\nEqs. (5) and (6) lead to Kawabata’s extended Landau-\nLifshitz equation [20] derived by the projection operator\nmethod [19]. In the Markovian limit, i.e.,τc≪1/(γH),\nwe can obtain the LLG equation (3) and the correspond-\ning LL equation (4) with α=γτcχsM[19].\nIn order to consider the flow of spins, i.e., spin cur-\nrent, we introduce the positional dependence. Since we\nare interested in the average motion, it is convenient to\nintroducethemeanvelocityofthecarrier, v. Theaveragemagnetization, /angbracketleftm(x,t)/angbracketright, is obtained by introducing the\npositional dependence and taking the ensemble average\nof Eq. (5). In terms of the mean velocity, the ensemble\naverage of the left-hand side of Eq. (5) leads to\n∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright. (7)\nAssuming /angbracketleftδH×m/angbracketright ≈ /angbracketleftδH/angbracketright×/angbracketleftm/angbracketright, which is applicable\nwhen the thermal fluctuation is small compared to the\nmean value, we obtain\n∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright=γ/angbracketleftHtot(x,t)/angbracketright×/angbracketleftm(x,t)/angbracketright.(8)\nThe mean magnetization density is expressed as\n/angbracketleftM(x,t)/angbracketright=ρ(x,t)/angbracketleftm(x,t)/angbracketright,i.e., by the product of the\nscalar and vectorial components both of which depend\non the position of the spin carrier at time t. The spin\ncarrier density satisfies the continuity equation,\n∂tρ(x,t)+∇·(vρ(x,t)) = 0. (9)\nBy multiplying the left-hand side of Eq. (8) by ρ(x,t)\nand using the continuity equation (9), the closed expres-\nsion for the mean magnetization is obtained as [21]\nρ(∂t/angbracketleftm/angbracketright+v·∇/angbracketleftm/angbracketright) =∂tρ/angbracketleftm/angbracketright+/angbracketleftm/angbracketright∇·vρ+ρv·∇/angbracketleftm/angbracketright\n=∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright, (10)\nwhere Div v/angbracketleftM/angbracketrightis defined by\nDivv/angbracketleftM/angbracketright=3/summationdisplay\ni=1∂vi/angbracketleftM/angbracketright\n∂xi=/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright.(11)\nBy multiplying the right-hand side of Eq. (8) by ρ(x,t)\nand using Eq. (10), we obtain\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γ(H+/angbracketleftδH/angbracketright)×/angbracketleftM/angbracketright.(12)\nEquation (12) takes the standard form of a time-\nevolution equation for extensive thermodynamical vari-\nables under flow [21]. The average of Eq. (6) with the\npositional dependence is given by\n∂t/angbracketleftδH(x,t)/angbracketright=−1\nτc[/angbracketleftδH(x,t)/angbracketright−χ/angbracketleftM(x(t),t)/angbracketright],(13)\nwherex(t) is the mean position at time tof the spin car-\nrier, which flows with velocity v=∂tx(t) andχ=χs/ρ\nis assumed to be a constant independent of the position.\nEquations (12) and (13) constitute the basis for the sub-\nsequent study of magnetization dynamics in the presence\nof spin-transfer torque.\nThe formal solution of Eq. (13) is expressed as\n/angbracketleftδH(x,t)/angbracketright=χ\nτc/integraldisplayt\n−∞ψ(t−t′)/angbracketleftM(x(t′),t′)/angbracketrightdt′,(14)\nwhere the memory kernel is given by ψ(t) = exp[−t/τc].\nUsing partial integration, we obtain\n/angbracketleftδH(x,t)/angbracketright=χ/angbracketleftM/angbracketright−/integraldisplayt\n−∞ψ(t−t′)χ/angbracketleft˙M(t′)/angbracketrightdt′,(15)3\nwhere the explicit expression for ˙M(t) =˙M(x(t),t) is\ngiven by the convective derivative,\n˙M(t) =∂tM(x(t),t)+(v·∇)M(x(t),t).(16)\nSubstituting Eq. (15) into Eq. (12), we obtain the equa-\ntion of motion for the mean magnetization density,\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+γ/integraldisplayt\n−∞dt′ψ(t−t′)χ/angbracketleftM(t)/angbracketright×/angbracketleft˙M(t′)/angbracketright.(17)\nEquation (17) supplemented by Eq. (16) is the principal\nresultof this paper.\nWhen the relaxation time of the fluctuating field, τc, is\nvery short compared to the time scale of the magnetiza-\ntion dynamics, the memory kernel is decoupled and Eq.\n(17) can be written in the form of an LLG-type equation\nas\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright+α\nM/angbracketleftM/angbracketright×˙/angbracketleftM/angbracketright,(18)\nwhereα=γτcχMis the Gilbert damping constant. Sub-\nstituting the explicit form of the convective derivative,\nEq. (16), into Eq. (18) and using Eq.(11) we obtain the\nfollowing LLG-type equation:\n∂t/angbracketleftM/angbracketright+/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+α\nM/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+α\nM/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright].(19)\nIf∇·v= 0, Eq. (19) reduces to Eq. (14) of Ref. [9],\nwhich is derived by replacing the time derivative of mag-\nnetization∂tMon both sides of the LLG equation (3)\nby the convective derivative ∂tM+v·∇·M. The term\n/angbracketleftM/angbracketright(∇·v) appears not on the right-hand side ofEq. (19)\nbut on the left-hand side, which means we cannot obtain\nEq. (19) using the same procedure used in Ref. [9]. As\nshown in Refs. [9, 22], Eq. (19) with /angbracketleftM/angbracketright(∇·v) = 0\nleads to a steady-state solution in the comoving frame,\n/angbracketleftM(t)/angbracketright=/angbracketleftM0(x−vt)/angbracketright, where /angbracketleftM0(x)/angbracketrightdenotes the\nstationary solution in the absence of domain wall mo-\ntion. However, if /angbracketleftM/angbracketright(∇·v)/negationslash= 0, the steady-state so-\nlution may break the Galilean invariance. The situa-\ntion/angbracketleftM/angbracketright(∇·v)/negationslash= 0 can be realized, for example, in\nmagnetic semiconductors [23, 24], where the spin carrier\ndensity is spatially inhomogeneous, i.e.,∇ρ/negationslash= 0.\nThe last term of Eq. (19) represents the non-adiabatic\ncomponent of the current-induced torque, which is also\nknown as the “ βterm”. By comparing Eq. (19) with\nEq. (1), one can see that the coefficient of the last term\nisequaltotheGilbert dampingconstant α. However,Eq.\n(19) is valid when the relaxation time of the fluctuating\nfield,τc, is very short compared to the time scale of the\nmagnetization dynamics. It should be noted that the\ngeneralformoftheequationdescribingthemagnetization\ndynamics is given by Eq. (17) where the last term on theright-hand side is the origin of the αandβterms. It is\npossible to projectthe torque representedby the memory\nfunction onto the direction of the αandβterms. This\nprojection leads to α/negationslash=βin general.\nIn order to observe the effect of τcon the magneti-\nzation dynamics we applied our theory to the current-\ninduced magnetization switching in the magnetic multi-\nlayer shown in Fig.1 (b). We assumed that the fixed and\nfree layers are single-domain magnetic layers acting as a\nlarge spin characterized by the total magnetization vec-\ntor defined as Si=/integraltext\ndV/angbracketleftMi/angbracketright, wherei= 1(2) for the\nfixed (free) layer and/integraltext\ndVdenotes the volume integra-\ntion over the fixed (free) layer. Both the magnetization\nvector of the fixed layer S1and the effective magnetic\nfield,H, acting on the free layer lie in the plane.\nIntegrating Eqs. (12) and (13) over the volume of the\nfree layer, we obtain the equations,\n∂tS2+/integraldisplay\ndSˆn·J=γ(H+/angbracketleftδH/angbracketright)×S2,(20)\n∂t/angbracketleftδH/angbracketright=−1\nτc(/angbracketleftδH/angbracketright−χVS2), (21)\nwhereJ=v⊗/angbracketleftM/angbracketrightis the spin current tensor/integraltext\ndSrep-\nresents the surface integration over the free layer, ˆnis\nthe unit normal vector of the surface, and χV=χ/Vis\ndefined by the volume of the free layer V.\nThe same procedure used to derive Eq. (17) yields\n∂tS2+/integraldisplay\ndSˆn·J=γH×S2\n+γ/integraldisplayt\n−∞dt′ψ(t−t′)χVS2(t)×∂t′S2(t′),(22)\nwhereψ(t) = exp[−t/τc].\nWhen the relaxation time of the fluctuating field is\nshort compared to the time scale of magnetization dy-\nnamics, the LLG-type equation in the presence of the\nspin-transfer torque is obtained as\n∂tS2+/integraldisplay\ndSˆn·J=γH×S2+α\nS2S2×∂tS2,(23)\nwhereα=γτcχVS2. By introducing the conventional\nform of the spin-transfer torque [1], we obtain the follow-\ning LLG-type equation:\n∂tS2−I\neg/planckover2pi1(S2×S1)×S2=γH×S2+α\nS2S2×∂tS2.(24)\nHowever, Eq. (24) is valid only when τc<1/(γH). As\nmentioned before, the torque represented by using the\nmemory function generally has a component parallel to\nthe non-adiabatic torque. In order to observe the ef-\nfect of the non-adiabatic torque induced by the memory\nfunction on the magnetization dynamics, we performed\nnumerical simulation using Eqs. (20) and (21).4\nFIG. 2: The z-component of the magnetization S2is plotted\nas a function of time for various values of τc. The initial\ndirection of the free layer is taken to lie in the direction of\nthe effective magnetic field, which is aligned to the zaxis.\nThe initial angle between S1andS2is taken to be 45◦. The\nGilbert damping constant αis 0.01.\nFor the simulation, we used the following conditions.\nAt the initial time of t= 0, we assumed that the mag-\nnetization of the free layer is aligned parallel to the ef-\nfective magnetic field Hand the angle between the mag-\nnetizations of the fixed and the free layers is 45◦. This\narrangement corresponds to the recent experiment on a\nmagnetic tunnel junction system [18]. We also assumed\nthat the fluctuation field has zero mean value at t= 0,\ni.e.,/angbracketleftδH(0)/angbracketright=0.\nIn Fig. 2, we plot the time dependence of the zcom-\nponent of the magnetization of the free layer, S2, under\nthe large-enough spin current to flip the magnetization\nof the free layer, Ig/planckover2pi1S2\n2S1/(eαγH) =−10. The value of\nτcis varied while the value of α= 0.01 is maintained.\nThe solid, dotted, and dot-dashed lines correspond to\nγHτc= 0.1,1.0, and 10.0, respectively. As shown in\nFig. 2, the time required for the magnetization of the\nfree layer to flip decreases with increasing τc, which can\nbe understood by considering the non-adiabatic torque\ninduced by the spin current. The non-adiabatic torque\ninduced by the spin current is obtained by projecting\nthe torque given by the last term of Eq. (22) onto the\ndirection of S2×S1, which results in the positive con-\ntribution to the spin-flip motion of S2. Since the last\nterm of Eq. (22) includes a memory function, the non-\nadiabatic torque induced by the spin current increases\nwith increasing τc. Therefore, the time required for S2\nto flip decreases with increasing τc. ForγHτc>10 we\nobserve no further decrease of the time required for S2\nto flip because the memory function is an integral of the\nvectorS2(t)×∂t′S2(t′) and the contributions from the\nmemory at t−t′≫1/(γH) is eliminated.\nIn conclusion, we derived the equation for the motion\nof magnetization in the presence of a spin current by us-\ning the local equilibrium assumption in non-equilibriumthermodynamics. We demonstrated that the value of the\ncoefficientβis not equal to that of the Gilbert damping\nconstantαin general. However, we also show that the\nequalityα=βholds ifτc≪1/(γH). We then applied\nour theory to current-induced magnetization reversal in\nmagnetic multilayersand showed that the switching time\nis a decreasing function of τc.\nThe authors would like to acknowledge the valuable\ndiscussions they had with S.E. Barnes, S. Maekawa, P.\nM. Levy, K. Kitahara, K. Matsushita, J. Sato and T.\nTaniguchi. This work was supported by NEDO.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[4] M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer,\nG. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys.\nLett.83, 105 (2003).\n[5] 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[6] L. LandauandE. Lifshitz, Phys. Z.Sowjet. 8, 153(1935).\n[7] T. L. Gilbert, Armour Research Foundation Project No.\nA059, Supplementary Report, May 1, 1959; IEEE Trans.\nMagn.40, 3443 (2004).\n[8] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n[9] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[10] H. Kohno, G. Tatara, andJ. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006).\n[11] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W.\nBauer, Phys. Rev. B 74, 144405 (2006).\n[12] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn.\nMagn. Mater 320, 1282 (2008).\n[13] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73,\n054428 (2006).\n[14] M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zang-\nwill, Phys. Rev. B 75, 214423 (2007).\n[15] R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDon-\nald, Phys. Rev. B 75, 214420 (2007).\n[16] J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys.\nRev. Lett. 93, 256602 (2004).\n[17] A. A. Tulapurkar et al., Nature 438, 339 (2005).\n[18] H. Kubota et al., Nature Physics 4, 37 (2008).\n[19] K. Miyazaki and K. Seki, J. Chem. Phys. 108, 7052\n(1998).\n[20] A. Kawabata, Prog. Theor. Phys 48, 2237 (1972).\n[21] S. R. de Groot and P. Mazur, Nonequilibrium thermody-\nnamics(North-Holland, Amsterdam, 1962).\n[22] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn.\nMagn. Mater 8, 153 (1935).\n[23] H. Ohno, Science 281, 951 (1998).\n[24] T. Dietl and H. Ohno, Materials Today 9, 18 (2006)."    },    {        "title": "0805.3495v1.Intrinsic_and_non_local_Gilbert_damping_in_polycrystalline_nickel_studied_by_Ti_Sapphire_laser_fs_spectroscopy.pdf",        "content": "Intrinsic and non-local Gilbert damping in\npolycrystalline nickel studied by Ti:Sapphire laser fs\nspectroscopy\nJ Walowski1, M Djordjevic Kaufmann1, B Lenk1, C Hamann2\nand J McCord2, M M unzenberg1\n1Universit at G ottingen, Friedirch-Hund-Platz 1, 37077 G ottingen, Germany\n2IFW Dresden, Helmholtzstra\u0019e 20, 01069 Dresden\nE-mail: walowski@ph4.physik.uni-goettingen.de\nAbstract. The use of femtosecond laser pulses generated by a Ti:Sapphire laser\nsystem allows us to gain an insight into the magnetization dynamics on time scales from\nsub-picosecond up to 1 ns directly in the time domain. This experimental technique is\nused to excite a polycrystalline nickel (Ni) \flm optically and probe the dynamics\nafterwards. Di\u000berent spin wave modes (the Kittel mode, perpendicular standing\nspin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are\nidenti\fed as the Ni thickness is increased. The Kittel mode allows determination of the\nGilbert damping parameter \u000bextracted from the magnetization relaxation time \u001c\u000b.\nThe non-local damping by spin currents emitted into a non-magnetic metallic layer\nof vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied\nfor wedge-shaped Ni \flms 1 nm \u000030 nm. The damping parameter increases from\n\u000b= 0:045 intrinsic for nickel to \u000b > 0:10 for the heavy materials, such as Pd and\nDy, for the thinnest Ni \flms below 10 nm thickness. Also, for the thinnest reference Ni\n\flm thickness, an increased magnetic damping below 4 nm is observed. The origin of\nthis increase is discussed within the framework of line broadening by locally di\u000berent\nprecessional frequencies within the laser spot region.arXiv:0805.3495v1  [cond-mat.other]  22 May 2008Gilbert damping in Nickel thin \flms 2\n1. Introduction\nThe understanding of picosecond-pulsed excitation of spin packets, spin wave modes\nand spin currents is of importance in developing a controlled magnetic switching concept\nbeyond the hundred picosecond timescale and to test the speed of magnetic data storage\nmedia heading to the physical limits. Over the last years profound progress has been\nmade within that \feld by using femtosecond laser spectroscopy. The recent discoveries\nin ultrafast magnetization dynamics are heading to a new understanding [1{5] and\nnew all-optical switching concepts have been discovered [6]. In addition, the all-optical\nmethod has developed into a valuable tool to study the magnetization dynamics of\nthe magnetic precession and thereby access magnetocrystalline anisotropies and the\nmagnetic damping [7{11] or the dynamics of magnetic modes in nanometer sized arrays\nof magnetic structures [12, 13] and single magnetic nanostructures [14, 15]. Naturally,\none \fnds similarities and di\u000berences as compared to magnetic resonance techniques\nin frequency space (FMR) [16], optical techniques such as Brillouin light scattering\n(BLS) [17,18] and time-resolved techniques, for example pulsed inductive magnetometry\n(PIMM) [19]. Advantages and disadvantages of the di\u000berent techniques have already\nbeen compared in previous work [20{22]. The same concepts can be applied to the\nfemtosecond-laser-based all-optical spectroscopy techniques. Here we discuss their\nabilities, highlighting some aspects and peculiarities [11,23{27]:\ni. After excitation within the intense laser pulse, the nature of the magnetic relaxation\nmechanisms determine the magnetic modes observed on the larger time scale [5].\nFor a Ni wedge di\u000berent modes are found as the thickness is increased: coherent\nprecession (Kittel mode), standing spin waves (already found in [28]) and dipolar\nsurface spin waves (Damon-Eshbach modes) appear and can be identi\fed.\nii. Magnetic damping has been extracted by the use of fs spectroscopy experiments\nalready in various materials, epitaxial \flms, as a function of the applied \feld\nstrength, \feld orientation and laser excitation power [7{11]. Using the Kittel\nmode, we study the energy dissipation process caused by non-local damping by spin\ncurrents [29] in Ni by attaching a transition metal \flm (vanadium (V), palladium\n(Pd) and a rare earth \flm (dysprosium (Dy)) as a spin sink material and compare\nthem to a Ni reference sample. The present advantages and disadvantages of the\nmethod are discussed.\niii. A modi\fcation of the magnetic damping is found for the thinnest magnetic layers\nbelow 4 nm. The understanding of this e\u000bect is of high interest because of the\nincrease in methods used to study magnetic damping processes in the low \feld\nregion in the current literature. We present a simple model of line broadening\nknown from FMR [30{32] and adapted to the all-optical geometry that pictures\nthe e\u000bect of the increased intrinsic apparent damping observed. Therein a spread\nlocal magnetic property within the probe spot region is used to mimic the increased\napparent damping for the low \feld region.Gilbert damping in Nickel thin \flms 3\na)\nb)Side view:\nFigure 1. a) Schematics of the pump probe experiment to determine the change in\nKerr rotation as a function of the delay time \u001c. b) Experimental data on short and\nlong time scales. On top a schematic on the processes involved is given.\n2. Experimental Technique\nThe all-optical approach to measuring magnetization dynamics uses femtosecond laser\npulses in a pump-probe geometry. In our experimental setup a Ti:Sapphire oscillator\ngenerates the fs laser pulses which are then ampli\fed by a regenerative ampli\fer (RegA\n9050). This systems laser pulse characteristics are 815 nm central wave length, a\nrepetition rate of 250 kHz, a temporal length of 50 \u000080 fs and an energy of \u00181\u0016J\nper pulse. The beam is split into a strong pump beam (95% of the incoming power),\nwhich triggers the magnetization dynamics by depositing energy within the spot region,Gilbert damping in Nickel thin \flms 4\nand a weaker probe pulse (5% of the incoming power) to probe the magnetization\ndynamics via the magneto-optical Kerr e\u000bect delayed by the time \u001c, in the following\nabbreviated as time-resolved magneto-optic Kerr e\u000bect (TRMOKE). The schematic\nsetup and sample geometry is given in \fgure 1a). The spot diameters of the pump\nand probe beam are 60 \u0016m and 30\u0016m respectively. A double-modulation technique is\napplied to detect the measured signal adapted from [33]: the probe beam is modulated\nwith a photo-elastic modulator (PEM) at a frequency f1= 250 kHz and the pump\nbeam by a mechanic chopper at a frequency f2= 800 Hz. The sample is situated in\na variable magnetic \feld (0 \u0000150 mT), which can be rotated from 0\u000e(in-plane) to 90\u000e\n(out of plane) direction. The degree of demagnetization can be varied by the pump\n\ruence (10 mJ =cm2\u000060 mJ=cm2) to up to 20% for layer thicknesses around 30 nm and\nup to over 80% for layers thinner than 5 nm. The samples studied were all grown on\nSi(100) substrates by e-beam evaporation in a UHV chamber at a base pressure of\n\u00185\u000210\u000010mbar. For a variation of the thickness, the layers are grown as wedges with\na constant gradient on a total wedge length of 15 mm.\n3. Results and discussion\n3.1. Kittel mode, standing spin waves and Damon-Eshbach surface modes\nTo give an introduction to the TRMOKE signals \u0001 \u0012Kerr(\u001c) measured on the timescale\nfrom picoseconds to nanoseconds \frst, the ultrafast demagnetization on a characteristic\ntime scale\u001cMand the magnetic precessional motion damped on a time scale \u001c\u000bis shown\nfor a Ni \flm in \fgure 1b); the schematics of the processes involved on the di\u000berent time\nscales are given on the top. The change in Kerr rotation \u0001 \u0012Kerr(\u001c) shows a sudden drop\nat\u001c= 0 ps. This mirrors the demagnetization within a timescale of \u0018200 fs [34{36]. For\nthe short time scale the dynamics are dominated by electronic relaxation processes, as\ndescribed phenomenologically in the three temperature model [34] or by connecting the\nelectron-spin scattering channel with Elliot-Yafet processes, as done by Koopmans [36]\nand Chantrell [4] later. At that time scale the collective precessional motion lasting up\nto the nanosecond scale is initiated [28, 37]: the energy deposited by the pump pulse\nleads to a change in the magnetic anisotropy and magnetization, and thus the total\ne\u000bective \feld. Within \u001810 ps the total e\u000bective \feld has recovered to the old value\nand direction again. However, the magnetization, which followed the e\u000bective \feld,\nis still out of equilibrium and starts to relax by precessing around the e\u000bective \feld.\nThis mechanism can be imagined as a magnetic \feld pulse a few picoseconds long, and\nis therefore sometimes called an anisotropy \feld pulse. The resulting anisotropy \feld\npulse is signi\fcantly shorter than standard \feld pulses [38]. This makes the TRMOKE\nexperiment di\u000berent to other magnetization dynamics experiments.\nThe fact that the situation is not fully described by the model can be seen in the\nfollowing. Already van Kampen et al. [28] not only observed the coherent precessional\nmode, they also identi\fed another mode at a higher frequency than the coherentGilbert damping in Nickel thin \flms 5\nprecession mode, shifted by !k;n\u00182Ak2= 2An\u0019=t Ni2, the standing spin wave (PSSW)\nmode. It originates from the con\fnement of the \fnite layer thickness, where Ais the\nexchange coupling constant and nis a given order. Here we also present the \fnding of\ndipolar propagating spin waves. For all three, the frequency dependence as a function\nof the applied magnetic \feld will be discussed, a necessity for identifying them in the\nexperiments later on.\nFor the coherent precession the frequency dependence is described by the Kittel\nequation. It is derived by expressing the e\u000bective \feld in the Landau-Lifshitz-Gilbert\n(LLG equation) as a partial derivative of the free magnetic energy [39, 40]. Assuming\nnegligible in-plane anisotropy in case of the polycrystalline nickel (Ni) \flm and small\ntilting angles of the magnetization out of the sample plane (\feld is applied 35\u000eout of\nplane \fgure 1a)), it is solved as derived in [41]:\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs\u0013\n; (1)\nFor the standing spin waves (PSSW) a similar equation is given. For the geometry\nwith the \feld applied 35\u000eout of plane (\fgure 1a) the frequencies !and!k;ndo not\nsimply add as in the \feld applied in plane geometry [41]:\n!=\r\n\u00160s\n(\u00160Hx+2Ak2\nMs)\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+2Ak2\nMs\u0013\n; (2)\nWhile the exchange energy dominates in the limit of small length scales, the\nmagnetic dipolar interaction becomes important at larger length scales. Damon and\nEshbach [42] derived by taking into account the dipolar interactions in the limit of\nnegligible exchange energy, the solution of the Damon-Eshbach (DE) surface waves\npropagating with a wave vector qalong the surface, decaying within the magnetic layer.\nThe wavelengths are found to be above the >\u0016m range for Ni [27].\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+M2\nS\n4[1\u0000exp(\u00002qtNi)]\u0013\n; (3)\nThe depth of the demagnetization by the femtosecond laser pulse is given by the\noptical penetration length \u0015opt\u001915 nm (\u0015= 800 nm). From the nature of the excitation\nprocess in the TRMOKE experiment one can derive that for di\u000berent thicknesses tNiit\nwill change from an excitation of the full \flm for a \u001810 nm \flm to a thin excitation\nlayer only for a few 100 nm thick \flm; thus the excitation will be highly asymmetric.\nThe model of the magnetic anisotropy \feld pulse fails to explain these e\u000bects since it is\nbased on a macrospin picture.\nAnother way to look at the excitation mechanism has been discussed by Djordjevic\net al. [5]. When the magnetic system is excited, on a length scale of the optical\npenetration depth short wavelength (high kvector) spin-wave excitations appear. As\ntime evolves, two processes appear: the modes with high frequency owning a fastGilbert damping in Nickel thin \flms 6\noscillation in space are damped very fast by giving part of the deposited energy to\nthe lattice. In addition, through multiple magnon interaction lower k-vector states are\npopulated, resulting in the highest occupation of the lowest energy modes at the end\n(e.g. the PSSW and DE modes here). As the Ni thickness is increased, the excitation\npro\fle becomes increasingly asymmetric, favoring inhomogeneous magnetic excitations,\nas the PSSW mode. The DE modes, due to their nature based on a dipolar interaction,\nare expected to be found only for higher thicknesses.\nFigure 2. Change in Kerr rotation after excitation on the long time scale for Cu 2nm =\nNi tNinm=Si(100) with tNi= 20 and b) their Fourier transform for di\u000berent applied\n\felds 0\u0000150 mT, (35\u000eout of plane (blue)). In c) the Fourier power spectra as color\nmaps for three Ni thicknesses tNi= 20, 40 and 220 nm are given. The data overlaid is\ndetermined form the peak positions. The straight lines are the analysis of the di\u000berent\nmodes and are identi\fed in the graph (Kittel model), perpendicular standing spin wave\n(PSSW) and dipolar surface spin wave (Damon Eshbach mode).Gilbert damping in Nickel thin \flms 7\nThe identi\fcation of the mode is important in determining a value for the magnetic\ndamping\u000b. Figure 2 pictures the identi\fcation of the di\u000berent modes and their\nappearance for di\u000berent Ni thicknesses. The data are handled as follows: for a\ntNi= 20 nm \flm on Si(100), covered with a 2 nm Cu protection layer, in a) the original\ndata after background subtraction and in b) its corresponding Fourier transform, shown\nfor increasing applied magnetic \feld. The evolution of the mode frequency and its\namplitude increase can be followed. An exponentially decaying incoherent background\nis subtracted from the data. This has to be done very carefully, to avoid a step-\nlike background which will be evident after Fourier transform as a sum of odd higher\nharmonics. The frequency resolution is limited by the scan range of 1 ns corresponding\nto \u0001!=2\u0019= 1 GHz. However, since the oscillation is damped within the scan range,\nthe datasets have been extended before Fourier transform to increase their grid points.\nA color map of the power spectrum is shown in \fgure 2c), where the peak positions are\nmarked by the data points overlaid. For the 20 nm thick \flm with tNi= 20 nm\u0018\u0015opt\nonly a single mode is observed. The mode is analyzed by 1 indicating the Kittel mode\nbeing present (data points and line in \fgure 2c), top) using Kz= 3:03\u0001104J=m3. With\nincreasing nickel thickness tNi= 40 nm> \u0015 opt, the perpendicular standing spin waves\n(PSSW) of \frst order are additionally excited and start to appear in the spectra (\fgure\n2c), middle). An exchange constant A= 9:5\u00011012J=m is extracted. In the limit of\ntNi= 220 nm\u001d\u0015opt(\fgure 1c)) the excitation involves the surface only. Hence, modes\nwith comparable amplitude pro\fle, e.g. with their amplitude decaying into the Ni layer,\nare preferred. Consequently DE surface waves are identi\fed as described by 3 and\ndominate the spectra up to critical \felds as high as \u00160Hcrit= 100 mT. For tNi= 220 nm\nthe wave factor is k= 2\u0016m (data points and line in \fgure 2c), bottom). For larger\n\felds than 100 mT the DE mode frequency branch merges into the Kittel mode [27].\nTo resume the previous \fndings for the \frst subsection, we have shown that in\nfact the DE modes, though they are propagating spin-wave modes, can be identi\fed\nin the spectra and play a very important role for Ni thicknesses above tNi= 80 nm.\nThey appear for thicknesses much thinner than the wavelength of the propagating\nmode. Perpendicular standing spin waves (PSSW) give an important contribution to\nthe spectra for Ni thicknesses above tNi= 20 nm. For thicknesses below tNi= 20 nm we\nobserve the homogeneously precessing Kittel mode only. This thickness range should\nbe used to determine the magnetic damping in TRMOKE experiments.\n3.2. Data analysis: determination of the magnetic damping\nFor the experiments carried out in the following with tNi<25 nm the observed dynamics\ncan be ascribed to the coherent precession of the magnetization (Kittel mode). The\nanalysis procedure is illustrated in the following using the data given in \fgure 3a). A\nPd layer is attached to a Ni \flm with the thicknesses (Ni 10 nm =Pd 5 nm=Si(100)) to\nstudy the non-local damping by spin currents absorbed by the Pd. The di\u000berent spectra\nwith varying the magnetic \feld strength from 0 mT \u0000150 mT are plotted from bottomGilbert damping in Nickel thin \flms 8\nto top (with the magnetic \feld tilted 35\u000eout of the sample plane).\n0 .0 4 5 0 .0 5 0 0 .0 5 5  \nα\n0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 02468\nν [GHz]\nµ0 He x t  [m T ]0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 8 - 4 048\n  \n∆θk  [a.u.]\nτ  [p s ]µ0He x t =\n1 5 0 m T \n1 4 0 m T \n1 3 0 m T \n1 2 0 m T \n1 1 0 m T \n1 0 0 m T \n 9 0 m T \n 8 0 m T \n 7 0 m T \n 6 0 m T \n 5 0 m T \n 4 0 m T \n 3 0 m T \n  0 m T a) b)5 nm Pd/ 10 n m Ni\n5 nm Pd/ 10 n m Ni\nFigure 3. a) Kerr rotation spectra for a Cu 2nm =Ni 10 nm=Pd 5 nm=Si(100) layer,\nmeasured for \felds applied from 0 \u0000150 mT (35\u000eout of plane, (blue)) and the \ftted\nfunctions (white, dashed). b) The magnetic damping \u000band precession frequencies\nextracted from the \fts to the measured spectra. The line is given by the Kittel mode\n(gray).\nThe data can be analyzed using the harmonic function with an exponential decay\nwithin\u001c\u000b:\n\u0001\u0012k\u0018exp\u0012\n\u0000\u001c\n\u001c\u000b\u0013\n\u0001sin(2\u0019(\u001c\u0000\u001c0)\u0017) +B(\u001c); (4)\nThe precession frequency \u0017=!=2\u0019and the exponential decay time \u001c\u000bof the\nprecession amplitude is extracted, where the function B(\u001c) stands for the background\narising from the uncorrelated magnetic and phonon excitations. To determine the\nGilbert damping parameter \u000bas given in the ansatz by Gilbert, the exponential decay\ntime\u001c\u000bhas to be related with \u000b. The LLG equation is solved under the same\npreconditions as for equation 1 using an exponential decay of the harmonic precession\nwithin\u001c\u000bfrom 4. Then the damping parameter \u000band can be expressed by the followingGilbert damping in Nickel thin \flms 9\nequation [41]:\n\u000b=1\n\u001c\u000b\r\u0010\nHx\u0000Kz\n\u00160Ms+Ms\n2\u0011: (5)\nIt is evident from 5 that in order to determine the Gilbert damping \u000bfrom the decay\nof the Kittel mode \u001c\u000b, the variables \r,MsandKzhave to be inserted, and therefore Kz\nhas to be determined beforehand.\nIn \fgure 3a), the background B(\u001c) is already subtracted. The \fts using 4 are\nplotted with the dashed lines on top of the measured spectra. The results are presented\nin b). The frequencies range from 3 GHz for 30 mT to 7 :5 GHz for the 150 mT applied\nmagnetic \feld. They increase linearly with the strength of the applied magnetic \feld for\nhigh \feld values. The extrapolated intersection with the ordinate is related to the square\nroot of the dipolar and anisotropy \feld. Using the Kittel equation (1), one determines\nthe out-of-plane anisotropy constant KzofKz= 6:8\u0001104J=m3. The calculated magnetic\ndamping\u000bas a function of the applied \feld is given in the graph below: this is mostly\nconstant but increases below 60 mT. Within the ansatz given by Gilbert, the damping\nconstant\u000bis assumed to be \feld-independent. We \fnd that this is ful\flled for most\nof the values: the average value of \u000b= 0:0453(4), consistent with earlier \fndings by\nBhagat and Lubitz from FMR experiments [43], is indicated by the line in the plot. The\n\u000bgiven in the following will always be averaged over a \feld region where the damping\nis Gilbert-like. A deviation from this value occurs for the small external \feld strengths.\nIt originates for two reasons: \ftting 5 with a few periods only does not determine a\nreliable value of the exponential precession decay time \u001c\u000band leads to a larger error.\nSecond, magnetic inhomogeneities mapping a spread in anisotropy energies within the\nprobe spot region can also be a source, and this becomes generally more important for\neven thicker \flms below 4 nm [31]. This will be discussed in more detail in the last\nsection of the manuscript.\n3.3. Intrinsic damping: nickel wedge\nFor our experiments Ni was chosen instead of Fe or Py as a ferromagnetic layer. The\nlatter would be preferable because of their lower intrinsic damping \u000bint, which make\nthe \flms more sensitive for detecting the non-local contribution to the damping. The\nreason for using Ni for our experiments is the larger signal excited in the TRMOKE\nexperiments. The magnetic damping \u000bintis used as a reference later on. The di\u000berent\nspectra with varying the Ni thickness tNiNixnm=Si(100) from 2 nm \u0014x\u001422 nm are\nplotted from bottom to top (with the constant magnetic \feld 150 mT and tilted 30\u000e\nout of plane) in \fgure 4a). The measurements were performed immediately after the\nsample preparation, in order to prevent oxidation on the nickel surface caused by the\nlack of a protection layer (omitted on purpose). The spectra show similar precession\nfrequency and initial excitation amplitude. However, the layers with tNi<10 nm show a\nfrequency shift visually recognized in the TRMOKE data. Furthermore, the precessionGilbert damping in Nickel thin \flms 10\namplitude decreases faster for the thinner layers. Figure 5 shows the frequencies and\nthe damping parameter extracted from the measured data in the intrinsic case for the\nnickel wedge sample (black squares). While the precession frequency given for 150 mT\nis almost constant above 8 nm Ni thickness, it starts to drop by about 25% for the\nthinnest layer. The magnetic damping \u000b(black squares) is found to increase to up to\n\u000b= 0:1, an indication that in addition to the intrinsic there are also extrinsic processes\ncontributing. It has to be noted that the change in \u000bis not correlated with the decrease\nof the precession frequency. The magnetic damping \u000bis found to increase below a\nthickness of 4 nm, while the frequency decrease is observed below a thickness of 10 nm.\nA priori\r,MsandKzcan be involved in the observed frequency shift, but they can\nnot be disentangled within a \ft of our \feld-dependent experiments. However, from our\nmagnetic characterization no evidence of a change of \randMsis found. A saturation\nmagnetization \u00160Ms= 0:659 T and g-factor of 2.21 for Ni are used throughout the\nmanuscriptzandKzis determined as a function of the Ni thickness, which shows a\n1=tNibehavior, as expected for a magnetic interface anisotropy term [44].\nThe knowledge of the intrinsic Gilbert damping \u000bintof the Ni \flm of a constant\nvalue for up to 3 nm thickness allows us to make a comparative study of the non-local\ndamping\u000b0, introduced by an adjacent layer of vanadium (V) and palladium (Pd) as\nrepresentatives for transition metals, and dysprosium (Dy) as a representative of the\nrare earths. Both damping contributions due to intrinsic \u000bintand non-local spin current\ndamping\u000b0are superimposed by:\n\u000b=\u000bint+\u000b0: (6)\nThey have to be disentangled by a study of the thickness dependence and compared\nto the theory of spin-current pumping, plus a careful comparison to the intrinsic value\n\u000binthas to be made.\n3.4. Non-local spin current damping: theory\nDynamic spin currents excited by a precessing moment in an adjacent nonmagnetic\nlayer (NM) are the consequence of the fact that static spin polarization at the interface\nfollows a dynamic movement of a collective magnetic excitation. The e\u000bect has already\nbeen proposed in the seventies [45,46] and later calculated within a spin reservoir model\nwith the spins pumped through the interfaces of the material by Tserkovnyak [29, 47].\nFor each precession, pumping of the spin current results in a corresponding loss in\nmagnetization, and thus in a loss of angular momentum. The spin information is lost\nand the backward di\u000busion damps the precession of the magnetic moment. In addition to\nthe \frst experiments using ferromagnetic resonance (FMR) [48{54] it has been observed\nin time- resolved experiments using magnetic \feld pulses for excitation [55,56]. In fact\nzAn altered g-factor by interface intermixing can not decrease its value below \u00182. Also, there is no\nevidence for a reduced Msfor lower thicknesses found in the Kerr rotation versus Ni thickness data.\nMore expected is a change in the magnetic anisotropy Kz. For the calculation of \u000blater on, the in\nboth cases (assuming a variation of Kzor an altered \r) the di\u000berences are negligible.Gilbert damping in Nickel thin \flms 11\na) b)\n0 250 500 750 10 00- 10 - 8 - 6 - 4 - 2 0\n2 2  n m 1 8  n m 1 0  n m 1 4  n m 8  n m 7  n m 6  n m 5  n m 4  n m 3  n m 2  n m \n \n∆θk  [a.u.]\nτ  [ p s ] 0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 1 0 -8-6-4-20\n1 7  n m \n2 3  n m 1 4  n m 1 0  n m 8  n m 7  n m 6  n m 5  n m 4  n m 2  n m \n3  n m \n  \n∆θk [a.u.]\nτ  [p s]Ni r efer enc e x Ni/ 5 nm DydNi = dNi =\nFigure 4. a) Kerr rotation spectra for nickel layers from tNi= 2 nm\u000022 nm, measured\non the nickel wedge tNi= nm Ni=Si(100) and opposed in b) by a nickel wedge Al 2nm =\nDy 5nm= tNi= nm Ni=Si(100) with a 5 nm Dy spin-sink layer.\nthe non-local spin current damping is very closely related to the damping by spin-\rip\nscattering described within the s-d current model [57, 58] that uses the approximation\nof strongly localized d-states and delocalized s-states [59].\nA review describes the underlying circuit theory and dynamics of the spin currents\nat interfaces in detail [60]. The outcome of the theoretical understanding is that the\nadditional Gilbert damping is proportional to the angular momentum Ar;ltransmitted\nthrough the interface. Since each interface owns a characteristic re\rection and\ntransmission, the size of Ar;ldepends on the matching of the Fermi surfaces. The\nabsolute value is given by the total balance between transmitted angular momentum\nand the back \row. For the non-local damping \u000b0one \fnds:\n\u000b0=\r~G\"#\n4\u0019MstFM1\n1 +q\n\u001csf\n\u001celtanh\u0010\ntNM\n\u0015sd\u0011\u00001: (7)\nThe tanh function stems thereby from the di\u000busion pro\fle of the spin currents\ndetermined by the spin di\u000busion length \u0015sdwithin the non-magnetic material withGilbert damping in Nickel thin \flms 12\nthicknesstNM. Also, one \fnds from the analysis the ratio of the electron scattering\nrate\u001celversus the spin \rip rate \u001csf. The total amount of spin current through the\ninterfaces is determined by the interface spin mixing conductance G\"#. It is related\nto the magnetic volume. It is therefore that scales with the thickness of the magnetic\nlayertFM. The e\u000bective gyromagnetic ratio altered by the spin-current implies that in\naddition to an increased damping a small frequency shift will be observed. The non-local\nGilbert damping becomes important when it exceeds the intrinsic damping \u000bint.\n3.5. Non-local damping: vanadium, palladium and dysprosium\nDi\u000bering from other techniques, TRMOKE experiments require optical access for\nexcitation and detection, setting some restrictions to the layer stack assembly that can\nbe investigated with this method: a thick metallic layer on top of the magnetic layer is\nnot practical. Placing the damping layer below the magnetic layer is also unfavorable:\nby increasing the spin sink thickness the roughness of the metal \flm will increase with\nthe metals layer thickness and introduce a di\u000berent defects density, altering \u000bint. In the\nfollowing the nickel thickness will be varied and the spin sink thickness will be kept \fxed\nat 5 nm. To warrant that the nickel \flms magnetic properties are always comparable\nto the reference experiment ( Kz,\u000bint), they are always grown \frst on the Si(100). For\nthe Pd case the damping layer is below the Ni layer. Here the excitation mechanism\ndid not work and the oscillations were too weak in amplitude to analyze the damping\n\u000b, probably due to the high re\rectivity of Pd.\nThe results are presented in \fgure 4b) for the nickel wedge sample Ni xnm=Si(100)\nwith a 5 nm dysprosium (Dy) as a spin sink layer, covered by an aluminum protection\nlayer, as opposed to the nickel wedge sample data without this in a). The nickel layer\nthickness is varied from 2 nm \u0014x\u001422 nm. All spectra were measured in an external\nmagnetic \feld set to 150 mT and tilted 30\u000eout of plane. For the thinnest Ni thickness,\nthe amplitude of the precession is found to be smaller due to the absorption of the Dy\nlayer on the top. While the precession is equally damped for the Ni thicknesses ranging\nfrom 7 to 23 nm, an increased damping is found for smaller thicknesses below this. The\ndi\u000berence in damping of the oscillations is most evident for tNi= 4 and 5 nm.\nThe result of the analysis as described before is summarized in \fgure 5. In this\ngraph the data are shown for the samples with the 5 nm V, Pd, Dy spin-sink layer and\nthe Ni reference. While for the Ni reference, and Ni with adjacent V and Dy layer,\nthe frequency dependence is almost equal, indicating similar magnetic properties for\nthe di\u000berent wedge-like shaped samples, the frequency for Pd is found to be somewhat\nhigher and starts to drop faster than for the others. The most probable explanation is\nthat this di\u000berence is due to a slightly di\u000berent anisotropy for the Ni grown on top of\nPd in this case. Nevertheless, the magnetic damping found for larger thicknesses tNiis\ncomparable with the Ni reference. In the upper graph of \fgure 5 the Gilbert damping\nas a function of the Ni layer thickness is shown. While for the Pd and Dy as a spin\nsink material a additional increase below 10 nm contributing to the damping can beGilbert damping in Nickel thin \flms 13\nidenti\fed, for V no additional damping contribution is found.\n0 . 05 0 . 10 0 . 15 \n0 5 1 0 1 5 20 685 1 0 1 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 \ndN i  [ nm]  α\n   \nν [GHz]\n x  N i \nw i th : \n 5 n m  V  \n 5 n m  P d \n 5 n m  D y α−αint\ndN i  [n m ]\nFigure 5. Gilbert damping parameters \u000band frequency \u0017as a function of the nickel\nlayer thickness for the intrinsic case and for di\u000berent damping materials of 5 nm V,\nPd, and Dy adjacent to the ferromagnet. \u000bis extracted from experiments over a large\n\feld region. The \fts are made using equation 5 and equation 7. In the inset the data\nis shown on a reciprocal scale. Below, the frequency is given (150 mT). The lines are\nguides for the eye.\nFor the adjacent V layer, since it is a transition metal with a low spin orbit-\nscattering (light material with low atomic number Z), with a low spin-\rip scattering rate\nand thus a spin di\u000busion length larger than the thickness tNM(d\u001c\u0015sd), no additional\ndamping will occur. For Pd and Dy the situation is di\u000berent: whereas the heavier Pd\nbelongs to the transition metals with a strong orbit-scattering (heavy material with\nhigh atomic number Z), Dy belongs to the rare earth materials. It owns a localized 4fGilbert damping in Nickel thin \flms 14\nmagnetic moment: therefore, both own a high spin-\rip scattering rate and we expect\nthe latter two to be in the region where ( t\u001d\u0015sd). In their cases the thickness of 5 nm\nof the spin-sink layer is chosen to be larger than the spin di\u000busion length ( tNM\u001d\u0015sd).\nIn this limit the spin current emitted from the magnetic layer through the interface is\ntotally absorbed within the non-magnetic layer. One can simplify 6 to:\n\u000b0(1) =\r~G\"#\n4\u0019Mst\u00001\nFM: (8)\nThis is called the limit of a perfect spin sink. The additional non-local spin current\ndamping is expected to behave inversely proportional with the nickel layer thickness\n\u0018t\u00001\nFM. The inset gives the analysis and the data point on a reciprocal scale. The slope\nshows a linear increase for thinner nickel layers, as expected for an inverse proportionality\nfor both the Pd and the Dy. Since the value for the intrinsic damping of the nickel \flm\nincreases below 4 nm this contribution has to be subtracted to reveal the spin-current\ncontribution. The value for \u000b0is then found to be 0 :07 for the 2 nm Ni =5 nm Pd \flm,\nwhich is in the order found by Mizukami by FMR for sputtered Permalloy \flms with\na Pd spin sink ( \u000b0= 0:04 for 2 nm Py =5 nm Pd) [49, 50]. A further analysis of the\nthickness dependence of \u000byields values for the prefactor in 7 for Pd (0 :33(3) nm) and Dy\n(0:32(3) nm) with the \ft given in the graph. From that value the real part of the interface\nspin mixing conductance in 7 can be calculated. It is found to be G\"#= 4:5(5)\u00011015\ncm1\nfor the Ni/Pd and Ni/Dy interface. The increase of the intrinsic damping \u000binthas\nbeen analyzed using an inverse thickness dependence (prefactor of 0 :1 nm). While it\ndescribes the data in the lower thickness range, it can be seen that it does not describe\nthe thickness dependence for the thicker range and thus, probably the increase does not\noriginate from an interface e\u000bect.\n3.6. Increased damping caused by anisotropy \ructuations: consequences for the\nall-optical approach\nIn this last part we want to focus on the deviation from the intrinsic damping \u000bintfor\nthe thin nickel layers itself ( tNi<4 nm). In the low \feld range (10 \u000050 mT) small\nmagnetization inhomogeneities can build up even when the magnetization appears to\nbe still saturated from the hysteresis curve (the saturation \felds are a few mT). For\nthese thin layers the magnetization does not align parallel in an externally applied \feld\nany more, but forms ripples. The in\ruence of the ripples on the damping is discussed\nin reference [32]. In the following we adopt this ansatz to the experimental situation\nof the TRMOKE experiment. We deduce a length scale on which the magnetization\nreversal appears for two di\u000berent Ni thicknesses and relate it to the diameter of our\nprobe spot. Lateral magnetic inhomogeneities were studied using Kerr microscopy at\ndi\u000berent applied magnetic \felds [44]. Magnetization reversal takes place at low \felds\nof a -0.5 to 2 mT. The resolution of the Kerr microscopy for this thin layer thickness\ndoes not allow us to see the extent of the ripple e\u000bect in the external \feld where the\nincrease of \u000band its strong \feld dependence is observed. However, the domains in theGilbert damping in Nickel thin \flms 15\ndemagnetized state also mirror local inhomogeneities. For our Ni xnm=Si(100) sample\nthis is shown in \fgure 6a) and b). The domains imaged using Kerr microscopy are\nshown for a 3 nm and a 15 nm nickel layer in the demagnetized state. The domains of\nthe 15 nm layer are larger than the probe spot diameter of 30 \u0016m, whereas the domains\nof the 3 nm layer are much smaller.\nd =15nmNi\nd =3nmNi\ndemagnetized\ndemagnetizeda)\nb)c)\nd)20µm\n20µm\nFigure 6. a) and b) Kerr microscopy images for the demagnetized state for 15 nm\nand 3 nm. c) and d) corresponding model representing the areas with slightly varying\nanisotropy\nFrom that observation, the model of local anisotropy \ructuations known from\nFMR [30, 31] is schematically depicted in \fgure 6c) and d). A similar idea was also\ngiven by McMichael [61] and studied using micromagnetic simulations. While for the\nthick \flm the laser spot probes a region of almost homogeneous magnetization state, for\nthe thin layer case the spot averages over many di\u000berent regions with slightly di\u000berent\nmagnetic properties and their magnetization slightly tilted from the main direction\naveraging over it. The TRMOKE signal determined mirrors an average over the probed\nregion. It shows an increased apparent damping \u000band a smaller \u001c\u000bresulting from the\nline broadening and di\u000berent phase in frequency space. While for the thick layer the\ntypical scale of the magnetic inhomogeneity is as large as the probe laser spot given\nand only 1-2 regions are averaged, for the thinner \flm of dNi= 3 nm many regions\nare averaged within a laser spot, as can be seen in 6b) and d). Because the magnetic\ninhomogeneity mapping local varying anisotropies becomes more important for smaller\n\felds, it also explains the strong \feld dependence of \u000bobserved within that region.\nFigure 7 shows data calculated based on the model, in which the upper curve (i)Gilbert damping in Nickel thin \flms 16\nis calculated from the values extracted from the experimental data for the 10 nm nickel\nlayer, curve (ii) is calculated by a superposition of spectra with up to 5% deviation\nfrom the central frequency at maximum and curve (iii) is calculated by a superposition\nof spectra of 7% deviation from the central frequency at maximum to mimic the line\nbroadening. The corresponding amplitudes of the superposed spectra related to di\u000berent\nKzvalues is plotted in the inset of the graph to the given frequencies. The apparent\ndamping is increased by 0.01 (for 5%) and reaches the value given in \fgure 3b) for\nthe 10 nm \flm determined for the lowest \feld values of 30 mT. These e\u000bects generally\nbecome more important for thinner \flms, since the anisotropy \ructuations arising from\nthickness variations are larger, as shown by the Kerr images varying on a smaller length\nscale. These \ructuations can be vice versa determined by the analysis.\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s50/s52\n/s55/s46/s48 /s55/s46/s53 /s56/s46/s48\n/s32/s32/s77/s32/s91/s97/s46/s117/s46/s93\n/s32/s91/s112/s115/s93/s40/s105/s105/s105/s41/s40/s105/s41\n/s40/s105/s105/s41/s40/s105/s105/s105/s41/s40/s105/s41/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s65\n/s32/s91/s71/s72/s122/s93/s40/s105/s105/s41\nFigure 7. a) Datasets generated by superposing the spectra with the frequency spread\naccording to the inset: (i) is calculated from the values extracted from the experimental\ndata for the 10 nm Ni layer, (ii) by a superposition of spectra with up to 5% and (iii) is\ncalculated by a superposition of spectra owing 7% variation from the central frequency\nat maximum. The average precession amplitude declines faster if a higher spread of\nfrequencies (i.e. di\u000berent anisotropies) are involved.Gilbert damping in Nickel thin \flms 17\n4. Conclusion\nTo conclude, we have shown that all-optical pump-probe experiments are a powerful\ntool to explore magnetization dynamics. Although the optical access to the magnetic\nlayer allows an access to the surface only, magnetization dynamics can be explored\ndirectly in the time domain, resolving di\u000berent types of spin-wave modes (Kittel mode,\nperpendicular standing spin waves and Damon-Eshbach dipolar surface waves). This is\nin contrast to FMR experiments, where the measured data is a response of the whole\nsample. The obtained data can be similar to the \feld-pulsed magnetic excitations and\nthe Gilbert damping parameter \u000b, needed for the analysis of magnetization dynamics\nand the understanding of microscopic energy dissipation, can be determined from these\nexperiments. We have evaluated the contributions of non-local spin current damping\nfor V, Pd and Dy. Yet there are limits, as shown for the nickel layer thicknesses\nbelow 4 nm, where the magnetic damping increases and may overlay other contributions.\nThis increase is attributed to an inhomogeneous line broadening arising from a strong\nsensitivity to local anisotropy variations. Further experiments will show whether the\nreduction of the probe spot diameter will improve the results by sensing smaller areas\nand thus reducing the line broadening.\nAcknowledgments\nThe \fnancial support by the Deutsche Forschungsgemeinschaft within the SPP 1133\nprogramme is greatfully acknowledged.Gilbert damping in Nickel thin \flms 18\nReferences\n[1] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge. Phys. Rev. Lett. ,\n95(26):267207, 2005.\n[2] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu,\nE. F. Aziz, M. Wietstruk, H. A. Durr, and W. Eberhardt. Nat. Mater. , 6:740{743, 2007.\n[3] U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D. 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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. 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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": "0807.2901v1.Current_induced_dynamics_of_spiral_magnet.pdf",        "content": "arXiv:0807.2901v1  [cond-mat.str-el]  18 Jul 2008Current-induced dynamics of spiral magnet\nKohei Goto,1,∗Hosho Katsura,1,†and Naoto Nagaosa1,2,‡\n1Department of Applied Physics, The University of Tokyo,\n7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n2Cross-Correlated Materials Research Group (CMRG), ASI, RI KEN, Wako 351-0198, Japan\nWe study the dynamics of the spiral magnet under the charge cu rrent by solving the Landau-\nLifshitz-Gilbert equation numerically. In the steady stat e, the current /vectorjinduces (i) the parallel shift\nof the spiral pattern with velocity v= (β/α)j(α,β: the Gilbert damping coefficients), (ii) the\nuniform magnetization Mparallel or anti-parallel to the current depending on the ch irality of the\nspiral and the ratio β/α, and (iii) the change in the wavenumber kof the spiral. These are ana-\nlyzed by the continuum effective theory using the scaling arg ument, and the various nonequilibrium\nphenomena such as the chaotic behavior and current-induced annealing are also discussed.\nPACS numbers: 72.25.Ba, 71.70.Ej, 71.20.Be, 72.15.Gd\nThe current-induced dynamics of the magnetic struc-\ntureisattractingintensiveinterestsfromtheviewpointof\nthe spintronics. A representative example is the current-\ndriven motion of the magnetic domain wall (DW) in fer-\nromagnets [1, 2]. This phenomenon can be understood\nfrom the conservation of the spin angular momentum,\ni.e., spin torque transfer mechanism [3, 4, 5, 6, 7]. The\nmemorydevicesusing this current-inducedmagneticDW\nmotion is now seriously considered [8]. Another example\nis the motion of the vortex structure on the disk of a\nferromagnet, where the circulating motion of the vortex\ncore is sometimes accompanied with the inversion of the\nmagnetization at the core perpendicular to the disk [9].\nTherefore, the dynamics of the magnetic structure in-\nduced by the current is an important and fundamental\nissue universal in the metallic magnetic systems. On the\notherhand, thereareseveralmetallicspiralmagnetswith\nthe frustrated exchange interactions such as Ho metal\n[10, 11], and with the Dzyaloshinskii-Moriya(DM) inter-\naction suchas MnSi [12, 13, 14], (Fe,Co)Si [15], and FeGe\n[16]. Thequantumdisorderingunderpressureorthenon-\ntrivial magnetic textures have been discussed for the lat-\nter class of materials. An important feature is that the\ndirection of the wavevector is one of the degrees of free-\ndom in addition to the phase of the screw spins. Also the\nnon-collinear nature of the spin configuration make it an\ninteresting arena for the study of Berry phase effect [17],\nwhichappearsmostclearlyinthecouplingtothecurrent.\nHowever, the studies on the current-induced dynamics of\nthe magnetic structures with finite wavenumber, e.g., an-\ntiferromagnet and spiral magnet, are rather limited com-\npared with those on the ferromagnetic materials. One\nreason is that the observation of the magnetic DW has\nbeen difficult in the case of antiferromagnets or spiral\nmagnets. Recently, the direct space-time observation of\nthe spiral structure by Lorentz microscope becomes pos-\nsible for the DM induced spiralmagnets [15, 16] since the\nwavelength of the spiral is long ( ∼100nm). Therefore,\nthe current-induced dynamics of spiral magnets is now\nan interesting problem of experimental relevance.In this paper, we study the current-induced dynamics\nof the spiral magnet with the DM interaction as an ex-\nplicit example. One may consider that the spiral magnet\ncan be regarded as the periodic array of the DW’s in fer-\nromagnet,butithasmanynontrivialfeaturesunexpected\nfrom this naive picture as shown below.\nThe Hamiltonian we consider is given by [18]\nH=/integraldisplay\nd/vector r/bracketleftBigJ\n2(/vector∇/vectorS)2+γ/vectorS·(/vector∇×/vectorS)/bracketrightBig\n,(1)\nwhereJ >0 is the exchange coupling constant and γis\nthe strength of the DM interaction. The ground state of\nHis realized when /vectorS(/vector r) is a proper screw state such that\n/vectorS(/vector r) =S(/vector n1cos/vectork·/vector r+/vector n2sin/vectork·/vector r), (2)\nwhere the wavenumber /vectork=/vector n3|γ|/J, and/vector ni(i= 1,2,3)\nform the orthonormal vector sets. The ground state en-\nergy is given by −VS2γ2/2JwhereVis the volume of\nthe system. The sign of γis equal to that of( /vector n1×/vector n2)·/vector n3,\ndetermining the chirality of the spiral.\nThe equation of motion of the spin under the current\nis written as\n˙/vectorS=gµB\n¯h/vectorBeff×/vectorS−a3\n2eS(/vectorj·/vector∇)/vectorS+a3\n2eSβ/vectorS×(/vectorj·/vector∇)/vectorS+α\nS/vectorS×˙/vectorS\n(3)\nwhere/vectorBeff=−δH/δ/vectorSis the effective magnetic field and\nα,βare the Gilbert damping constants introduced phe-\nnomenologically [19, 20].\nWe discretize the Hamiltonian Eq.(1) and the equation\nof motion Eq.(3) by putting spins on the chain or the\nsquare lattice with the lattice constant a, and replacing\nthe derivative by the difference. The length of the spin\n|/vectorSi|is a constant of motion at each site i, and we can\neasily derive ˙H=δH\nδ/vectorS·˙/vectorS=−α|˙/vectorS|2from Eq.(3), i.e., the\nenergy continues to decrease as the time evolution.\nWe start with the one-dimensional case along x-axis\nas shown in Fig.1. The discretization means replacing\n∂x/vectorS(x) by (/vectorSi+1−/vectorSi−1)/2a, and∂2\nx/vectorS(x) by (/vectorSi+1−2/vectorSi+2\n/vectorSi−1)/a2. We note that the wavenumber which mini-\nmizes Eq.(1) is k=k0= arcsin( γ/J) on the discretized\none-dimensional lattice. Numerical study of Eq.(3) have\nbeen done with gµB/¯h= 1, 2e= 1,S= 1,a= 1\nJ= 2, and γ= 1.2. In this condition, the wavelength\nof the spiral λ= 2π/k0≈11.6 is long compared with\nthe lattice constant a= 1, and we choose the time scale\n∆t/(1 +α2) = 10−2. We have confirmed that the re-\nsults do not depend on ∆ teven if it is reduced by the\nfactor 10−1or 10−2. The sample size Lis 104with the\nopen boundary condition. As we will show later, the\ntypical value of the current is j∼2γand in the real\nsituation with the wavelength λ[nm], the exchange cou-\npling constant J[eV] and the lattice constant a[nm], it is\nj≈3.2×1015J/(λa)[A/m2]. Substituting J= 0.02,\nλ= 100,a= 0.5 into above estimatation, the typi-\ncal current is 1012[A/m2], and the unit of the time is\n∆t=J/¯h≈30[ps].\nThe Gilbert damping coefficients α,βare typically\n10−3∼10−1in the realistic systems. In most of the cal-\nculations, however,wetake α= 5.0toaccelaratethecon-\nvergence to the steady state. The obtained steady state\ndepends only on the ratio β/αexcept the spin configura-\ntions near the boundaries as confirmed by the simlations\nwithα= 0.1. We employ the two types of initial condi-\ntion, i.e., the ideal proper screw state with the wavenum-\nberk0, and the random spin configurations. The differ-\nence of the dynamics in these two cases are limited only\nin the early stage ( t <5000∆t).\nNow we consider the steady state with the constant\nvelocity for the shift of the spiral pattern obtained after\nthe time of the order of 105∆t. One important issue here\nis the current-dependence of the velocity, which has been\ndiscussed intensivelyfor the DW motion in ferromagnets.\nIn the latter case, there appears the intrinsic pinning in\nthe case of β= 0 [4], while the highly nonlinear behavior\nforβ/α/negationslash= 0 [20]. In the special case of β=α, the trivial\nsolution corresponding to the parallel shift of the ground\nstate configuration of Eq.(1) with the velocity v=jis\nconsidered to be realized [5]. Figure 2 shows the results\nfor the velocity, the induced uniform magnetization Sx\nalongx-axis, and the wavevector kof the spiral in the\nsteady state. The current-dependence of the velocity for\nthe cases of β= 0.1,0.5α,αand 2αis shown in Fig.\n2(a). Figure 2(b) shows the β/α-dependence of the ve-\nlocity for the fixed current j= 1.2. It is seen that the\nvelocity is almost proportional to both the current jand\nthe ratio β/α. Therefore, we conclude that the velocity\nv= (β/α)jwithout nonlinear behavior up to the current\nj∼2γ, which is in sharp contrast to the case of the DW\nmotion in ferromagnets. The unit of the velocity is given\nbya/∆t, which is of the order of 20[m /s] fora≈5[˚A]\nand ∆t≈30[ps]. In Fig. 2(c) shown the wavevector k\nof the spiral under the current j= 1.2 for various values\nofβ/α. It shows a non-monotonous behavior with the\nmaximum at β/α≈0.2, and is always smaller than the\nFIG. 1: Spin configurations in the spiral magnet (a) in\nthe equilibrium state, and (b) under the current. Under\nthe current /vectorj, the uniform magnetization Sxalong the spi-\nral axis/current direction is induced together with the ro-\ntation of the spin, i.e., the parallel shift of the spiral pat -\ntern with the velocity v. Note that the magnetization\nis anti-parallel/parallel to the current direction with po si-\ntive/negative γforβ < α, while it is reversed for β > α,\nand the wavenumber kchanges from the equilibrium value.\n 0  0.5  1  1.5  2  2.5  3  3.5  4 \n 0  0.5  1  1.5  2  2.5v\njβ/α=2.0 \nβ/α=1.0 \nβ/α=0.5 \nβ/α=0.1 (a) \n 0  0.5  1  1.5  2 \n 0  0.5  1  1.5  2 v / j \nβ / α(b) \n 0  0.2  0.4  0.6  0.8  1 \n 0  0.5  1  1.5  2 k\nβ / α(c) \n-1 -0.5  0  0.5  1 \n 0  0.5  1  1.5  2 Sx\nβ / α(d) \nFIG. 2: For the case of γ= 1.2, the numerical result is shown.\n(a) The steady state velocity vas a function of the current\njwith the fixed values of β= 0.1,0.5α,α, and 2α. (b) The\nvelocityvas a function of β/αfor a fixed value of the current\nj= 1.2.visalmost proportional to β/α. (c)Thewavenumber\nkas a function of β/α. The dotted line shows k0in the\nequilibrium. (d) The uniform magnetization Sxalong the\ncurrent direction as a function of β/αfor a fixed value of\nj= 1.2.\nwavenumber k0in the equilibrium shown in the dotted\nline. Namely, the period of the spiral is elongated by the\ncurrent. As shown in Fig. 2(d), there appears the uni-\nform magnetization Sxalong the x-direction. Sxis zero\nand changes the sign at β/α= 1. With the positive γ\n(as in the case of Fig. 2(d)), Sxis anti-parallel to the\ncurrentj//xwithβ < αand changes its direction for\nβ > α. For the negative γ, the sign of Sxis reversed. As\nfor the velocity /vector v, on the other hand, it is always parallel\nto the current /vectorj.3\nNow we present the analysis of the above results in\nterms of the continuum theory and a scaling argument.\nFor one-dimensional case, the modified LLG Eq.(3) can\nbe recast in the following form:\n˙/vectorS=−J/vectorS×∂2\nx/vectorS−(2γSx+j)∂x/vectorS+/vectorS×(α˙/vectorS+βj∂x/vectorS).(4)\nIt is convenientto introduceamovingcoordinates ˆξ(x,t),\nˆη, andˆζ(x,t) (see Fig.1) [21]. They are explicitly defined\nthrough ˆ x, ˆyand ˆzas\nˆζ(x,t) = cos( k(x−vt)+φ)ˆy+sin(k(x−vt)+φ)ˆz,\nˆξ(x,t) =−sin(k(x−vt)+φ)ˆy+cos(k(x−vt)+φ)ˆz,\nand ˆη= ˆx. We restrict ourselves to the following ansatz:\n/vectorS(x,t) =Sxˆη+/radicalbig\n1−S2xˆζ(x,t), (5)\nwhereSxis assumed to be constant.\nBy substituting Eq.(5) into Eq.(4), we obtain vas\nv=β\nαj, (6)\nby requiring that there is no force along ˆ η- andˆζ-\ndirections acting on each spin. In contrast to the DW\nmotioninferromagnets,thevelocity vbecomeszerowhen\nβ→0 even for large value of the current. The numerical\nresults in Fig. 2(a), (b) show good agreement with this\nprediction Eq.(6).\nOn the other hand, the magnetization Sxalongx-axis\nis given by\nSx=β/α−1\n2γ−Jkj, (7)\nonce the wavevector kis known. Here we note that the\nabove solution is degenerate with respect to k, which\nneeds to be determined by the numerical solution. From\nthe dimensional analysis, the spiral wavenumber kis\ngiven by the scaling form, k=k0g(j/(2γ),β/α) with\nthe dimensionless function g(x,y) and also is Sxthrough\nEq.(7).\nMotivated by the analysis above, we study the γ-\ndependence of the steady state properties. In Fig.3,\nwe show the numerical results for k/k0andSxas the\nfunctions of j/2γin the cases of β/α= 0.1,0.5 and\n2. Roughly speaking, the degeneracies of the data are\nobtained approximately for each color points (the same\nβ/αvalue) with different γvalues. The deviation from\nthe scaling behavior is due to the discrete nature of the\nlattice model, which is relevant to the realistic situation.\nForβ/α= 0.1(black points in Fig.3), kremainsconstant\nandSxis induced almost proportional to the current up\nj/2γ≈0.4, where the abrupt change of koccurs. For\nβ/α= 0.5 (blue points) and β/α= 2.0 (red points), the\nchanges of kandSxare more smooth. A remarkable\nresult is that the spin Sion the lattice point iis well 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 \n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 k / k 0\nj / 2γβ/α=2.0, γ=1.2 \nβ/α=2.0, γ=1.0 \nβ/α=2.0, γ=0.8 \nβ/α=2.0, γ=0.6 \nβ/α=0.5, γ=1.2 \nβ/α=0.5, γ=1.0 \nβ/α=0.5, γ=0.8 \nβ/α=0.5, γ=0.6 \nβ/α=0.1, γ=1.2 \nβ/α=0.1, γ=1.0 \nβ/α=0.1, γ=0.8 \nβ/α=0.1, γ=0.6 (a) \n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 \n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 Sx \nj / 2γβ/α=2.0, γ=1.2 \nβ/α=2.0, γ=1.0 \nβ/α=2.0, γ=0.8 \nβ/α=2.0, γ=0.6 \nβ/α=0.5, γ=1.2 \nβ/α=0.5, γ=1.0 \nβ/α=0.5, γ=0.8 \nβ/α=0.5, γ=0.6 \nβ/α=0.1, γ=1.2 \nβ/α=0.1, γ=1.0 \nβ/α=0.1, γ=0.8 \nβ/α=0.1, γ=0.6 (b) \nFIG.3: Thescalingplotfor (a) k/k0wherek0isthewavenum-\nber in the equilibrium without the current, and (b) Sxin the\nsteady state as the function of j/2γ. The black, blue, and\nred color points correspond to β/α= 0.1, 0.5 and 2.0, respec-\ntively. The curves in (b) indicate Eq.(7) calculated from th e\nkvalues in (a), showing the good agreement with the data\npoints.\ndescribed by Eq.(5) at x=xi, and hence the relation\nEq.(7) is well satisfied as shown by the curves in Fig.\n3(b), even though the scaling relation is violated to some\nextent. For larger values of jbeyond the data points,\ni.e.,j/2γ >0.75 forβ/α= 0.1,j/2γ >1.5 forβ/α= 0.5\nandj/2γ >0.9 forβ/α= 2.0, the spin configuration is\ndisordered from harmonic spiral characterized by a sin-\ngle wavenumber k. The spins are the chaotic funtion of\nboth space and time in this state analogousto the turbu-\nlance. This instability is triggered by the saturated spin\nSx=±1, occuring near the edge of the sample.\nNext, we turn to the simulations on the two-\ndimensional square lattice in the xy-plane. In this case,\nthe direction of the spiral wavevector becomes another\nimportant variable because the degeneracy of the ground\nstate energy occurs.\nStarting with the random spin configuration, we sim-\nulate the time evolution of the system without and with\nthe current as shown in Fig.4. Calculation has been done\nwith the same parameters as in the one-dimensional case4\n(a)/vectorj= 0\n(b)/vectorj= (0.3,0)\n(c)/vectorj= (0.3/√\n2,0.3/√\n2)\ncolor box of Sz(r)\ncolor box of |Sz(k)|2\nFIG. 4: The time evolution of the zcomponent Szof the\nspin from the random initial configuration of the 102×102\nsection in the middle of the sample is shown in the case of\n(a)j= 0, (b) j= 0.3 along the x-axis, (c) j= 0.3 along\nthe (1,1)-direction. From the left, t= 102∆t, 1.7×103∆t,\n5×103∆t. The rightmost panels show the spectral intensity\n|Sz(/vectork,5×103∆t)|2from the whole sample of the size 210×210\nin the momentum space /vectork= (kx,ky).\nwhereγ= 1.2,β= 0, and the system size is 210×210.\nIn the absence of the current, the relaxation of the\nspins into the spiral state is very slow, and many dislo-\ncations remain even after a long time. Correspondingly,\nthe energy does not decrease to the ground state value\nbut approaches to the higher value with the power-law\nlike long-time tail. The momentum-resolved intensity is\ncircularly distributed with the broad width as shown in\nFig. 4(a) corresponding to the disordered direction of\n/vectork. This glassy behavior is distinct from the relaxation\ndynamics of the ferromagnet where the large domain for-\nmation occurs even though the DW’s remain. Now we\nput the current along the ˆ x(Fig. 4(b)) and (ˆ x+ˆy) (Fig.\n4(c)) directions. It is seen that the direction of /vectorkis con-\ntrolled by the current also with the radial distribution\nin the momentum space being narrower than that in the\nabsence of j(Fig. 4(a)). This result suggests that the\ncurrentjwith the density ∼1012[A/m2] of the time du-\nration∼0.1[µsec] can anneal the directional disorder of\nthe spiral magnet. After the alignment of /vectorkis achieved,\nthe simulations on the one-dimensional model described\nabove are relevant to the long-time behavior.To summarize, we have studied the dynamics of the\nspiral magnet with DM interaction under the current j\nby solving the Landau-Lifshitz-Gilbert equation numeri-\ncally. In the steady state under the charge current j, the\nvelocityvis given by ( β/α)j(α,β: the Gilbert-damping\ncoefficients), the uniform magnetization is induced par-\nallel or anti-parallel to the current direction, and period\nof the spiral is elongated. The annealing effect especially\non the direction of the spiral wavevector /vectorkis also demon-\nstrated.\nTheauthorsaregratefultoN.FurukawaandY.Tokura\nfor fruitful discussions. This work was supported in part\nby Grant-in-Aids (Grant No. 15104006, No. 16076205,\nand No. 17105002) and NAREGI Nanoscience Project\nfromtheMinistryofEducation, Culture, Sports, Science,\nandTechnology. HK wassupported bythe JapanSociety\nfor the Promotion of Science.\n∗Electronic address: goto@appi.t.u-tokyo.ac.jp\n†Electronic address: katsura@appi.t.u-tokyo.ac.jp\n‡Electronic address: nagaosa@appi.t.u-tokyo.ac.jp\n[1] L. Berger, J. Appl. Phys. 49, 2156 (1978).\n[2] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[3] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972).\n[4] G. Tatara and H. Kohno, Phys. Rev. Lett. 92,\n086601(2004).\n[5] E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204\n(2005).\n[6] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nand T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[7] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno,\nNature428,539 (2004).\n[8] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S.\nS. P. Parkin, Nat. Phys. 3, 21 (2007).\n[9] K. Yamada et al., Nat. Mat. 6, 269 (2007).\n[10] W. C. Koehler, J. Appl. Phys 36, 1078 (1965).\n[11] R. A. Cowley et al., Phys. Rev. B 57, 8394 (1998).\n[12] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S.\nHeinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S.\nBlugel, and R. Wiesendanger, Nature (London) 447, 190\n(2007).\n[13] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Nature\n(London) 414, 427 (2001).\n[14] N. Doiron-Leyraud, I. R. Waker, L. Taillefer, M. J.\nSteiner, S. R. Julian, and G. G. Lonzarich, Nature (Lon-\ndon) 425, 595 (2003).\n[15] M. Uchida, Y. Onose, Y. Matsui, and Y. Tokura, Science\n311, 359 (2006).\n[16] M. Uchida et al., Phys. Rev. B 77, 184402 (2008).\n[17] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n[18] L. D. Landau, in Electrodynamics of Continuous Media\n(Pergamon Press, 1984), p178.\n[19] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[20] A. Thiaville et al., Europhys. Lett. 69, 990 (2005).\n[21] T. Nagamiya, in Solid State Physics , edited by F. Seitz,\nD. Turnbull, and H. Ehrenreich (Academic Press, New\nYotk, 1967), Vol. 20, p. 305."    },    {        "title": "0807.3715v1.Damped_driven_coupled_oscillators__entanglement__decoherence_and_the_classical_limit.pdf",        "content": "arXiv:0807.3715v1  [quant-ph]  23 Jul 2008Damped driven coupled oscillators: entanglement,\ndecoherence and the classical limit\nR. D. Guerrero Mancilla, R. R. Rey-Gonz´ alez and\nK. M. Fonseca-Romero\nGrupo de ´Optica e Informaci´ on Cu´ antica, Departamento de F´ ısica,\nUniversidad Nacional de Colombia - Bogot´ a\nE-mail:rdguerrerom@unal.edu.co\nE-mail:rrreyg@unal.edu.co\nE-mail:kmfonsecar@unal.edu.co\nAbstract. The interaction of (two-level) Rydberg atoms with dissipative QED\ncavity fields can be described classically or quantum mechanically, eve n for very\nlow temperatures and mean number of photons, provided the damp ing constant is\nlarge enough. We investigate the quantum-classical border, the e ntanglement and\ndecoherence of an analytically solvable model, analog to the atom-ca vity system, in\nwhich the atom (field) is represented by a (driven and damped) harm onic oscillator.\nThe maximum value of entanglement is shown to depend on the initial st ate and\nthe dissipation-rate to coupling-constant ratio. While in the original model the\natomic entropy never grows appreciably (for large dissipation rate s), in our model it\nreaches a maximum before decreasing. Although both models predic t small values of\nentanglement and dissipation, for fixed times of the order of the inv erse of the coupling\nconstant and large dissipation rates, these quantities decrease f aster, as a function of\nthe ratio of the dissipation rate to the coupling constant, in our mod el.\n‡This research was partially funded by DIB and Facultad de Ciencias, U niversidad\nNacional de Colombia.\nPACS numbers: 03.65.Ud, 03.67.Mn, 42.50.Pq, 89.70.CfDamped Driven Coupled Oscillators 2\n1. Introduction\nOne expects quantum theory to approach to the classical theory , for example in\nthe singular limit of a vanishing Planck’s constant, /planckover2pi1→0, or for large quantum\nnumbers. However, dissipative systems can bring forth some surp rises: for example,\nQED (quantum electrodynamics) cavity fields interacting with two-le vel systems, may\nexhibitclassicalorquantumbehavior, evenifthesystemiskeptatv erylowtemperatures\nand if the mean number of photons in the cavity is of the order of one [1, 2], depending\non the strength of the damping constant. Classical behavior, in th is context refers\nto the unitary evolution of one of the subsystems, as if the other s ubsystem could be\nreplaced by a classical driving. In QED cavities, the atom, which ente rs in one of the\nrelevant Rydberg states (almost in resonance with the field sustain ed in the cavity),\nconserves its purity and suffers a unitary rotation inside the cavity – exactly as if it\nwere controlled by a classical driving field – without entangling with the electromagnetic\nfield. This unexpected behavior was analyzed in reference [1] emplo ying several short-\ntime approximations, and it was found that in the time needed to rota te the atom, its\nstate remains almost pure.\nOther driven damped systems, composed by two (or more) subsys tems can be\nreadily identified. Indeed, in the last years there has been a fast de velopment of quite\ndifferent physical systems and interfaces between them, including electrodynamical\ncavities [3, 4], superconducting circuits [5, 6], confined electrons [7, 8, 9] and\nnanoresonators [10, 11, 12], on which it is possible to explore genuine quantum effects\nat the level of a few excitations and/or in individual systems. For ins tance, the\ninteraction atom-electromagnetic field is exploited in experiments wit h trapped ions\n[13, 14], cavity electrodynamics and ensembles of atoms interacting with coherent states\nof light [15], radiation pressure over reflective materials in experime nts coupling the\nmechanical motion of nanoresonators to light [12], and the coupling o f cavities with\ndifferent quality factors in the manufacturing of more reliable Ramse y zones [16]. In\nmany of these interfaces it is possible to identify a system which coup les strongly to the\nenvironment and another which couples weakly. For example, in the e xperiments of S.\nHaroche the electromagnetic field decays significantly faster [17] (or significantly slower\n[16]) than the atoms, the quality factor Qof the nanoresonators is much smaller than\nthat of the cavity, and the newest Ramsey zones comprise two cou pled cavities of quite\ndifferent Q. Several of these systems therefore, can be modelled as coupled harmonic\noscillators, one which can be considered dissipationless.\nIn this contribution we study an exactly solvable system, composed of two\noscillators, which permits the analysis of large times, shedding additio nal light on the\nclassical-quantum border. Entanglement and entropy, as measur ed by concurrence and\nlinear entropy, are used to tell “classical” from quantum effects.Damped Driven Coupled Oscillators 3\n2. The model\nThe system that we consider in this manuscript comprises two oscillat ors of natural\nfrequencies ω1andω2, coupled through an interaction which conserves the (total)\nnumber of excitations and whose coupling constant abruptly chang es from zero to g\nat some initial time, and back to zero at some final time. We take into a ccount that the\nsecond oscillator loses excitations at the rate γ, through a phenomenological Liouvillian\nof Lindblad form, corresponding to zero temperature, in the dyna mical equation of\nmotion [18]. Lindblad superoperators are convenient because they preserve important\ncharacteristics of physically realizable states, namely hermiticity, c onservation of the\ntrace and semi-positivity [19]. In order to guarantee the presence of excitations, the\nsecond oscillator is driven by a classical resonant field.\nThe interaction can be considered to be turned on (off) in the remot e past (remote\nfuture) if it is always present (coupled Ramsey zones or nanoreson ators coupled to\ncavity fields), or can really be present for a finite time interval (for example in atoms\ntravelling through cavities). The initial states of the coupled oscillat ors also depend\non the experimental setup, varying from the base state of the co mpound system to\na product of the steady state of the coupled damped oscillator with the state of the\nother oscillator. Since we want to make comparisons with Ramsey zon es, the choices in\nthe formulation of this model have been inspired by the analogy with t he atom-cavity\nsystem, for a cavity –kept at temperatures of less than 1K– whos e lifetime is much\nshorter than the lifetime of Rydberg states, allowing us to ignore th e Lindblad operator\ncharacterizing the atomic decay process. The first oscillator ther efore is a cartoon of\nthe atom, at least in the limit where only its first two states are signific antly occupied,\nwhile the second oscillator corresponds to the field.\nAll the ingredients detailed before can be summarily put into the Liouv ille-von\nNeumann equation for the density matrix ˆ ρof the total system\ndˆρ\ndt=−i\n/planckover2pi1[ˆH,ˆρ]+γ(2ˆaˆρˆa†−ˆa†ˆaˆρ−ˆρˆa†ˆa) (1)\nwhereˆHis the total Hamiltonian of the system and the second term of the rh s of (1)\nis the Lindblad superoperator which accounts for the loss of excita tions of the second\noscillator. In absence of the coupling with the first oscillator, the inv erse of twice the\ndissipation rate γgives the mean lifetime of the second oscillator. The first two terms\nof the total Hamiltonian\nˆH=/planckover2pi1ω1ˆb†ˆb+/planckover2pi1ω2ˆa†ˆa+/planckover2pi1g(Θ(t)−Θ(t+T))(ˆa†ˆb+ˆaˆb†)+i/planckover2pi1ǫ(e−iωDtˆa†−eiωDtˆa),(2)\nare the free Hamiltonians of the two harmonic oscillators; the next t erm, which is\nmodulated by the step function Θ( t), is the interaction between them and the last is\nthe driving. The bosonic operators of creation ˆb(ˆa) and annihilation ˆb†(ˆa†) of one\nexcitation of the first (second) oscillator, satisfy the usual comm utation relations. From\nhere on we focus on the case of resonance, ω1=ω2=ωD=ω. The interaction time\nTis left undefinite until the end of the manuscript, where we compare our results with\nthose of the atom-cavity system.Damped Driven Coupled Oscillators 4\n3. Dynamical evolution\nThe solution of the dynamical equation (1) can be written as\nˆρ(t) =D(β(t),α(t))˜ρ(t)D†(β(t),α(t)), (3)\nwhereD(β(t),α(t)) is the two-mode displacement operator,\nD(β(t),α(t)) =D1(β(t))D2(α(t)) =eβ(t)ˆb†−β∗(t)ˆbeα(t)ˆa†−α∗(t)ˆa,\nand ˜ρ(t) is the total density operator in the interaction picture defined by equation (3).\nBy replacing (3) into (1), and employing the operator identities\nd\ndtD(α) =/parenleftbigg\n−α∗˙α−˙α∗α\n2+ ˙αˆa†−˙α∗ˆa/parenrightbigg\nD(α) =D(α)/parenleftbiggα∗˙α−˙α∗α\n2+ ˙αˆa†−˙α∗ˆa/parenrightbigg\n,\nwith the dot designating the time derivative as usual, we are able to de couple the\ndynamics of the displacement operators, obtaining the following dyn amical equations\nfor the labels αandβ\nd\ndt/parenleftigg\nα\nβ/parenrightigg\n=/parenleftigg\n−γ−iω−ig\n−ig−iω/parenrightigg/parenleftigg\nα\nβ/parenrightigg\n+/parenleftigg\nǫe−iωt\n0/parenrightigg\n, (4)\nfor times between zero and T. On the other hand, the Ansatz (3) also provides the\nequation of motion for ˜ ρ(t), which turns out to be very similar to (1) but with the\nhamiltonian ˜H=ˆH(ǫ= 0), that is, without driving. The separation provided by our\nAnsatz is also appealing from the point of view of its possible physical in terpretation,\nbecause the effect of the driving has been singled out, and quantum (entangling and\npurity) effects are extracted from the displaced density operato r ˜ρ(t).\nThe two oscillators interact after the second oscillator reaches its stationary\ncoherent state\nˆρ2(t) = tr1ˆρ(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleǫ\nγe−iωt/angbracketrightbigg/angbracketleftbiggǫ\nγe−iωt/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)\nas can be verified by solving (4) with the interaction turned off. If we want a mean\nnumber of excitations of the order of one then the driving amplitude must satisfy ǫ≈γ,\nand thereby the larger the dissipation is, the larger the driving is to b e chosen. At zero\ntime, when the oscillators begin to interact, the state of the total system is separable\nwith the second oscillator state given by (5). The first oscillator, on the other hand,\nbegins in a pure state which we choose as a linear combination of its gro und and first\nexcited states (again inspired on the analogy with the atom-cavity s ystem). Thus, the\ninitial state ˆ ρ(0) given by\nD/parenleftbigg\n0,ǫ\nγ/parenrightbigg\n(cos(θ)|0∝angbracketright+sin(θ)|1∝angbracketright)(cos(θ)∝angbracketleft0|+sin(θ)∝angbracketleft1|)⊗|0∝angbracketright∝angbracketleft0|/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n˜ρ(0)D†/parenleftbigg\n0,ǫ\nγ/parenrightbigg\n,(6)\ncorresponds to a state of the form described by equation (3) with β(0) = 0 and\nα(0) =ǫ/γ. At later times, the solution maintains the same structure, but –as canDamped Driven Coupled Oscillators 5\nbe seen from the solution of (4) – the labels of the displacement oper ators evolve as\nfollows\nα(t) =ǫe−1\n2(γ+2iω)t/braceleftbigg1\nγcos(˜gt)+sin(˜gt)\n2˜g/bracerightbigg\n, (7)\nβ(t) =−ie−iωtǫ\ng+iǫ\nge−1\n2(γ+2iω)t/braceleftbigg\ncos(˜gt)+/parenleftbig\n−2g2+γ2/parenrightbigsin(˜gt)\n2γ˜g/bracerightbigg\n, (8)\nwhere we have defined the new constant ˜ g=1\n2/radicalbig\n4g2−γ2.We employ ˜ g, which also\nappears in the solution of the displaced density operator, to define three different\nregimes: underdamped (˜ g2>0), critically damped (˜ g2= 0) and overdamped (˜ g2<0)\nregime. It is important to notice that there is no direct connection w ith the quality\nfactorofthedampedoscillator: itispossibletohavephysical syste ms intheoverdamped\nregime defined here even with relatively large quality factors, if the in teraction constant\ngis much smaller than ω, the frequency of the oscillators.\nThe inspection of the equations (7) and (8), allows one to clearly iden tify the time\nscale 2/γ, after which the stationary state is reached and the state of the first oscillator\njust rotates with frequency ωand have a mean number of excitations equal to ǫ2/g2.\nThe doubling of the damping time of the second oscillator, from 1 /γin the absence\nof interaction to 2 /γ, in the underdamped regime, can be seen as an instance of the\nshelving effect [20]. The first oscillator, which in absence of interactio n, suffers no\ndamping, it is now driven and damped. It can be thought that the exc itations remain\nhalf of the time on each oscillator, and that they decay with a damping constant γ,\nthereby leading to an effective damping constant of γ/2. An interesting feature of the\nsolution is that the displacement of the second oscillator goes to zer o, in the stationary\nstate. In the stationary state, the first oscillator evolves as if it w ere driven by a classical\nfield−i/planckover2pi1ǫexp(−iωt) and damped with a damping rate g, without any interaction with\na second oscillator. More generally speaking, we remark that from t he point of view of\nthe first oscillator, the evolution of its displacement operator happ ens as if there were\ndamping but no coupling, and the driving were of the form /planckover2pi1g(β−iα), or, in terms of\nthe parameters of the problem,\nF(t) =−i/planckover2pi1ǫe−iωt−i/planckover2pi1ǫe−(γ/2+iω)t/parenleftbigg/parenleftbiggg\nγ−1/parenrightbigg\ncos(˜gt)+2g2+gγ−γ2\n2γ˜gsin(˜gt)/parenrightbigg\n.(9)\nThis behavior is particularly relevant in the following extreme case, wh ose complete\nsolution depends only on the displacement operators. If the initial s tate of the first\noscillator is the ground state then ˜ ρdoes not evolve in time, i. e. it remains in the state\n|00∝angbracketright, and the total pure and separable joint state is\nρ(t) =|β(t)∝angbracketright∝angbracketleftβ(t)|⊗|α(t)∝angbracketright∝angbracketleftα(t)|. (10)\nEven in the more general case considered here, corresponding to the initial state\n(6), the solution of ˜ ρ(t) possesses only a few non-zero elements. If we write the total\ndensity operator as\n˜ρ(t) =/summationdisplay\ni1i2j1j2˜ρj1j2\ni1i2|i1i2∝angbracketright∝angbracketleftj1j2|, (11)Damped Driven Coupled Oscillators 6\nwe canarrangethe elements corresponding to zero and one excita tions ineach oscillator,\nas the two-qubit density matrix\n˜ρ00\n00(t) ˜ρ01\n00(t) ˜ρ10\n00(t) 0\n˜ρ00\n01(t) ˜ρ01\n01(t) ˜ρ10\n01(t) 0\n˜ρ00\n10(t) ˜ρ01\n10(t) ˜ρ10\n10(t) 0\n0 0 0 0\n. (12)\nIf we measure time in units of gby defining t′=gtwe have only two free parameters\nΓ =γ\ngand Ω =ω\ng. The nonvanishing elements of the density matrix, written in the\nunderdamped case ( |Γ|<2), are given by (hermiticity of the density operator yields the\nmissing non-zero elements)\n˜ρ00\n00(t′) = 1−sin2θe−Γt′/parenleftbigg4−Γ2cos(√\n4−Γ2t′)\n4−Γ2−Γsin(√\n4−Γ2t′)√\n4−Γ2/parenrightbigg\n˜ρ01\n01(t′) = 2sin2(θ)e−Γt′1−cos/parenleftbig√\n4−Γ2t′/parenrightbig\n4−Γ2\n˜ρ10\n10(t′) = sin2(θ)e−Γt′/parenleftigg\n(2−Γ2)cos/parenleftbig√\n4−Γ2t′/parenrightbig\n+2\n4−Γ2−Γsin/parenleftbig√\n4−Γ2t′/parenrightbig\n√\n4−Γ2/parenrightigg\n˜ρ00\n01(t′) =isin(2θ)eiΩt′−Γt′\n2sin/parenleftbig√\n4−Γ2t′\n2/parenrightbig\n√\n4−Γ2\n˜ρ00\n10(t′) =sin(2θ)\n2eiΩt′−Γt′\n2/parenleftigg\ncos/parenleftbigg√\n4−Γ2t′\n2/parenrightbigg\n−Γsin/parenleftbig√\n4−Γ2t′\n2/parenrightbig\n√\n4−Γ2/parenrightigg\n˜ρ01\n10(t′) = 2isin2(θ)e−Γt′sin/parenleftbig√\n4−Γ2t′\n2/parenrightbig\n√\n4−Γ2/parenleftigg\nΓsin/parenleftbig√\n4−Γ2t′\n2/parenrightbig\n√\n4−Γ2−cos/parenleftbigg√\n4−Γ2t′\n2/parenrightbigg/parenrightigg\nThe expressions of the elements of the density matrix in the critically damped case\nΓ = 2 and in the overdamped case Γ >2 can be obtained from those given in the text\nfor the underdamped case Γ <2.\n4. Entanglement\nAlthough quantities like quantum discord [21] have been proposed to extract the\nquantum content of correlations between two systems, we prese ntly quantify the\nquantum correlations between both oscillators employing a measure of entanglement.\nDue to the dynamics of the system, and the initial states chosen, t he whole system\nbehaves as a couple of qubits and therefore its entanglement can b e measured by\nWootters’ concurrence [22]. One of the most important characte ristics of the form\nof the solution given by (3) is that concurrence, as well as linear ent ropy, depend only\non the displaced density operator ˜ ρ(t′). In our case the concurrence reduces to\nC(t′) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalig\n˜ρ01\n10(t′)˜ρ10\n01(t′)+/radicalig\n˜ρ01\n01(t′)˜ρ10\n10(t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalig\n˜ρ01\n10(t′)˜ρ10\n01(t′)−/radicalig\n˜ρ01\n01(t′)˜ρ10\n10(t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= 2/radicalig\n˜ρ01\n10(t′)˜ρ10\n01(t′) = 2|˜ρ10\n01(t′)|, (13)Damped Driven Coupled Oscillators 7\nwhere the positivity and hermiticity of the density matrix were used. The explicit\nexpressions for the concurrence in the underdamped (UD), critic ally damped (CD) and\noverdamped (OD) regimes are\nCUD(t′) = 4sin2(θ)e−Γt′sin/parenleftig√\n4−Γ2t′\n2/parenrightig\n√\n4−Γ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓsin/parenleftig√\n4−Γ2t′\n2/parenrightig\n√\n4−Γ2−cos/parenleftbigg√\n4−Γ2t′\n2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nCCD(t′) = 2sin2(θ)e−2t′t′|t′−1|, (14)\nCOD(t′) = 4sin2(θ)e−Γt′sinh/parenleftig√\nΓ2−4t′\n2/parenrightig\n√\nΓ2−4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓsinh/parenleftig√\nΓ2−4t′\n2/parenrightig\n√\nΓ2−4−cosh/parenleftbigg√\nΓ2−4t′\n2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nAllthedependence ontheinitialstateiscontainedonthesquared n ormofthecoefficient\nof the state |1∝angbracketrightof the displaced density operator. In all regimes the concurrence vanishes\nat zero time, because the initial state considered is separable. How ever, while in the\nunderdamped case the concurrence vanishes periodically (see equ ation (15) below), in\nthe other two cases it crosses zero once ( t >0) and reaches zero assymptotically as time\ngrows. This shows a markedly different qualitative behavior (see figu res 1 and 2).\nIn the underdamped regime the zeroes of the concurrence are fo und at times\nτ1n=2nπ√\n4−Γ2,andτ2n=2πn+2arccos/parenleftbigΓ\n2/parenrightbig\n√\n4−Γ2, (15)\nwherenis a non-negative integer. In this contribution, the inverse sine and cosine\nfunctions are chosen to take values in the interval [0 ,π/2]. The time τ10corresponds to\nthe initial state. The sequence of concurrence zeroes is thereby 0 =τ10< τ20< τ11<\nτ21...As the critical damping is approached, the time τ11is pushed towards infinity,\nwhileτ20approaches the finite time 2 /Γ (see figure 2). For the initial states considered\nin this manuscript we do not observe the sudden death of the entan glement since the\nconcurrence is zero only for isolated instants of time.\nIf one writes the concurrence in the underdamped regime in the alte rnative form\nCUD(t′) =sin2θ\n2(1−Γ2/4)e−Γt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ\n2−sin/parenleftigg\narcsin/parenleftbiggΓ\n2/parenrightbigg\n+2/radicalbigg\n1−Γ2\n4t′/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(16)\nit is easy to verify that at the times τ±n, given by\nτ±n=1√\n4−Γ2/parenleftbigg\n(2n+1)π±arccos/parenleftbiggΓ2\n4/parenrightbigg\n−2arcsin/parenleftbiggΓ\n2/parenrightbigg/parenrightbigg\n>0, n= 0,1,...(17)\nthe concurrence reaches the local maxima\nC±n= sin2θ/parenleftigg/radicalbigg\n1+Γ2\n4±Γ\n2/parenrightigg\nexp\n−Γ/parenleftig\n(2n+1)π±arccos(Γ2\n4)−2arcsin(Γ\n2)/parenrightig\n2/radicalig\n1−Γ2\n4\n.\nWe observe these maxima to lie on the curves sin2θK±exp(−Γt), where the constants\nK±=/radicalig\n1+Γ2\n4±Γ\n2satisfy the inequalities√\n2−1≤K−≤1≤K+≤√\n2+1. Maxima\nof concurrence depend on both the initial state and the value of th e rescaled dampingDamped Driven Coupled Oscillators 8\nconstant, and reach the maximum available value of one only in the non -dissipative case\nfor a particular initial state. In order to have negligible values of con currence (except for\nsmall time intervals around the zeroes of concurrence) it is necess ary to have times much\nlarger than 1 /γ. From the point of view of classical-like behavior, the most favourab le\nscenario corresponds to zero or almost zero concurrence, which are obtained for short\ntime intervals around τ1n, τ2nand for large values of time.\nIn the overdamped regime, the concurrence presents two maxima ,τ−andτ+> τ−\nτ±=2arccosh(Γ /2)±arccosh(Γ2/4)\n2/radicalbig\n1−Γ2/4, (18)\nboth of which go to zero as the rescaled dissipation rate grows, τ+→4ln(Γ)/Γ and\nτ−→ln(2)/Γ (see figure 2). The function arccosh( x) is chosen as to return nonnegative\nvalues for x≥1. Since the global maximum of concurrence, which corresponds to the\nlater time, scales like 1 /(2Γ) for large values of Γ, in the highly overdamped regime\nquantum correlations are not developed at any time.\nFigure 1. Concurrence as a function of time and the rescaled damping consta nt in\nthe (a) underdamped, (b) critically damped, and (c) overdamped c ase.\nThe behavior of concurrence in the different regimes is shown in figur e 1. It is\napparent that small values of concurrence are obtained for very small times and for large\ntimes in the underdamped case and for all times for a highly overdamp ed oscillator. In\nfigure2we depict thetimesatwhichconcurrence attainsamaximum, andthemaximum\nvalues of concurrence, as a function of the rescaled damping cons tant. One can see how\nthe first two times of maximum concurrence go to zero, while the oth er times diverge,\nas the critically damped regime is reached. The first two maxima of con currence vanish\nmore slowly than the rest of maxima, which hit zero at Γ = 2.\n5. Entropy\nThe entropy is analized employing the linear entropy of the first oscilla tor, the system of\ninterest. As remarked before, the first oscillator behaves like a tw o-level system, whereDamped Driven Coupled Oscillators 9\nFigure 2. (a)Concurrence local maxima at times τ+(−)nin black (gray) color for\nn= 0 full line, n= 1 dashed line and n= 2 dashed-dot line and (b) times of maximum\nvalues of concurrence, as a function of the rescaled damping cons tant.\nthe maximum value of the linear entropy, 0.5, is obtained when the pop ulationof each of\nthe two states is one half. The type of “classical” behavior which allow s the interaction\nwith Ramsey zones to be modelled like a classical driving force occurs w hen the linear\nentropy is very small, and hence the state of the first oscillator is (a lmost) pure and\nuncorrelated with the state of the second oscillator. The linear ent ropy for the first\noscillator can be computed as\nδ1(t′) = 1−tr1(tr2ρtr2ρ) = 1−tr1(tr2˜ρtr2˜ρ) = 2det(tr 2˜ρ),(19)\nwhere thelast equality holdsfortwo-level systems. Inequation(1 9) thedensity operator\nof the first oscillator is assumed to be represented by a 2 ×2 matrix. Employing the\nexpressions we have found for the elements of ˜ ρwe obtain\nδ1(t′) = 2sin4θ x(t′)(1−x(t′)), (20)\nwherex, in the underdamped regime, is given by\nxUD(t) =e−Γtsin2/parenleftig/radicalbig\n1−Γ2/4t−arccos(Γ/2)/parenrightig\n1−Γ2/4. (21)\nSurprisingly, as in the case of concurrence, the influence of the init ial state factors out\nin the expression of the linear entropy of the first oscillator, which t urns out to be\nproportional to the square of the population of |1∝angbracketrightin the initial displaced operator.\nAs it is well known, in the limit of zero dissipation, the linear entropy of t he reduced\ndensity matrix is equal to one fourth of the square of the concurr ence. At times τ2n\n(see eq.(15)), when both concurrence and linear entropy vanish, the total state of the\nsystem is separable, ρ(gτ2n) =|β(gτ2n)∝angbracketright∝angbracketleftβ(gτ2n)| ⊗ρ2(gτ2n); that is, from the point\nof view of the first oscillator the evolution is unitary like. Since the linea r entropy\nbegins at zero, because the initial state is pure and separable, the re is a maximum in\nthe interval (0 ,τ20), which turns out to give a linear entropy of exactly 0.5 (we treat\nthe case sin θ= 1, because —due to the scaling property discussed before— a simp leDamped Driven Coupled Oscillators 10\nmultiplication by sin4θgives the result for other cases). Indeed, as the function x(t)\nchanges continuously from x(t= 0) = 1 to x(t=τ20) = 0, it crosses 0.5 at some time\nτ30in between, giving the maximum value possible of the linear entropy. Alt hough the\nexact value of τ30can be obtained only numerically, good analytical approximations can\nbe readily obtained. For example, τ30≈π/(4+ 4g+ 2g2), gives an error smaller than\n0.5%.\nFor small values of the rescaled damping constant, Γ /lessorapproxeql0.237, there are several\nsolutions to the equation x(t) = 0.5 in the interval (0 ,log(2)/Γ), which give absolute\nmaxima of the linear entropy, while the times\nτ4n=2arccos(Γ /2)+nπ/radicalbig\n1−Γ2/4, n= 0,1,2,··· (22)\ncorrespond to local minima. In the interval (log(2) /Γ,∞) the times τ4ngive local\nmaxima. All of the local maxima and minima given by eq. (22) belong to th e curve\n2e−Γt(1−e−Γt). The large time behavior of the local maxima of linear entropy and\nconcurrence is, thereby, of the same form constant ×exp(−Γt′). For values of Γ >0.237\nall times τ4ngive local maxima. The maxima of concurrence and linear entropy coin cide\nonly in the weakly damped case, because concurrence and linear ent ropy are not\nindependent for pure bipartite states.\nAt times τ1n(see eq.(15)), where the total state ρ(gτ1n) =ρ1(gτ1n)⊗\n|α(gτ1n)∝angbracketright∝angbracketleftα(gτ1n)|, is separable, the reduced state of the first oscillator is mixed. The\nlinear entropy is small for short ( τ1n≪log(2)/Γ) and large ( τ1n≫log(2)/Γ) times.\nIn the overdamped regime the function x(t), which appers on the expression for\nlinear entropy (20) and is given by\nxOD(t′) =e−Γt′sinh2(/radicalbig\nΓ2/4−1t′−arccosh(Γ /2))\nΓ2/4−1, (23)\nbegins at one for t′= 0, and goes down to zero for large values of time. The time at\nwhich it crosses one half can be calculated to be τ0.5≈0.16557<1/6 for Γ = 2 and\nfor large values of Γ it goes as τ0.5≈ln2/(2Γ). It is easy to find interpolating functions\nwith small error for the time of crossing,\n˜ρ10\n10\nt′=1\n6+4\nln2sinh/parenleftbigg\narccosh(Γ\n2)tanh/parenleftbigg\narccosh (Γ\n2)\n1.6/parenrightbigg/parenrightbigg\n= 0.5(1+∆)\nwhere|∆|<2.5%. It is interesting to notice that for large values of the damping th is\ntime (ln(2) /(2Γ)) is half the time needed to obtain the maximum value of concurre nce,\nand that, at the later time, the linear entropy is 3/4 of the maximum v alue of entropy, a\nrelatively large value. The state of the first oscillator always become s maximally mixed\nbefore becoming pure again, no matter how large the value of the da mping. We show\nthe behavior of linear entropy in figure 3. In the underdamped regim e there are infinite\nmaxima and minima, while for critical damping and for the overdamped r egime there\nare only two maxima. The first maximum always corresponds to a linear entropy of one\nhalf.Damped Driven Coupled Oscillators 11\nFigure 3. Linear entropy of the first oscillator as a function of time and the re scaled\ndamping constant in the (a) underdamped, (b) critically damped, an d (c) overdamped\ncase.\n6. Conclusions\nIn the present contribution we have shown that the classical quan tum border in this\nmodel depends mainly on the initial state and on damping constant to interaction\ncoupling ratio, and that quantum effects, characteristic of the un derdamped regime,\ncan be seen in the other regimes for small times. In order to make co nnection with\nRamsey zones we remember in that physical system ω≈1010Hz,Q≈104,g≈104\nHz andTR≈10−5s, which was chosen as to produce π/2 pulse, that is a pulse that\ncan rotate the state of the two-level system, as represented in a Bloch sphere, by an\nangleπ/2. These numbers place thesystem into the highly overdamped (reg ime because\nΓ =ω/(Qg)≈102≫2) and give a rescaled evolution time of the order of gT≈10−1.\nHere we use the same values of ω,γandg, and an evolution time of order 1/g. Indeed,\nthe hamiltonian /planckover2pi1ωˆb†ˆb+/planckover2pi1g(Θ(t)−Θ(t+T))(α0e−iωtˆb†+α∗\n0eiωtˆb), with∝bardblα0∝bardbl ≈1 —\nwhich would model the interaction of the first oscillator with a classica l driving field of\nan average number of excitations of the order of one — has a chara cteristic time 1 /g,\ncorresponding to T′≈1.\nThe dynamical behavior of the linear entropy obtained here, is quite different\nfrom that of ref. [1]: there the linear entropy was never large for t he relevant time\ninterval, here it grows to the maximum possible for a two-level syste m, and then goes\nto zero very quickly. Therefore, in this model dissipation produces relaxation also, and\na description obviating the second oscillator still needs a dissipation p rocess. Although,\nat the evolution time T, both models predict a small atomic entropy, in Ramsey zones\nit decreases as δ1(TR′≈0.1)≈4/Γ, while in the present model it goes to zero as\nδ1(T′≈1)∝1/Γ4. Qualitative and quantitative differences notwithstanding, at the\nevolution time the linear entropy is very small, in both cases, due to th e smallness of\nthe ratio g/γ. As remarked before the quality factor of the damped oscillator do es\nnot appear directly in either case; it can be perfectly possible to hav e a very weaklyDamped Driven Coupled Oscillators 12\ndamped oscillator and a highly overdamped interaction. However, as the first oscillator\nquality factor is improved, the damping constant will eventually be co mparable with\nthe interaction constant, and there will be considerable entanglem ent between both\noscillators. 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We recover the Landau-Lift shitz-Gilbert equation and express the\neffective fields and Gilbert damping tensor in terms of the sca ttering matrix. Dissipation of magnetic\nenergy equals energy current pumped out of the system by the t ime-dependent magnetization, with\nseparable spin-relaxation induced bulk and spin-pumping g enerated interface contributions. In\nlinear response, our scattering theory for the Gilbert damp ing tensor is equivalent with the Kubo\nformalism.\nMagnetization relaxation is a collective many-body\nphenomenon that remains intriguing despite decades of\ntheoretical and experimental investigations. It is im-\nportant in topics of current interest since it determines\nthe magnetization dynamics and noise in magnetic mem-\nory devices and state-of-the-art magnetoelectronic ex-\nperiments on current-induced magnetization dynamics\n[1]. Magnetization relaxation is often described in terms\nof a damping torque in the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n, (1)\nwhereMis the magnetization vector, γ=gµB//planckover2pi1is the\ngyromagnetic ratio in terms of the gfactor and the Bohr\nmagnetonµB, andMs=|M|is the saturation magneti-\nzation. Usually, the Gilbert damping ˜G(M) is assumed\nto be a scalar and isotropic parameter, but in general it\nis a symmetric 3 ×3 tensor. The LLG equation has been\nderived microscopically [2] and successfully describes the\nmeasured response of ferromagnetic bulk materials and\nthin films in terms of a few material-specific parameters\nthatareaccessibletoferromagnetic-resonance(FMR) ex-\nperiments [3]. We focus in the following on small fer-\nromagnets in which the spatial degrees of freedom are\nfrozen out (macrospin model). Gilbert damping pre-\ndicts a striclylinear dependence ofFMR linewidts on fre-\nquency. This distinguishes it from inhomogenous broad-\nening associated with dephasing of the global precession,\nwhich typically induces a weaker frequency dependence\nas well as a zero-frequency contribution.\nThe effective magnetic field Heff=−∂F/∂Mis the\nderivative of the free energy Fof the magnetic system\nin an external magnetic field Hext, including the classi-\ncal magnetic dipolar field Hd. When the ferromagnet is\npart of an open system as in Fig. 1, −∂F/∂Mcan be\nexpressed in terms of a scattering S-matrix, quite anal-\nogous to the interlayer exchange coupling between ferro-\nmagnetic layers [4]. The scattering matrix is defined in\nthe space of the transport channels that connect a scat-\ntering region (the sample) to thermodynamic (left andleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath via metallic normal metal leads (N).\nright) reservoirs by electric contacts that are modeled by\nideal leads. Scattering matrices also contain information\nto describe giant magnetoresistance, spin pumping and\nspin battery, and current-induced magnetization dynam-\nics in layered normal-metal (N) |ferromagnet (F) systems\n[4, 5, 6].\nIn the following we demonstrate that scattering the-\nory can be also used to compute the Gilbert damping\ntensor˜G(M).The energy loss rate of the scattering re-\ngion can be described in terms of the time-dependent\nS-matrix. Here, we generalize the theory of adiabatic\nquantum pumping to describe dissipation in a metallic\nferromagnet. Our idea is to evaluate the energy pump-\ningoutoftheferromagnetandtorelatethistotheenergy\nloss of the LLG equation. We find that the Gilbert phe-\nnomenology is valid beyond the linear response regime of\nsmall magnetization amplitudes. The only approxima-\ntion that is necessary to derive Eq. (1) including ˜G(M)\nis the (adiabatic) assumption that the frequency ωof the\nmagnetization dynamics is slow compared to the relevant\ninternal energy scales set by the exchange splitting ∆.\nThe LLG phenomenology works so well because /planckover2pi1ω≪∆\nsafely holds for most ferromagnets.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from spin-orbit interaction in\ncombinationwith impurityscatteringthattransfersmag-\nnetic energy to itinerant quasiparticles [3]. The subse-\nquent drainage of the energy out of the electronic sys-\ntem,e.g.by inelastic scattering via phonons, is believed\nto be a fast process that does not limit the overall damp-\ning. Our key assumption is adiabaticiy, meaning that\nthe precession frequency goes to zero before letting the\nsample size become large. The magnetization dynam-\nics then heats up the entire magnetic system by a tiny2\namount that escapes via the contacts. The leakage heat\ncurrent then equals the total dissipation rate. For suf-\nficiently large samples, bulk heat production is insensi-\ntive to the contact details and can be identified as an\nadditive contribution to the total heat current that es-\ncapes via the contacts. The chemical potential is set\nby the reservoirs, which means that (in the absence of\nan intentional bias) the sample is then always very close\nto equilibrium. The S-matrix expanded to linear order\nin the magnetization dynamics and the Kubo linear re-\nsponse formalisms should give identical results, which we\nwill explicitly demonstrate. The role of the infinitesi-\nmal inelastic scattering that guarantees causality in the\nKubo approach is in the scattering approach taken over\nby the coupling to the reservoirs. Since the electron-\nphonon relaxation is not expected to directly impede the\noverall rate of magnetic energy dissipation, we do not\nneed to explicitly include it in our treatment. The en-\nergy flow supported by the leads, thus, appears in our\nmodel to be carried entirely by electrons irrespective of\nwhethertheenergyisactuallycarriedbyphonons, incase\nthe electrons relax by inelastic scattering before reaching\nthe leads. So we are able to compute the magnetization\ndamping, but not, e.g., how the sample heats up by it .\nAccording to Eq. (1), the time derivative of the energy\nreads\n˙E=Heff·dM/dτ= (1/γ2)˙ m/bracketleftBig\n˜G(m)˙ m/bracketrightBig\n,(2)\nin terms of the magnetization direction unit vector m=\nM/Msand˙ m=dm/dτ. We now develop the scatter-\ning theory for a ferromagnet connected to two reservoirs\nby normal metal leads as shown in Fig. 1. The total\nenergy pumping into both leads I(pump)\nEat low tempera-\ntures reads [11, 12]\nI(pump)\nE= (/planckover2pi1/4π)Tr˙S˙S†, (3)\nwhere˙S=dS/dτandSis the S-matrix at the Fermi\nenergy:\nS(m) =/parenleftbiggr t′\nt r′/parenrightbigg\n. (4)\nrandt(r′andt′) are the reflection and transmissionma-\ntrices spanned by the transport channels and spin states\nfor an incoming wave from the left (right). The gener-\nalization to finite temperatures is possible but requires\nknowledge of the energy dependence of the S-matrix\naround the Fermi energy [12]. The S-matrix changes\nparametrically with the time-dependent variation of the\nmagnetization S(τ) =S(m(τ)). We obtain the Gilbert\ndamping tensor in terms of the S-matrix by equating the\nenergy pumping by the magnetic system (3) with the en-\nergy loss expression (2), ˙E=I(pump)\nE. Consequently\nGij(m) =γ2/planckover2pi1\n4πRe/braceleftbigg\nTr/bracketleftbigg∂S\n∂mi∂S†\n∂mj/bracketrightbigg/bracerightbigg\n,(5)which is our main result.\nThe remainder of our paper serves three purposes. We\nshow that (i) the S-matrix formalism expanded to linear\nresponseis equivalentto Kubolinearresponseformalism,\ndemonstrate that (ii) energy pumping reduces to inter-\nface spin pumping in the absence ofspin relaxationin the\nscattering region, and (iii) use a simple 2-band toy model\nwith spin-flip scattering to explicitly show that we can\nidentify both the disorder and interface (spin-pumping)\nmagnetization damping as additive contributions to the\nGilbert damping.\nAnalogous to the Fisher-Lee relation between Kubo\nconductivity and the Landauer formula [15] we will now\nprove that the Gilbert damping in terms of S-matrix (5)\nis consistent with the conventionalderivation of the mag-\nnetization damping by the linear response formalism. To\nthis end we chose a generic mean-field Hamiltonian that\ndepends on the magnetization direction m:ˆH=ˆH(m)\ndescribes the system in Fig. 1. ˆHcan describe realistic\nband structures as computed by density-functional the-\nory including exchange-correlation effects and spin-orbit\ncouplingaswell normaland spin-orbitinduced scattering\noff impurities. The energy dissipation is ˙E=/angb∇acketleftdˆH/dτ/angb∇acket∇ight,\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value for the non-\nequilibriumstate. Inlinearresponse,weexpandthemag-\nnetization direction m(t) around the equilibrium magne-\ntization direction m0,\nm(τ)=m0+u(τ). (6)\nThe Hamiltonian can be linearized as ˆH=ˆHst+\nui(τ)∂iˆH, where ˆHst≡ˆH(m0) is the static Hamilto-\nnian and∂iˆH≡∂uiˆH(m0), where summation over re-\npeated indices i=x,y,zis implied. To lowest order\n˙E= ˙ui(τ)/angb∇acketleft∂iˆH/angb∇acket∇ight, where\n/angb∇acketleft∂iˆH/angb∇acket∇ight=/angb∇acketleft∂iˆH/angb∇acket∇ight0+/integraldisplay∞\n−∞dτ′χij(τ−τ′)uj(τ′).(7)\n/angb∇acketleft.../angb∇acket∇ight0denotes equilibrium expectation value and the re-\ntarded correlation function is\nχij(τ−τ′) =−i\n/planckover2pi1θ(τ−τ′)/angbracketleftBig\n[∂iˆH(τ),∂jˆH(τ′)]/angbracketrightBig\n0(8)\nin the interaction picture for the time evolution. In order\nto arrive at the adiabatic (Gilbert) damping the magne-\ntization dynamics has to be sufficiently slow such that\nuj(τ)≈uj(t) + (τ−t) ˙uj(t). Since m2= 1 and hence\n˙ m·m= 0 [7]\n˙E=i∂ωχij(ω→0)˙ui˙uj, (9)\nwhereχij(ω) =/integraltext∞\n−∞dτχij(τ)exp(iωτ). Next, we use\nthe scattering states as the basis for expressing the\ncorrelation function (8). The Hamiltonian consists of\na free-electron part and a scattering potential: ˆH=\nˆH0+ˆV(m). We denote the unperturbed eigenstates of3\nthe free-electron Hamiltonian ˆH0=−/planckover2pi12∇2/2mat en-\nergyǫby|ϕs,q(ǫ)/angb∇acket∇ight, wheres=l,rdenotes propagation\ndirection and qtransverse quantum number. The po-\ntentialˆV(m) scatters the particles between these free-\nelectron states. The outgoing (+) and incoming wave\n(-) eigenstates |ψ(±)\ns,q(ǫ)/angb∇acket∇ightof the static Hamiltonian ˆHst\nfulfill the completeness conditions /angb∇acketleftψ(±)\ns,q(ǫ)|ψ(±)\ns′,q′(ǫ′)/angb∇acket∇ight=\nδs,s′δq,q′δ(ǫ−ǫ′) [10]. These wave functions can be ex-\npressed as |ψ(±)\ns(ǫ)/angb∇acket∇ight= [1 +ˆG(±)\nstˆVst]|ϕs(ǫ)/angb∇acket∇ight, where the\nstatic retarded (+) and advanced (-) Green functions are\nˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1andηis a positive infinites-\nimal. By expanding χij(ω) in the basis of the outgo-\ning wave functions |ψ(+)\ns/angb∇acket∇ight, the low-temperature linear re-\nsponse leads to the followingenergydissipation (9) in the\nadiabatic limit\n˙E=−π/planckover2pi1˙ui˙uj/angbracketleftBig\nψ(+)\ns,q|∂iˆH|ψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′|∂jˆH|ψ(+)\ns,q/angbracketrightBig\n,\n(10)\nwith wave functions evaluated at the Fermi energy ǫF.\nIn order to compare the linear response result, Eq.\n(10), withthat ofthe scatteringtheory, Eq. (5), weintro-\nduce the T-matrix ˆTasˆS(ǫ;m) = 1−2πiˆT(ǫ;m), where\nˆT=ˆV[1 +ˆG(+)ˆT] in terms of the full Green function\nˆG(+)(ǫ,m) = [ǫ+iη−ˆH(m)]−1. Although the adiabatic\nenergy pumping (5) is valid for any magnitude of slow\nmagnetization dynamics, in order to make connection to\nthe linear-response formalism we should consider small\nmagnetization changes to the equilibrium values as de-\nscribed by Eq. (6). We then find\n∂τˆT=/bracketleftBig\n1+ˆVstˆG(+)\nst/bracketrightBig\n˙ui∂iˆH/bracketleftBig\n1+ˆG(+)\nstˆVst/bracketrightBig\n.(11)\ninto Eq. (5) and using the completeness of the scattering\nstates, we recover Eq. (10).\nOur S-matrix approach generalizes the theory of (non-\nlocal) spin pumping and enhanced Gilbert damping in\nthin ferromagnets [5]: by conservation of the total an-\ngular momentum the spin current pumped into the\nsurrounding conductors implies an additional damping\ntorque that enhances the bulk Gilbert damping. Spin\npumping is an N |F interfacial effect that becomes impor-\ntant in thin ferromagnetic films [14]. In the absence of\nspin relaxation in the scattering region, the S-matrix can\nbe decomposed as S(m) =S↑(1+ˆσ·m)/2+S↓(1−ˆσ·\nm)/2, where ˆσis a vector of Pauli matrices. In this case,\nTr(∂τS)(∂τS)†=Ar˙ m2, whereAr= Tr[1−ReS↑S†\n↓]\nand the trace is over the orbital degrees of freedom only.\nWe recover the diagonal and isotropic Gilbert damping\ntensor:Gij=δijGderived earlier [5], where\nG=γMsα=(gµB)2\n4π/planckover2pi1Ar. (12)\nFinally, we illustrate by a model calculation that\nwe can obtain magnetization damping by both spin-\nrelaxationandinterfacespin-pumpingfromtheS-matrix.We consider a thin film ferromagnet in the two-band\nStoner model embedded in a free-electron metal\nˆH=−/planckover2pi12\n2m∇2+δ(x)ˆV(ρ), (13)\nwhere the in-plane coordinate of the ferromagnet is ρ\nand the normal coordinate is x.The spin-dependent po-\ntentialˆV(ρ) consists of the mean-field exchange interac-\ntion oriented along the magnetization direction mand\nmagnetic disorder in the form of magnetic impurities Si\nˆV(ρ) =νˆσ·m+/summationdisplay\niζiˆσ·Siδ(ρ−ρi),(14)\nwhich are randomly oriented and distributed in the film\natx= 0. Impurities in combination with spin-orbit cou-\npling will give similar contributions as magnetic impuri-\nties to Gilbert damping. Our derivation of the S-matrix\nclosely follows Ref. [8]. The 2-component spinor wave\nfunction can be written as Ψ( x,ρ) =/summationtext\nk/bardblck/bardbl(x)Φk/bardbl(ρ),\nwhere the transverse wave function is Φ k/bardbl(ρ) = exp(ik/bardbl·\nρ)/√\nAfor the cross-sectional area A. The effective one-\ndimensional equation for the longitudinal part of the\nwave function is then\n/bracketleftbiggd2\ndx2+k2\n⊥/bracketrightbigg\nck/bardbl(x) =/summationdisplay\nk′\n/bardbl˜Γk/bardbl,k′\n/bardblck/bardbl(0)δ(x),(15)\nwhere the matrix elements are defined by ˜Γk/bardbl,k′\n/bardbl=\n(2m//planckover2pi12)/integraltext\ndρΦ∗\nk/bardbl(ρ)ˆV(ρ)Φk′\n/bardbl(ρ)and the longitudinal\nwave vector k⊥is defined by k2\n⊥= 2mǫF//planckover2pi12−k2\n/bardbl. For\nan incoming electron from the left, the longitudinal wave\nfunction is\nck/bardbls=χs√k⊥/braceleftBigg\neik⊥xδk/bardbls,k′\n/bardbls′+e−ik⊥xrk/bardbls,k′\n/bardbls′,x<0\neik⊥xtk/bardbls,k′\n/bardbls′,x>0,\n(16)\nwheres=↑,↓andχ↑= (1,0)†andχ↓= (0,1)†. Inver-\nsion symmetry dictates that t′=tandr=r′. Continu-\nity of the wave function requires 1+ r=t. The energy\npumping (3) then simplifies to I(pump)\nE=/planckover2pi1Tr/parenleftbig˙t˙t†/parenrightbig\n/π.\nFlux continuity gives t= (1 +iˆΓ)−1, whereˆΓk/bardbls,k′\n/bardbls′=\nχ†\nsˆΓk/bardbls,k′\n/bardbls′χs′(4k⊥k⊥)−1/2.\nIn the absence of spin-flip scattering, the transmis-\nsion coefficient is diagonal in the transverse momentum:\nt(0)\nk/bardbl= [1−iη⊥σ·m]/(1+η2\n⊥), whereη⊥=mν/(/planckover2pi12k⊥).\nThe nonlocal (spin-pumping) Gilbert damping is then\nisotropic,Gij(m) =δijG′,\nG′=2ν2/planckover2pi1\nπ/summationdisplay\nk/bardblη2\n⊥\n(1+η2\n⊥)2. (17)\nIt can be shown that G′is a function of the ratio be-\ntween the exchange splitting versus the Fermi wave vec-\ntor,ηF=mν/(/planckover2pi12kF).G′vanishes in the limits ηF≪14\n(nonmagnetic systems) and ηF≫1 (strong ferromag-\nnet).\nWe include weak spin-flip scattering by expanding the\ntransmission coefficient tto second order in the spin-\norbit interaction, t≈/bracketleftbigg\n1+t0iˆΓsf−/parenleftBig\nt0iˆΓsf/parenrightBig2/bracketrightbigg\nt0, which\ninserted into Eq. (5) leads to an in general anisotropic\nGilbert damping. Ensemble averaging over all ran-\ndom spin configurations and positions after considerable\nbut straightforward algebra leads to the isotropic result\nGij(m) =δijG\nG=G(int)+G′(18)\nwhereG′is defined in Eq. (17). The “bulk” contribution\nto the damping is caused by the spin-relaxation due to\nthe magnetic disorder\nG(int)=NsS2ζ2ξ, (19)\nwhereNsis the number of magnetic impurities, Sis the\nimpurity spin, ζis the average strength of the magnetic\nimpurity scattering, and ξ=ξ(ηF) is a complicated ex-\npression that vanishes when ηFis either very small or\nvery large. Eq. (18) proves that Eq. (5) incorporates the\n“bulk” contribution to the Gilbert damping, which grows\nwith the number of spin-flip scatterers, in addition to in-\nterface damping. We could have derived G(int)[Eq. (19)]\nas well by the Kubo formula for the Gilbert damping.\nThe Gilbert damping has been computed before based\non the Kubo formalism based on first-principles elec-\ntronic band structures [9]. However, the ab initio appeal\nis somewhat reduced by additional approximations such\nas the relaxation time approximation and the neglect of\ndisorder vertex corrections. An advantage of the scatter-\ningtheoryofGilbertdampingisitssuitabilityformodern\nab initio techniques of spin transport that do not suffer\nfrom these drawbacks [16]. When extended to include\nspin-orbit coupling and magnetic disorder the Gilbert\ndamping can be obtained without additional costs ac-\ncording to Eq. (5). Bulk and interface contributions can\nbe readily separated by inspection of the sample thick-\nness dependence of the Gilbert damping.\nPhononsareimportantforthe understandingofdamp-\ning at elevated temperatures, which we do not explic-\nitly discuss. They can be included by a temperature-\ndependent relaxation time [9] or, in our case, structural\ndisorder. A microscopic treatment of phonon excitations\nrequires extension of the formalism to inelastic scatter-\ning, which is beyond the scope of the present paper.\nIn conclusion, we hope that our alternative formal-\nism of Gilbert damping will stimulate ab initio electronic\nstructure calculations as a function of material and dis-\norder. By comparison with FMR studies on thin ferro-\nmagnetic films this should lead to a better understanding\nof dissipation in magnetic systems.This work was supported in part by the Re-\nsearch Council of Norway, Grants Nos. 158518/143 and\n158547/431, and EC Contract IST-033749 “DynaMax.”\n∗Electronic address: Arne.Brataas@ntnu.no\n[1] For a review, see M. D. Stiles and J. Miltat, Top. Appl.\nPhys.101, 225 (2006) , and references therein.\n[2] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta-\ntus Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys.\n48, 2906 (1970); V. Korenman, and R. E. Prange, Phys.\nRev. B6, 2769 (1972); V. S. 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Bauer, Phys. Rev. B 49, 14684\n(1994).\n[9] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[10] P. A. Mello and N. Kumar, Quantum Transport in Meso-\nscopic Systems , Oxford University Press (New York,\n2005).\n[11] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys.\nRev. Lett., 87, 236601 (2001).\n[12] M. Moskalets and M. B¨ uttiker, Phys. Rev. B 66, 035306\n(2002);Phys. Rev. B 66, 205320 (2002).\n[13] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002); X. Wang,\nG. E. W. Bauer, B. J. van Wees, A. Brataas, and Y.\nTserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[14] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataa s,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett 90,\n187601 (2003);M. V. Costache, M. Sladkov, S. M. Watts,\nC. H. van der Wal, and B. J. van Wees, Phys. Rev.\nLett.97, 216603 (2006); G. Woltersdorf, O. Mosendz, B.\nHeinrich, and C. H. Back, Phys. Rev. Lett 99, 246603\n(2007).\n[15] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).\n[16] M. Zwierzycki et al., Phys. Stat. Sol. B 245, 623 (2008)"    },    {        "title": "0808.1373v1.Gilbert_Damping_in_Conducting_Ferromagnets_I__Kohn_Sham_Theory_and_Atomic_Scale_Inhomogeneity.pdf",        "content": "arXiv:0808.1373v1  [cond-mat.mes-hall]  9 Aug 2008Gilbert Damping in Conducting Ferromagnets I:\nKohn-Sham Theory and Atomic-Scale Inhomogeneity\nIon Garate and Allan MacDonald\nDepartment of Physics, The University of Texas at Austin, Au stin TX 78712\n(Dated: October 27, 2018)\nWe derive an approximate expression for the Gilbert damping coefficient αGof itinerant electron\nferromagnets which is based on their description in terms of spin-density-functional-theory (SDFT)\nand Kohn-Sham quasiparticle orbitals. We argue for an expre ssion in which the coupling of mag-\nnetization fluctuations to particle-hole transitions is we ighted by the spin-dependent part of the\ntheory’s exchange-correlation potential, a quantity whic h has large spatial variations on an atomic\nlength scale. Our SDFT result for αGis closely related to the previously proposed spin-torque\ncorrelation-function expression.\nPACS numbers:\nI. INTRODUCTION\nThe Gilbert parameter αGcharacterizes the damping\nof collective magnetization dynamics1. The key role of\nαGin current-driven2and precessional3magnetization\nreversal has renewed interest in the microscopic physics\nof this important material parameter. It is generally\naccepted that in metals the damping of magnetization\ndynamics is dominated3by particle-hole pair excitation\nprocesses. The main ideas which arise in the theory of\nGilbert damping have been in place for some time4,5. It\nhas however been difficult to apply them to real materi-\nals with the precision required for confident predictions\nwhich would allow theory to play a larger role in design-\ning materials with desired damping strengths. Progress\nhas recently been achieved in various directions, both\nthrough studies6of simple models for which the damp-\ning can be evaluated exactly and through analyses7of\ntransition metal ferromagnets that are based on realis-\ntic electronic structure calculations. Evaluation of the\ntorquecorrelationformula5forαGusedinthelatercalcu-\nlations requires knowledge only of a ferromagnet’s mean-\nfield electronic structure and of its Bloch state lifetime,\nwhich makes this approach practical.\nRealistic ab initio theories normally employ spin-\ndensity-functional theory9which has a mean-field theory\nstructure. In this article we use time-dependent spin-\ndensity functional theory to derive an explicit expression\nfor the Gilbert damping coefficient in terms of Kohn-\nSham theory eigenvalues and eigenvectors. Our final\nresult is essentially equivalent to the torque-correlation\nformula5forαG, but has the advantages that its deriva-\ntion is fully consistent with density functional theory,\nthat it allows for a consistent microscopic treatments of\nboth dissipative and reactive coefficients in the Landau-\nLiftshitz Gilbert (LLG) equations, and that it helps\nestablish relationships between different theoretical ap-\nproaches to the microscopic theory of magnetization\ndamping.\nOur paper is organized as follows. In Section II\nwe relate the Gilbert damping parameter αGof a fer-\nromagnet to the low-frequency limit of its transversespin response function. Since ferromagnetism is due\nto electron-electron interactions, theories of magnetism\nare always many-electron theories, and it is necessary to\nevaluate the many-electron response function. In time-\ndependent spin-density functional theory the transverse\nresponse function is calculated using a time-dependent\nself- consistent-field calculation in which quasiparticles\nrespond both to external potentials and to changesin the\ninteraction-induced effective potential. In Section III we\nuse perturbation theory and time-dependent mean-field\ntheory to express the coefficients which appear in the\nLLG equations in terms of the Kohn-Sham eigenstates\nand eigenvaluesof the ferromagnet’sground state. These\nformal expressions are valid for arbitrary spin-orbit cou-\npling, arbitrary atomic length scale spin-dependent and\nscalarpotentials, and arbitrarydisorder. By treating dis-\norder approximately, in Section IV we derive and com-\npare two commonly used formulas for Gilbert damping.\nFinally, in Section V we summarize our results.\nII. MANY-BODY TRANSVERSE RESPONSE\nFUNCTION AND THE GILBERT DAMPING\nPARAMETER\nThe Gilbert damping parameter αGappears in the\nLandau-Liftshitz-Gilbert expression for the collective\nmagnetization dynamics of a ferromagnet:\n∂ˆΩ\n∂t=ˆΩ×Heff−αGˆΩ×∂ˆΩ\n∂t. (1)\nIn Eq.( 1) Heffis an effective magnetic field which\nwe comment on further below and ˆΩ = (Ω x,Ωy,Ωz) is\nthe direction of the magnetization. This equation de-\nscribes the slow dynamics of smooth magnetization tex-\ntures and is formally the first term in an expansion in\ntime-derivatives.\nThe damping parameter αGcan be measured by per-\nforming ferromagnetic resonance (FMR) experiments in\nwhich the magnetization direction is driven weakly away\nfrom an easy direction (which we take to be the ˆ z-\ndirection.). To relate this phenomenological expression2\nformally to microscopic theory we consider a system in\nwhich external magnetic fields couple only11to the elec-\ntronic spin degree of freedom and associate the magneti-\nzation direction ˆΩ with the direction of the total electron\nspin. Forsmalldeviationsfromtheeasydirection,Eq.(1)\nreads\nHeff,x= +∂ˆΩy\n∂t+αG∂ˆΩx\n∂t\nHeff,y=−∂ˆΩx\n∂t+αG∂ˆΩy\n∂t. (2)\nThe gyromagnetic ratio has been absorbed into the unitsof the field Heffso that this quantity has energy units\nand we set /planckover2pi1= 1 throughout. The corresponding formal\nlinear response theory expression is an expansion of the\nlong wavelength transverse total spin response function\nto first order12in frequency ω:\nS0ˆΩα=/summationdisplay\nβ[χst\nα,β+ωχ′\nα,β]Hext,β (3)\nwhereα,β∈ {x,y},ω≡i∂tis the frequency, S0is the to-\ntal spin ofthe ferromagnet, Hextis the external magnetic\nfield and χis the transverse spin-spin response function:\nχα,β(ω) =i/integraldisplay∞\n0dtexp(iωt)/an}bracketle{t[Sα(t),Sβ(t)]/an}bracketri}ht=/summationdisplay\nn/bracketleftbigg/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht\nωn,0−ω−iη+/an}bracketle{tΨ0|Sβ|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sα|Ψ0/an}bracketri}ht\nωn,0+ω+iη/bracketrightbigg\n(4)\nHere|Ψn/an}bracketri}htis an exact eigenstate of the many-body Hamiltonian and ωn,0is the excitation energy for state n. We use\nthis formal expression below to make some general comments abou t the microscopic theory of αG. In Eq.( 3) χst\nα,βis\nthe static ( ω= 0) limit of the response function, and χ′\nα,βis the first derivative with respect to ωevaluated at ω= 0.\nNotice that we have chosen the normalization in which χis the total spin response to a transverse field; χis therefore\nextensive.\nThe keystep in obtainingthe Landau-Liftshitz-Gilbert\nform for the magnetization dynamics is to recognize that\nin the static limit the transverse magnetization responds\nto an external magnetic field by adjusting orientation to\nminimize the total energy including the internal energy\nEintand the energy due to coupling with the external\nmagnetic field,\nEext=−S0ˆΩ·Hext. (5)\nIt follows that\nχst\nα,β=S2\n0/bracketleftBigg\n∂2Eint\n∂ˆΩαˆΩβ/bracketrightBigg−1\n. (6)\nWe obtain a formal equation for Heffcorresponding to\nEq.( 2) by multiplying Eq.( 3) on the left by [ χst\nα,β]−1and\nrecognizing\nHint,α=−1\nS0/summationdisplay\nβ∂2Eint\n∂ˆΩα∂ˆΩβˆΩβ=−1\nS0∂Eint\n∂ˆΩα(7)\nas the internal energy contribution to the effective mag-\nnetic field Heff=Hint+Hext. With this identification\nEq.( 3) can be written in the form\nHeff,α=/summationdisplay\nβLα,β∂tˆΩβ (8)\nwhere\nLα,β=−S0[i(χst)−1χ′(χst)−1]α,β=iS0∂ωχ−1\nα,β.(9)According to the Landau-Liftshitz Gilbert equation then\nLx,y=−Ly,x= 1 and\nLx,x=Ly,y=αG. (10)\nExplicit evaluation of the off-diagonal components of L\nwill in general yield very small deviation from the unit\nresult assumed by the Landau-Liftshitz-Gilbert formula.\nThe deviation reflects mainly the fact that the magneti-\nzation magnitude varies slightly with orientation. We do\nnot comment further on this point because it is of little\nconsequence. Similarly Lx,xis not in general identical\ntoLy,y, although the difference is rarely large or impor-\ntant. Eq.( 10) is the starting point we use later to derive\napproximate expressions for αG.\nIn Eq.( 9) χα,β(ω) is the correlation function for an\ninteracting electron system with arbitrary disorder and\narbitrary spin-orbit coupling. In the absence of spin-\norbit coupling, but still with arbitrary spin-independent\nperiodic and disorder potentials, the ground state of a\nferromagnet is coupled by the total spin-operator only to\nstates in the same total spin multiplet. In this case it\nfollows from Eq.( 4) that\nχst\nα,β= 2/summationdisplay\nnRe/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht]\nωn,0=δα,βS0\nH0\n(11)\nwhereH0is a static external field, which is necessary\nin the absence of spin-orbit coupling to pin the magne-\ntization to the ˆ zdirection and splits the ferromagnet’s3\nground state many-body spin multiplet. Similarly\nχ′\nα,β= 2i/summationdisplay\nnIm[/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht]\nω2\nn,0=iǫα,βS0\nH2\n0.\n(12)\nwhereǫx,x=ǫy,y= 0 and ǫx,y=−ǫy,x= 1, yielding\nLx,y=−Ly,x= 1 and Lx,x=Ly,y= 0. Spin-orbit\ncoupling is required for magnetization damping8.\nIII. SDF-STONER THEORY EXPRESSION FOR\nGILBERT DAMPING\nApproximate formulas for αGin metals are inevitably\nbased on on a self-consistent mean-field theory (Stoner)\ndescriptionofthemagneticstate. Ourgoalistoderivean\napproximate expression for αGwhen the adiabatic local\nspin-densityapproximation9isused forthe exchangecor-\nrelation potential in spin-density-functional theory. The\neffective Hamiltonian which describes the Kohn-Sham\nquasiparticle dynamics therefore has the form\nHKS=HP−∆(n(/vector r),|/vector s(/vector r)|)ˆΩ(/vector r)·/vector s,(13)\nwhereHPis the Kohn-Sham Hamiltonian of a paramag-\nnetic state in which |/vector s(/vector r)|(the local spin density) is set to\nzero,/vector sis the spin-operator, and\n∆(n,s) =−d[nǫxc(n,s)]\nds(14)\nis the magnitude of the spin-dependent part of the\nexchange-correlation potential. In Eq.( 14) ǫxc(n,s) is\nthe exchange-correlation energy per particle in a uni-\nform electron gas with density nand spin-density s.\nWe assume that the ferromagnet is described using\nsome semi-relativistic approximation to the Dirac equa-\ntion like those commonly used13to describe magnetic\nanisotropy or XMCD, even though these approximations\nare not strictly consistent with spin-density-functional\ntheory. Within this framework electrons carry only a\ntwo-componentspin-1/2degreeoffreedomandspin-orbit\ncoupling terms are included in HP. Sincenǫxc(n,s)∼\n[(n/2 +s)4/3+ (n/2−s)4/3], ∆0(n,s)∼n1/3is larger\nclosertoatomic centersand farfrom spatiallyuniform on\natomic length scales. This property figures prominently\nin the considerations explained below.\nIn SDFT the transverse spin-response function is ex-\npressed in terms of Kohn-Sham quasiparticle response to\nboth external and induced magnetic fields:\ns0(/vector r)Ωα(/vector r) =/integraldisplayd/vectorr′\nVχQP\nα,β(/vector r,/vectorr′) [Hext,β(/vectorr′)+∆0(/vectorr′)Ωβ(/vectorr′)].\n(15)\nIn Eq.( 15) Vis the system volume, s0(/vector r) is the magni-\ntude of the ground state spin density, ∆ 0(/vector r) is the mag-\nnitude of the spin-dependent part of the ground stateexchange-correlation potential and\nχQP\nα,β(/vector r,/vectorr′) =/summationdisplay\ni,jfj−fi\nωi,j−ω−iη/an}bracketle{ti|/vector r/an}bracketri}htsα/an}bracketle{t/vector r|j/an}bracketri}ht/an}bracketle{tj|/vectorr′/an}bracketri}htsβ/an}bracketle{t/vectorr′|i/an}bracketri}ht,\n(16)\nwherefiis the ground state Kohn-Sham occupation fac-\ntor for eigenspinor |i/an}bracketri}htandωij≡ǫi−ǫjis a Kohn-\nSham eigenvalue difference. χQP(/vector r,/vectorr′) has been normal-\nized so that it returns the spin-density rather than total\nspin. Like the Landau-Liftshitz-Gilbert equation itself,\nEq.( 15) assumes that only the direction of the mag-\nnetization, and not the magnitudes of the charge and\nspin-densities, varies in the course of smooth collective\nmagnetization dynamics14. This property should hold\naccurately as long as magnetic anisotropies and exter-\nnal fields are weak compared to ∆ 0. We are able to use\nthis property to avoid solving the position-space integral\nequation implied by Eq.( 15). Multiplying by ∆ 0(/vector r) on\nboth sides and integrating over position we find15that\nS0Ωα=/summationdisplay\nβ1\n¯∆0˜χQP\nα,β(ω)/bracketleftbig\nΩβ+Hext,β\n¯∆0/bracketrightbig\n(17)\nwhere we have taken advantage of the fact that in FMR\nexperiments Hext,βandˆΩ are uniform. ¯∆0is a spin-\ndensity weighted average of ∆ 0(/vector r),\n¯∆0=/integraltext\nd/vector r∆0(/vector r)s0(/vector r)/integraltext\nd/vector rs0(/vector r), (18)\nand\n˜χQP\nα,β(ω) =/summationdisplay\nijfj−fi\nωij−ω−iη/an}bracketle{tj|sα∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sβ∆0(/vector r)|j/an}bracketri}ht\n(19)\nis the response function of the transverse-part of the\nquasiparticleexchange-correlationeffective field response\nfunction, notthe transverse-part of the quasiparticle\nspin response function. In Eq.( 19), /an}bracketle{ti|O(/vector r)|j/an}bracketri}ht=/integraltext\nd/vector rO(/vector r)/an}bracketle{ti|/vector r/an}bracketri}ht/an}bracketle{t/vector r|j/an}bracketri}htdenotes a single-particle matrix ele-\nment. Solving Eq.( 17) for the many-particle transverse\nsusceptibility (the ratio of S0ˆΩαtoHext,β) and inserting\nthe result in Eq.( 9) yields\nLα,β=iS0∂ωχ−1\nα,β=−S0¯∆2\n0∂ωIm[˜χQP−1\nα,β].(20)\nOur derivation of the LLG equation has the advantage\nthat the equation’s reactive and dissipative components\nare considered simultaneously. Comparing Eq.( 15) and\nEq.( 7) we find that the internal anisotropy field can also\nbe expressed in terms of ˜ χQP:\nHint,α=−¯∆2\n0S0/summationdisplay\nβ/bracketleftbig\n˜χQP−1\nα,β(ω= 0)−δα,β\nS0¯∆0/bracketrightbig\nΩβ.(21)\nEq.( 20) and Eq.( 21) provide microscopic expressions\nfor all ingredients that appear in the LLG equations4\nlinearized for small transverse excursions. It is gener-\nally assumed that the damping coefficient αGis inde-\npendent of orientation; if so, the present derivation is\nsufficient. The anisotropy-field at large transverse ex-\ncursions normally requires additional information about\nmagnetic anisotropy. We remark that if the Hamiltonian\ndoes not include a spin-dependent mean-field dipole in-\nteraction term, as is usually the case, the above quantity\nwill return only the magnetocrystalline anisotropy field.\nSince the magnetostatic contribution to anisotropy is al-\nways well described by mean-field-theory it can be added\nseparately.\nWe conclude this section by demonstrating that the\nStoner theory equations proposed here recover the exact\nresults mentioned at the end of the previous section for\nthe limit in which spin-orbit coupling is neglected. We\nconsider a SDF theory ferromagnet with arbitrary scalar\nand spin-dependent effective potentials. Since the spin-\ndependent part of the exchange correlation potential is\nthen the only spin-dependent term in the Hamiltonian it\nfollows that\n[HKS,sα] =−iǫα,β∆0(/vector r)sβ (22)\nand hence that\n/an}bracketle{ti|sα∆0(/vector r)|j/an}bracketri}ht=−iǫα,βωij/an}bracketle{ti|sβ|j/an}bracketri}ht.(23)\nInserting Eq.( 23) in one of the matrix elements of\nEq.( 19) yields for the no-spin-orbit-scattering case\n˜χQP\nα,β(ω= 0) =δα,βS0¯∆0. (24)The internal magnetic field Hint,αis therefore identically\nzero in the absence of spin-orbit coupling and only exter-\nnal magnetic fields will yield a finite collective precession\nfrequency. Inserting Eq.( 23) in both matrix elements of\nEq.( 19) yields\n∂ωIm[˜χQP\nα,β] =ǫα,βS0. (25)\nUsing both Eq.( 24) and Eq.( 25) to invert ˜ χQPwe re-\ncover the results proved previously for the no-spin-orbit\ncase using a many-body argument: Lx,y=−Ly,x= 1\nandLx,x=Ly,y= 0. The Stoner-theory equations de-\nrived here allow spin-orbit interactions, and hence mag-\nnetic anisotropy and Gilbert damping, to be calculated\nconsistently from the same quasiparticle response func-\ntion ˜χQP.\nIV. DISCUSSION\nAs long as magnetic anisotropy and external magnetic\nfields are weak compared to the exchange-correlation\nsplitting in the ferromagnet we can use Eq.( 24) to ap-\nproximate ˜ χQP\nα,β(ω= 0). Using this approximation and\nassuming that damping is isotropic we obtain the follow-\ning explicit expression for temperature T→0:\nαG=Lx,x=−S0¯∆2\n0∂ωIm[˜χQP−1\nx,x] =π\nS0/summationdisplay\nijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|sx∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sx∆0(/vector r)|j/an}bracketri}ht\n=π\nS0/summationdisplay\nijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|[HP,sy]|i/an}bracketri}ht/an}bracketle{ti|[HP,sy]|j/an}bracketri}ht.(26)\nThe second form for αGis equivalent to the first and follows from the observation that for m atrix elements between\nstates that have the same energy\n/an}bracketle{ti|[HKS,sα]|j/an}bracketri}ht=−iǫα,β/an}bracketle{ti|∆0(/vector r)sβ|j/an}bracketri}ht+/an}bracketle{ti|[HP,sα]|j/an}bracketri}ht= 0 (for ωij= 0). (27)\nEq. ( 26) is valid for any scalar and any spin-dependent potential. It is clear however that the numerical value of αG\nin a metal is very sensitive to the degree of disorder in its lattice. To s ee this we observe that for a perfect crystal\nthe Kohn-Sham eigenstates are Bloch states. Since the operator ∆0(/vector r)sαhas the periodicity of the crystal its matrix\nelements are non-zero only between states with the same Bloch wav evector label /vectork. For the case of a perfect crystal\nthen\nαG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht\n=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht. (28)\nwherenn′are band labels and s0is the ground state\nspin per unit volume and the integral over /vectorkis over theBrillouin-zone (BZ).5\nClearly αGdiverges16in a perfect crystal since\n/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn/an}bracketri}htis generically non-zero. A theory of\nαGmust therefore always account for disorder in a crys-\ntal. The easiest way to account for disorder is to replace\ntheδ(ǫ/vectorkn−ǫF) spectral function of a Bloch state by a\nbroadened spectral function evaluated at the Fermi en-\nergyA/vectorkn(ǫF). If disorder is treated perturbatively this\nsimpleansatzcan be augmented17by introducing impu-\nrity vertex corrections in Eq. ( 28). Provided that the\nquasiparticlelifetimeiscomputedviaFermi’sgoldenrule,these vertex corrections restore Ward identities and yield\nan exact treatment of disorder in the limit of dilute im-\npurities. Nevertheless, this approach is rarely practical\noutside the realm of toy models, because the sources of\ndisorder are rarely known with sufficient precision.\nAlthough appealing in its simplicity, the δ(ǫ/vectorkn−ǫF)→\nA/vectorkn(ǫF) substitution is prone to ambiguity because it\ngives rise to qualitatively different outcomes depending\non whether it is applied to the first or second line of Eq.\n( 28):\nα(TC)\nG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht,\nα(SF)\nG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht.\n(29)\nα(TC)\nGis the torque-correlation (TC) formula used in\nrealistic electronic structure calculations7andα(SF)\nGis\nthe spin-flip (SF) formula used in certain toy model\ncalculations18. The discrepancy between TC and SF ex-\npressions stems from inter-band ( n/ne}ationslash=n′) contributions\nto damping, which may now connect states with dif-\nferentband energies due to the disorder broadening of\nthe spectral functions. Therefore, /an}bracketle{t/vectorkn|[HKS,sα]|/vectorkn′/an}bracketri}htno\nlonger vanishes for n/ne}ationslash=n′and Eq. ( 27) indicates that\nα(TC)\nG≃α(SF)\nGonly if the Gilbert damping is dominated\nby intra-band contributions and/or if the energy differ-\nence between the states connected by inter-band transi-\ntions is small compared to ∆ 0. When α(TC)\nG/ne}ationslash=α(SF)\nG,\nit isa priori unclear which approach is the most accu-\nrate. One obvious flaw of the SF formula is that it pro-\nducesaspuriousdampinginabsenceofspin-orbitinterac-\ntions; this unphysical contribution originates from inter-\nband transitions and may be cancelled out by adding\nthe leading order impurity vertex correction19. In con-\ntrast, [HP,sy] = 0 in absence of spin-orbit interaction\nandhencetheTCformulavanishesidentically, evenwith-\nout vertex corrections. From this analysis, TC appears\nto have a pragmatic edge over SF in materials with weak\nspin-orbitinteraction. However, insofarasit allowsinter-\nband transitions that connect states with ωi,j>∆0,\nTC is not quantitatively reliable. Furthermore, it canbe shown17that when the intrinsic spin-orbit coupling\nis significant (e.g. in ferromagnetic semiconductors), the\nadvantage of TC over SF (or vice versa) is marginal, and\nimpurity vertex corrections play a significant role.\nV. CONCLUSIONS\nUsing spin-density functional theory we have derived\na Stoner model expression for the Gilbert damping co-\nefficient in itinerant ferromagnets. This expression ac-\ncounts for atomic scale variations of the exchange self\nenergy, as well as for arbitrary disorder and spin-orbit\ninteraction. By treating disorder approximately, we have\nderived the spin-flip and torque-correlationformulas pre-\nviously used in toy-model and ab-initio calculations, re-\nspectively. Wehavetracedthediscrepancybetweenthese\nequations to the treatment of inter-band transitions that\nconnect states which are not close in energy. A better\ntreatment of disorder, which requires the inclusion of im-\npurity vertex corrections, will be the ultimate judge on\nthe relativereliabilityofeitherapproach. Whendamping\nis dominated by intra-band transitions, a circumstance\nwhich we believe is common, the two formulas are identi-\ncal and both arelikely to provide reliable estimates. This\nwork was suported by the National Science Foundation\nunder grant DMR-0547875.\n1For a historical perspective see T.L. Gilbert, IEEE Trans.\nMagn.40, 3443 (2004).\n2Foranintroductoryreviewsee D.C. RalphandM.D.Stiles,\nJ. Magn. Mag. Mater. 320, 1190 (2008).3J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag-\nnetic Structures III: Fundamentals of Nanomagnetism\n(Springer-Verlag, New York, 2005).\n4V. Korenman and R. E. Prange, Phys. Rev. B 6, 27696\n(1972).\n5V. Kambersky, Czech J. Phys. B 26, 1366 (1976).\n6Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004); E.M. Hankiewicz, G. Vig-\nnale and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007);\nY. Tserkovnyak et al., Phys. Rev. B 74, 144405 (2006) ;\nH.J. Skadsem, Y. Tserkovnyak, A. Brataas, G.E.W. Bauer,\nPhys. Rev. B 75, 094416 (2007); H. Kohno, G. Tatara\nand J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006);\nR.A. Duine et al., Phys. Rev. B 75, 214420 (2007). Y.\nTserkovnyak, A. Brataas, and G.E.W. Bauer, J. Magn.\nMag. Mater. 320, 1282 (2008).\n7K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett.\n99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416\n(2007).\n8For zero spin-orbit coupling αGvanishes even in presence\nof magnetic impurities, provided that their spins follow th e\ndynamics of the magnetization adiabatically.\n9O. Gunnarsson, J. Phys. F 6, 587 (1976).\n10Z. Qian, G. Vignale, Phys. Rev. Lett. 88, 056404 (2002).\n11In doing so we dodge the subtle difficulties which compli-\ncate theories of orbital magnetism in bulk metals. See for\nexample J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys.\nRev. Lett. 99, 197202 (2007); I. Souza and D. Vanderbilt,\nPhys. Rev. B 77, 054438 (2008) and work cited therein.\nThis simplification should have little influence on the the-\nory of damping because the orbital contribution to the\nmagnetization is relatively small in systems of interest an dbecause it in any event tends to be collinear with the spin\nmagnetization.\n12For most materials the FMR frequency is by far the small-\nest energy scale in the problem. Expansion to linear order\nis almost always appropriate.\n13See for example A.C. Jenkins and W.M. Temmerman,\nPhys. Rev. B 60, 10233 (1999) and work cited therein.\n14This approximation does not preclude strong spatial varia-\ntions of|s0(/vector r)|and|∆0(/vector r)|at atomic lenghtscales; rather it\nis assumed that such spatial profiles will remain unchanged\nin the course of the magnetization dynamics.\n15For notational simplicity we assume that all magnetic\natoms are identical. Generalizations to magnetic com-\npounds are straight forward.\n16Eq. ( 26) is valid provided that ωτ <<1. While this con-\ndition is normally satisfied in cases of practical interest, it\ninvariably breaks down as τ→ ∞. Hence the divergence\nof Eq. ( 26) in perfectcrystals is spurious.\n17I. Garate and A.H. MacDonald (in preparation).\n18J. Sinova et al., Phys. Rev. B 69, 85209 (2004). In order to\nget the equivalence, trade hzby ∆0and use ∆ 0=JpdS0,\nwhereJpdis the p-d exchange coupling between GaAs va-\nlence band holes and Mn d-orbitals. In addition, note that\nour spectral function differs from theirs by a factor 2 π.\n19H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006)."    },    {        "title": "0808.3923v1.Gilbert_Damping_in_Conducting_Ferromagnets_II__Model_Tests_of_the_Torque_Correlation_Formula.pdf",        "content": "arXiv:0808.3923v1  [cond-mat.mtrl-sci]  28 Aug 2008Gilbert Damping in Conducting Ferromagnets II:\nModel Tests of the Torque-Correlation Formula\nIon Garate and Allan MacDonald\nDepartment of Physics, The University of Texas at Austin, Au stin TX 78712\n(Dated: October 29, 2018)\nWe report on a study of Gilbert damping due to particle-hole p air excitations in conducting\nferromagnets. We focus on a toy two-band model and on a four-b and spherical model which provides\nan approximate description of ferromagnetic (Ga,Mn)As. Th ese models are sufficiently simple that\ndisorder-ladder-sum vertex corrections to the long-wavel ength spin-spin response function can be\nsummed to all orders. An important objective of this study is to assess the reliability of practical\napproximate expressions which can be combined with electro nic structure calculations to estimate\nGilbert damping in more complex systems.\nPACS numbers:\nI. INTRODUCTION\nThe key role of the Gilbert parameter αGin current-\ndriven1and precessional2magnetization reversal has led\nto a renewed interest in this important magnetic ma-\nterial parameter. The theoretical foundations which re-\nlate Gilbert damping to the transversespin-spinresponse\nfunction of the ferromagnet have been in place for some\ntime3,4. It has nevertheless been difficult to predict\ntrends as a function of temperature and across mate-\nrials systems, partly because damping depends on the\nstrength and nature of the disorder in a manner that re-\nquires a more detailed characterization than is normally\navailable. Two groups have recently5reported success-\nful applications to transition metal ferromagets of the\ntorque-correlation formula4,5,6forαG. This formula has\nthe important advantage that its application requires\nknowledge only of the band structure, including its spin-\norbit coupling, and of Bloch state lifetimes. The torque-\ncorrelation formula is physically transparent and can be\napplied with relative ease in combination with modern\nspin-density-functional-theory7(SDFT) electronic struc-\nture calculations. In this paper we compare the pre-\ndictions of the torque correlation formula with Kubo-\nformula self-consistent-Born-approximation results for\ntwo different relatively simple model systems, an ar-\ntificial two-band model of a ferromagnet with Rashba\nspin-orbit interactions and a four-band model which cap-\ntures the essential physics of (III,Mn)V ferromagnetic\nsemiconductors8. The self-consistent Born approxima-\ntion theory for αGrequires that ladder-diagram vertex\ncorrections be included in the transverse spin-spin re-\nsponse function. Since the Born approximation is ex-\nact for weak scattering, we can use this comparison to\nassess the reliability of the simpler and more practical\ntorque-correlationformula. Weconcludethat the torque-\ncorrelationformulaisaccuratewhentheGilbertdamping\nis dominated by intra-band excitations of the transition\nmetal Fermi sea, but that it can be inaccurate when it is\ndominated by inter-band excitations.\nOur paper is organized as follows. In Section II we ex-\nplain how we evaluate the transverse spin-spin responsefunction for simple model ferromagnets. Section III dis-\ncusses our result for the two-band Rashba model while\nSection IV summarizes our findings for the four-band\n(III,Mn)V model. We conclude in Section V with a sum-\nmary of our results and recommended best practices for\nthe use of the torque-correlation formula.\nII. GILBERT DAMPING AND TRANSVERSE\nSPIN RESPONSE FUNCTION\nA. Realistic SDFT vs.s-d and p-d models\nWe view the two-band s−dand four band p−dmod-\nels studied in this paper as toy models which capture the\nessential features of metallic magnetism in systems that\nare, at least in principle9, more realistically described\nusing SDFT. The s−dandp−dmodels correspond\nto the limit of ab initio SDFT in which i) the majority\nspind-bands are completely full and the minority spin\nd-bands completely empty, ii) hybridization between s\norpandd-bands is relatively weak, and iii) there is ex-\nchange coupling between dandsorpmoments. In a\nrecent paper we have proposed the following expression\nfor the Gilbert-damping contribution from particle-hole\nexcitations in SDFT bands:\nαG=1\nS0∂ωIm[˜χQP\nx,x] (1)\nwhere ˜χQP\nx,xis a response-function which describes how\nthe quasiparticle bands change in response to a spatially\nsmooth variation in magnetization orientation and S0is\nthe total spin. Specifically,\n˜χQP\nα,β(ω) =/summationdisplay\nijfj−fi\nωij−ω−iη∝an}bracketle{tj|sα∆0(/vector r)|i∝an}bracketri}ht∝an}bracketle{ti|sβ∆0(/vector r)|j∝an}bracketri}ht.\n(2)\nwhereαandβlabel the xandytransverse spin direc-\ntions and the easy direction for the magnetization is as-\nsummed to be the ˆ zdirection. In Eq.( 2) |i∝an}bracketri}ht,fiandωij\nare Kohn-Sham eigenspinors, Fermi factors, and eigenen-\nergy differences respectively, sαis a spin operator, and2\n∆0(/vector r) is the difference between the majorityspin and mi-\nnority spin exchange-correlation potential. In the s−d\nandp−dmodels ∆ 0(/vector r) is replaced by a phenomeno-\nlogical constant, which we denote by ∆ 0below. With\n∆0(/vector r) replaced by a constant ˜ χQP\nx,xreduces to a standard\nspin-response function for non-interacting quasiparticles\nin a possibly spin-dependent random static external po-\ntential. The evaluation of this quantity, and in particu-\nlar the low-frequency limit in which we are interested, is\nnon-trivial only because disorder plays an essential role.\nB. Disorder Perturbation Theory\nWe start by writing the transverse spin response func-\ntion of a disordered metallic ferromagnet in the Matsub-ara formalism,\n˜χQP\nxx(iω) =−V∆2\n0\nβ/summationdisplay\nωnP(iωn,iωn+iω) (3)\nwhere the minus sign originates from fermionic statistics,\nVis the volume of the system and\nP(iωn,iωn+iω)≡/integraldisplaydDk\n(2π)DΛα,β(iωn,iωn+iω;k)Gβ(iωn+iω,k)sx\nβ,α(k)Gα(iωn,k). (4)\nIn Eq. ( 4) |αk∝an}bracketri}htis a band eigenstate at momentum k,Dis the dimensionality of the system, sx\nα,β(k) =∝an}bracketle{tαk|sx|βk∝an}bracketri}ht\nis the spin-flip matrix element, Λ α,β(k) is its vertex-corrected counterpart (see below), and\nGα(iωn,k) =/bracketleftbigg\niωn+EF−Ek,α+i1\n2τk,αsign(ωn)/bracketrightbigg−1\n. (5)\nWe have included disorder within the Born approximation by incorpora ting a finite lifetime τfor the quasiparticles\nand by allowing for vertex corrections at one of the spin vertices.\nα,kΛβ,kβ,k\nα,ksxβ,k\nk'\nα,kΛ\nα'k''β\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n+ =\nFIG. 1: Dyson equation for the renormalized vertex of the tra nsverse spin-spin response function. The dotted line denot es\nimpurity scattering.\nThe vertex function in Eq.( 4) obeys the Dyson equation (Fig. ( 1)):\nΛα,β(iωn,iωn+iω;k) =sx\nα,β(k)+\n+/integraldisplaydDk′\n(2π)Dua(k−k′)sa\nα,α′(k,k′)Gα′(iωn,k′)Λα′,β′(iωn,iωn+iω;k′)Gβ′(iωn+iω,k′)sa\nβ′,β(k′,k),(6)\nwhereua(q)≡naV2a(q)(a= 0,x,y,z),nais the den-\nsity of scatterers, Va(q) is the scattering potential (di-\nmensions: (energy) ×(volume)) and the overline stands\nfor disorder averaging10,11. Ward’s identity requires thatua(q) andτk,αbe related via the Fermi’s golden rule:\n1\nταk= 2π/integraldisplay\nk′ua(k−k′)/summationdisplay\nα′sa\nα,α′sa\nα′,αδ(Ekα−Ek′α′),(7)\nwhere/integraltext\nk≡/integraltext\ndDk/(2π)D. In this paper we restrict\nourselves to spin-independent ( a= 0) disorder and3\nspin-dependent disorder oriented along the equilibrium-\nexchange-field direction( a=z)12. Performing theconventional13integration around the branch cuts of P,\nwe obtain\n˜χQP\nxx(iω) =V∆2\n0/integraldisplay∞\n−∞dǫ\n2πif(ǫ)[P(ǫ+iδ,ǫ+iω)−P(ǫ−iδ,ǫ+iω)+P(ǫ−iω,ǫ+iδ)−P(ǫ−iω,ǫ−iδ)] (8)\nwheref(ǫ) is the Fermi function. Next, we perform an\nanalytical continuation iω→ω+iηand take the imag-\ninary part of the resulting retarded response function.\nAssuming low temperatures, this yields\nαG=∆2\n0\n2πs0{Re[P(−iδ,iδ)]−Re[P(iδ,+iδ)]}\n=∆2\n0\n2πs0Re(PA,R−PR,R) (9)\nwheres0=S0/V,\nPR(A),R=/integraldisplay\nkΛR(A),R\nα,β(k)GR\nβ(0,k)sx\nβ,α(k)GR(A)\nα(0,k)\n(10)\nandGR(A)(0,k) is the retarded (advanced) Green’s func-\ntion at the Fermi energy. The principal difficulty of\nEq.( 9) resides in solving the Dyson equation for the ver-\ntex function. We first discuss our method of solution in\ngeneral terms before turning in Sections III and IV to its\napplication to the s−dandp−dmodels.\nC. Evaluation of Impurity Vertex Corrections for\nMulti-Band Models\nEq.( 6) encodes disorder-induced diffusive correlations\nbetween itinerant carriers, and is an integral equation\nof considerable complexity. Fortunately, it is possible to\ntransform it into a relatively simple algebraic equation,\nprovided that the impurity potentials are short-rangedin\nreal space.Referring back at Eq.( 6) it is clear that the solution of\nthe Dysonequationwouldbe trivialifthevertexfunction\nwasindependent ofmomentum. That is certainlynot the\ncase in general, because the matrix elements of the spin\noperators may be momentum dependent. Yet, for short-\nrange scatterers the entire momentum dependence of the\nvertex matrix elements comes from the eigenstates alone:\nsa\nα,α′(k,k′) =/summationdisplay\nm,m′∝an}bracketle{tαk|m∝an}bracketri}ht∝an}bracketle{tm′|α′k′∝an}bracketri}htsa\nm,m′(11)\nThis property motivates our solution strategy which\ncharacterizes the momentum dependence of the vertex\nfunction by expanding it in terms of the eigenstates of sz\n(sxorsybases would work equally well):\nΛα,β(k) =∝an}bracketle{tαk|Λ|βk∝an}bracketri}ht\n=/summationdisplay\nm,m′∝an}bracketle{tαk|m∝an}bracketri}htΛm,m′∝an}bracketle{tm′|βk∝an}bracketri}ht(12)\nwhere|m∝an}bracketri}htisaneigenstateof sz, witheigenvalue m. Plug-\nging Eqs.( 11) and ( 12) into Eq.( 6) demonstrates that,\nas expected, Λ m,m′isindependent of momentum. After\ncancelling common factors from both sides of the result-\ning expressionand using ∂qua(q) = 0 (a= 0,z)we arrive\nat\nΛR(A),R\nm,m′=sx\nm,m′+/summationdisplay\nl,l′UR(A),R\nm,m′:l,l′ΛR(A),R\nl,l′ (13)\nwhere\nUR(A),R\nm,m′:l,l′≡/parenleftbig\nu0+uzmm′/parenrightbig/integraldisplay\nk∝an}bracketle{tm|αk∝an}bracketri}htGR(A)\nα(0,k)∝an}bracketle{tαk|l∝an}bracketri}ht∝an}bracketle{tl′|βk∝an}bracketri}htGR\nβ(0,k)∝an}bracketle{tβk|m′∝an}bracketri}ht (14)\nEqs. ( 12),(13)and(14)provideasolutionforthevertex\nfunction that is significantly easier to analyse than the\noriginal Dyson equation.III. GILBERT DAMPING FOR A MAGNETIC\n2DEG\nThe first model we consider is a two-dimensional elec-\ntrongas(2DEG)model withferromagnetismandRashba\nspin-orbit interactions. We refer to this as the magnetic4\n2DEG (M2DEG) model. This toy model is almost never\neven approximately realistic14, but a theoretical study\nof its properties will prove useful in a number of ways.\nFirst, it is conducive to a fully analytical evaluation of\nthe Gilbert damping, which will allow us to precisely un-\nderstand the role of different actors. Second, it enables\nus to explain in simple terms why higher order vertex\ncorrections are significant when there is spin-orbit inter-\naction in the band structure. Third, the Gilbert damping\nof a M2DEG has qualitative features similar to those of\n(Ga,Mn)As.\nThe band Hamiltonian of the M2DEG model is\nH=k2\n2m+bk·σ (15)\nwherebk= (−λky,λkx,∆0), ∆0is the difference be-\ntween majority and minority spin exchange-correlation\npotentials, λis the strength of the Rashba SO couplingand/vector σ= 2/vector sis avectorofPaulimatrices. Thecorrespond-\ning eigenvalues and eigenstates are\nE±,k=k2\n2m±/radicalBig\n∆2\n0+λ2k2 (16)\n|αk∝an}bracketri}ht=e−iszφe−isyθ|α∝an}bracketri}ht (17)\nwhere φ=−tan−1(kx/ky) and θ=\ncos−1(∆0//radicalbig\n∆2\n0+λ2k2) are the spinor angles and\nα=±is the band index. It follows that\n∝an}bracketle{tm|α,k∝an}bracketri}ht=∝an}bracketle{tm|e−iszφe−isyθ|α∝an}bracketri}ht\n=e−imφdm,α(θ) (18)\nwheredm,α=∝an}bracketle{tm|e−isyθ|α∝an}bracketri}htis a Wigner function for\nJ=1/2 angular momentum15. With these simple spinors,\nthe azimuthal integral in Eq.( 14) can be performed an-\nalytically to obtain\nUR(A),R\nm,m′:l,l′=δm−m′,l−l′(u0+uzmm′)/summationdisplay\nα,β/integraldisplaydkk\n2πdmαGR(A)\nα(k)dlα(θ)dm′β(θ)GR\nβ(k)dl′β(θ), (19)\nwhere the Kronecker delta reflects the conservation of\nthe angular momentum along z, owing to the azimuthal\nsymmetry of the problem. In Eq.( 19)\ndm,m′=/parenleftbigg\ncos(θ/2)−sin(θ/2)\nsin(θ/2) cos(θ/2)/parenrightbigg\n,(20)\nand the retarded and advanced Green’s functions are\nGR(A)\n+=1\n−ξk−bk+(−)iγ+\nGR(A)\n−=1\n−ξk+bk+(−)iγ−, (21)\nwhereξk=k2−k2\nF\n2m,bk=/radicalbig\n∆2\n0+λ2k2, andγ±is (half)the golden-rulescatteringrate ofthe band quasiparticles.\nIn addition, Eq. ( 13) is readily inverted to yield\nΛR(A),R\n+,+= ΛR(A),R\n−,−= 0\nΛR(A),R\n+,−=1\n21\n1−UR(A),R\n+,−:+,−\nΛR(A),R\n−,+=1\n21\n1−UR(A),R\n−,+:−,+(22)\nIn order to make further progress analytically we as-\nsumethat (∆ 0,λkF,γ)<< EF=k2\nF/2m. It then follows\nthatγ+≃γ−≡γand that γ=πN2Du0+πN2Duz\n4≡\nγ0+γz. Eqs. ( 19) and ( 20) combine to give\nUR,R\n−,+:−,+=UR,R\n+,−:+,−= 0\nUA,R\n−,+:−,+= (γ0−γz)/bracketleftbiggi\n−b+iγcos4/parenleftbiggθ\n2/parenrightbigg\n+i\nb+iγsin4/parenleftbiggθ\n2/parenrightbigg\n+2\nγcos2/parenleftbiggθ\n2/parenrightbigg\nsin2/parenleftbiggθ\n2/parenrightbigg/bracketrightbigg\nUA,R\n+,−:+,−= (UA,R\n−,+:−,+)⋆(23)\nwhereb≃/radicalbig\nλ2k2\nF+∆2\n0and cosθ≃∆0/b. The first\nand second terms in square brackets in Eq.( 23) emerge\nfrom inter-band transitions ( α∝ne}ationslash=βin Eq. ( 19)), while\nthe last term stems from intra-band transitions ( α=β).Amusingly, Uvanishes when the spin-dependent scatter-\ning rate equals the Coulomb scattering rate ( γz=γ0); in\nthis particular instance vertex corrections are completely\nabsent. On the other hand, when γz= 0 and b << γ5\nwe have UA,R\n−,+:−,+≃UA,R\n+,−:+,−≃1, implying that vertex\ncorrections strongly enhance Gilbert damping (recall Eq.\n( 22)). We will discuss the role of vertexcorrectionsmore\nfully below.\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0050.0100.015αG∆0=0.3 εF  ;  λ kF = 0 ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG. 2: M2DEG : Gilbert damping in the absence of spin-\norbit coupling. When the intrinsic spin-orbit interaction is\nsmall, the 1st vertex correction is sufficient for the evalua-\ntion of Gilbert damping, provided that the ferromagnet’s ex -\nchange splitting is large compared to the lifetime-broaden ing\nof the quasiparticle energies. For more disordered ferroma g-\nnets (EFτ0<5 in this figure) higher order vertex corrections\nbegin to matter. In either case vertex corrections are signi fi-\ncant. In this figure 1 /τ0stands for the scattering rate off spin-\nindependent impurities, defined as a two-band average at the\nFermi energy, and the spin-dependent and spin-independent\nimpurity strengths are chosen to satisfy u0= 3uz.\nAfter evaluating Λ( k) from Eqs. ( 12),( 22)and ( 23),\nthe last step is to compute\nPR(A),R=/integraldisplay\nkΛR(A),R\nα,β(k)sx\nβ,α(k)GR(A)\nα(k)GR\nβ(k).(24)\nSinceweareassumingthattheFermienergyisthelargest\nenergy scale, the integrand in Eq. ( 24) is sharply peaked\nat the Fermi surface, leading to PR,R≃0. In the case of\nspin-independent scatterers ( γz= 0→γ=γ0), tedious\nbut straightforward algebra takes us to\nαG(uz= 0) =N2D∆2\n0\n4s0γ0(λ2k2\nF)(b2+∆2\n0+2γ2\n0)\n(b2+∆2\n0)2+4∆2\n0γ2\n0.(25)\nEq. (29) agrees with results published in the recent\nliterature16. We note that αG(uz= 0) vanishes in the\nabsence of SO interactions, as expected. It is illustrative\nto expand Eq. ( 25) in the b >> γ 0regime:\nαG(uz= 0)≃N2D∆2\n0\n2s0/bracketleftbiggλ2k2\nF\n2(b2+∆2\n0)1\nγ0+λ4k4\nF\n(b2+∆2\n0)3γ0/bracketrightbigg\n(26)\nwhich displays intra-band ( ∼γ−1\n0) and inter-band ( ∼\nγ0) contributions separately. The intra-band damp-\ning is due to the dependence of band eigenenergies on0.00 0.05 0.10 0.15 0.20\n1/(εFτ0)0.00.20.40.60.81.0αG∆0=εF/3  ;  λ kF=1.2 εF  ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG.3:M2DEG :Gilbert dampingforstrongSOinteractions\n(λkF= 1.2EF≃4∆0). In this case higher order vertex cor-\nrections matter (up to 20 %) even at low disorder. This sug-\ngests that higher order vertex corrections will be importan t\nin real ferromagnetic semiconductors because their intrin sic\nSO interactions are generally stronger than their exchange\nsplittings.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ0)0.0000.0050.0100.0150.020αG∆0=0.3 εF  ;  λkF=∆0/5  ;  u0=3 uz\nintra−band\ninter−band\ntotal\nFIG. 4: M2DEG : Gilbert damping for moderate SO inter-\nactions ( λkF= 0.2∆0). In this case there is a crossover be-\ntween the intra-band dominated and the inter-band domi-\nnated regimes, which gives rise to a non-monotonic depen-\ndence of Gilbert damping on disorder strength. The stronger\nthe intrinsic SO relative to the exchange field, the higher th e\nvalue of disorder at which the crossover occurs. This is why\nthe damping is monotonically increasing with disorder in Fi g.\n( 2) and monotonically decreasing in Fig. ( 3).\nmagnetization orientation, the breathing Fermi surface\neffect4which produces more damping when the band-\nquasiparticles scatter infrequently because the popula-\ntion distribution moves further from equilibrium. The\nintra-band contribution to damping therefore tends to\nscale with the conductivity. For stronger disorder,\nthe inter-band term in which scattering relaxes spin-6\norientations takes over and αGis proportional to the\nresistivity. Insofar as phonon-scattering can be treated\nas elastic, the Gilbert damping will often show a non-\nmonotonic temperature dependence with the intra-band\nmechanism dominating at low-temperatures when the\nconductivity is largeand the inter-band mechanism dom-\ninatingathigh-temperatureswhentheresistivityislarge.\nFor completeness, we also present analytic results for\nthe case γ=γzin theb >> γ zregime:\nαG(u0= 0)≃N2D∆2\n0\n2s0/bracketleftbigg1\nγzλ2k2\nF\n6b2−2∆2\n0+γz3b4+6b2∆2\n0−∆4\n0\n(3b2−∆2\n0)3/bracketrightbigg\n(27)\nThis expression illustrates that spin-orbit (SO) interac-\ntions in the band structure are a necessary condition for\nthe intra-band transition contribution to αG. The in-\nterband contribution survives in absence of SO as long\nas the disorder potential is spin-dependent. Interband\nscatteringis possiblefor spin-dependent disorderbecause\nmajority and minority spin states on the Fermi surface\nare not orthogonal when their potentials are not identi-\ncal. Note incidentally the contrast between Eq.( 26) and\nEq. ( 27): in the former the inter-bandcoefficient is most\nsuppressed at weak intrinsic SO interaction while in the\nlatter it is the intra-band coefficient which gets weakest\nfor small λkF.\nMore general cases relaxing the (∆ 0,λkF,γ)<< E F\nassumption must be studied numerically; the results are\ncollected in Figs. ( 2), ( 3) and ( 4). Fig ( 2) highlights\nthe inadequacy of completely neglecting vertex correc-\ntions in the limit of weak spin-orbit interaction; the in-\nclusion of the the leading order vertex correction largely\nsolves the problem. However, Fig. ( 2) and ( 3) together\nindicate that higher order vertex corrections are notice-\nable when disorder or spin-orbit coupling is strong. In\nthe light of the preceding discussion the monotonic de-\ncay in Fig.( 3) may appear surprising because the inter-\nband contribution presumably increases with γ. Yet,\nthis argument is strictly correct only for weakly spin-\norbit coupled systems, where the crossover betwen inter-\nband and intra-band dominated regimes occurs at low\ndisorder. For strongly spin-orbit coupled systems the\ncrossover may take place at a scattering rate that is (i)\nbeyondexperimentalrelevanceand/or(ii)largerthanthe\nband-splitting, in which case the inter-band contribution\nbehaves much like its intra-band partner, i.e. O(1/γ).\nNon-monotonic behavior is restored when the spin-orbit\nsplitting is weaker, as shown in Fig. ( 4).\nFinally, our analysis opens an opportunity to quan-\ntify the importance of higher order impurity vertex-\ncorrections. Kohno, Shibata and Tatara11claim that the\nbare vertex along with the firstvertex correction fully\ncaptures the Gilbert damping of a ferromagnet, provided\nthat ∆ 0τ >>1. To first order in Uthe vertex function\nis\nΛR(A),R\nm,m′=sx\nm,m′+/summationdisplay\nll′UR(A),R\nm,m′:l,l′sx\nl,l′(28)Takingγ=γzfor simplicity, we indeed get\nlim\nλ→0αG≃Aγ+O(γ2)\nA(1)\nA(∞)= 1 (29)\nwhereA(1) contains the first vertex correction only, and\nA(∞) includes all vertex corrections. However, the state\nof affairs changes after turning on the intrinsic SO inter-\naction, whereupon Eq. ( 29) transforms into\nαG(λ∝ne}ationslash= 0)≃Bγ+C1\nγ\nB(1)\nB(∞)=∆2\n0(3b2−∆2\n0)3(3b2+∆2\n0)\n4b6(3b4+6b2∆2\n0−∆4\n0)\nC(1)\nC(∞)=(b2+∆2\n0)(3b2−∆2\n0)\n4b4(30)\nWhen ∆ 0<< λk F,bothintra-bandand inter-bandratios\nshow a significant deviation from unity17, to which they\nconverge as λ→0. In order to understand this behavior,\nlet us look back at Eq. ( 22). There, we can formally ex-\npand the vertex function as Λ =1\n2/summationtext∞\nn=0Un, where the\nn-th order term stems from the n-th vertex correction.\nFrom Eq. ( 23) we find that when λ= 0,Un∼O(γn)\nand thus n≥2 vertex corrections will not matter for the\nGilbert damping, which is O(γ)18whenEF>> γ. In\ncontrast, when λ∝ne}ationslash= 0 the intra-band term in Eq. ( 23)\nis no longer zero, and consequently allpowers of Ucon-\ntainO(γ0) andO(γ1) terms. In other words, all vertices\ncontribute to O(1/γ) andO(γ) in the Gilbert damping,\nespecially if λkF/∆0is not small. This conclusion should\nprove valid beyond the realm of the M2DEG because it\nrelies only on the mantra “intra-band ∼O(1/γ); inter-\nband∼O(γ)”. Our expectation that higher order vertex\ncorrectionsbe importantin (Ga,Mn)As will be confirmed\nnumerically in the next section.\nIV. GILBERT DAMPING FOR (Ga,Mn)As\n(Ga,Mn)As and other (III,Mn)V ferromagnets are like\ntransition metals in that their magnetism is carried\nmainly by d-orbitals, but unlike transition metals in that\nneither majority nor minority spin d-orbitals are present\nat the Fermi energy. The orbitals at the Fermi energy\nare very similar to the states near the top of the valence\nband states of the host (III,V) semiconductor, although\nthey are of course weakly hybridized with the minority\nand majority spin d-orbitals. For this reason the elec-\ntronic structure of (III,Mn)V ferromagnets is extremely\nsimple and can be described reasonably accurately with\nthe phenomenologicalmodel whichwe employin this sec-\ntion. Becausethe top ofthe valence band in (III,V) semi-\nconductors is split by spin-orbit interactions, spin-orbit\ncoupling plays a dominant role in the bands of these fer-\nromagnets. An important consequence of the strong SO7\ninteraction in the band structure is that diffusive vertex\ncorrections influence αGsignificantly at allorders; this\nis the central idea of this section.\nUsing a p-d mean-field theory model8for the ferro-\nmagnetic groundstate and afour-band sphericalmodel19\nfor the host semiconductor band structure, Ga 1−xMnxAs\nmay be described by\nH=1\n2m/bracketleftbigg/parenleftbigg\nγ1+5\n2γ2/parenrightbigg\nk2−2γ3(k·s)2/bracketrightbigg\n+∆0sz,(31)\nwheresis the spin operator projected onto the J=3/2\ntotal angular momentum subspace at the top of the va-\nlence band and {γ1= 6.98,γ2=γ3= 2.5}are the Lut-\ntinger parameters for the spherical-band approximation\nto GaAs. In addition, ∆ 0=JpdSNMnis the exchange\nfield,Jpd= 55meVnm3is the p-d exchange coupling,\nS= 5/2 is the spin of the Mn ions, NMn= 4x/a3is\nthe density of Mn ions, and a= 0.565nm is the lattice\nconstant of GaAs.\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0200.0400.060αGp=0.6 nm−3 (εF=500 meV)  ;  x=0.04  ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG. 5: GaMnAs : Higher order vertex corrections make a\nsignificant contributionto Gilbert damping, dueto theprom i-\nnent spin-orbit interaction in the band structure of GaAs.\nxis the Mn fraction, and pis the hole concentration that\ndetermines the Fermi energy EF. In this figure, the spin-\nindependent impurity strength u0was taken to be 3 times\nlarger than the magnetic impurity strength uz. 1/τ0corre-\nsponds to the scattering rate off Coulomb impurities and is\nevaluated as a four-band average at the Fermi energy.\nThe ∆ 0= 0 eigenstates of this model are\n|˜α,k∝an}bracketri}ht=e−iszφe−isyθ|˜α∝an}bracketri}ht (32)\nwhere|˜α∝an}bracketri}htis an eigenstate of szwith eigenvalue ˜ α. Un-\nfortunately, the analytical form of the ∆ 0∝ne}ationslash= 0 eigen-\nstates is unknown. Nevertheless, since the exchange field\npreserves the azimuthal symmetry of the problem, the\nφ-dependence of the full eigenstates |αk∝an}bracketri}htwill be iden-\ntical to that of Eq. ( 32). This observation leads to\nUm,m′:l,l′∝δm−m′,l−l′, which simplifies Eq. ( 14). αG\ncanbe calculatednumericallyfollowingthe stepsdetailed0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0020.0040.0060.0080.010αGp=0.2 nm−3 (εF=240 meV)  ;  x=0.08  ;  u0=3 uz\nintra−band\ninter−band\ntotal\nγ3=γ2=0.5\nFIG. 6: GaMnAs : When the spin-orbit splitting is reduced\n(in this case by reducing the hole density to 0 .2nm−3and\nartificially taking γ3= 0.5), the crossover between inter-\nand intra-band dominated regimes produces a non-monotonic\nshape of the Gilbert damping, much like in Fig. ( 4). When\neitherγ2orpis made larger or xis reduced, we recover the\nmonotonic decay of Fig.( 5).\nin the previous sections; the results are summarized in\nFigs. ( 5) and ( 6). Note that vertex corrections mod-\nerately increase the damping rate, as in the case of a\nM2DEG model with strong spin-orbit interactions. Fig.\n( 5) underlines both the importance of higher order ver-\ntex corrections in (Ga,Mn)As and the monotonic decay\nof the damping as a function of scattering rate. The lat-\nter signals the supremacy of the intra-band contribution\nto damping, accentuated at larger hole concentrations.\nHadtheintrinsicspin-orbitinteractionbeensubstantially\nweaker20,αGwould have traced a non-monotonic curve\nas shown in Fig. ( 6). The degree to which the intraband\nbreathing Fermi surface model effect dominates depends\non the details of the band-structure and can be influ-\nenced by corrections to the spherical model which we\nhaveadoptedheretosimplifythevertex-correctioncalcu-\nlation. The close correspondence between Figs. ( 5)-( 6)\nand Figs. ( 3)-( 4) reveals the success of the M2DEG\nas a versatile gateway for realistic models and justifies\nthe extensive attention devoted to it in this paper and\nelsewhere.\nV. ASSESSMENT OF THE\nTORQUE-CORRELATION FORMULA\nThus far we have evaluated the Gilbert damping for\na M2DEG model and a (Ga,Mn)As model using the\n(bare) spin-flip vertex ∝an}bracketle{tα,k|sx|β,k∝an}bracketri}htand its renormal-\nized counterpart ∝an}bracketle{tα,k|Λ|β,k∝an}bracketri}ht. The vertex corrected re-\nsults are expected to be exact for 1 /τsmall compared\nto the Fermi energy. For practical reasons, state-of-the-\nart band-structure calculations5forgo impurity vertex8\ncorrections altogether and instead employ the torque-\ncorrelation matrix element, which we shall denote as\n∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht(see below for an explicit expression). In\nthis section we compare damping rates calculated using\nsx\nα,βvertices with those calculated using Kα,βvertices.\nWe also compare both results with the exact damping\nrates obtained by using Λ α,β. The ensuing discussion\noverlaps with and extends our recent preprint6.\nWe shall begin by introducing the following identity4:\n∝an}bracketle{tα,k|sx|β,k∝an}bracketri}ht=i∝an}bracketle{tα,k|[sz,sy]|β,k∝an}bracketri}ht\n=i\n∆0(Ek,α−Ek,β)∝an}bracketle{tα,k|sy|β,k∝an}bracketri}ht\n−i\n∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(33)\nIn Eq. ( 33) we have decomposed the mean-field quasi-\nparticle Hamiltonian into a sum of spin-independent, ex-\nchange spin-splitting, and other spin-dependent terms:\nH=Hkin+Hso+Hex, whereHkinis the kinetic (spin-\nindependent) part, Hex= ∆0szis the exchange spin-\nsplitting term and Hsois the piece that contains the in-\ntrinsic spin-orbit interaction. The last term on the right\nhand side of Eq. ( 33) is the torque-correlation matrix\nelement used in band structure computations:\n∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht ≡ −i\n∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(34)\nEq. ( 33) allows us to make a few general remarks on\nthe relation between the spin-flip and torque-correlation\nmatrix elements. For intra-band matrix elements, one\nimmediately finds that sx\nα,α=Kα,αand hence the two\napproaches agree. For inter-band matrix elements the\nagreement between sx\nα,βandKα,βshould be nearly iden-\ntical when the first term in the final form of Eq.( 33)\nis small, i.e.when21(Ek,α−Ek,β)<<∆0. Since this\nrequirement cannot be satisfied in the M2DEG, we ex-\npect that the inter-band contributions from Kandsx\nwill always differ significantly in this model. More typi-\ncalmodels,likethefour-bandmodelfor(Ga,Mn)As, have\nband crossings at a discrete set of k-points, in the neigh-\nborhood of which Kα,β≃sx\nα,β. The relative weight of\nthese crossing points in the overall Gilbert damping de-\npends on a variety of factors. First, in order to make\nan impact they must be located within a shell of thick-\nness 1/τaround the Fermi surface. Second, the contri-\nbution to damping from those special points must out-\nweigh that from the remaining k-points in the shell; this\nmight be the case for instance in materials with weak\nspin-orbit interaction and weak disorder, where the con-\ntribution from the crossing points would go like τ(large)\nwhile the contribution from points far from the cross-\nings would be ∼1/τ(small). Only if these two con-\nditions are fulfilled should one expect good agreement\nbetween the inter-band contribution from spin-flip and\ntorque-correlationformulas. When vertexcorrectionsare\nincluded, of course, the same result should be obtained\nusing either form for the matrix element, since all matrixelements are between essentially degenerate electronic\nstates when disorder is treated non-perturbatively6,16.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.0000.0020.0040.0060.0080.010αG∆0=0.8 εF ;  λ kF=0.05 εF  ;  uz =0\nK\nsx\nΛ\nFIG. 7:M2DEG : Comparison of Gilbert damping predicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. In this figure the intrins ic\nspin-orbit interaction is relatively weak ( λkF= 0.05EF≃\n0.06∆0) and we have taken uz= 0. The torque correla-\ntionformula does notdistinguish between spin-dependenta nd\nspin-independent disorder.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.000.050.100.150.20αG∆0=0.1 εF  ;  λ kF=0.5 εF  ;  uz=0\nK\nΛ\nsx\nFIG. 8:M2DEG : Comparison of Gilbert damping predicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. In this figure the intrins ic\nspin-orbit interaction is relatively strong ( λkF= 0.5EF=\n5∆0) and we have taken uz= 0\nIn the remining part of this section we shall focus on a\nmore quantitative comparison between the different for-\nmulas. For the M2DEG it is straightforward to evaluate\nαGanalytically using Kinstead of sxand neglecting ver-\ntex corrections; we obtain\nαK\nG=N2D∆0\n8s0/bracketleftBigg\nλ2k2\nF\nb2∆0\nγ+/parenleftbiggλ2k2\nF\n∆0b/parenrightbigg2γ∆0\nγ2+b2/bracketrightBigg\n(35)9\nwhere we assumed ( γ,λkF,∆0)<< ǫF. By compar-\ning Eq. ( 35) with the exact expression Eq. ( 25), we\nfind that the intra-band parts are in excellent agreement\nwhen ∆ 0<< λk F, i.e. when vertex corrections are rela-\ntively unimportant. In contrast, the inter-band parts dif-\nfer markedly regardless of the vertex corrections. These\ntrendsarecapturedby Figs. (7) and( 8), which compare\nthe Gilbert damping obtained from sx,Kand Λ matrix\nelements. Fig. ( 7) corresponds to the weak spin-orbit\nlimit, whereitisfoundthatindisorderedferromagnets sx\nmaygrosslyoverestimatetheGilbertdampingbecauseits\ninter-band contributiondoes not vanish even as SO tends\nto zero. As explained in Section III, this flaw may be re-\npaired by adding the leading order impurity vertex cor-\nrection. The torque-correlation formula is free from such\nproblem because Kvanishes identically in absence of SO\ninteraction. Thus the main practical advantage of Kis\nthat it yields a physically sensible result without having\nto resort to vertex corrections. Continuing with Fig.( 7),\nat weak disorder the intra-band contributions dominate\nand therefore sxandKcoincide; even Λ agrees, because\nfor intra-band transitions at weak spin-orbit interaction\nthe vertex corrections are unimportant. Fig. ( 8) cor-\nresponds to the strong spin-orbit case. In this case, at\nlow disorder sxandKagree well with each other, but\ndiffer from the exact result because higher order vertex\ncorrections alter the intra-band part substantially. For a\nsimilar reason, neither sxnorKagree with the exact Λ\nat higher disorder. Based on these model calculations,\nwe do not believe that there are any objective grounds to\nprefer either the Ktorque-correlation or the sxspin-flip\nformula estimate of αGwhen spin-orbit interactions are\nstrong and αGis dominated by inter-band relaxation. A\nprecise estimation of αGunder these circumstances ap-\npears to require that the character of disorder, incud-\ning its spin-dependence, be accounted for reliably and\nthat the vertex-correction Dyson equation be accurately\nsolved. Carrying out this program remains a challenge\nboth because of technical complications in performing\nthe calculation for general band structures and because\ndisorder may not be sufficiently well characterized.\nAnalogous considerations apply for Figs. ( 9) and\n( 10), which show results for the four-band model re-\nlated to (Ga,Mn)As. These figures show results similar\nto those obtained in the strong spin-orbit limit of the\nM2DEG (Fig. 8). Overall, our study indicates that\nthetorque-correlation formula captures the intra-band\ncontributions accurately when the vertex corrections are\nunimportant, while it is less reliable for inter-band con-\ntributions unless the predominant inter-band transitions\nconnect states that are close in energy. The torque-\ncorrelation formula has the practical advantage that it\ncorrectly gives a zero spin relaxation rate when there is\nno spin-orbit coupling in the band structure and spin-\nindependent disorder. The damping it captures derives\nentirely from spin-orbit coupling in the bands. It there-\nfore incorrectly predicts, for example, that the damp-\ning rate vanishes when spin-orbit coupling is absent in0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.000.100.200.300.400.50αGp=0.4nm−3  (εF=380 meV) ;  x=0.08 ; uz=0\nsx\nΛ\nK\nFIG.9:GaMnAs : Comparison ofGilbertdampingpredicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. pis the hole concentration\nthat determines the Fermi energy EFandxis the Mn frac-\ntion. Due to the strong intrinsic SO, this figure shows simila r\nfeatures as Fig.( 8).\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ)0.000.050.100.150.20αGp=0.8nm−3  (εF=605 meV) ; x=0.04 ;  uz=0\nsx\nΛ\nK\nFIG. 10: GaMnAs : Comparison of Gilbert damping pre-\ndicted using spin-flip and torque matrix element formulas, a s\nwell as the exact vertex corrected result. In relation to Fig .\n( 9) the effective spin-orbit interaction is stronger, due to a\nlargerpand a smaller x.\nthe bands and the disorder potential is spin-dependent.\nNevertheless, assuming that the dominant disorder is\nnormally spin-independent, the K-formula may have a\npragmatic edge over the sx-formula in weakly spin-orbit\ncoupled systems. In strongly spin-orbit coupled systems\nthere appears to be little advantage of one formula over\nthe other. We recommend that inter-band and intra-\nband contributions be evaluated separately when αGis\nevaluated using the torque-correlation formula. For the\nintra-band contribution the sxandKlife-time formulas\nare identical. The model calculations reported here sug-10\ngest that vertex corrections to the intra-band contribu-\ntion do not normally have an overwhelming importance.\nWe conclude that αGcan be evaluated relatively reliably\nwhen the intra-band contribution dominates. When the\ninter-band contribution dominates it is important to as-\nsess whether or not the dominant contributions are com-\ning from bands that are nearby in momentum space, or\nequivalently whether or not the matrix elements which\ncontribute originate from pairs of bands that are ener-\ngetically spaced by much less than the exchange spin-\nsplitting at the same wavevector. If the dominant con-\ntributions are from nearby bands, the damping estimate\nshould have the same reliability as the intra-band contri-\nbution. If not, we conclude that the αGestimate should\nbe regarded with caution.\nTo summarize, this article describes an evaluation\nof Gilbert damping for two simple models, a two-\ndimensionalelectron-gasferromagnetmodelwith Rashba\nspin-orbit interactions and a four-band model which pro-\nvides an approximate description of (III, Mn)V of fer-\nromagnetic semiconductors. Our results are exact in\nthe sense that they combine time-dependent mean field\ntheory6with an impurity ladder-sum to all orders, hence\ngiving us leverage to make the following statements.First, previously neglected higher order vertex correc-\ntions become quantitatively significant when the intrin-\nsic spin-orbit interaction is larger than the exchange\nsplitting. Second, strong intrinsic spin-orbit interaction\nleads to the the supremacy of intra-band contributions in\n(Ga,Mn)As, with the corresponding monotonic decay of\nthe Gilbert damping as a function ofdisorder. Third, the\nspin-torque formalism used in ab-initio calculations of\nthe Gilbert damping is quantitatively reliable as long as\nthe intra-band contributions dominate andthe exchange\nfield is weaker than the spin-orbit splitting; if these con-\nditions are not met, the use of the spin-torque matrix\nelement in a life-time approximation formula offers no\nsignificant improvement overthe originalspin-flip matrix\nelement.\nAcknowledgments\nThe authors thank Keith Gilmore and Mark Stiles for\nhelpful discussions and feedback. This work was sup-\nported by the Welch Foundation and by the National\nScience Foundation under grant DMR-0606489.\n1Foranintroductoryreviewsee D.C. RalphandM.D.Stiles,\nJ. Magn. Mag. Mater. 320, 1190 (2008).\n2J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag-\nnetic Structures III: Fundamentals of Nanomagnetism\n(Springer-Verlag, New York, 2005).\n3V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n4V. Kambersky, Czech J. Phys. B 26, 1366 (1976); V. Kam-\nbersky, Czech J. Phys. B 34, 1111 (1984).\n5K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett.\n99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416\n(2007).\n6Ion Garate and A.H. MacDonald, arXiv:0808.1373.\n7O. Gunnarsson, J. Phys. F 6, 587 (1976).\n8For reviews see T. Jungwirth et al., Rev. Mod. Phys. 78,\n809 (2006); A.H. MacDonald, P. Schiffer and N. Samarth,\nNature Materials 4, 195 (2005).\n9These simplified models sometimes have the advantage\nthat their parameters can be adjusted phenomenologically\nto fit experiments, compensating for inevitable inaccura-\ncies inab initio electronic structure calculations. This ad-\nvantage makes p−dmodels of (III,Mn)V ferromagnets\nparticularly useful. s−dmodels of transition elements are\nless realistic from the start because they do not account for\nthe minority-spin hybridized s−dbands which are present\nat the Fermi energy.\n10This is not the most general type of disorder for quasi-\nparticles with spin >1/2, but it will be sufficient for the\npurpose of this work.\n11H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006).\n12We assume that the spins of magnetic impurities are frozenalong the staticpart of the exchange field. In reality, the\ndirection of the impurity spins is a dynamical variable that\nis influenced by the magnetization precession.\n13G.D. Mahan, Many-Particle Physics (3rd Ed.), Physics of\nSolids and Liquids Series (2000)\n14A possible exception is the ferromagnetic 2DEG recently\ndiscovered in GaAs/AlGaAs heterostructures with Mn δ-\ndoping; see A. Bove et. al, arXiv:0802.3871v3.\n15J.J. Sakurai, Modern Quantum Mechanics , Addison-\nWesley (1994).\n16E.M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys.\nRev. B 75, 174434 (2007). In their case the inter-band\nsplitting in the Green’s function is Ω, while in our case it\nis 2b. In addition, we neglect interactions between band\nquasiparticles.\n17C(1) and C(∞) differ by as much as 25%; the disparity\nbetween B(1) andB(∞) may be even larger.\n18The disorder dependence in αGoriginates not only from\nthe vertex part, but from the Green’s functions as well.\nIt is useful to recall thatR\nGσG−σ∝1/(b+isg(σ)γ) andR\nGσGσ∝1/γ.\n19P. Yu, M. Cardona, Fundamentals of Semiconductors (3rd\nEd.), Springer (2005).\n20Notwithstanding that the four-band model is a SO → ∞\nlimit of the more general six-band model, we shall tune the\neffective spin-orbit strength via p(hole concentration) and\nγ3.\n21Strictly speaking, it is |sx\nα,β|2≃ |Kα,β|2what is needed,\nrather than sx\nα,β≃Kα,β. The former condition is less de-\nmanding, and can occasionally be satisfied when Eα−Eβ\nis of the order of the exchange splitting."    },    {        "title": "0809.2611v1.Stochastic_dynamics_of_magnetization_in_a_ferromagnetic_nanoparticle_out_of_equilibrium.pdf",        "content": "arXiv:0809.2611v1  [cond-mat.mes-hall]  16 Sep 2008Stochastic dynamics of magnetization in a ferromagnetic na noparticle out of\nequilibrium\nDenis M. Basko1and Maxim G. Vavilov2\n1International School of Advanced Studies (SISSA), via Beir ut 2-4, 34014 Trieste, Italy\n2Department of Physics, University of Wisconsin, Madison, W I 53706, USA\n(Dated: September 15, 2008)\nWe consider a small metallic particle (quantumdot) where fe rromagnetism arises as a consequence\nof Stoner instability. When the particle is connected to ele ctrodes, exchange of electrons between\nthe particle and the electrodes leads to a temperature- and b ias-driven Brownian motion of the\ndirection of the particle magnetization. Under certain con ditions this Brownian motion is described\nby the stochastic Landau-Lifshitz-Gilbert equation. As an example of its application, we calculate\nthe frequency-dependentmagnetic susceptibility of the pa rticle in a constant external magnetic field,\nwhich is relevant for ferromagnetic resonance measurement s.\nPACS numbers: 73.23.-b, 73.40.-c, 73.50.Fq\nI. INTRODUCTION\nThe description of fluctuations of the magnetization\nin small ferromagnetic particles pioneered by Brown1is\nbased on the Landau-Lifshitz-Gilbert (LLG) equation2,3\nwith a phenomenologically added stochastic term. This\napproach has been widely used: just a few recent appli-\ncations are a numerical study of the dynamic response\nof the magnetization to the oscillatory magnetic field,4\na numerical study of ferromagnetic resonance spectra,5\nstudy of resistance noise in spin valves,6and a study of\nthe magnetization switching and relaxation in the pres-\nence of anisotropy and a rotating magnetic field.7\nIn equilibrium the statistics of stochastic term in\nthe LLG equation can be simply written from the\nfluctuation-dissipation theorem.1However, out of equi-\nlibrium a proper microscopic derivation is required. Mi-\ncroscopic derivations of the stochastic LLG equation out\nof equilibrium, available in the literature, use the model\nof a localized spin coupled to itinerant electrons,8,9,10,11\nor deal with non-interacting electrons.12In contrast to\nthis approach, we start from a purely electronic system\nwhere the magnetization arises as a consequence of the\nStoner instability. Our derivation has certain similarity\nwith that of Ref. 13 for a bulk ferromagnet, where the di-\nrection of magnetization is fixed and cannot be changed\nglobally, so its local fluctuations are small and their de-\nscription by a gaussianaction is sufficient. This situation\nshould be contrasted to the case of a nanoparticle where\nthe direction ofthe magnetizationcan be completely ran-\ndomized by the fluctuations, so that the effective action\nfor the direction of the classicalmagnetization has a non-\ngaussian part. The bias-driven Brownian motion of the\nmagnetization with a fixed direction (due to and easy-\naxis anisotropy and ferromagnetic electrodes) has been\nalso studied in Ref. 14 using rate equations.\nWe assume that the single-electron spectrum of the\nparticle, which is also called a quantum dot in the litera-\nture, to be chaotic and described by the random-matrix\ntheory15,16. Totakeintoaccounttheelectron-electronin-teractions in the dot we use the universal Hamiltonian,17\nwith a generalized spin part, corresponding to a ferro-\nmagnetic particle. Electrons occupy the quantum states\nof the full Hamiltonian and form a net spin of the par-\nticle of order of S0≫1; throughout the paper we use\n¯h= 1. The dot is coupled to two leads, see Fig. 1, which\nwe assumed to be non-magnetic. The approach can be\neasily extended to the case of magnetic leads. The num-\nberNchof the transversechannels in the leads, which are\nwell coupled to the dot, is assumed to be large, Nch≫1.\nEquivalently, the escape rate 1 /τof electrons from the\ndot into the leads is largecompared to the single-electron\nmean level spacing δ1in the dot. This coupling to the\nleads is responsible for tunnelling processes of electrons\nbetween states in the leads and in the dot with random\nspin orientation. As a result of such tunnelling events,\nthe net spin of the particle changes. We show that this\nexchangeofelectronsgivesrisebothtotheGilbert damp-\ning and the magnetization noise in the presented model,\nand under conditions specified below, the time evolution\nof the particle spin is described by the stochastic LLG\nequation.\nWe study in detail the conditions for applicability of\nthe stochastic approach. We find that these limits are\nset by three independent criteria. First, the contact re-\nsistance should be low compared to the resistance quan-\ntum, which is equivalent to Nch≫1. If this condi-\ntion is broken, the statistics of the noise cannot be con-\nsidered gaussian. Physically, this condition means that\neach channel can be viewed as an independent source\nof noise, so the contribution of many channels results in\nthe gaussian noise by virtue of the central limit theo-\nrem ifNch≫1. Second, the system should not be too\nclose to the Stoner instability: the mean-field value of\nthe total spin S0≫√Nch. If this condition is violated,\nthe fluctuations of the absolute value of the magneti-\nzation become of the order of the magnetization itself.\nThird,S2\n0≫Teff/δ1, whereTeffis the effective tempera-\nture of the system, which is the energy scale of the elec-\ntronicdistribution function determined bya combination2\nNLNRB~\nz \nB~\nB0Drain Source\nµ+eVµFerromagnetic\nnanoparticle\nµ\nFIG. 1: (Color online). Device setup considered in this work :\na small ferromagnetic particle (quantum dot) coupled to two\nnon-magnetic leads (see text for details).\noftemperature andbiasvoltage(the Boltzmannconstant\nkB= 1throughoutthe paper). Otherwise, theseparation\nof the degrees of freedom into slow (the direction of the\nmagnetization) and fast (the electron dynamics and the\nfluctuations of the absolute value of the magnetization)\nis not possible.\nIn the present model we completely neglect the spin-\norbit interaction inside the particle, whose effect is as-\nsumedtobeweakascomparedtotheeffectoftheleads.18\nThe effects of the electron-electron interaction in the\ncharge channel (weak Coulomb blockade) are suppressed\nforNch≫1,15so we do not consider it.\nAs an application of the formalism, we consider themagnetic susceptibility in the ferromagnetic resonance\nmeasurements, which is a standardcharacteristicofmag-\nnetic samples. Recently, a progress was reported in mea-\nsurements of the magnetic susceptibility on small spatial\nscales in response to high-frequency magnetic fields.19\nMeasurements of the ferromagnetic resonance were also\nreportedfornanoparticles, connected to leadsfor asome-\nwhat different setup in Ref. 20.\nThe paper is organized as follows. In Sec. II we intro-\nduce the model for electrons in a small metallic particle\nsubject to Stoner instability. In Sec. III we analyze the\neffective bosonic action for the magnetization of the par-\nticle. In Sec. IV we obtain the equation of motion for the\nmagnetization with the stochastic Langevin term, which\nhas the form of the stochastic Landau-Lifshitz-Gilbert\nequation, and derive the associated Fokker-Planck equa-\ntion. In Sec. V we discuss the conditions for the applica-\nbility of the approach. In Sec. VI we calculate the mag-\nnetic susceptibility from the stochastic LLG equation.\nII. MODEL AND BASIC FORMALISM\nWithin the random matrix theory framework, elec-\ntrons in a closed chaotic quantum dot are described by\nthe following fermionic action:\nS[ψ,ψ∗] =/contintegraldisplay\ndt\nN/summationdisplay\nn,n′=1ψ†\nn(t)(δnn′i∂t−Hnn′)ψn′(t)−E(S(t))\n, Si≡N/summationdisplay\nn=1ψ†\nnˆσi\n2ψn. (2.1)\nHereψnis a two-component Grassmann spinor, truns along the Keldysh contour, as marked by/contintegraltext\n; ˆσx,y,zare the Pauli\nmatrices (we use the hat to indicate matrices in the spin space and us e the notation ˆ σ0for the 2 ×2 unit matrix).\nHnn′is anN×Nrandom matrix from a gaussian orthogonal ensemble, described by the pair correlators:\nHmnHm′n′=Nδ2\n1\nπ2[δmn′δnm′+δmm′δnn′]. (2.2)\nHereδ1is the mean single-particle level spacing in the dot.\nThe magnetization energy E(S) is the generalizationof the JsS2term in the universal Hamiltonian for the electron-\nelectron interaction in a chaotic quantum dot.17Since we are going to describe a ferromagnetic state with a largevalu e\nof the total spin on the dot, we must go beyond the quadratic term ; in fact, all terms should be included. E(S) can be\nviewed as the sum of all irreducible many-particle vertices in the spin c hannel, obtained after integrating out degrees\nof freedom with high energies (above Thouless energy); the corre sponding term in the action is thus local in time, and\ncan be written as the time integral of an instantaneous function E(S(t)). This functional can be decoupled using the\nHubbard-Stratonovich transformation with a real vector field h(t), which we call below the internal magnetic field:\nexp/parenleftbigg\n−i/contintegraldisplay\ndtE(S)/parenrightbigg\n=/integraldisplay\nDh(t) exp/parenleftbigg\ni/contintegraldisplay\ndt(2h·S−˜E(h))/parenrightbigg\n. (2.3)\nWe rewrite the action S[ψ,ψ∗] in the form\nS[ψ,ψ∗,h] =/contintegraldisplay\ndt\nN/summationdisplay\nn,n′=1ψ†\nn(t)/parenleftBig\nˆG−1/parenrightBig\nnn′ψn′(t)−˜E(h(t))\n, (2.4)3\nwhere the inverse Green’s function\n/parenleftBig\nˆG−1/parenrightBig\nnn′= (iˆσ0∂t+h·ˆσ)δnn′−Hnn′ˆσ0(2.5)\nis a matrix in time variables t,t′, in orbital indices nand\nn′with 1≤n,n′≤N, in spin indices, and in forward(+)\nand backward ( −) directions on the Keldysh contour.\nIntegration over fermionic fields ψn,ψ†\nnyields the purely\nbosonic action:\nS[h] =−iTr/braceleftBig\nln(−iˆG−1)/bracerightBig\n−/contintegraldisplay\ndt˜E(h(t)),(2.6)\nwhere the trace is taken over allindices of the Green’s\nfunction, listed above.\nIn the space of forward and backward directions on\nthe Keldysh contour, we perform the standard Keldysh\nrotation, introducing the retarded ( GR), advanced ( GA),\nKeldysh (GK), and zero ( GZ) components of the Green’s\nfunction:\n/parenleftbiggˆGRˆGK\nˆGZˆGA/parenrightbigg\n=1\n2/parenleftbigg\n1 1\n1−1/parenrightbigg/parenleftbiggˆG++ˆG+−\nˆG−+ˆG−−/parenrightbigg/parenleftbigg\n1 1\n−1 1/parenrightbigg\n,\n(2.7)\nas well as the classical ( hcl) and quantum ( hq) compo-\nnents of the field:\n/parenleftbigg\nhclhq\nhqhcl/parenrightbigg\n=1\n2/parenleftbigg\n1−1\n1 1/parenrightbigg/parenleftbigg\nh+0\n0−h−/parenrightbigg/parenleftbigg\n1 1\n1−1/parenrightbigg\n.\n(2.8)\nWe will also write this matrix as h=hclτcl+hqτq,\nwhereτclandτqare 2×2 matrices in the Keldysh space\ncoinciding with the unit 2 ×2 matrix and the first Pauli\nmatrix, ˆσx, respectively.\nThe saddle point of the bosonic action Eq. (2.6) is\nfound by the first order variation with respect to hcl,q(t),\nwhich gives the self-consistency equation:\nhq(t) = 0,−∂˜E(hcl(t))\n∂hcl\nj(t)=i\n2Tr\nn,σ/braceleftBig\nˆσjˆGK\nnn(t,t)/bracerightBig\n.(2.9)\nWe also note that the right-hand side of this equation is\nproportional to the total spin of electrons of the particle\nfor a given trajectory of hcl(t):\nS(t) =i\n4Tr\nn,σ/braceleftBig\nˆσjˆGK(t,t)/bracerightBig\n. (2.10)\nIn Eqs. (2.9) and (2.10), the trace is taken over orbital\nand spin indices only.\nIn the limit N→ ∞, one can obtain a closed equation\nfor the Green’s function traced over the orbital indices:21\nˆg(t,t′) =iδ1\nπN/summationdisplay\nn=1ˆGnn(t,t′). (2.11)\nThe matrix ˆ g(t,t′) satisfies the following constraint:\n/integraldisplay\nˆg(t,t′′)ˆg(t′′,t′)dt′′=τclˆσ0δ(t−t′),(2.12)where the right-hand side is just the direct product of\nunit matrices in the spin, Keldysh, and time indices.\nThe Wigner transform of ˆ gK(t,t′) is related to the spin-\ndependent distribution function ˆf(ε,t) of electrons in the\ndot:\n∞/integraldisplay\n−∞ˆgK(t+˜t/2,t−˜t/2)eiε˜td˜t= 2ˆf(ε,t).(2.13)\nIn equilibrium, ˆf(ε) = ˆσ0tanh(ε/2T).\nThe self-consistency condition (2.9) takes the form\n−∂˜E(hcl(t))\n∂hcl\ni(t)=π\n2δ1lim\nt′→tTr\nσ/braceleftbig\nˆσiˆgK(t,t′)/bracerightbig\n−2hcl\ni(t)\nδ1.\n(2.14)\nThe last term takes care of the anomaly arising from\nnon-commutativity of the limits N→ ∞andt′→t.\nIn this paper we consider the dot coupled to two leads,\nidentified as left ( L) and right ( R). The leads have\nNLandNRtransverse channels, respectively, see Fig 1.\nFor non-magnetic leads and spin-independent coupling\nbetween the leads and the particle, we can characterize\neach channel by its transmission Tnwith 0< Tn≤1\nand by the distribution function of electrons in the chan-\nnelFn(t−t′), assumed to be stationary. We consider\nthe limit of strong coupling between the leads and the\nparticle,/summationtextNch\nn=1Tn≫1.\nThe coupling to the leads gives rise to a self-energy\nterm, which should be included in the definition of the\nGreen’s function, Eq. (2.5). Without going into details\nof the derivation, presented in Ref. 21, we give the final\nform of the equation for the Green’s function traced over\nthe orbital states, Eq. (2.11):\n[∂t−ih·ˆσ,ˆg]\n=Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbigg\n−FnˆgZˆgRFn−FnˆgA−ˆgK\nˆgZ−ˆgZFn/parenrightbigg\n×/bracketleftbigg\nˆ1+Tn\n2/parenleftbigg\nˆgR−ˆ1+FnˆgZˆgRFn+FnˆgA\n0 −ˆgA−ˆ1+ ˆgZFn/parenrightbigg/bracketrightbigg−1\n.\n(2.15)\nHere the products of functions include convolution in\ntime variables. This equation is analogous to the Usadel\nequation used in the theory of dirty superconductors.22\nTo conclude this section, we discuss the dependence\n˜E(h). Deep in the ferromagnetic state, i.e.far from the\nStoner critical point, we expect the mean-field approach\nto give a good approximation for the total spin of the\ndot. Namely, the mean field acting on the electron spins,\nis given by 2 h0=dE(S)/dS≡E′(S). We then require\nthat the response of the system to this field gives the\nsame average value for the spin:\nS0=2h0\n2δ1=E′(S0)\n2δ1. (2.16)\nHere we evaluated S0from Eq. (2.10) and applied the\nself-consistency equation (2.14) to equilibrium state with4\nˆgK∝ˆσ0, when the contribution of the first term in the\nright hand side of Eq. (2.14) vanishes.\nNot expecting strong deviations of the magnitude of\nthe spin from the mean-field value, we focus on the form\nof˜E(h) when|h| ≈h0. The inverse Fourier transform of\nEq. (2.3) and angular integration for the isotropic E(S)\ngives\ne−i˜E(h)∆t= const∞/integraldisplay\n0sin2Sh∆t\n2Sh∆te−iE(S)∆tS2dS,(2.17)\nwhere ∆tis the infinitesimal time increment used in the\nconstruction of the functional integral in Eq. (2.3).\nExpanding E(S) near the mean-field value S0,\nE(S)≈E(S0)+E′(S−S0)+E′′\n2(S−S0)2,(2.18)\nperforming the integration in the stationary phase ap-\nproximation and using S0=h0/δ1=−E′/(2δ1), we ob-\ntain\n˜E(h) =−2(h−h0−E′′S0/2)2\nE′′+˜E0,(2.19)\nwhere˜E0ish−independent term. This expression for\n˜E(h) defines the action S[h], Eq. (2.6).\nThe energy E(S) does not contain the energy Eorb(S),\nassociated with the orbital motion of electrons in the\nparticle. Namely, to form a total spin Sof the par-\nticle, we have to redistribute Selectrons over orbital\nstates, which changes the orbital energy of electrons by\nEorb(S)≃δ1S2. The total energy Etot(S) of the particle\nis the sum of two terms: Etot(S) =E(S)+Eorb(S). Sim-\nilarly, we obtain the total energy of the system in terms\nof internal magnetic field\n˜Etot(h) =˜E(h)−h2\nδ1\n=−2/parenleftbigg1\nE′′+1\n2δ1/parenrightbigg\n(h−h0)2+˜E1,(2.20)\nwhere˜E1does not depend on h. We notice that the\nextremumof ˜Etot(h)correspondsto h=h0anddescribes\nthe expectation value of the internal magnetic field in an\nisolated particle. The energy cost of fluctuations of the\nmagnitude of the internal magnetic field is characterized\nby the coefficient 1 /E′′+1/2δ1.\nIII. KELDYSH ACTION\nIn this Section we analyze the action Eq. (2.6) for the\ninternal magnetic field hα. We expect that the classical\ncomponent hcl(t) of this field contains fast and small os-\ncillations of its magnitude around the mean-field value\nh0. We further expect that the orientation of hcl(t)\nchanges slowly in time, but is not restricted to smalldeviations from some specific direction. Based on this\npicture, we introduce a unit vector n(t), assumed to de-\npend slowly on time, and write\nhcl(t) = (h0+hcl\n/bardbl(t))n(t), (3.1)\nwherehcl\n/bardbl(t) is assumed to be fast and small. We expand\nthe action (2.6) to the second order in small fluctuations\nof the quantum component hq(t) and the radial classical\ncomponent hcl\n/bardbl(t):\nS[h]≈ −2π\nδ1/integraldisplay\ndtgK(t,t)hq(t)+\n+8\nE′′/integraldisplay\ndthcl\n/bardbl(t)n(t)hq(t)\n−/integraldisplay\ndtdt′ΠR\nij(t,t′)hq\ni(t)hcl\n/bardbl(t′)nj(t′)\n−/integraldisplay\ndtdt′ΠA\nij(t,t′)hcl\n/bardbl(t)ni(t)hq\nj(t′)\n−/integraldisplay\ndtdt′ΠK\nij(t,t′)hq\ni(t)hq\nj(t′).(3.2)\nThe applicability of this quadratic expansion is discussed\nin Sec. VB.\nIn Eq. (3.2) we introduced the polarization operator,\ndefined as the kernel of the quadratic part of the action\nof the fluctuating bosonic fields:\n/parenleftbigg\nΠZΠA\nΠRΠK/parenrightbigg\n≡/parenleftbigg\nΠcl,clΠcl,q\nΠq,clΠq,q/parenrightbigg\n,(3.3a)\nΠαβ\nij(t,t′) =i\n2δ2Tr/braceleftbig\nlnG−1/bracerightbig\nδhβ\nj(t′)δhα\ni(t),(3.3b)\nwhereα,β=cl,qandi,j=x,y,z. The short time\nanomaly is explicitly taken into account in the definition\nof the polarization operators, see Eq. (3.14) below.\nThefirsttermofEq.(3.2)containsthevector gKofthe\nKeldysh component of the Green function ˆ gK= ˆσ0gK\n0+\nˆσ·gK. We emphasize that the Green’s function and\nthe polarization operator in Eq. (3.2) are calculated at\nhcl\n/bardbl(t) = 0 and hq(t) = 0 for a given trajectory of the\nclassical field h0n(t).\nA. Keldysh component of the Green function\nFor the Green’s function in the classical field we have\nˆgR(t,t′) =−ˆgA(t,t′) = ˆσ0δ(t−t′),(3.4)\nwhile the Keldysh component satisfies the equation\n/bracketleftbig\n∂t−ih0n·ˆσ,ˆgK/bracketrightbig\n=Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbig\n2Fn−ˆgK/parenrightbig\n.(3.5)\nWe introduce the notation\n1\nτ=Nch/summationdisplay\nn=1Tnδ1\n2π=1\nτL+1\nτR, (3.6)5\nThen the scalar gK\n0and vector gKcomponents of ˆ gK=\nˆσ0gK\n0+ˆσ·gKsatisfy two coupled equations:\n/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\ngK\n0(t,t′) =Nch/summationdisplay\nn=1Tnδ1\n2π2Fn(t−t′)\n+ih0[n(t)−n(t′)]·gK(t,t′), (3.7a)/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\ngK(t,t′)+h0[n(t)+n(t′)]×gK(t,t′)\n=ih0[n(t)−n(t′)]gK\n0(t,t′). (3.7b)\nAs a zero approximation, we can consider the station-\nary situation: gK\n0(t,t′) =gK\n0(t−t′) andn(t) = const. In\nthis case, we have\ngK\n0(t,t′) =τ\nτL2FL(t−t′)+τ\nτR2FR(t−t′) (3.8)\nandgK= 0.\nFor an arbitrarytime dependence n(t), Eq. (3.7b) can-\nnot be solved analytically. However, if the variation of\nn(t) is slow enough, we can make a gradient expansion:\n/parenleftbigg\n∂t+1\nτ/parenrightbigg\ngK+2h0n×gK=i˜th0˙ngK\n0.(3.9)\nHere we introduced t= (t+t′)/2,˜t=t−t′,∂t+∂t′=∂t.\nThe dependence on ˜tis split off and remains unchanged,\nwhile for the dependence on tthe solution is determined\nby a linear operator L+\nn:\nL±\nn=/parenleftbigg1\nτ±∂t±2h0n×/parenrightbigg−1\n, (3.10a)\nL+\nn(ω)X(ω) =n(n·X(ω))\n−iω+1/τ+\n+1\n2/summationdisplay\n±−n×[n×X(ω)]±i[n×X(ω)]\n−i(ω±2h0)+1/τ.\n(3.10b)\nHere we assume that the direction of the internal mag-\nnetic field nchanges slowly in time, and |˙n|τ≪1.\nThus, all perturbations of gKdecay with the charac-\nteristic time τ. In particular, the solution of Eq. (3.9)\nhas the form\ngK=i˜th0L+\nn˙ngK\n0≈i˜th0τgK\n0˙n+2h0τn×˙n\n(2h0τ)2+1.(3.11)\nExpression for the first term in Eq. (3.2) can be easily\nobtained from Eq. (3.11) by taking the limit ˜t→0 and\ntaking into account that any fermionic distribution func-\ntion in the time representation has the following equal-\ntime asymptote:\ngK\n0(t,t′)≈2\niπ1\nt−t′,(t→t′).(3.12)\nWe have\ngK(t,t) =2h0τ\nπ˙n+2h0τn×˙n\n(2h0τ)2+1. (3.13)We notice that n·gK(t,t) = 0, and therefore the first\nterm in the action Eq. (3.2) is coupled only to the tan-\ngential fluctuations of hq(t)⊥n(t).\nB. Polarization operator\nWe express the polarization operator in terms of the\nunit vector n(t). The polarization operator can be rep-\nresented as the response of the Green’s functions to a\nchange in the field, as follows directly from the defini-\ntion (3.3b) and the expression (2.6) for the action:\nΠαβ\nij(t,t′) =π\n2δ1Tr4×4/braceleftbig\nταˆσiδˆg(t,t)/bracerightbig\nδhβ\nj(t′)−2\nδ1τq\nαβδijδ(t−t′).\n(3.14)\nHere the Green function δˆg(t,t) can be calculated as the\nfirst-order response of the solution of Eq. (2.15) to small\narbitrary (in all three directions) increments of δhcl(t)\nandδhq(t). The zero-order solution of Eq. (2.15) in the\nfieldhcl=h0nandhq= 0 is\nˆg(t,t) = ˆσ0/parenleftbigg\nδ(t−t′)gK\n0(t−t′)\n0−δ(t−t′)/parenrightbigg\n.(3.15)\nFirst, we calculate δˆgZ, which responds only to δhq:\n/bracketleftbigg\n∂t+∂t′−1\nτ/bracketrightbigg\nδˆgZ(t,t′)−ih0ˆσ·n(t)δˆgZ(t,t′)\n+δˆgZ(t,t′)ih0ˆσ·n(t′) = 2iˆσ·δhq(t)δ(t−t′).\n(3.16)\nSince∂t+∂t′=∂t, the solution always remains propor-\ntional toδ(t−t′):\nδˆgZ(t,t′) =−2iˆσ(L−\nnδhq)(t)δ(t−t′).(3.17)\nGivenδˆgZ, components δˆgR,Acan be found either from\nEq. (2.15), or, equivalently, using the constraint ˆ g2=ˆ1:\nˆgδˆg+δˆgˆg= 0⇒δˆgR=−ˆgKδˆgZ\n2, δˆgA=δˆgZˆgK\n2.\n(3.18)\nWe notice that both δˆgR,Arespond only to hq(t) and,\ntherefore,\nΠZ\nij(t,t′)∝Tr{ˆσi(δˆgR(t,t)+δˆgA(t,t))}\nδhcl\nj(t′)≡0.(3.19)\nThis equation ensures that the action along the Keldysh\ncontour vanishes for hq≡0.\nTo evaluate the remaining three components of the po-\nlarization operator, we can apply the variational deriva-\ntives to the sum of δˆgK(t,t) +δˆgZ(t,t) with respect to\neither classical δhcl(t′) or quantum δhq(t′) field, which\ngive ΠR\nij(t,t′) and ΠK\nij(t,t′), respectively. Then, the ad-\nvanced component ΠA\nij(t,t′) = [ΠR\nji(t′,t)]∗.6\nThe equation for δˆgK=ˆσ·δgKreads as\n/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\nδgK(t,t′)\n+h0/bracketleftbig\nn(t)×δgK(t,t′)−δgK(t,t′)×n(t′)/bracketrightbig\n=i/bracketleftbig\nδhcl(t)−δhcl(t′)/bracketrightbig\ngK\n0(t−t′)\n−2iδhq(t)δ(t−t′)−Q(t,t′),\n(3.20a)\nwhere\nQ(t,t′) =Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbigggK\n0\n2δgZgK\n0\n2+FnδgZFn/parenrightbigg\n−Nch/summationdisplay\nn=1Tn(1−Tn)δ1\n2π/parenleftbigggK\n0\n2−Fn/parenrightbigg\nδgZ/parenleftbigggK\n0\n2−Fn/parenrightbigg\n(3.20b)\nandδgZ= Tr{ˆσδˆgZ}/2 withδˆgZgiven by Eq. (3.17).\nTo calculate the retarded component ΠR\nijof the polar-\nization operator, we calculate the response of δgK(t,t′)\ntoδhqin the limit t′→t. Using the asymptotic behavior\nof the Fermi function, Eq. (3.12), we obtain:\nδgK(t,t) =/integraldisplaydω\n2π−2iω\nπL+\nn(ω)δhcl(ω)e−iωt.(3.21)Substituting this expression for δgK(t,t) to Eq. (3.14),\nwe obtain\nΠR\nij(ω) = ΠR\n/bardbl,ij(ω)+ΠR\n⊥,ij(ω),(3.22a)\nwith\nΠR\n/bardbl,ij(ω) =−2\nδ1ninj\n1−iωτ, (3.22b)\nΠR\n⊥,ij(ω) =−2\nδ1/summationdisplay\n±δij−ninj±ieijknk\n2(3.22c)\n×(1±2ih0τ)\n1−i(ω∓2h0)τ.\nHere we represented the polarization operator ΠR\nij(ω) as\na sum of the radial, ΠR\n/bardbl,ij(ω), and tangential, ΠR\n⊥,ij(ω),\nterms. We note that the action Eq. (3.2) contains only\ntheradialcomponentoftheretardedandadvancedpolar-\nization operators because we do not perform expansion\nin terms of the tangential fluctuations of the classical\ncomponent of the field hcl(t).\nIn response to δhq, both corrections δgK(t,t′) and\nδgZ(t,t′) contain terms ∝δ(t−t′), However, their sum\nδgK(t,t′)+δgZ(t,t′) remains finite in the limit t→t′:\nδgK(t,t′)+δgZ(t,t′) =−2i/integraldisplay\ndt′′/integraldisplay\ndt1dt2L+\nn(¯t−t1)Q(t1−t2+˜t/2;t2−t1+˜t/2)L−\nn(t2−t′′)δhq(t′′),(3.23a)\nQ(τ1;τ2) =2\nτδ(τ1)δ(τ2)−Nch/summationdisplay\nn=1Tnδ1\n2π/bracketleftbigggK\n0(τ1)\n2gK\n0(τ2)\n2+Fn(τ1)Fn(τ2)/bracketrightbigg\n(3.23b)\n−Nch/summationdisplay\nn=1Tn(1−Tn)δ1\n2π/bracketleftbigggK\n0(τ1)\n2−Fn(τ1)/bracketrightbigg/bracketleftbigggK\n0(τ2)\n2−Fn(τ2)/bracketrightbigg\nwith¯t= (t+t′)/2 and˜t=t−t′.\nFrom Eq. (3.23) we obtain the following expression for\nthe Keldysh component of the polarization operator:\nΠK\nij(ω) = ΠK\n/bardbl,ij(ω)+ΠK\n⊥,ij(ω), (3.24a)\nΠK\n/bardbl,ij(ω) =−ininj\nω2+1/τ2R(ω), (3.24b)\nΠK\n⊥,ij(ω) =−i\n2/summationdisplay\n±δij−ninj±ieijknk\n(ω∓2h0)2+1/τ2R(ω).(3.24c)\nHere function R(ω) coincides with the noise power of\nelectric current through a metallic particle in the approx-imation of non-interacting electrons\nR(ω) =Nch/summationdisplay\nn=1/integraldisplaydε\n8πTn\n×/braceleftBig/bracketleftbig\n8−gK\n0(ε)gK\n0(ε+ω)−4Fn(ε)Fn(ε+ω)/bracketrightbig\n+(1−Tn)/bracketleftbig\ngK\n0(ε)−2Fn(ε)/bracketrightbig/bracketleftbig\ngK\n0(ε+ω)−2Fn(ε+ω)/bracketrightbig/bracerightBig\n.\n(3.25)\nIn principle, electron-electron interaction in the charge\nchannel can be taken into account. The interaction mod-\nifies the expression Eq. (3.25) for R(ω) to the higher\norder23inτδ1≪1 and we neglect this correction here.\nIn this paper we consider a particle connected to\nelectron leads at temperature Twith the applied bias\nV. In this case, FL,R(ε) = tanh(ε−µL,R)/(2T) with7\nµL−µR=V, and the integration over εgives\n2πτR(ω) = 4ωcothω\n2T+ΞΥT(V,ω),(3.26)\nwhere\nΥT(V,ω)≡/summationdisplay\n±2(ω±V)cothω±V\n2T−4ωcothω\n2T\n(3.27)\nand Ξ is the ”Fano factor” for a dot\nΞ =τ2\nτLτR+τ3δ1\n2πτ2\nR/summationdisplay\nn∈LTn(1−Tn)\n+τ3δ1\n2πτ2\nL/summationdisplay\nn∈RTn(1−Tn).(3.28)\nAt|V| ≫Tthe function Υ T(V,ω) has two scales of ω:\n(i)Tsmears the non-analyticity at ω→0, but the value\nof ΥT(V,ω) deviates from Υ T(V,0) at|ω| ∼ |V|. Thus,\nthe typical time scale above which one can approximate\nΠK(ω) by a constant is at least ω≪max{T,|V|}. In the\nlimitω→0 we have\nΠK\nij(ω= 0) =−i8τTeff\nδ1/parenleftbigg\nninj+δij−ninj\n(2h0τ)2+1/parenrightbigg\n.(3.29)\nThe effective temperature Teffis given by\nTeff≡T+Ξ/parenleftbiggV\n2cothV\n2T−T/parenrightbigg\n.(3.30)\nC. Final form of the action\nWe can rewrite the action for magnetization field h=\n{hcl;hq}withhclin the form of Eq. (3.1) as a sum of\nthe radial and tangential terms:\nS[h] =S/bardbl[hcl\n/bardbl,hq\n/bardbl]+S⊥[n(t),hq\n⊥]. (3.31)\nThe radial term in the action has the form\nS/bardbl[hcl\n/bardbl,hq\n/bardbl] = (D−1\n/bardbl)αβ(t,t′)hα\n/bardbl(t)hβ\n/bardbl(t′),(3.32)\nwhere the inverse function of the internal magnetic field\npropagator is given by\n(D−1\n/bardbl)αβ(t,t′) =4\nE′′/parenleftbigg\n0 1\n1 0/parenrightbigg\nδ(t−t′)\n−/parenleftBigg\n0 ΠR\n/bardbl(t,t′)\nΠA\n/bardbl(t,t′) ΠK\n/bardbl(t,t′)/parenrightBigg(3.33)\nand Παβ\n/bardbl(t,t′) =ninjΠαβ\n/bardbl,ij(t,t′). From this equation we\nfind\nDR\n/bardbl(ω) =Dq,cl\n/bardbl(ω) =E′′\n4−iω+1/τ\n−iω+(δ1+E′′/2)/(τδ1),\n(3.34)andDA\n/bardbl(ω) = [DR\n/bardbl(ω)]∗. The Keldysh component is\nDK\n/bardbl(ω) =Dq,q\n/bardbl(ω) =DR\n/bardbl(ω)ΠK\n/bardbl(ω)DA\n/bardbl(ω).(3.35)\nThe tangential term in the action is\nS⊥[n(t),hq\n⊥] =−4h0τ\nδ1/integraldisplay\ndt(˙n+2h0τn×˙n)hq\n⊥\n(2h0τ)2+1\n−4\nδ1/integraldisplay\ndt[n(t)×[n(t)×B]]·hq\n⊥(t)\n−/integraldisplay\ndtdt′hq\n⊥,i(t)ΠK\n⊥,ij(t−t′)hq\n⊥,j(t′).\n(3.36)\nHere we recovered the external magnetic field B(t). The\npolarization operator ΠK\n⊥,ijis given by Eq. (3.24c).\nIV. LANGEVIN EQUATION\nA. Langevin equation for the direction of the\ninternal magnetic field\nIn this section we consider evolution of the direction\nvectorn, described by the tangential terms in the action,\nEq. (3.36). We neglect fluctuations of the magnitude\nof the internal magnetic field, h/bardbl, the conditions when\nthese fluctuations can be neglected are listed in the next\nsection.\nWe decouple the quadratic in hq\n⊥component of the\nactionin Eq. (3.36) by introducingan auxiliaryfield w(t)\nwith the probability distribution\nP[w(t)]∝exp/braceleftbigg4i\nδ2\n1/integraldisplay\ndtdt′(ΠK\n⊥)−1\nij(t,t′)wi(t)wj(t′)/bracerightbigg\n,\n(4.1)\nand the correlation function\n∝an}bracketle{twi(t)wj(t′)∝an}bracketri}ht=δ2\n1\n8iΠK\n⊥,ij(t,t′).(4.2)\nThe field w(t) plays the role of the gaussian random\nLangevin force. Integration of the tangential part of\nthe action, Eq. (3.36), over hq\n⊥produces a functional\nδ-function, whose argument determines the equation of\nmotion:\n˙n+2τh0[n×˙n]\n4τ2h2\n0+1−1\nτh0(w−n×[n×B]) = 0.(4.3)\nThe above equation can be resolved with respect to ˙n:\n˙n=−2[n×(w+B)]−1\nh0τ[n×[n×(w+B)]].(4.4)\nThis equation is the Langevin equation for the direc-\ntionn(t) of the internal magnetic field in the presence\nof the external magnetic field B(t) and the Langevin\nstochastic forces w(t).8\nB. The Fokker-Plank equation\nNext, we follow the standard procedure of derivation\nof the Fokker-Plank equation for the distribution P(n)\nof the probability for the internal magnetic field to point\nin the direction n. The probability distribution satisfies\nthe continuity equation:\n∂P\n∂t+∂Ji\n∂ni= 0, (4.5)\nwhere the probability current is defined as\nJ=−/parenleftbigg\n2n×B+1\nh0τn×[n×B]/parenrightbigg\nP\n+1\n2/angbracketleftBig\nξ/parenleftbigg\nξ·∂P\n∂n/parenrightbigg/angbracketrightBig (4.6)\nand the stochastic velocity ξis introduced in terms of\nthe field was\nξ=−2[n×w]−1\nh0τ[n×[n×w]].(4.7)\nThe derivative ∂/∂nis understood as the differentiation\nwithrespecttolocalEuclideancoordinatesinthetangent\nspace. Performing averaging over fluctuations of win\nEq. (4.6), we obtain\n∂P\n∂t=∂\n∂n/braceleftbigg(2h0τ)[n×B]+[n×[n×B]]\nh0τP/bracerightbigg\n+1\nT0∂2P\n∂n2,(4.8)\nwhere the time constant T0is defined as\nT0=2(h0τ)2\nτTeffδ1. (4.9)\nBelow we use the polar coordinates for the\ndirection of the internal magnetic field, n=\n{sinθcosϕ,sinθsinϕ,cosθ}. In this case the Fokker-\nPlank equation can be rewritten in the form\n∂P\n∂t=1\nsinθ∂\n∂ϕ/bracketleftbigg\nFϕP+1\nT01\nsinθ∂P\n∂ϕ/bracketrightbigg\n+1\nsinθ∂\n∂θ/bracketleftbigg\nsinθFθP+sinθ\nT0∂P\n∂θ/bracketrightbigg\n,(4.10)\nwhere\nFϕ=Bxsinϕ−Bycosϕ\nh0τ\n+ 2cosθ(Bxcosϕ+Bysinϕ)−2sinθBz,(4.11)\nFθ= 2(Bxsinϕ−Bycosϕ)\n−cosθ\nh0τ(Bxcosϕ+Bysinϕ)+sinθ\nh0τBz.(4.12)It should be supplemented by the normalization condi-\ntion:\n2π/integraldisplay\n0dϕπ/integraldisplay\n0sinθdθP(ϕ,θ) = 1, (4.13)\nwhich is preserved if the boundary conditions at θ= 0,π\nare imposed:\nlim\nθ→0,πsinθ2π/integraldisplay\n0dϕ∂P\n∂θ= 0. (4.14)\nBelow we apply the Fokker Plank equation for calcu-\nlations of the magnetization of a particle\nM=/integraldisplaydΩn\n4πnP(n) (4.15)\nV. APPLICABILITY OF THE APPROACH\nIn this section we discuss the conditions of validity\nof the stochastic LLG equation, see Eq. (4.10), for the\nmodel of ferromagnetic metallic particle connected to\nleads at finite bias. We briefly listed these conditions\nin the Introduction. Here we present their more detailed\nquantitative analysis.\nA. Fluctuations of the radial component of the\ninternal magnetic field\nWe represented the classical component of the internal\nmagnetic field hclin terms of a slowly varying direction\nn(t) and fast oscillations hcl\n/bardblof its magnitude around\nthe average value h0. Now, we evaluate the amplitude of\noscillations of the radial component hcl\n/bardblof the field, using\nthe radial term in the action, see Eqs. (3.31) and (3.32).\nThe typicalfrequencies fortime evolutionofsmallfluc-\ntuations of the internal magnetic field in the radial direc-\ntion are of order of\nω∼δ1+E′′/2\nδ11\nτ(5.1)\nas one can conclude from the explicit form of the prop-\nagatorDR\n/bardbl(ω), Eq. (3.34), of these fluctuations. This\nscale has the meaning of the inverse RC-time in the\nspin channel. Deep in the ferromagnetic state (i. e., far\nfrom the Stoner critical point E′′+2δ1= 0) we estimate\nδ1+E′′/2∼δ1(which is equivalent to E′′∼h0/S0),\nso this spin-channel RC-time is of the same order as the\nescape time τ. This estimate for the frequency range is\nconsistent with the simple picture, which describes the\nevolution of the internal magnetic field of the grain as\na response to a changing value of the total spin of the\nparticle due to random processes of electron exchange9\nbetween the dot and the leads. The electron exchange\nhappens with the characteristic rate 1 /τ.\nThe correlation function ∝an}bracketle{thcl\n/bardbl(t)hcl\n/bardbl(t′)∝an}bracketri}htcan be evalu-\nated by performing the Gaussian integration with the\nquadratic action in hcl\n/bardblandhq\n/bardbl. Using Eq. (3.35), we ob-\ntain the equal-time correlation function\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht=i\n2/integraldisplaydω\n2πDK\n/bardbl(ω)\n=(E′′)2\n32τδ1/integraldisplaydω\n2π2πR(ω)\nω2+[1+E′′/(2δ1)]2/τ2.(5.2)\nThis equation gives the value of fluctuations of the radial\ncomponent of the internal magnetic field of the particle.\nThese fluctuations survive even in the limit T= 0 and\nV= 0, when R(ω) = 2|ω|/πτ. We have the following\nestimate\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht=(E′′)2\n16πτδ1lnETτ\n1+E′′/(2δ1),(5.3)\nthe upper cutoff ETis the Thouless energy, ET=vF/L\nfor a ballistic dot with diameter Land electron Fermi\nvelocityvF.\nThe separation of the internal magnetic field into the\nradial and tangential components is justified, provided\nthat the fluctuations/radicalBig\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}htof the radial component\nare much smaller than the average value of the field h0,\ni.e.∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht ≪h2\n0. Using the estimate Eq. (5.3), we\nobtain the necessary requirement for the applicability of\nequations for the slow evolution of the vector of the in-\nternal magnetic field of a particle:\nS0≫/radicalbigg\n1\nτδ1ln(ETτ), (5.4)\nwhereS0is the spin of a particle in equilibrium and\nwe again used the estimate E′′∼h0/S0. Condition of\nEq. (5.4) requires that the system is not close to the\nStoner instability.\nB. Applicability of the gaussian approximation\nLet us discuss the applicability of the gaussian approx-\nimation for the action in hcl\n/bardblandhq. The coefficients in\nfront of terms hq(t)h/bardbl(t1)...hcl\n/bardbl(tn) are obtained by tak-\ning thenth variational derivative of δgK(t,t)+δgZ(t,t),\nor, equivalently, byiteratingthe Usadelequation ntimes.\nSince the typical frequencies of h/bardblareω∼1/τ, the left-\nhand side of the equation is ∼δ(n+1)gK/τ, while the\nright-hand side is hcl\n/bardblδ(n)gK. Since the only time scale\nhereisτ, allthe coefficientsofthe expansionofthe action\ninhcl\n/bardbl(ω) atω∼1/τare of the same order:\nSn+1∼τn−1\nE′′/integraldisplaydω1...dωn\n(2π)n×\n×hcl\n/bardbl(ω1)...hcl\n/bardbl(ωn)hq(−ω1−...−ωn).(5.5)At the same time, the typical value of hcl\n/bardbl(ω∼1/τ),\nas determined by the gaussian part of the action, was\nestimated in the previous subsection to be of the order of/radicalBig\nτDK\n/bardbl(ω∼τ)∼√τδ1≪1, so the higher-order terms\nare indeed not important.\nFor the quantum component of the field the quadratic\nand quartic terms in the action are estimated as\nSn+1∼Teffτn\nδ1/integraldisplaydω1...dωn\n(2π)n×\n×hq(ω1)...hq(ωn)hq(−ω1−...−ωn).(5.6)\nIfTeff≫1/τ, then the typicalfrequencyscaleis ω∼1/τ,\nsothe quadraticterm gives hq(ω∼1/τ)∼/radicalbig\nδ1/Teff, and\nSn∼(δ1/Teff)n/2−1∼[τδ1/(τTeff)]n/2−1. IfTeff≪1/τ,\nat the typical scale ω∼Teffwe obtainhq(ω∼Teff)∼/radicalbig\nδ1/(T2\neffτ), so again Sn∼(τδ1)n/2−1≪1 forn>2.\nPhysically, the parameter 1 /(τδ1) =Nch(orTeff/δ1,\nif it is larger) can be identified with the number of the\nindependent sources of the noise acting on the magne-\ntization field. Thus, the smallness of the non-gaussian\npart of the action is nothing but the manifestation of the\ncentral limit theorem.\nC. Applicability of the Fokker-Plank equation\nFrom the above analysis we found that evolution of\nthe direction of the internal magnetic field in time is\ndescribed by a characteristic time T0, introduced in\nEq. (4.9). From the analysis of the fluctuations of the\nmagnitude of the internal magnetic field, see Eq. (5.1),\nwe obtain the following condition when the separation\ninto slow and fast variables is legitimate. The criterium\ncan be formulated as T0≫τ, which can be presented as\nTeff\nδ1≪/parenleftbiggh0\nδ1/parenrightbigg2\n=S2\n0. (5.7)\nVI. MAGNETIC SUSCEPTIBILITY OF\nMETALLIC PARTICLES OUT OF EQUILIBRIUM\nThe LLG equation derived in this paper for a ferro-\nmagnetic particle with finite bias between the leads can\nbe applied to a number of experimental setups. More-\nover, the derivation of the equation can be generalized\nto spin-anisotropic contacts with leads or Hamiltonian of\nelectron states in the particle. In this paper we apply\nthe stochastic equation for spin distribution function to\nthe analysis of the magnetic susceptibility at finite fre-\nquency. The susceptibility is the basic characteristic of\nmagnetic systems, it can often be measured directly, and\ndetermines other measurable quantities.\nBelow, we calculate the susceptibility of an ensemble\nof particles placed in constant magnetic field of an ar-\nbitrary strength and oscillating weak magnetic field, see10\nFig. 1. We consider the oscillating magnetic field with\nits components in directions parallel and perpendicular\nto the constant magnetic field.\nA. Solution at zero noise power\nAtTeff= 0 when w(t) = 0, and at fixed direction\nof the field, B(t) =ezB(t), equation of motion (4.4) is\neasily integrated for an arbitrary time dependence B(t):\nϕ=ϕ0+t/integraldisplay\n02B(t′)dt′, (6.1)\ntanθ\n2= tanθ0\n2exp\n−t/integraldisplay\n0B(t′)\nh0τdt′\n.(6.2)\nHere the direction of magnetic field correspondsto θ= 0.\nB. Constant magnetic field\nAt finiteTeffin constant magnetic field B0the Fokker-\nPlank equation has a simple solution\nP0(θ) =b\nsinhbebcosθ\n4π, (6.3)\nwhere the strength of constant magnetic field is written\nin terms of the dimensionless parameter\nb≡(2h0τ)B0\nτδ1Teff. (6.4)\nSubstituting this probability function to Eq. (4.15), we\nobtain the classical Langevin expression for the magne-\ntization of a particle in a magnetic field\nMz= cothb−1\nb, Mx=My= 0.(6.5)\nThis expression for the magnetization coincides with the\nmagnetization of a metallic particle in thermal equilib-\nrium, provided that the temperature is replaced by the\neffective temperature Teffdefined by Eq. (3.30).\nThe differential dc susceptibility is equal to\nχdc\n/bardbl=dMz(b)\ndb=1\nb2−1\nsinh2b. (6.6)\nC. Longitudinal susceptibility\nWe now consider the response of the magnetization to\nweakoscillations ˜Bz(t)oftheexternalmagneticfieldwith\nfrequencyωin direction parallel to the fixed magnetic\nfieldB0. We write the oscillatory component of the field\nin terms of the dimensionless field strength:\nb/bardble−iωt+b∗\n/bardbleiωt=2h0˜Bz(t)\nδ1Teff. (6.7)The linear correction to the probability distribution can\nbe cast in the form\nP(θ,t) =/bracketleftBig\n1+b/bardblu/bardbl(θ)e−iωt+b∗\n/bardblu∗\n/bardbl(θ)eiωt/bracketrightBig\nP0(θ),\n(6.8)\nwithP0(θ) defined by Eq. (6.3). The magnetic ac sus-\nceptibility can be evaluated from Eq. (6.8) as\nχ/bardbl(ω,b) = 2ππ/integraldisplay\n0u/bardbl(θ)P0(θ)cosθsinθdθ. (6.9)\nTheequationfor u/bardbl(θ) isobtainedfromEq.(4.10)with\nBz=B0+˜Bz(t):\n∂2u/bardbl\n∂θ2+cosθ−bsin2θ\nsinθ∂u/bardbl\n∂θ+iΩu/bardbl=bsin2θ−2cosθ,\n(6.10)\nwhere we introduced the dimensionless frequency\nΩ =ωT0, (6.11)\nand the time constant T0is defined in Eq. (4.9).\nNote the symmetry of Eq. (6.10) with respect to the\nsimultaneous change b→ −bandθ→π−θ. Also, the\nnormalization condition for the probability function re-\nquires that\nπ/integraldisplay\n0u/bardbl(θ)P0(θ)sinθdθ= 0. (6.12)\nThe latter holds if the boundary conditions Eq. (4.14)\nare satisfied, which in the case of axial symmetry can be\nwritten as\nlim\nθ→0,π/braceleftbigg\nsinθ∂u/bardbl(θ)\n∂θ/bracerightbigg\n= 0. (6.13)\nThedifferentialequation(6.10)withtheboundarycon-\ndition Eq. (6.13) can be solved numerically and then the\nsusceptibility is evaluated according to Eq. (6.9). The\nresult is shown in Figs. 2 and 3, where the susceptibility\nis shown as a function of frequency ωor magnetic field b,\nrespectively. We also consider various asymptotes for the\nacsusceptibility, obtainedfromthe solutionofEq.(6.10).\nAt zero constant magnetic field, b= 0, we find the\nexact solution of Eq. (6.10) explicitly:\nu/bardbl(θ) =cosθ\n1−iΩ/2. (6.14)\nThis solution allows us to calculate the ac susceptibility\nin the form\nχ/bardbl(Ω,b= 0) =1\n31\n1−iΩ/2. (6.15)\nForb≫1 only cosθ∼1/bmatter, and we can find a\nspecific solution of the inhomogeneous equation:\nu/bardbl(θ) =1−b(1−cosθ)\nb−iΩ/2, b≫1. (6.16)11\n0.000.100.200.30\n02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b)\nΩb= 0.1\nb= 0.5\nb= 1\nb= 2\nb= 5\nFIG. 2: (Color online). Plot of the real and imaginary parts\nof the susceptibility χ/bardbl(Ω,b) as a function of the dimension-\nless frequency Ω = ωT0. The oscillatory field at frequency ω\nis parallel to the constant magnetic field with strength b. The\nreal part of the susceptibility decreases monotonically fr om\nits dc value, Eq. (6.6), as frequency increases, while the im ag-\ninary part increases linearly at small Ω ≪1, see Eq. (6.20),\nand decreases at higher frequencies.\nThe requirement of regularity at the opposite end can\nbe replaced by the probability normalization condition,\nEq. (6.12), which is satisfied by this solution. Substitut-\ning this solution to Eq. (6.9), we obtain the strong field\nasymptote for the ac susceptibility\nχ/bardbl(Ω,b) =1\nb(b−iΩ/2). (6.17)\nFor Ω≫1 and Ω ≫b, we can neglect the derivatives\nin Eq. (6.10) and find the solution in the form\nu/bardbl(θ)≈bsin2θ−2cosθ\niΩ, (6.18)\nThis solution u/bardbl(θ) also satisfies Eq. (6.12). For the sus-\nceptibility, Eq. (6.9), we obtain\nχ/bardbl(Ω→ ∞,b) =2i\nΩ/parenleftbiggcothb\nb−1\nb2/parenrightbigg\n.(6.19)\nFinally, the low frequency limit can be also analyzed\nanalytically. The real part of the susceptibility coincides\nwith the differential susceptibility in dc magnetic field,\nEq. (6.6), for the imaginary part to the first order in\nfrequency we obtain, see Appendix,\nImχ/bardbl(Ω,b) = Ωf/bardbl(b). (6.20)\nThe function f/bardbl(b) has a complicated analyticalform and\nis not presented here, but its plot is shown in Fig. 4.\nIn all considered four limiting cases, the asymptotic\napproximations hold regardless the order in which the\nlimits are taken. Indeed, the asymptote of the expression\nfor the susceptibility in the zero field, Eq. (6.15), has the\nasymptote at Ω → ∞consistent with Eq. (6.19) at b= 0.0.000.100.200.30\n02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b)\nbΩ = 0 .1\nΩ = 0 .5\nΩ = 1\nΩ = 2\nΩ = 5\nFIG. 3: (Color online). Plot of the real and imaginary parts\nof theacsusceptibility χ/bardbl(Ω,b) at several values of the dimen-\nsionless frequency Ω of the oscillating magnetic field along the\nconstant magnetic field with strength b. In general, magnetic\nfield suppresses both real and imaginary parts of the suscep-\ntibility.\nSimilarly, the high frequency limit of Eq. (6.17) coincides\nwith the limit b→ ∞of Eq. (6.19). Both limits of weak\nand strong magnetic field of the imaginary part of the\nsusceptibilityatlowfrequencies, Eq.(6.20), coincidewith\nthe imaginarypartof χ/bardbl(Ω,b), calculatedfromEq.(6.15)\nand Eq. (6.19), respectively.\nIn general, we make a conjecture that the ac suscepti-\nbility is given by the following expression:\nχ/bardbl(Ω,b) =/summationdisplay\nnχ/bardbl\nn(b)\n1−iΩ/Γ/bardbl\nn(b),(6.21)\nwhere functions χ/bardbl\nn(b) and Γ/bardbl\nn(b) are real and describe\nthe degeneracy points of the homogeneous differential\nequation Eq. (6.10) with real iΩ. This expansion is re-\nlated to the expansion of time-dependent Fokker-Plank\nequations in the spherical harmonics, analyzed in Ref. 1.\nIn particular, χ/bardbl\nn>1(b→0) =O(b) and Γ/bardbl\nn(b→0) =\nn(n+1)+O(b).\nFor practical purposes, we found from a numerical\nanalysis that even the single-pole approximation,\nχapp\n/bardbl(ω,b) =/parenleftbigg1\nb2−1\nsinh2b/parenrightbigg/bracketleftbig\n1−iωT/bardbl(b)/bracketrightbig−1,(6.22)\ngives a very good estimate of the susceptibility for all\nvalues ofωandb. The analysis shows that the suscep-\ntibility, Eq. (6.9), obtained from a numerical solution of\nEq. (6.10), is within a few per cent of the estimate given\nby Eq. (6.22). The characteristic time constant, T/bardbl(b),\nas a function of magnetic field bis chosen from the high\nfrequency asymptote Eq. (6.19):\nT/bardbl(b) =T0\n2sinhbsinh2b−b2\nbcoshb−sinhb.(6.23)\nTo evaluate the accuracy of the above approximation,\nEq. (6.22), we consider the opposite limit of low frequen-12\n012345670.00.050.100.15\nPSfrag replacementsf/bardbl(b), fapp\n/bardbl(b)\nbf/bardbl(b)\nfapp\n/bardbl(b)\nΩ = 5\nFIG. 4: (Color online). Dependence on magnetic field bof\nthe slope of the imaginary part of the linear in frequency\nsusceptibility χ/bardbl(Ω,b) at low frequencies Ω ≪1, calculated\naccording toEq. (6.20). For comparison, we also plot functi on\nfapp\n/bardbl(b), see Eq. (6.24).\ncies, Ω≪1, and compare the exact result for the imagi-\nnary part of the susceptibility, Eq. (6.20), with\nImχapp\n/bardbl(ω,b) =ωT0fapp\n/bardbl(b),\nfapp\n/bardbl(b) =1\n2b2sinh3b(sinh2b−b2)2\nbcoshb−sinhb.(6.24)\nFor visual comparison of functions f/bardbl(b) andfapp\n/bardbl(b), we\nplot both functions in Fig. 4, where these curves are\nnearly indistinguishable. The difference between these\ntwo curves vanishes at b→0 andb→ ∞, and has a\nmaximal difference at b≈2, which constitutes only tiny\nfraction off/bardbl(b).\nD. Transverse susceptibility\nNext, we consider the response of the magnetization\nto weak oscillations ˜B⊥(t) of the external magnetic field\nwith frequency ωin direction perpendicular to the fixed\nmagnetic field B0. We write the oscillatory component\nof the field in the form:\n˜B⊥(t) =δ1Teff\n2h0/bracketleftbig\nb⊥(ex+iey)e−iωt+b∗\n⊥(ex−iey)eiωt/bracketrightbig\n.\n(6.25)\nThis field represents a circular polarization of an ac\nmagnetic field in the ( x,y) plane, perpendicular to\nthe fixed magnetic field in the z-direction: B=\n{B⊥cosωt;B⊥sinωt;B0}. We look for the linear cor-\nrection to the probability distribution in the form\nP(ϕ,θ,t) =P0(θ)\n×/bracketleftbig\n1+b⊥u⊥(θ)eiϕ−iωt+b∗\n⊥u∗\n⊥(θ)e−iϕ+iωt/bracketrightbig\n.(6.26)\nThe equation for u⊥(θ) is obtained from the Fokker-\nPlank equation Eq. (4.10), linearized in the parame-terb⊥:\n∂2u⊥\n∂θ2+cosθ−bsin2θ\nsinθ∂u⊥\n∂θ+/parenleftbigg\niΩ⊥−1\nsin2θ/parenrightbigg\nu⊥\n=−sinθ(2+2ih0τb+bcosθ).\n(6.27)\nHere the dimensionless frequency is a difference between\nthe drive frequency ωand the precession frequency in\nexternal field B0:\nΩ⊥= (ω−2B0)T0= Ω−2(h0τ)b, (6.28)\nwhereT0is defined in Eq. (4.9) and the right equality is\nwritten in terms of dimensionless variables Ω, Eq. (6.11),\nandb, Eq. (6.4). Equation (6.27) is symmetric with re-\nspect to the simultaneous change θ→π−θ,b→ −b,\ni→ −i, Ω⊥→ −Ω⊥(“parity”). The function P(ϕ,θ,t)\nis single-valued at the poles θ= 0 andθ=π, only if\nu⊥(θ= 0) = 0, u⊥(θ=π) = 0.(6.29)\nThe latter equations establish the boundary conditions\nfor the differential equation (6.27). We also note that\nthe normalization condition is satisfied for any function\nu⊥(θ).\nWe define the susceptibility in response to the ac mag-\nnetic field, Eq. (6.25), as\nχ⊥(Ω,b) = 2ππ/integraldisplay\n0u⊥(θ)P0(θ)sin2θdθ. (6.30)\nThis expression for the susceptibility can be used to cal-\nculate the magnetization of a particle\nM(t) =/integraldisplay\nn(ϕ,θ)P(ϕ,θ,t)sinθdθdϕ (6.31)\ntothe lowestorderin the acmagnetic field. Inparticular,\nMx(t) = Re(χ⊥b⊥e−iωt), My(t) = Im(χ⊥b⊥e−iωt).\n(6.32)\nSolving numerically the differential equation (6.27)\nwith the corresponding boundary conditions, Eq. (6.29),\nwe obtain the transverse susceptibility, Eq. (6.30), shown\nin Figs. 5 and 6. Below we analyze several limiting cases.\nIn zero fixed magnetic field, b= 0, we have the exact\nsolution of Eq. (6.27):\nu⊥(θ) =sinθ\n1−iΩ⊥/2. (6.33)\nThis solution corresponds to the solution in the longitu-\ndinal case, rotated by 90◦, cf. Eq. (6.14).\nAtω= 0 and Ω ⊥=−2bh0τ, the solution of Eq. (6.27)\nhas a simple form u/bardbl(θ) = sinθand corresponds to a tilt\nof the external field. The susceptibility due to such tilt\nis\nχ⊥(Ω = 0,b) =2\nb2(bcothb−1).(6.34)13\nt\n−0.20.00.20.40.6\n−5 0 5 10 15−0.20.00.20.40.6PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b)\nΩbb= 0.4;h0τ= 5\nb= 1;h0τ= 2\nb= 2.5;h0τ= 0.8\nΩ = 2; h0τ= 2\nΩ = 2; h0τ= 0.5\nΩ = 0 .5;h0τ= 2\nΩ = 0 .5;h0τ= 0.5\nΩ = 2\nΩ = 5\nFIG. 5: (Color online). Plot of the real and imaginary parts\nof the transverse susceptibility χ⊥(Ω,b) as a function of the\ndimensionless frequency Ω. Negative frequency correspond s\nto the opposite sense of the circular polarization of the ac\nmagnetic field in a plane, perpendicular to the constant mag-\nnetic field with strength b. The parameters of the three shown\ncurves are chosen so that h0τb= 2.\nIn strong fixed magnetic field, b≫1, we need to con-\nsider small angles θ∼1/√\nb, therefore, we can approxi-\nmate cosθ≈1 in Eq. (6.27) and obtain:\nu⊥(θ) =b+2+2ih0τb\nb+2−iΩ⊥sinθ. (6.35)\nThe susceptibility in the limit b≫1 is given by\nχ⊥(Ω,b) =1+2ih0τ\nb(b(1+2ih0τ)−iΩ). (6.36)\nAt Ω⊥≫1,bwe can disregard the terms in Eq. (6.27)\nwith derivatives. Moreover, the contribution to the sus-\nceptibility, Eq. (6.30), from the vicinity of θ= 0 and\nθ=πis suppressed as sin2θ. This observation allows us\nto write the solution in the form\nu⊥(θ) = sinθ2+2ih0τb+bcosθ\n−iΩ⊥, (6.37)\nConsequently, we obtain the following high frequency,\nΩ⊥≫1, asymptote for the susceptibility:\nχ⊥(Ω,b) =i\nΩ−2h0τb/bracketleftbiggbcothb−1\nb2(2ih0τb−1)+1/bracketrightbigg\n.\n(6.38)\nWe can use the approximate expression for the suscep-\ntibility in response to the transverse oscillating magnetic\nfield\nχapp\n⊥(ω) =bcothb−1\nb2[1+2iB0T⊥(b)]\n×[1+i(2B0−ω)T⊥(b)]−10.00.10.20.30.4\n−10 −5 0 5 100.00.10.20.3PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b)\nΩ\nbΩ = 2; h0τ= 2\nΩ = 2; h0τ= 0.5Ω = 0 .5;h0τ= 2\nΩ = 0 .5;h0τ= 0.5\nΩ = 2\nΩ = 5\nFIG. 6: (Color online). Plot of the real and imaginary parts\nof the transverse susceptibility χ⊥(Ω,b) as a function of the\nstrength bof a constant magnetic field, shown for two values\nof frequency Ω and two values of the “damping factor” h0τ.\nNegative values of bcorresponds to the opposite sense of the\ncircular polarization of the ac magnetic field in a plane, per -\npendicular to the constant magnetic field. The real part of\nthe susceptibility exhibits a strong non-monotonic behavi or\nat weak magnetic fields.\nThecorrespondingcharacteristictimeconstant T⊥(b)can\nbe found for any bfrom the asymptotic behavior of\nχ⊥(Ω,b) at Ω ⊥≫1,b:\nT⊥(b) =T0bcothb−1\nb2+1−bcothb. (6.39)\nVII. CONCLUSIONS\nWe have studied the slow dynamics of magnetization\nin a small metallic particle (quantum dot), where the fer-\nromagnetism has arisen as a consequence of Stoner insta-\nbility. Theparticleisconnectedto non-magneticelectron\nreservoirs. A finite bias is applied between the reservoirs,\nthus bringing the whole electron system away from equi-\nlibrium. The exchange of electrons between the reser-\nvoirs and the particle results in the Gilbert damping3of\nthe magnetization dynamics and in a temperature- and\nbias-driven Brownian motion of the direction of the par-\nticle magnetization. Analysis of magnetization dynam-\nics and transport properties of ferromagnetic nanoparti-\ncles is commonly performed4,5,6,7,11within the stochas-\ntic Landau-Lifshitz-Gilbert (LLG) equation2,3, which is\nan analogue of the Langevin equation written for a unit\nthree-dimensional vector.\nWe derived the stochastic LLG equation from a mi-\ncroscopicstarting point and established conditions under\nwhich the description ofthe magnetizationofaferromag-\nnetic metallic particle by this equation is applicable. We\nconcluded that the applicability of the LLG equation for14\na ferromagneticparticle is set by three independent crite-\nria. (1)Thecontactresistanceshouldbelowcomparedto\nthe resistance quantum, which is equivalent to Nch≫1.\nOtherwise the noise cannot be consideredgaussian. Each\nchannel can be viewed as an independent source of noise\nand only the contribution of many channels results in\nthe gaussian noise by virtue of the central limit theorem\nforNch≫1. (2) The system should not be too close\nto the Stoner instability: the mean-field value of the to-\ntal spinS2\n0≫Nch. Otherwise, the fluctuations of the\nabsolute value of the magnetization become of the or-\nder of the magnetization itself. (3) S2\n0≫Teff/δ1, where\nTeff≃max{T,|eV|}is the effective temperature of the\nsystem, which is the energy scale of the electronic dis-\ntribution function. Otherwise, the separation into slow\n(the direction of the magnetization) and fast (the elec-\ntron dynamics and the magnitude of the magnetization)\ndegrees of freedom is not possible.\nUnder the above conditions, the dynamics of the mag-\nnetization is described in terms of the stochastic LLG\nequation with the power of Langevin forces determined\nby the effective temperature of the system. The effective\ntemperature is the characteristic energy scale of the elec-\ntronic distribution function in the particle determined by\na combination of the temperature and the bias voltage.\nIn fact, for a considered here system with non-magnetic\ncontacts between non-magnetic reservoirs and a ferro-\nmagnetic particle the power of the Langevin forces is\nproportional to the low-frequency noise of total charge\ncurrent through the particle. We further reduced the\nstochastic LLG equation to the Fokker-Planck equation\nfor a unit vector, corresponding to the direction of the\nmagnetization of the particle. The Fokker-Plank equa-\ntion can be used to describe time evolution of the distri-\nbution of the direction of magnetization in the presence\nof time-dependent magnetic fields and voltage bias.\nAs an example of application of the Fokker-Plank\nequation for the magnetization, we have calculated the\nfrequency-dependent magnetic susceptibility of the par-\nticle in a constant external magnetic field (i. e., linear\nresponse of the magnetization to a small periodic mod-\nulation of the field, relevant for ferromagnetic resonance\nmeasurements). We have not been able to obtain an ex-\nplicit analytical expression for the susceptibility at ar-\nbitrary value of the applied external field and frequency;\nhowever, analysisofdifferent limiting caseshas lead us to\na simple analytical expression which gives a good agree-\nment with the numerical solution of the Fokker-Planck\nequation.\nAcknowledgements\nWe acknowledge discussions with I. L. Aleiner, G.\nCatelani, A. Kamenev and E. Tosatti. M.G.V. is grate-\nful to the International Centre for Theoretical Physics(Trieste, Italy) for hospitality.\nAPPENDIX A: LONGITUDINAL\nSUSCEPTIBILITY AT LOW FREQUENCIES\nWe find the linearin frequency Ω ≪1 correctionto the\ndc susceptibility. For this purpose, we look for a solution\nto Eq. (6.10) in the form\nu/bardbl(θ) =u(0)\n/bardbl(θ)+u(1)\n/bardbl(θ), (A1)\nwhereu(0)\n/bardbl(θ) is the solution of Eq. (6.10) at Ω = 0 and\nu(1)\n/bardbl(θ)∝Ω. We choose\nu(0)\n/bardbl(θ) =1\nb−cothb+cosθ, (A2)\nsincethis formof u(0)\n/bardbl(θ) preservesthe normalizationcon-\ndition (6.12). This function can be found directly as a\nsolution of Eq. (6.10) with Ω = 0 or as a variational\nderivative of function P0(θ), defined in Eq. (6.3), with\nrespect tob.\nThe linear in Ω correction u(1)\n/bardbl(θ) is the solution to the\ndifferential equation\n∂2u(1)\n/bardbl(θ)\n∂θ2+cosθ−bsin2θ\nsinθ∂u(1)\n/bardbl(θ)\n∂θ=−iΩu(0)\n/bardbl(θ).(A3)\nFrom this equation, we can easily find\n∂u(1)\n/bardbl(θ)\n∂θ=−iΩ\nbsinθ/bracketleftbigg\ncothb−cosθ−e−bcosθ\nsinhb/bracketrightbigg\n.(A4)\nWe notice that the solution to the latter equation will\nautomatically satisfy the boundary conditions, given by\nEq. (6.13). Integrating Eq. (A4) once again, we obtain\nthe following expression for function u(1)\n/bardbl(θ):\nu(1)\n/bardbl(θ) =C(b)\n−iΩ\nb/integraldisplayθ\n0/bracketleftBigg\ncothb−cosθ′−e−bcosθ′\nsinhb/bracketrightBigg\ndθ′\nsinθ′.(A5)\nHere the integration constant C(b) has to be chosen to\nsatisfy the normalization condition, Eq. (6.12), which re-\nsults in complicated expression for the final form of the\nfunctionu(1)\n/bardbl(θ).\nTo obtain function f/bardbl(b), introduced in Eq. (6.20), we\nhave to perform the final integration\nf/bardbl(b) =2π\nΩ/integraldisplayπ\n0u(1)\n/bardbl(θ)P0(θ)sinθcosθdθ. (A6)\nThe result of integration is shown in Fig. 4.15\n1W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n2L. Landau and E. Lifshitz, Phys. Z. Sowietunion 8, 153\n(1935).\n3T. Gilbert, Phys. Rev. 100, 1243 (1955).\n4J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n5K. D. Usadel, Phys. Rev. B 73, 212405 (2006).\n6J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak,\nPhys. Rev. B 75, 092405 (2007).\n7S. I. Denisov, K. Sakmann, P. Talkner, and H¨ anggi, Phys.\nRev. B75, 184432 (2007).\n8A. Rebei and M. Simoniato, Phys. Rev. B 71, 174415\n(2005).\n9H. Katsura, A. V. Balatsky, Z. Nussinov, and N. Nagaosa,\nPhys. Rev. B 73, 212501 (2006).\n10A. S. N´ u˜ nez and R. A. Duine, Phys. Rev. B 77, 054401\n(2008).\n11A. L. Chudnovskiy, S. Swiebodzinski, and A. Kamenev,\nPhys. Rev. Lett. 101, 066601 (2008).\n12J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,Phys. Rev. Lett. 95, 016601 (2005).\n13R.A.Duine, A.S.N´ u˜ nez, J.Sinova, andA.H.MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n14X. Waintal and P. W. Brouwer, Phys. Rev. Lett. 91,\n247201 (2003).\n15I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys.\nReports 358, 309 (2002).\n16C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).\n17I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys.\nRev. B62, 14886 (2000).\n18Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n19S. Tamaru et al., J. Appl. Phys. 91, 8034 (2002).\n20J. C. Sankey et al., Phys. Rev. Lett. 96, 227601 (2006).\n21Y. Ahmadian, G. Catelani, and I. L. Aleiner, Phys. Rev.\nB72, 245315 (2005).\n22K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n23G. Catelani and M. G. Vavilov, Phys. Rev. B 76, 201303\n(2007)."    },    {        "title": "0809.2910v1.Spin_transfer_torque_induced_reversal_in_magnetic_domains.pdf",        "content": "arXiv:0809.2910v1  [cond-mat.other]  17 Sep 2008Spin-transfer torque induced reversal in\nmagnetic domains\nS. MurugeshaM. Lakshmananb\naDepartment of Physics & Meteorology, IIT-Kharagpur, Kharagpu r 721 302, India\nbCentre for Nonlinear Dynamics, School of Physics, Bharathid asan University,\nTiruchirappalli 620024, India\nAbstract\nUsing the complex stereographic variable representation f or the macrospin, from\na study of the nonlinear dynamics underlying the generalize d Landau-Lifshitz(LL)\nequation with Gilbert damping, we show that the spin-transf er torque is effectively\nequivalent to an applied magnetic field. We study the macrosp in switching on a\nStoner particle due to spin-transfer torque on application of a spin polarized cur-\nrent. We find that the switching due to spin-transfer torque i s a more effective\nalternative to switching by an applied external field in the p resence of damping. We\ndemonstrate numerically that a spin-polarized current in t he form of a short pulse\ncan be effectively employed to achieve the desired macro-spin switching.\nKey words: Nonlinear spin dynamics, Landau-Lifshitz equation, Spin- transfer\ntorque, Magnetization reversal\nPACS:75.10.Hk, 67.57.Lm, 75.60.Jk, 72.25.Ba\n1 Introduction\nIn recent times the phenomenon of spin-transfer torque has gained much at-\ntention in nanoscale ferromagnets[1,2,3]. Electromigration refers to the recoil\nlinearmomentumimpartedontheatomsofametalorsemiconductor asalarge\ncurrent is conducted across. Analogously, if the current is spin-p olarized, the\ntransfer of a strong current across results in a transfer of spin angular momen-\ntum to the atoms. This has lead to the possibility of current induced s witch-\ning of magnetization in nanoscale ferromagnets. With the success o f GMR,\n∗Corresponding author. Tel: +91 431 2407093, Fax:+91 431 240 7093\nEmail address: lakshman@cnld.bdu.ac.in (M. Lakshmanan).\nPreprint submitted to Chaos, Solitons and Fractals 7 Septem ber 2021this has immense application potential in magnetic recording devices s uch as\nMRAMs[3,4,5,6]. The phenomenon has been studied in several nanomag netic\npile geometries. The typical set up consists of a nanowire[3,7,8,9,10,11 ], or a\nspin-valve pillar, consisting of two ferromagnetic layers, one a long f erromag-\nneticpinnedlayer, and another small ferromagnetic layer or film, separated\nby a spacer conductor layer (see Figure 1). The pinned layer acts a s a reser-\nvoir for spin polarized current which on passing through the conduc tor and\non to the thin ferromagnetic layer induces an effective torque on th e spin\nmagnetization in the thin film ferromagnet. A number of experiments have\nbeen conducted on this geometry and the phenomenon has been co nvincingly\nconfirmed [12,13,14,15]. Although the microscopic quantum theory of the phe-\nnomenon is fairly well understood, interestingly the behavior of the average\nspin magnetization vector can be described at the semi-classical lev el by the\nLL equation with an additional term[16].\nj\nxyz\nPinned layer Conductor Thin film ConductorS S p\nFig. 1. A schematic diagram of the spin-valve pillar. A thin fi lm ferromagnetic layer\nwith magnetization Sis separated from long ferromagnetic layer by a conductor. ˆSp\nis the direction of magnetization in the pinned region, whic h also acts as a reservoir\nfor spin polarized current.\nFrom a different point of view, several studies have focused on mag netic pulse\ninduced switching of the macro-magnetization vector in a thin nanod ot un-\nder different circumstances [17,18,19,20]. Several experimental st udies have\nalso focussed on spin-current induced switching in the presence of a magnetic\nfield, switching behavior for different choices of the angle of the app lied field,\nvariation in the switching time, etc., [12,21,22,23,24,25]. A numerical stu dy\non the switching phenomenon induced by a spin current in the presen ce of\na magnetic field pulse has also been investigated very recently in [26]. A s an\nextension to two dimensional spin configurations, the switching beh avior on a\nvortex has been studied in [27].\nIn this article, by investigating the nonlinear dynamics underlying the gener-\nalized Landau-Lifshitz equation with Gilbert damping, we look at the ex citing\npossibility of designing solid state memory devices at the nanoscale, w herein\nmemory switching is induced using a spin polarized current alone, witho ut\nthe reliance on an external magnetic field. We compare earlier studie d switch-\ning behavior for the macro-magnetization vector in a Stoner partic le [17] in\nthe presence of an external magnetic field, and the analogous cas e wherein\nthe applied field is now replaced by a spin polarized current induced spin -\n2transfer torque, i.e., with the thin film in the first case replaced by a s pin\nvalve pillar. It will be shown that a pulse of spin polarized current is mor e\neffective in producing a switching compared to an applied field. In doing so we\nrewrite the system in terms of a complex stereographic variable inst ead of the\nmacro-magnetization vector. This brings a significant clarity in unde rstanding\nthe nonlinear dynamics underlying the macrospin system. Namely, it w ill be\nshown that, in the complex system, the spin-transfer torque is eff ectively an\nimaginary applied magnetic field. Thus the spin-transfer term can ac complish\nthe dual task of precession of the magnetization vector and dissip ation.\nThe paper is organized as follows: In Section 2 we discuss briefly the m odel\nsystem and the associated extended LL equation. In Section 3 we in troduce\nthe stereographic mapping of the constant spin magnetization vec tor to a\ncomplex variable, and show that the spin-transfer torque is effect ively an\nimaginary applied magnetic field. In Section 4 we present results from our\nnumerical study on spin-transfer torque induced switching pheno menon of the\nmacro-magnetization vector, for a Stoner particle. In particular , we study two\ndifferent geometries for the free layer, namely, (a) an isotropic sp here and (b)\nan infinite thin film. In applications to magnetic recording devices, the typ-\nical read/write time period is of the order of a few nano seconds. We show\nthat, in order to achieve complete switching in these scales, the spin -transfer\ntorque induced by a short pulse of sufficient magnitude can be affirma tively\nemployed. We conclude in Section 5 with a discussion of the results and their\npractical importance.\n2 The extended LL equation\nThe typical set up of the spin-valve pillar consists of a long ferromag netic\nelement, or wire, with magnetization vector pinned in a direction indica ted by\nˆSp, as shown in Figure 1. It also refers to the direction of spin polarizat ion of\nthe spin current. A free conduction layer separates the pinned ele ment from\nthe thin ferromagnetic film, or nanodot, whose average spin magne tization\nvectorS(t) (of constant magnitude S0) is the dynamical quantity of interest.\nThe cross sectional dimension of the layers range around 70 −100nm, while\nthe thickness of the conduction layer is roughly 2 −7nm[3,20]. The free layer\nthus acts as the memory unit, separated from the pinned layer cum reservoir\nby the thin conduction layer. It is well established that the dynamics of the\nmagnetization vector Sin the film in the semiclassical limit is efficiently de-\nscribed by an extended LL equation[16]. If ˆ m(={m1,m2,m3}=S/S0) is the\nunit vector in the direction of S, then\ndˆ m\ndt=−γˆ m×/vectorHeff+λˆ m×dˆ m\ndt−γag(P,ˆ m·ˆSp)ˆ m×(ˆ m׈Sp),(1)\n3a≡ℏAj\n2S0Ve. (2)\nHere,γis the gyromagnetic ratio (= 0 .0176Oe−1ns−1) andS0is the satura-\ntion magnetization (Henceforth we shall assume 4 πS0= 8400, the saturation\nmagnetization value for permalloy). The second term in (1) is the phe nomeno-\nlogical dissipation term due toGilbert[28] with damping coefficient λ. The last\nterm is the extension to the LL equation effecting the spin-transfe r torque,\nwhereAis the area of cross section, jis the current density, and Vis the\nvolume of the pinned layer.′a′, as defined in (2), has the dimension of Oe, and\nis proportional to the current density j.g(P,ˆ m·ˆSP) is given by\ng(P,ˆ m·ˆSp) =1\nf(P)(3+ˆ m·ˆSp)−4;f(P) =(1+P)3\n(4P3/2),(3)\nwheref(P)isthepolarizationfactorintroducedbySlonczewski [1],and P(0≤\nP≤1) is the degree of polarization of the pinned ferromagnetic layer. F or\nsimplicity, we take this factor gto be a constant throughout, and equal to 1.\n/vectorHeffisthe effective fieldacting onthespin vector due toexchange intera ction,\nanisotropy, demagnetization and applied fields:\n/vectorHeff=/vectorHexchange+/vectorHanisotropy+/vectorHdemagnetization +/vectorHapplied,(4)\nwhere\n/vectorHexchange=D∇2ˆ m, (5)\n/vectorHanisotropy=κ(ˆ m·ˆ e/bardbl)ˆ e/bardbl, (6)\n∇·/vectorHdemagnetization =−4πS0∇·ˆ m. (7)\nHere,κis the strength of the anisotropy field. ˆ e/bardblrefers to the direction of\n(uniaxial) anisotropy, In what follows we shall only consider homogen eous\nspin states on the ferromagnetic film. This leaves the exchange inte raction\nterm in (4) redundant, or D= 0, while (7) for /vectorHdemagnetization is readily solved\nto give\n/vectorHdemagnetization =−4πS0(N1m1ˆ x+N2m2ˆ y+N3m3ˆ z), (8)\nwhereNi,i= 1,2,3 are constants with N1+N2+N3= 1, and {ˆ x,ˆ y,ˆ z}are the\northonormal unit vectors. Equation (1) now reduces to a dynamic al equation\nfor a representative macro-magnetization vector ˆ m.\nIn this article we shall be concerned with switching behavior in the film p urely\ninduced by the spin-transfer torque term, and compare the resu lts with earlier\nstudies on switching due to an applied field [17] in the presence of dissip ation.\nConsequently, it will be assumed that /vectorHapplied= 0 in our analysis.\n43 Complex representation using stereographic variable\nIt proves illuminating to rewrite (1) using the complex stereographic variable\nΩ defined as[29,30]\nΩ≡m1+im2\n1+m3, (9)\nso that\nm1=Ω+¯Ω\n1+|Ω|2;m2=−i(Ω−¯Ω)\n1+|Ω|2;m3=1−|Ω|2\n1+|Ω|2.(10)\nFor the spin valve system, the direction of polarization of the spin-p olarized\ncurrentˆSpremains a constant. Without loss of generality, we chose this to be\nthe direction ˆ zin the internal spin space, i.e., ˆSp=ˆ z. As mentioned in Sec.\n2, we disregard the exchange term. However, for the purpose of illustration,\nwe choose /vectorHapplied={0,0,ha3}for the moment but take ha3= 0 in the later\nsections. Defining\nˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl} (11)\nand upon using (9) in (1), we get\n(1−iλ)˙Ω =−γ(a−iha3)Ω+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−\nΩ2e−iφ/bardbl)/bracketrightBig\n−iγ4π S0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω−N1\n2(1−Ω2−|Ω|2)Ω\n−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n,(12)\nwherem/bardbl=ˆ m·ˆ e/bardbl. Using (10) and (11), m/bardbl, and thus (12), can be written\nentirely in terms of Ω.\nIt is interesting to note that in this representation the spin-trans fer torque\n(proportional to the parameter a) appears only in the first term in the right\nhand side of (12) as an addition to the applied magnetic field ha3but with a\nprefactor −i. Thus the spin polarization term can be considered as an effective\napplied magnetic field. Letting κ= 0, and N1=N2=N3in (12), we have\n(1−iλ)˙Ω =−γ(a−iha3)Ω, (13)\nwhich on integration leads to the solution\nΩ(t) = Ω(0) exp( −(a−iha3)γt/(1−iλ))\n= Ω(0) exp( −a+λha3\n1+|λ|2γt) exp(−iaλ−ha3\n1+|λ|2γt). (14)\n5The first exponent in (14) describes relaxation, or switching, while t he second\nterm describes precession. From the first exponent in (14), we no te that the\ntime scale ofswitching is given by 1 /(a+λha3).λbeing small, thisimplies that\nthe spin-torque term is more effective in switching the magnetization vector.\nFurther, letting ha3= 0, we note that in the presence of the damping term\nthe spin transfer produces the dual effect of precession and diss ipation.\nTo start with we shall analyze the fixed points of the system for the two cases\nwhich we shall be concerned with in this article: (i) the isotropic spher e char-\nacterized by N1=N2=N3= 1/3, and (ii) an infinite thin film characterized\nbyN1= 0 =N3,N2= 1.\n(i) First we consider the case when the anisotropy field is absent, or κ= 0.\nFrom (12) we have\n(1−iλ)˙Ω =−aγΩ−iγ4πS0\n1+|Ω|2/bracketleftBig\nN3(1−|Ω|2)Ω−N1\n2(1−Ω2−|Ω|2)Ω\n−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n.(15)\nIn the absence of anisotropy ( κ= 0), we see from (15) that the only fixed\npoint is Ω 0= 0.To investigate the stability of this fixed point we expand (15)\nup to a linear order in perturbation δΩ around Ω 0. This gives\n(1−iλ)δ˙Ω =−aγδΩ−iγ4πS0[N3−1\n2(N1+N2)]δΩ+iγ2πS0(N1−N2)δ¯Ω.(16)\nFor the isotropic sphere, N1=N2=N3= 1/3, (16) reduces to\n(1−iλ)δ˙Ω =−aγδΩ. (17)\nWe find the fixed point is stable since a >0. For the thin film, N1= 0 =\nN3,N2= 1. (16) reduces to\n(1−iλ)δ˙Ω =−aγδΩ+iγ2πS0δΩ−iγπS0δ¯Ω. (18)\nThis may be written as a matrix equation for Ψ ≡(δΩ,δ¯Ω)T,\n˙Ψ =MΨ, (19)\nwhereMis a matrix obtained from (18) and its complex conjugate, whose\ndeterminant and trace are\n|M|=(a2+3π2S2\n0)γ2\n1+λ2;Tr(M) =(−2a−4πS0λ)γ\n1+λ2.(20)\nSince|M|is positive, the fixed point Ω 0= 0 is stable if Tr|M|<0, or,\n(a+2πS0λ)>0.\n6The equilibrium point (a), Ω 0= 0, corresponds to ˆ m=ˆ z. Indeed this holds\ntrue even in the presence of an applied field, though we have little to d iscuss\non that scenario here.\n(ii) Next we consider the system with a nonzero anisotropy field in the ˆ z\ndirection. (12) reduces to\n(1−iλ)˙Ω =−aγΩ+iκγ(1−|Ω|2)\n(1+|Ω|2)Ω−iγ4πS0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω\n−N1\n2(1−Ω2−|Ω|2)Ω−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n.(21)\nHere again the only fixed point is Ω 0= 0.As in (i), the stability of the\nfixed point is studied by expanding (21) about Ω 0to linear order. Following\nthe same methodology in (i) we find the criteria for stability of the fixe d\npoint for the isotropic sphere is ( a+λκ)>0, while for the thin film it is\n(a+λ(κ+2πS0))>0.\n(iii) With nonzero κin an arbitrary direction the fixed point in general moves\naway from ˆ z.\nFinally, it is also of interest to note that a sufficiently large current lea ds to\nspin wave instabilities induced through spin-transfer torque [31,32]. In the\npresent investigation, however, we have assumed homogeneous m agnetization\nover the free layer, thus ruling out such spin wave instabilities. Rece ntly we\nhaveinvestigated spinwave instabilitiesoftheSuhltypeinduced bya napplied\nalternating field in thin film geometries using stereographic represen tation[30].\nIt will be interesting to investigate the role of a spin-torque on such instabil-\nities in the spin valve geometry using this formulation. This will be pursu ed\nseparately.\n4 Spin-transfer torque induced switching\nWenowlookattheinterestingpossibilityofeffectingcompleteswitchin g ofthe\nmagnetization using spin-transfer torque induced by a spin curren t. Numerical\nstudies on switching effected on a Stoner particle by an applied magne tic field,\nor in the presence of both a spin-current and applied field, in the pre sence of\ndissipation and axial anisotropy have been carried out recently and switching\nhas been demonstrated [17,26]. However, the intention here is to ind uce the\nsame using currents rather than the applied external fields. Also, achieving\nsuch localized magnetic fields has its technological challenges. Spin-t ransfer\ntorque proves to be an ideal alternative to accomplish this task sinc e, as we\nhave pointed out above, it can be considered as an effective (albeit c omplex)\n7magnetic field. In analogy with ref. [17], where switching behavior due to an\napplied magnetic field has been studied we investigate here switching b ehav-\nior purely due to spin-transfer torque, on a Stoner particle. Nume rical results\nin what follows have been obtained by directly simulating (12) and makin g\nuse of the relations in (10), for appropriate choice of parameters . It should\nbe remembered that (12) is equivalent to (1), and so the numerical results\nhave been further confirmed by directly numerically integrating (1) also for\nthe corresponding parameter values. We consider below two sample s differing\nin their shape anisotropies, reflected in the values of ( N1,N2,N3) in the de-\nmagnetization field: a) isotropic sphere, N1=N2=N3= 1/3 and b) a thin\nfilmN1= 0 =N3,N2= 1. The spin polarization ˆSpof the current is taken\nto be in the ˆ zdirection. The initial orientation of ˆ mis taken to be close to\n−ˆ z. In what follows this is taken as 170◦fromˆ zin the (z−x) plane. The\norientation of uniaxial anisotropy ˆ e/bardblis also taken to be the initial direction\nofˆ m. With these specified directions for ˆSpandˆ e/bardblthe stable fixed point is\nslightly away from ˆ z, the direction where the magnetization ˆ mis expected to\nswitch in time. A small damping is assumed, with λ= 0.008. The magnitude\nof anisotropy κis taken to be 45 Oe. As stated earlier, for simplicity we have\nconsidered the magnetization to be homogeneous.\n4.1 Isotropic sphere\nIt is instructive to start by investigating the isotropic sphere, whic h is char-\nacterized by the demagnetization field with N1=N2=N3= 1/3. With these\nvalues for ( N1,N2,N3), (12) reduces to\n(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n(22)\nA constant current of a= 10Oeis assumed. Using (2), for typical dimen-\nsions, this equals a current density of the order 108A/cm2. We notice that\nfor the isotropic sample the demagnetization field does not play any r ole in\nthe dynamics of the magnetization vector. In the absence of aniso tropy and\ndamping the spin-transfer torque term leads to a rapid switching of Sto the\nˆ zdirection. This is evident from (22), which becomes\n˙Ω =−aγΩ, (23)\nwith the solution Ω = Ω 0e−aγt, and the time scale for switching is given by\n1/aγ. Figure 2.a shows the trajectory traced out by the magnetization vector\nS, for 5 ns, initially close to the −ˆ zdirection, switching to the ˆ zdirection.\nFigure 2.b depicts the dynamics with anisotropy but no damping, all ot her\nparameters remaining same. While the same switching is achieved, this is\n8more smoother due to the accompanying precessional motion. Not e that with\nnonzero anisotropy, ˆ zis not the fixed point any more. The dynamics with\ndamping but no anisotropy (Figure 2.c) resembles Figure 2.a, while Figu re\n2.d shows the dynamics with both anisotropy and damping.\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)\nxyz-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 2. Trajectory of the magnetization vector m, obtained by simulating (12) for\nthe isotropic sphere ( N1=N2=N3= 1/3), and using the relations in (10), for\na= 10Oe(a)withoutanisotropyanddamping,(b)withanisotropybut nodamping,\n(c) without anisotropy but nonzero damping and (d) with both anisotropy and\ndamping nonzero. The results have also been confirmed by nume rically integrating\n(1). The arrows point in the initial orientation (close to −ˆ z) and the direction of the\nspincurrent ˆ z. Evolution shownis for aperiodof 5 ns.Note that thefinal orientation\nis not exactly ˆ zin the case of nonzero anisotropy ((b) and (d)).\nIt may be noticed that Figures 2.c and 2.d resemble qualitatively Figure s\n2.a and 2.b, respectively, while differing mainly in the time taken for the\nswitching. It is also noticed that switching in the absence of anisotro py is\nfaster. Precession assisted switching has been the favored reco rding process in\nmagnetic memory devices, as it helps in keeping the exchange interac tion at\na minimum[18,19]. The sudden switching noticed in the absence of anisot ropy\nessentially refers to a momentary collapse of order in the magnetic m edia.\n9This can possibly lead to strong exchange energy and a breakdown o f our\nassumption regarding homogeneity of the magnetization field. Howe ver, such\nrapid quenching assisted by short high intensity magnetic pulses has in fact\nbeen achieved experimentally [33].\nA comparison with reference [17] is in order. There it was noted that with an\napplied magnetic field, instead of a spin torque, a precession assiste d switch-\ning was possible only in the presence of a damping term. In Section 3 we\npointed out how the spin transfer torque achieves both precessio n and damp-\ning. Consequently, all four scenarios depicted in Figure 2 show switc hing of\nthe magnetization vector without any applied magnetic field.\n4.2 Infinite thin film\nNext we consider an infinite thin film, whose demagnetization field is give n by\nN1= 0 =N3andN2= 1. With these values (12) becomes\n(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n−iγ4πS0/parenleftbigg1−|Ω|2\n1+|Ω|2/parenrightbigg\nΩ.(24)\nHere againin the absence of anisotropy Ω = 0 is the only fixed point. Th us the\nspin vector switches to ˆ zin the absence of damping and anisotropy (Figure\n3a). In order to achieve this in a time scale of 5 ns, we find that the value of a\nhas to be of order 50 Oe. Again the behavior is in stark contrast to the case\ninduced purely by an applied field[17], wherein the spin vector traces o ut a\ndistorted precessional trajectory. As in Sec. 4.1, the trajecto ry traced out in\nthe presence of damping is similar to that without damping (Figure 3c) . The\ncorresponding trajectories traced out in the presence of anisot ropy are shown\nin Figures 3b and 3d.\n4.3 Switching of magnetization under a pulsed spin-polariz ed current\nWe noticed that in the absence of uniaxial anisotropy, the constan t spin polar-\nized current can effect the desired switching to the orientation of ˆSp(Figure\n2). This is indeed the fixed point for the system (with no anisotropy) . Fig-\nure 2 traces the dynamics of the magnetization vector in a period of 5 ns, in\nthe presence of a constant spin-polarized current. However, fo r applications in\nmagnetic media we choose a spin-polarized current pulseof the form shown\nin Figure 4. It may be recalled here that, as was observed in 4.2, with a spin\n10-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)\nxyz-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 3. Trajectory of the magnetization vector Sin a period of 5 ns, obtained as\nearlier by numerically simulating (12), and also confirming with (1), with demagne-\ntization field with by N1= 0 =N3,N2= 1 and a= 50Oe, and all the parameter\nvalues are as earlier. As in Figure 1, the ˆSand initial orientation are indicated\nby arrows. (a) Without anisotropy or damping, (b) nonzero an isotropy but zero\ndamping, (c) without anisotropy but nonzero damping and (d) both anisotropy and\ndamping nonzero. As earlier, in the presence of nonvanishin g anisotropy, the fixed\npoint is not the ˆ zaxis.\npolarized current of sufficient magnitude, the switching time can inde ed be\nreduced. We choose a pulse, polarized as earlier along the ˆ zdirection, with\nrise time and fall time of 1 .5ns, and a pulse width, defined as the time interval\nbetween half maximum, of 4 ns. We assume the rise and fall phase of the pulse\nto be of a sinusoidal form, though, except for the smoothness, t he switching\nphenomenon is independent of the exact form of the rise or fall pha se.\nIn Figures 5 and 6, we show trajectories of the spin vector for a pe riod of\n25ns, for the two different geometries, the isotropic sphere and a thin fi lm.\nThe action of the spin torque pulse, as in Figure 4, is confined to the fi rst\n5ns. We notice that, with the chosen value of a, this time period is enough\n11Rise time Fall time\nPulse width\n 0 20 40 60 80 100 120 140 160\n 0  1  2  3  4  5a (Oe)\ntime (ns)\nFig. 4. Pulse form showing the magnitude of a, or effectively the spin-polarized\ncurrent. The rise and fall phase are assumed to be of a sinusoi dal form. The rise\nand fall time are taken as 1 .5ns, and pulse width 4 ns. The maximum magnitude\nofais 150Oe.\nto effect the switching. In the absence of anisotropy, the directio n ofˆSpis\nthe fixed point. Thus a pulse of sufficient magnitude can effect a switc hing in\nthe desired time scale of 5 ns. From our numerical study we find that in order\nfor this to happen, the value of ahas to be of order 150 Oe, or, from (2),\na current density of order 109A/cm2, a magnitude achievable experimentally\n(see for example [34]). Comparing with sections 4.1 and 4.2, we note th at the\nextra oneorder ofmagnitude in current density isrequired due to t he duration\nof the rise and fall phases of the pulse in Figure(4). Here again we co ntrast\nthe trajectories with those induced by an applied magnetic field [17], w here\nthe switching could be achieved only in the presence of a uniaxial aniso tropy.\nIn Figure 5b for the isotropic sphere with nonzero crystal field anis otropy,\nwe notice that the spin vector switches to the fixed point near ˆ zaxis in the\nfirst 5ns. However the magnetization vector precesses around ˆ zafter the\npulse has been turned off. This is because in the absence of the spin- torque\nterm, the fixed point is along ˆ e/bardbl, the direction of uniaxial anisotropy. Due to\nthe nonzero damping term, the spin vector relaxes to the direction ofˆ e/bardblas\ntime progresses. The same behavior is noticed in Figure 6b for the th in film,\nalthough the precessional trajectory is a highly distorted one due to the shape\nanisotropy.\n12-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 5. Evolution of the magnetization vector Sin a period of 25 nsinduced by the\nspin-polarized current pulse in Figure 4, (a) with and (b) wi thout anisotropy for\nthe isotropic sample, with N1=N2=N3= 1/3 all other parameters remaining\nsame. A nonzero damping is assumed in both cases. The current pulse acts on the\nmagnetization vector for the first 5 ns. In both cases switching happens in the first\n5ns. In the presence of nonzero anisotropy field, (b), the magnet ization vector\nprecesses to the fixed point near ˆ z.\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 6. Evolution of the magnetization vector Sin a period of 25 nsinduced by\nthe spin-polarized current pulse in Figure 4, for a infinite t hin film sample, with\nN1= 0 =N3, andN2= 1, all other parameters remaining same, along with a\nnonzero damping. (a) Without anisotropy and (b) with anisot ropy. As in Figure 5,\nswitching happens in the first 5 ns.\n5 Discussion and conclusion\nWe have shown using analytical study and numerical analysis of the n onlinear\ndynamics underlying the magnetization behavior in spin-valve pillars th at a\nvery effective switching of macro-magnetization vector can be ach ieved by a\nspin transfer-torque, modeled using an extended LL equation. Re writing the\n13extended LL equation using the complex stereographic variable, we find the\nspin-transfer torque term indeed acts as an imaginary applied field t erm, and\ncan lead to both precession and dissipation. It has also been pointed out why\nthe spin-torque term is more effective in switching the magnetization vector\ncompared to the applied field. On application of a spin-polarized curre nt the\naverage magnetization vector in the free layer was shown to switch to the\ndirection of polarization of the spin polarized current. For a consta nt current,\nthe required current density was found to be of the order of 108A/cm2. For\nrecording in magnetic media, switching is achieved using a stronger po larized\ncurrent pulse of order 109A/cm2. Currents of these magnitudes have been\nachieved experimentally.\nAcknowledgements\nThe work forms part of a research project sponsored by the Dep artment of\nScience andTechnology, Government ofIndia anda DSTRamannaFe llowship\nto M. L.\nReferences\n[1] J. C. Slonczewski. Current-driven excitation of magnet ic multilayers J. Mag.\nMag. Mat. 1996; 159: L1-L7.\n[2] L. Berger. Emission of spin waves by a magnetic multilaye r traversed by a\ncurrent Phys. Rev. B 1996; 54: 9353-58.\n[3] M. D. Stiles and J. Miltat. Spin-Transfer Torque and Dyna mics Topics Appl.\nPhy. 2006; 101: 225-308.\n[4] S. A. Wolf, A. Y. Chtchelkanova and D. M. Treger. Spintron ics A retrospective\nand perspective IBM J. Res. Dev. 2006; 50: 101-110.\n[5] R. K. Nesbet. 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Rev. Lett 2004; 92: 02 6602.\n[32] S. Adam, M. L. Polianski and P. W. Brouwer. Current-indu ced transverse spin-\nwave instability in thin ferromagnets: Beyond linear stabi lity analysis Phys.\nRev. B 2006; 73: 024425.\n[33] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegman n, J. St¨ ohr, G.\nJu, B. Lu and D. Weller. The ultimate speed of magnetic switch ing in granular\nrecording media Nature 2004; 428: 831-33.\n[34] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, M . Tsoi and\nP. Wyder. Excitation of a Magnetic Multilayer by an Electric Current Phys.\nRev. Lett. 1998; 80: 4281-84.\n16"    },    {        "title": "0809.4311v1.The_theory_of_magnetic_field_induced_domain_wall_propagation_in_magnetic_nanowires.pdf",        "content": "arXiv:0809.4311v1  [cond-mat.other]  25 Sep 2008The theory of magnetic field induced domain-wall propagatio n in magnetic nanowires\nX. R. Wang, P. Yan, J. Lu, and C. He\nPhysics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China\nA global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic\nfield is obtained: A static DW cannot exist in a homogeneous ma gnetic nanowire when an external\nmagnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving\nDW must dissipate energy due to the Gilbert damping. As a resu lt, the wire has to release its\nZeeman energy through the DW propagation along the field dire ction. The DW propagation speed\nis proportional tothe energy dissipation rate that is deter mined bythe DW structure. Anoscillatory\nDW motion, either the precession around the wire axis or the b reath of DW width, should lead to\nthe speed oscillation.\nMagnetic domain-wall (DW) propagation in a\nnanowire due to a magnetic field[1, 2, 3, 4, 5] reveals\nmany interesting behaviors of magnetization dynamics.\nFor a tail-to-tail (TT) DW or a head-to-head (HH) DW\n(shown in Fig. 1) in a nanowire with its easy-axis along\nthe wire axis, the DW will propagate in the wire un-\nder an external magnetic field parallel to the wire axis.\nThe propagation speed vof the DW depends on the field\nstrength[3, 4]. There exists a so-called Walker’s break-\ndown field HW[6].vis proportional to the external field\nHforH < H WandH≫HW. The linear regimes\nare characterized by the DW mobility µ≡v/H. Ex-\nperiments showed that vis sensitive to both DW struc-\ntures and wire width[1, 2, 3]. DW velocity vdecreases as\nthe field increases between the two linear H-dependent\nregimes, leading to the so-called negative differential mo-\nbility phenomenon. For H≫HW, the DW velocity,\nwhose time-average is linear in H, oscillates in fact with\ntime [3, 6].\nIII II IMθ=0 θ=πH\nzxy∆\nADW\nFIG. 1: Schematic diagram of a HH DW of width ∆ in a\nmagnetic nanowire of cross-section A. The wire consists of\nthree phases, two domains and one DW. The magnetization\nin domains I and II is along +z-direction ( θ= 0) and -z-\ndirection ( θ=π), respectively. III is the DW region whose\nmagnetization structure could be very complicate. /vectorHis an\nexternal field along +z-direction.\nIt has been known for more than fifty years that\nthe magnetization dynamics is govern by the Landau-\nLifshitz-Gilbert (LLG)[7] equation that is nonlinear\nand can only be solved analytically for some special\nproblems[6, 8]. The field induced domain-wall (DW)propagation in a strictly one-dimensional wire has also\nbeen known for more than thirty years[6], but its exper-\nimental realization in nanowires was only achieved[1, 2,\n3, 4, 5] in recent years when we are capable of fabricating\nvarious nano structures. Although much progress[9, 10]\nhas been made in understanding field-induced DW mo-\ntion, it is still a formidable task to evaluate the DW\npropagation speed in a realistic magnetic nanowire even\nwhen the DW structure is obtained from various means\nlike OOMMF simulator and/or other numerical software\npackages. A global picture about why and how a DW\npropagates in a magnetic nanowire is still lacking.\nIn this report, we present a theory that reveals the\norigin of DW propagation. Firstly, we shall show that\nno static HH (TT) DW is allowed in a homogeneous\nnanowire in the presence of an external magnetic field.\nSecondly, energy conservation requires that the dissi-\npated energy must come from the energy decrease of the\nwire. Thus, the origin of DW propagation is as follows.\nA HH (TT) DW must move under an external field along\nthe wire. The moving DW must dissipate energy because\nof various damping mechanisms. The energy loss should\nbe supplied by the Zeeman energy released from the DW\npropagation. This consideration leads to a general re-\nlationship between DW propagation speed and the DW\nstructure. It is clear that DW speed is proportional to\nthe energy dissipation rate, and one needs to find a way\nto enhance the energy dissipation in order to increase the\npropagation speed. Furthermore, the present theory at-\ntributes a DW velocity oscillation for H≫HWto the\nperiodic motion of the DW, either the precession of the\nDW or oscillation of the DW width.\nIn a magnetic material, magnetic domains are formed\nin order to minimize the stray field energy. A DW that\nseparates two domains is defined by the balance between\nthe exchangeenergy and the magnetic anisotropyenergy.\nThe stray field plays little role in a DW structure. To\ndescribe a HH DW in a magnetic nanowire, let us con-\nsider a wire with its easy-axis along the wire axis (the\nshape anisotropy dominates other magnetic anisotropies\nand makes the easy-axis along the wire when the wire is\nsmall enough) which is chosen as the z-axis as illustrated2\nin Fig. 1. Since the magnitude of the magnetization /vectorM\ndoes not change in the LLG equation[8], the magnetic\nstate of the wire can be conveniently described by the\npolar angle θ(/vector x,t) (angle between /vectorMand the z-axis) and\nthe azimuthal angle φ(/vector x,t). The magnetization energy\nis mainly from the exchange energy and the magnetic\nanisotropy because the stray field energy is negligible in\nthis case. The wire energy can be written in general as\nE=/integraldisplay\nF(θ,φ,/vector∇θ,/vector∇φ)d3/vector x,\nF=f(θ,φ)+J\n2[(/vector∇θ)2+sin2θ(/vector∇φ)2]−MHcosθ,(1)\nwherefis the energydensity due to all kinds ofmagnetic\nanisotropies which has two equal minima at θ= 0 and\nπ(f(θ= 0,φ) =f(θ=π,φ)),J−term is the exchange\nenergy,Mis the magnitude of magnetization, and His\nthe external magnetic field along z-axis. In the absence\nofH, a HH static DW that separates θ= 0 domain and\nθ=πdomain (Fig. 1) can exist in the wire.\nNon-existence of a static HH (TT) DW in a magnetic\nfield-In order to show that no intrinsic static HH DW\nis allowed in the presence of an external field ( H/negationslash= 0),\none only needs to show that following equations have no\nsolution with θ= 0 at far left and θ=πat far right,\nδE\nδθ=J∇2θ−∂f\n∂θ−HMsinθ−Jsinθcosθ(/vector∇φ)2= 0,\nδE\nδφ=J/vector∇·(sin2θ/vector∇φ)−∂f\n∂φ= 0.\n(2)\nMultiply the first equation by ∇θand the second equa-\ntion by∇φ, then add up the two equations. One can\nshow a tensor Tsatisfying ∇·T= 0 with\nT=[f−HMcosθ+J\n2(|∇θ|2+sin2θ|∇φ|2)]1−\nJ(∇θ∇θ+sin2θ∇φ∇φ),\nwhere1is 3×3 unit matrix. A dyadic product ( ∇θ∇θ\nand∇φ∇φ) between the gradient vectors is assumed in\nT. If a HH DW exists with θ= 0 in the far left and\nθ=πin the far right, then it requires −f(0,φ)+HM=\n−f(π,φ)−HMthat holds only for H= 0 since f(0,φ) =\nf(π,φ). In other words, a DW in a nanowire under an\nexternal field must be time dependent that could be ei-\nther a local motion or a propagation along the wire. It\nshould be clear that the above argument is only true for\na HH DW in a homogeneous wire, but not valid with de-\nfect pinning that changes Eq. (2). Static DWs exist in\nfact in the presence of a weak field in reality because of\npinning.\nWhat is the consequence of the non-existence of a\nstatic DW? Generally speaking, a physical system un-\nder a constant driving force will first try a fixed point\nsolution[11]. It goes to other types of more complicatedsolutionsifafixedpointsolutionisnotpossible. Itmeans\nthat a DW has to move when an external magnetic field\nis applied to the DW along the nanowire as shown in\nFig. 1. It is well known[10] that a moving magnetiza-\ntion must dissipate its energy to its environments with a\nrate,dE\ndt=αM\nγ/integraltext+∞\n−∞(d/vector m/dt)2d3/vector x,where/vector mis the unit\nvector of /vectorM,αandγare the Gilbert damping constant\nand gyromagnetic ratio, respectively. Following the simi-\nlar method in Reference 12 for a Stoner particle, one can\nalso show that the energy dissipation rate of a DW is\nrelated to the DW structure as\ndE\ndt=−αγ\n(1+α2)M/integraldisplay+∞\n−∞/parenleftBig\n/vectorM×/vectorHeff/parenrightBig2\nd3/vector x,(3)\nwhere/vectorHeff=−δF\nδ/vectorMis the effective field. In regions I and\nII or inside a static DW, /vectorMis parallel to /vectorHeff. Thus no\nenergy dissipation is possible there. The energy dissipa-\ntion can only occur in the DW region when /vectorMis not\nparallel to /vectorHeff.\nDW propagation and energy dissipation- Foramagnetic\nnanowire in a static magnetic field, the dissipated energy\nmust come from the magnetic energy released from the\nDW propagation. The total energy of the wire equals\nthe sum of the energies of regions I, II, and III (Fig. 1),\nE=EI+EII+EIII.EIincreases while EIIdecreases\nwhen the DW propagates from left to the right along the\nwire. The net energy change of region I plus II due to\nthe DW propagation is\nd(EI+EII)\ndt=−2HMvA, (4)\nwherevis the DW propagating speed, and Ais the cross\nsection of the wire. This is the released Zeeman energy\nstored in the wire. The energy of region III should not\nchange much because the DW width ∆ is defined by the\nbalanceofexchangeenergyand magneticanisotropy,and\nis usually order of 10 ∼100nm. A DW cannot absorb\nor release too much energy, and can at most adjust tem-\nporarily energy dissipation rate. In other words,dEIII\ndt\nis either zero or fluctuates between positive and nega-\ntive values with zero time-average. Since energy release\nfrom the magnetic wire should be equal to the energy\ndissipated (to the environment), one has\n−2HMvA+dEIII\ndt=−αγ\n(1+α2)M/integraldisplay\nIII/parenleftBig\n/vectorM×/vectorHeff/parenrightBig2\nd3/vector x.\n(5)\nor\nv=αγ\n2(1+α2)HA/integraldisplay\nIII/parenleftBig\n/vector m×/vectorHeff/parenrightBig2\nd3/vector x+1\n2HMAdEIII\ndt.\n(6)\nVelocity oscillation- Eq. (6) is our central result that\nrelates the DW velocity to the DW structure. Obviously,\nthe right side of this equation is fully determined by the3\nDW structure. A DW can have two possible types of\nmotion under an external magnetic field. One is that a\nDW behaves like a rigid body propagating along the wire.\nThiscaseoccursoftenatsmallfield, anditisthebasicas-\nsumption in Slonczewski model[9] and Walker’s solution\nforH < H W. Obviously, both energy-dissipation and\nDW energy is time-independent,dEIII\ndt= 0. Thus,and\nthe DW velocity should be a constant. The other case\nis that the DW structure varies with time. For example,\ntheDWmayprecessaroundthewireaxisand/ortheDW\nwidth may breathe periodically. One should expect both\ndEIII\ndtandenergydissipationrateoscillatewith time. Ac-\ncording to Eq. (6), DW velocity will also oscillate. DW\nvelocity should oscillate periodically if only one type of\nDW motion (precession or DW breathing) presents, but\nit could be very irregular if both motions are present and\nthe ratio of their periods is irrational. Indeed, this os-\ncillation was observed in a recent experiment[3]. How\ncan one understand the wire-width dependence of the\nDW velocity? According to Eq. (6), the velocity is a\nfunctional of DW structure which is very sensitive to the\nwire width. For a very narrow wire, only transverse DW\nis possible while a vortex DW is preferred for a wide wire\n(large than DW width). Different vortexes yield differ-\nent values of |/vector m×/vectorHeff|, which in turn results in different\nDW propagation speed.\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s32/s32 \n/s68/s87 /s32/s118 /s32\n/s40/s109/s47/s115/s41\n/s72/s32/s40/s79/s101/s41/s32/s118/s40/s109/s47/s115/s41\n/s116/s32/s40/s110/s115/s41/s118/s40/s109/s47/s115/s41/s32\n/s116/s32/s40/s110/s115/s41\nFIG. 2: The time-averaged DW propagation speed versus\nthe applied magnetic field for a biaxial magnetic nanowire\nof cross section 4 nm×20nm. The wire parameters are\nK1=K2= 105J/m3,J= 4.×10−11J/m,M= 106A/m,\nandα= 0.1. Cross are for the calculated velocities from Eq.\n(7), and the open circles are for the simulated average veloc i-\nties. The dashed straight line is the fit to the small H < H W\nresults, and solid curve is the fit to a(H−H0)2/H+b/H.\nInsets: the instantaneous DW speed calculated from Eq. (6)\nforH= 50Oe < H W(left) and H= 1000Oe > H W(right).\nTime averaged velocity is\n¯v=αγ\n2(1+α2)HA/integraldisplay\nIII/parenleftBig\n/vector m×/vectorHeff/parenrightBig2\nd3/vector x,(7)\nwherebardenotestimeaverage. Itsaysthattheaveraged\nvelocity is proportional to the energy dissipation rate. In\norder to show that both Eqs. (6) and (7) are useful in\nevaluating the DW propagation speed from a DW struc-ture. We use OOMMF package to find the DW struc-\ntures and then use Eq. (7) to obtain the averagevelocity.\nFigure 2 is the comparison of such calculated velocities\n(cross) and numerical simulation (open circles with their\nerror bars smaller than the symbol sizes) for a magnetic\nnanowire of cross-section dimension 4 nm×20nmwith\na biaxial magnetic anisotropy f=−K1\n2M2\nz+K2\n2M2\nx.\nThe system parameters are K1=K2= 105J/m3,\nJ= 4.×10−11J/m,M= 106A/m, andα= 0.1. The\ngood overlap between the cross and open circles confirm\nthe correctness of Eq. (7). The ¯ v−Hcurve for H > H W\ncan be fit well by a∆(H−H0)2/H+b/H(see discus-\nsion later). The insets are instantaneous DW propaga-\ntion velocities for both H < H WandH > H W, by Eq.\n(6) from the instantaneous DW structures obtained from\nOOMMF. The left inset is the instantaneous DW speed\natH= 50Oe < H W, reaching its steady value in about\n1ns. The right inset is the instantaneous DW speed at\nH= 1000Oe > H W, showingclearlyanoscillation. They\nconfirm that the theory is capable of capturing all the\nfeatures of DW propagation.\nThe right side of Eq. (7) is positive and non-zero\nsince a time dependent DW requires /vector m×/vectorHeff/negationslash= 0,\nimplying a zero intrinsic critical field for DW propa-\ngation. If the DW keep its static structure, then the\nfirst term in the right side of Eq. (6) shall be pro-\nportional to a∆AH2, where ais a numerical number\nof order of 1 that depends on material parameters and\nthe DW structure. This is because the effective field\ndue to the exchange energy and magnetic anisotropy\nis parallel to /vectorM, and does not contribute to the en-\nergy dissipation. Thus, in this case, v=aαγ∆\n1+α2H\nwithµ=aαγ∆\n1+α2. Consider the Walker’s 1D model[6]\nin which f=−K1\n2M2cos2θ+K2\n2M2sin2θcos2φ,here\nK1andK2describe the easy and hard axes, respec-\ntively. From Walker’s trial function of a DW of width\n∆, lntanθ(z,t)\n2=1\n∆(t)/bracketleftBig\nz−/integraltextt\n0v(τ)dτ/bracketrightBig\nandφ(z,t) =φ(t),\none has (from Eq. (3)) the energy dissipation rate\ndE\ndt=−2αγA∆\n1+α2/bracketleftbig\nK2\n2M3sin2φcos2φ+H2M/bracketrightbig\n,(8)\nand DW energy change rate is\ndEIII\ndt=d\ndt/integraldisplay\nIIIF(θ,φ,/vector∇θ,/vector∇φ)d3/vector x=−4JA·˙∆\n∆2.(9)\nSubstituting Eqs. (8) and (9) into Eq. (6), one can easily\nreproduceWalker’sDWvelocityexpressionfor both H <\nHWand≫HW. For example, for H < H W=αK2M/2\nand ∆ = const., Eq. (6) gives\nv=α∆γ\n1+α2/bracketleftBigg\n1+/parenleftbiggK2Msinφcosφ\nH/parenrightbigg2/bracketrightBigg\nH.(10)\nThis velocity expression is the same as that of the\nSlonczewski model[9] for a one-dimensional wire. In4\nWalker’s analysis, φis fixed by K2andHthrough\nK2Msinφcosφ=H\nα. Using this φin the above ve-\nlocity expression, Walker’s mobility coefficient µ=γ∆\nα\nis recovered. This inverse damping relation is from the\nparticularpotentiallandscape in φ-direction. One should\nexpect different result if the shape of the potential land-\nscape is changed. Thus, this expression should not be\nused to extract the damping constant[1, 3].\nA DW may precess around the wire axis as well as\nbe substantially distorted from its static structure when\nH > H Was it was revealed in Walker’s analysis. Ac-\ncording to the minimum energy dissipation principle[13],\na DW will arrange itself as much as possible to satisfy\nEq. (2). Thus, the distortion is expected to absorb part\nofH. The precession motion shall induce an effective\nfieldg(φ) in the transverse direction, where gdepends\non the magnetic anisotropy in the transverse direction.\nOne may expect /vector m×/vectorHeff≃(H−H0)sinθˆz+sinθg(φ)ˆy,\nwhereH0istheDWdistortionabsorbedpartof H. Using\n|/vector m×/vectorHeff|2= (H−H0)2sin2θ+g2sin2θin Eq. (7), the\nDWpropagatingspeed takesthe followingh-dependence,\nv=aαγ∆(H−H0)2/H(1+α2)+bαγ∆/[H(1+α2)], linear\nin both ∆ and HforH≫H0, but a smaller DW mobil-\nity. This field-dependence is supported by the excellent\nfit in Fig. 2 for H > H W. The reasoning agrees also\nwith the minimum energy dissipation principle[13] since\n|/vector m×Heff|=Hsinθwhen/vectorMforH= 0 is used, and any\nmodification of /vectorMshould only make |/vector m×Heff|smaller.\nThe smaller mobility at H≫HW,H0leads naturally to\na negative differential mobility between H < H Wand\nH≫HW! In other words, the negative differential mo-\nbility is due to the transition of the DW from a high\nenergy dissipation structure to a lower one. This picture\ntells us that one should try to make a DW capable of\ndissipating as much energy as possible if one wants to\nachieve a high DW velocity. This is very different from\nwhat people would believefrom Walker’sspecial mobility\nformula of inverse proportion of the damping constant.\nTo increase the energy dissipation, one may try to re-\nduce defects and surface roughness. The reason is, by\nminimum energy dissipation principle, that defects are\nextra freedoms to lower |/vector m×/vectorHeff|because, in the worst\ncase, defects will not change |/vector m×/vectorHeff|when/vectorMwithout\ndefects are used.\nThe correctness of our central result Eq. (6) depends\nonly on the LLG equation, the general energy expres-\nsion of Eq. (1), and the fact that a static magnetic field\ncan be neither an energy source nor an energy sink of a\nsystem. It does not depend on the details of a DW struc-\nture aslong asthe DWpropagationis induced by a static\nmagnetic field. In this sense, our result is very general\nand robust, and it is applicable to an arbitrary magnetic\nwire. However, it cannot be applied to a time-dependent\nfield orthe current-inducedDWpropagation,atleastnotdirectly. Also, it may be interesting to emphasize that\nthere is no inertial in the DW motion within LLG de-\nscription since this equation contains only the first order\ntime derivative. Thus, there is no concept of mass in this\nformulation.\nIn conclusion, a global view of the field-induced DW\npropagation is provided, and the importance of energy\ndissipation in the DW propagation is revealed. A gen-\neral relationship between the DW velocity and the DW\nstructure is obtained. The result says: no damping, no\nDW propagation along a magnetic wire. It is shown that\nthe intrinsic critical field for a HH DW is zero. This\nzero intrinsic critical field is related to the absence of a\nstatic HH or a TT DW in a magnetic field parallel to\nthe nanowire. Thus, a non-zero critical field can only\ncome from the pinning of defects or surface roughness.\nThe observed negative differential mobility is due to the\ntransition of a DW from a high energy dissipation struc-\nture to a low energy dissipation structure. Furthermore,\nthe DW velocity oscillation is attributed to either the\nDW precession around wire axis or from the DW width\noscillation.\nThis work is supported by Hong Kong UGC/CERG\ngrants (# 603007 and SBI07/08.SC09).\n[1] T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito,\nand T. Shinjo, Science 284, 468 (1999).\n[2] D. Atkinson, D.A. Allwood, G. Xiong, M.D. Cooke, C.\nFaulkner, and R.P. Cowburn, Nat. Mater. 2, 85 (2003).\n[3] G.S.D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J.L.\nErskine, Nat. Mater. 4, 741 (2005); J. Yang, C. Nistor,\nG.S.D. Beach, and J.L. Erskine, Phys. Rev. B 77, 014413\n(2008).\n[4] M. Hayashi, L. Thomas, Y.B. Bazaliy, C. Rettner, R.\nMoriya, X. Jiang, and S.S.P. Parkin, Phys. Rev. Lett.\n96, 197207 (2006).\n[5] G.S.D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J.L.\nErskine, Phys. Rev. Lett. 97, 057203 (2006).\n[6] N.L. Schryer and L.R. Walker, J. of Appl. Physics, 45,\n5406 (1974).\n[7] T. L. Gilbert, Phys. Rev. 100, 1243 (1955).\n[8] Z.Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205\n(2006); X.R. Wang and Z.Z. Sun, ibid98, 077201 (2007).\n[9] A.P.Malozemoff andJ.C. Slonczewski, Magnetic Domain\nWalls in Bubble Material (Academica, New York, 1979).\n[10] A. Thiaville and Y. Nakatani in Spin Dynamics in Con-\nfined Magnetic Structures III Eds. B. Hillebrands and A.\nThiaville, Springer 2002.\n[11] X. R. Wang, and Q. Niu, Phys. Rev. B 59, R12755\n(1999); Z.Z. Sun, H.T. He, J.N. Wang, S.D. Wang, and\nX.R. Wang, Phys. Rev. B 69, 045315 (2004).\n[12] Z.Z. Sun, and X.R. Wang, Phys. Rev. B 71, 174430\n(2005);73, 092416 (2006); 74132401 (2006).\n[13] T. Sun, P. Meakin, and T. Jssang, Phys. Rev. E 51, 5353\n(1995)."    },    {        "title": "0809.4644v2.Damping_and_magnetic_anisotropy_of_ferromagnetic_GaMnAs_thin_films.pdf",        "content": "Anisotropic Magnetization Relaxa tion in Ferromagnetic GaMnAs \nThin Films \n \nKh.Khazen, H.J.von Bardeleben, M.Cubukcu, J.L.Cantin \nInstitut des Nanosciences de Paris,  \nUniversité Paris 6, UMR 7588 au CNRS \n140, rue de Lourmel, 75015 Paris, France \n \nV.Novak, K.Olejnik, M.Cukr \nInstitut of Physics, Academy of Sciences,  \nCukrovarnicka 10, 16253 Praha, Czech Republic \n \nL.Thevenard, A. Lemaître \nLaboratoire de Photonique et  des Nanostructures, CNRS \nRoute de Nozay, 91460 Marcoussis, France \n \n \nAbstract: \n \n The magnetic properties of annealed, epitaxial Ga 0.93Mn 0.07As layers under tensile and \ncompressive stress have been investigat ed by X-band (9GHz) and Q-band (35GHz) \nferromagnetic resonance (FMR) spectroscopy. From  the analysis of the linewidths of the \nuniform mode spectra the FMR Gilbert damping factor α has been determined. At T=4K we \nobtain a minimum damping factor of α = 0.003 for the compressively stressed layer. Its value \nis not isotropic. It has a minimum value for th e easy axes orientations of the magnetic field \nand increases with the measuring temperature. It s average value is for both type of films of \nthe order of 0.01 in spite of strong differences  in the inhomogeneous linewidth which vary \nbetween 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively .   \n \n \nPACS numbers: 75.50.Pp, 76.50.+g, 71.55.Eq  \nIntroduction: \n \nThe magnetic properties of ferromagnetic Ga 1-xMn xAs thin films with Mn \nconcentrations between x=0.03 and 0.08 have been studied in great detail in the recent years \nboth theoretically and experimentally. For recent reviews see references [1, 2].  A \nparticularity of GaMnAs ferro magnetic thin films as comp ared to conventional metal \nferromagnetic thin films is the predominance of  the magnetocrystalline anisotropy fields over \nthe demagnetization fields. The strong anisotropy fields are not directly related to the crystal \nstructure of GaMnAs but are induced by the la ttice mismatch between the GaMnAs layers and \nthe substrate material on which they are grow n. When grown on (100) GaAs substrates the \ndifference in the lattice constants induces biaxial strains  of ≈ 0.2% which give rise to \nanisotropy fields of several 103 Oe. The   low value of the de magnetization fields (~300Oe) is \nthe direct consequence of the small spin conc entration in diluted magnetic semiconductors \n(DMS) which for a 5% Mn doping leads to a saturation magnetization of only  40 emu/cm3. \nAs the strain is related to the lattice mismat ch it can be engineer ed by choosing different \nsubstrate materials. The two systems which have been investigated most often are (100) GaAs \nsubstrates and (100)GaInAs pa rtially relaxed buffer layers. These two cases correspond to \ncompressive and tensile strained Ga MnAs layers respectively [3].  \nThe static micro-magnetic pr operties of GaMnAs layers can be determined by \nmagnetization, transport, magneto-optical and ferromagnetic resonance techniques. For the \ninvestigation of the ma gnetocrystalline anisotropies the ferromagnetic resonance spectroscopy \n(FMR) technique has been shown to be partic ularly well adapted [2, 4]. The dynamics and \nrelaxation processes of the magnetization of such layers have hardly been investigated up to \nnow [5-7]. The previous FMR studies on this  subject concerned either unusually low doped \nGaMnAs layers [5, 7] or employed a single microwave frequency [6] which leads to an \noverestimation of the damping factor. The knowle dge and control of the relaxation processes \nis in particular important for device applications as they de termine for example the critical \ncurrents necessary for current induced magne tization switching. It is thus important to \ndetermine the damping factor for state of the art samples with high Curie temperatures of T C ≈ \n150K, such as those used in this work.  Anothe r motivation of this work is the search for a \npotential anisotropy of the ma gnetization relaxation in a dilu ted ferromagnetic semiconductor in which the m agnetocrystalline anisotro pies are strong and dom inant over the \ndemagnetization contribution.  \nThe intr insic sm all angle m agnetization re laxation is generally described by one \nparam eter, the Gilbert d amping coefficient α, which is defined by the Landau Lifshitz Gilbert \n(LLG) equation of m otion for the m agnetization: \n \n⎥⎦⎤\n⎢⎣⎡× +⎥⎦⎤\n⎢⎣⎡× −= ⋅dtsdMeffH MdtMdrr rrr\nγα\nγ1   eq.1 \nwith M the m agnetization, H eff the effective m agnetic field, α the dam ping fa ctor, γ the \ngyrom agnetic ratio and s the uni t vector parallel to M.   \nThe dam ping factor α is generally assum ed to be a scal ar quantity [8, 9] . It is defined \nfor sm all angle precess ion relaxatio n   which  is the case of FMR experim ents. This param eter \ncan be experim entally determ ined by FMR spectr oscopy either from  the angular variation of \nthe linewidth or from  the variati on of the uniform  mode linewidth ∆Hpp with th e microwave \nfrequency. In this second case the linewidth is given  by: \n \nω\nγω ⋅\n⋅⋅ + ∆= ∆+ ∆= ∆\nMGH H H Hin inpp\n2 hom hom hom32)(  eq.2 \nWith ∆Hpp the first derivative p eak-to -peak  linewid th of  the uniform mode of \nLorentzian lineshape, ω the angular m icrowave frequency an d G the Gil bert dam ping factor \nfrom which the m agnetization independent damping factor α can be deduced as α=G/γM. In  \neq. 2 it is assum ed that the m agnetiz ation and th e applied magnetic f ield are collin ear which is \nfulfilled for high symm etry direction s in GaMnAs  such as [001], [110] and [100]. Ot herwise a \n1/cos ( θ-θH) term  has to be added to eq.2 [8]. \n∆Hinhom is the inhom ogeneous, frequency indepe ndent linewidth; it can be further \ndecom posed in three con tributions, re lated to the crysta lline imperf ection of  the f ilm [10]: \n \nint\ninthom HHH H HHr\nH\nHr\nH\nHr\nin ∆⋅ + ∆⋅ + ∆⋅∂= ∆δδφδφδθθδ  eq.3 \n \nThese three term s were introduced to take in to account a slight m osaic structure of the \nmetallic thin f ilms def ined by the polar angles  (θ, φ) and their distributions ( ∆θ, ∆φ)  - \nexpressed in the first two term s in eq.3- and a distribution of  the internal anisotropy fields H int – the last term of eq.3. In the case of  homo epitaxial III-V films obtained by MBE growth like \nGaMnAs on GaAs, films of high crystalline qu ality are obtained [LPN] and only the third \ncomponent ( ∆Hint) is expected to play an important role.  \n Practically, the variation of the FMR linewid th with the microwave frequency can be   \nmeasured with resonant cavity systems at different frequencies between 9GHz and 35GHz; \nthe minimum requirement -used also in this work - is the use of two frequencies. We disposed \nin this work of 9GHz and 35GHz spectrometers. The linewidth is decomposed in a frequency independent inhomogeneous part and a linear fr equency dependent homogeneously broadened \npart. For most materials the inhomogeneous fr action of the linewidth is strongly sample \ndependent and depends further on the interface quali ty and the presence of cap layers. It can \nbe smaller but also much larger than the intrinsic linewidth. In Ga\n0.95Mn 0.05As single films \ntotal X-band linewidths be tween 100Oe and 1000Oe have been encountered. These \nobservations indicate already the impor tance of inhomogeneous broadening. The \nhomogeneous linewidth will depend on the intrinsic sample properties. This approach supposes that the inhomogeneous  linewidth is frequency independent and the homogenous \nlinewidth linear dependent on the frequency, two assumptions generally valid for high \nsymmetry orientations of the a pplied field for which the magne tization is parallel to the \nmagnetic field.  \nIt should be underlined that  in diluted magnetic semiconductor (DMS) materials like \nGaMnAs the damping parameter is not only determined by the sample composition x Mn [5]. It \nis expected to depend as we ll on (i) the magnetic compensati on which will vary with the \ngrowth conditions, (ii) the (hol e) carrier concentration respon sible for the ferromagnetic Mn-\nMn interaction which is influenced by the presence of native donor defects like arsenic \nantisite defects or Mn interstitial ions [11] and (iii) the valence bandstructure, sensitive to the \nstrain in the film. Due to the high out-of –plane and in-plane anisotropy of the magnetic \nparameters [12] which further vary with the applied field and the temperature a rather \ncomplex situation with an anisotropic and te mperature dependent da mping factor can be \nexpected in GaMnAs.  \nWhereas the FMR Gilbert damping factor has been determined for many metallic \nferromagnetic thin films [8] only three experimental FMR studies have been published for GaMnAs thin films up to now [5-7]. In ref.[ 5,7] low doped GaMnAs laye rs with a critical \ntemperature of 80K which do not correspond to the high quality, standa rd layers available \ntoday were studied. In the ot her work [6] higher doped layers  were investigated but the \nexperiments were limited to a single microw ave frequency (9GHz) and thus no frequency dependence could be studied.  In this work we present the results of FMR studies at 9GHz and \n35 GHz on two high quality GaMnAs layers with  optimum critical temperatures: one is a \ncompressively strained layer grown on a GaAs buffer layer and the othe r a tensile strained \nlayer grown on a (Ga,In)As buffer layer. Due to  the opposite sign of the strains the easy axis \nof magnetization is in-plane [ 100] in the first case and out-o f-plane [001] in the second. The \nGaMnAs layers have been annealed ex-situ after their growth in order to  reduce the electrical \nand magnetic compensation, to homogenize the laye rs and to increase the Curie temperature \nto ≈ 130K. Such annealings have become a st andard procedure for improving the magnetic \nproperties of low temperature molecular b eam epitaxy (LTMBE) grown GaMnAs films. \nIndeed, the low growth temperature required to incorporate the high Mn concentration \nwithout the formation of precipitates gives rise to native defect the conc entration of which can \nbe strongly reduced by the annealing.  \n \nExperimental details \nA first sample consisting of a Ga 0.93Mn 0.07As layer of 50nm thickness has been grown \nat 250° C by low temperature molecular beam epitaxy on a semi-insulating (100) oriented \nGaAs substrate. A thin GaAs buffer layer has been grown before the deposition of the \nmagnetic layer. The second sample, a 50 nm thick Ga 0.93Mn 0.07As layer have been grown \nunder very similar conditions on a partially relaxed (100) Ga 0.902In0.098As buffer layer; for \nmore details see ref. [13]. After the growth the structure was thermally annealed at 250° C for \n1h under air or nitrogen gas fl ow. The Curie temperatures were 157K and 130K respectively. \nBased on conductivity measuremen ts the hole concentratio n is estimated in the 1020cm-3 \nrange. \nThe FMR measurements were performed with Bruker X-band and Q-band \nspectrometers under standard conditions: mW microwave power and 100 KHz field \nmodulation. The samples were measured at te mperatures between 4K and 170K. The angular \nvariation of the FMR spectra was measured in  the two rotation planes (110) and (001). The \npeak-to peak linewidth of the first derivati ve spectra were obtained from a lineshape \nsimulation. The value of the st atic magnetization M(T) had been determined by a commercial \nsuperconducting quantum interference device (SQUID) magnetometer. A typical hysteresis \ncurve is shown in the inset of fig.8. \n \nExperimental results: The saturation magnetizations of the two laye rs and the magneto crystalline anisotropy \nconstants which had been previously de termined by SQUID and FMR measurements \nrespectively are given in table I. The anisotropy constants had been determined in the whole \ntemperature range but for clarity only its values  at T=55K and T=80K are given in table I. We \nsee that the dominant anisotropy constant K 2⊥ are of different sign with -55000 erg/cm3 to \n+91070 erg /cm3 and that the other three constants ha ve equally opposite signs in the two \ntypes of layers. The easy axes of magnetization are the in-pla ne [100] and the out-of-plane \n[001] direction respectively. Howe ver the absolute values of th e total effective perpendicular \nanisotropy constant Ku=K 2⊥ +K 4⊥ are less different for the two samples: -46517erg/cm3 and \n+57020erg/cm3 respectively. More detailed inform ation on the measurements of these \nmicromagnetic parameters will be published elsewhere.  \nFor the GaMnAs/GaAs layers the peak-to-peak linewidth of the first derivative \nuniform mode spectra has been strongly re duced by the thermal annealing; in the non \nannealed sample the X-band linewidth was highl y anisotropic with va lues between 150Oe and \n500Oe  at T=4K. After annealing it is reduced to  an quasi isotropic average value of 70Oe at \nX-band. Quite differently, for the GaMnAs/GaInA s system the annealing process decreases \nthe linewidth of the GaMnAs layers only marginally. Although full angular dependencies \nhave been measured by FMR we will analyze only the linewidth of the four high symmetry field orientations H//[001], H //[100], H//[1-10], H//[ 110] corresponding to the hard and easy \naxes of magnetization. As will be shown below, in spite of rather similar high critical temperatures (157K/130K) the linewidth are drastically di fferent for the two cases.  \n \n1. GaMnAs on GaAs \n In fig. 1a and 1b we show typical low te mperature FMR spectra at X-band and Q-band \nfrequencies for the hard [001] /intermediate  [100] axis orientation of the applied magnetic \nfield. The spectra are characterized by excelle nt signal to noise ra tios and well defined \nlineshapes.  We see that at both frequencies the lineshape is close to a Lorentzian. In addition \nto the main mode one low intensity spin wave  resonance is observed at both frequencies at \nlower fields (not shown). \n The linewidth at X-band (fig.2) is of  the order of 50Oe to 75Oe with a weak \norientation and temperature dependence. Above T>130K, close to the criti cal temperature, the \nlinewidth increase strongly. At Q-band we observe a systematic increase by a factor of two of \nthe total linewidth (fig.3) with an increase d temperature and orient ation dependence. As generally observed in GaMnAs, the easy axis orie ntation gives rise to th e lowest linewidth. At \nQ-band the lineshape is perfectly Lorentzian (f ig.1b). These linewidth are among the smallest \never reported for GaMnAs thin films, which re flects the high crystal line and magnetic quality \nof the film. \n To determine the damping factor α we have plotted the frequency dependence of the \nlinewidth for the different orientations and at various temperatures. An example is given in \nfig. 4 for T=80K; this allows us to determin e the inhomogeneous linewidth obtained from a \nlinear extrapolation to zero frequency and the damping factor from the slope. The \ninhomogeneous linewidth at T=80K is of the order of 30 Oe, i.e. 50% of the total linewidth at \nX-band. This shows that the approximation ∆Hinhom<< ∆Hhomo which had been previously \nused [5] to deduce the damping factor from a single (X-band) frequenc y measurement is not \nfulfilled here.  \n The temperature dependence of the inhomogeneous linewidth is shown in fig.5. \nSimilar trends as for the total linewidth in the non annealed films are observed: the linewidth \nis high at the lowest temperatures, decreases with increasing temperat ures up to 120K   and \nincreases again close to T C.   \n From the slope of the linewidth variati on with microwave frequency we obtain the \ndamping factor α (fig.6). Its high temperature value is of the order of 0.010 but we observe a \nsystematic, linear variation with the temperatur e and a factor two difference between the easy \naxis orientation [100] and the hard axis orientation [001].  \n \n \n2. GaMnAs on GaInAs \n Similar measurements have been performed on the annealed tensile strained layer. In \ntensile strained GaMnAs films the easy axis  of magnetization ([001]) coincides with the \nstrong uniaxial second order anisotropy directio n.  For that reason no FMR resonance can be \nobserved at temperatures below T=80K for the easy axis orientation H// [001] at X-band. For \nthe other three orientations the resonances  can be observed at X-band in the whole \ntemperature range 4K to T C. Due to the strong temperature dependence of the anisotropy \nconstants and the parallel decr ease of the internal anisotropy fields the easy axis resonance \nbecomes observable at X-band for temperatures above 80K. In the films on GaInAs much \nhigher linewidth are encountered th an in films on GaAs, the values are up to ten times higher \nindicating a strong inhomogeneity in this film. A second low fiel d resonance is systematically \nobserved at X-band and Q-band; it is equally attributed to a spin wave resonance.   Figures 7a and 7b show typical FMR spectra  at X- and Q-band re spectively. At both \nfrequencies the lineshape can no longer be simu lated by a Lorentzian but has changed into a \nGaussian lineshape. \n Contrary to the first cas e of GaMnAs/GaAs the X-band linewidth varies monotonously \nin the whole temperature region (fig.8). We observe a linewidth of ~600Oe at T=4K, which \ndecreases only slowly with temperature; the linewidth becomes minimal in the 100 K to 140K \nrange. The Curie temperature “s een” by the FMR spectroscopy is s lightly higher as compared \nto the one measured by SQUID due to th e presence of the applied magnetic field. \n At low temperature the Q-ba nd linewidth vary strongly w ith the orientation of the \napplied field with values be tween 500Oe and 700Oe. The lowest  value is observed for the \neasy axis orientation. They decrease as at X-band only slowly with increasing temperature \nand increase once again when approaching the Curie temperature. At Q-band the easy axis \nFMR spectrum, which is also accompanied by a str ong spin wave spectrum at lower fields, is \nobservable in the whole temperature range. \n For this sample we observe especially at Q-band a systematic difference between the cubic axes [100], [001] linewidth and the one for the in-plane [110] and [1-10] field \norientations (fig.8). The most surprising observation is that for temperatures below T<100K \nthe linewidth for H//[100] and H//[110] are co mparable at X-band and Q-band and thus an \nanalysis in the simple model discussed above is not possible. We attribute this to much higher \ncrystallographic/magnetic inhomogeneities, which mask the homogenous linewidth. The \norigin of the strong inhomogeneity is still unclear.  The only orientation for which in the whole temperature range a systematic increas e in the linewidth between X-and Q-band is \nobserved is the H//[1-10] orienta tion. We have thus analyzed th is variation (fig.10) according \nto eq.1.  \n In spite of important differences in the lin ewidth the slope varies only weakly which \nindicates that the inhomogeneous linewidth is  very temperature dependent and decreases \nmonotonously with increasing temperature from 570Oe to 350Oe.   \nIn the high temperature range (T ≥100K) the easy axis orientation could also be \nanalyzed (fig.11). The inhomogeneous  linewidth are lower than for the hard axis orientation at \nthe same temperatures and are in the 300Oe range (fig.12). The homogenous linewidth at \n9Ghz is in the 50Oe range which is close to the values determined in the first case of \nGaMnAs/GaAs.  \n From the slope (fig.13) we obtain the da mping factor which for the hard axis \norientation is α=0.010 in the whole temperature range. Th is value is comparable to the one measured for the GaMnAs/GaAs film for H//[110]. The damping factor for the easy axis \norientation is lower but   increases close to T C as in the previous case.  \n \nDiscussion: \n  An estimation of the FMR intrinsic damping factor in a ferromagnetic GaMnAs thin \nfilm has been made within a model of localiz ed Mn spins coupled by p-d kinetic exchange \nwith the itinerant-spin of holes treated by the 6-band Kohn-Luttinger Hamiltonian [5]. Note, that these authors take for the effective kinetic exchange field the value in the mean-\nfield approximation, i.e. H\neff=JN<S>, so that their calculation are made within the random \nphase approximation (RPA). RPA calculations of α have been made by Heinrich et al. [14] \nand have recently been used by Tserkovnyak et al .[15] for numerical app lications to the case \nof Ga 0.95Mn 0.05As.  Both models however, are phenomenological and include an \nadjustable parameter: the quasiparticle lifetime Γ for the holes in [5] and the spin-flip \nrelaxation T 2 in [15]. These models do not take into account neither multi-magnon \nscattering nor any damping beyond the RPA. It has been argued elsewhere [16], that in diluted \nmagnetic semiconductors such affects are only impor tant at high temperatur e (i.e. at T>Tc). In \nparticular, the increase of α in the vicinity of Tc may be attributed to such effects that are \nbeyond the scope of the models of references  [5] and [14]. At low temperatures T<<Tc \nhowever, where the corrections to the RPA are expected to be negligib le, the models of [5, \n14,15] provide us with a numerical value of α in agreement with our experiments if we \nintroduce reasonable values of   these parameters. For GaMnAs films with metallic \nconductivity, Mn concentrations of x= 0.05 and hole concentr ations of 0.5 nm-3 (5x1020cm-3) \nSinova et al [5] predict an isotropi c  low temperature damping factor α between 0.02 and 0.03 \ndepending on the quasiparticle life time broa dening. Tserkovnyak et al [15] found a similar \nvalue of α ≈ 0.01 for the isotropic damping factor fo r a typical GaMnAs film with 5% Mn \ndoping and full hole polarization. \n Both predicted values are of the same orde r of magnitude as the experimental values \ndetermined in this study. Our results for GaMn As/GaAs show further that the damping factor \nis not isotropic as generally a ssumed but is anisotropic with a lowest value for the in-plane  \neasy axis orientations of the applied magnetic field H//[100], H//[1-10] and an increase of up \nto a factor of two for the hardest axis orientat ion H// [001]. Intrinsic anisotropic damping is \nrelated to the fact that the free energy density depends on the orientation of the magnetization which in the case of GaMnAs is related to th e anisotropy of the p-hole Fermi surface. We have shown (table I) that the anisotropy of the magnetocristalline constants and the related \nfields are important in these strained layers and it is thus not surprisi ng to find also anisotropy \nof the damping factor. For further discussions on this subject see reference [9]. The system \nmight also contain extrinsic an isotropies related to the pres ence of lattice defects. Their \ninfluence can be deduced from the value and anisotropy of the inhomogeneous linewidth. In \nthe case of the compressively strained layers (G aMnAs/GaAs) we see that their value is small \nand rather isotropic quite differen tly from the tensile strained film. It is in the first case that \nour measurements show a factor of two anisot ropy of the Gilbert damping factors. A further \nindication for the intrinsic charac ter of the anisotropy is the fact that the damping factor for \nthe perpendicular orientation has the highest value. In this case any contribution from two \nmagnon scattering will be minimized. Anyway, such contributions are generally only \nimportant at low frequency measurements in the 2-6GHz range but even there they were \nfound to be negligible [7]. \n Additional material related parameters are expected to further influence the damping \nfactor. As the spin flip relaxation times will depend on the sample properties and in particular \nthe presence of scattering centers we will not expect to find a unique damping factor even for GaMnAs/GaAs samples with the same Mn composition x. More likely, different damping \nfactors are expected to be found in  real films and their values might be used to assess the film \nquality. In this sense the GaMnAs/GaAs film studi ed here is of course “better” than the one \non GaInAs in line with the strong difference in the sample inhomogeneities.  \n The inhomogeneous linewidth originates from spatial inhomogeneities in the local \nmagnetic anisotropy fields and inhomogeneities in the local exchange interactions. Given the particular growth conditions of these films, low temper ature molecular beam epitaxy, \ninhomogeneities can not be expected to be neglig ible in these materials.  If we had analyzed \nour X-band results of the GaMnAs/GaAs film in the spirit of ref. [5], i.e. assuming a \nnegligible inhomoge neous linewidth - ( ∆H\ninhom=0) -, we would have obtained artificially \nincreased damping factors. A further contribution might be expected from the intrinsic \ndisorder in these films: as GaMnAs is a diluted magnetic semiconductor with random \ndistribution of the Mn ions, this  disorder will even for crysta llographically perfect crystals \ngive some importance to this term.  \n In the previous studies of  the FMR damping factor in GaMnAs/GaAs single films \nhigher values have been reported. Matsuda et  al [6] found damping factors between 0.02 and \n0.06 in the T=10K to T=20K temperature range. They  observed the same tend encies as in this \nwork concerning the anisotropy and temperature dependence of α: the lowest damping factor is seen for the easy axis orientation and its va lues increases with increasing temperatures. The \nfilms of their study were however significantly different: (i) the Mn doping concentration was \nlower, x=0.03 and thus the hole concentratio n was equally lower and (ii) the critical \ntemperature of the annealed film was only T=80K. The anisotropy in the inhomogeneous \nlinewidth at T=20K was equally much higher, varying between 30Oe for the easy axis to \n250Oe for the hardest axis [110]. In a second study Sinova et al [5] have measured an \nannealed GaMnAs/GaAs sample with a similar composition (x= 0.08) and critical temperature \n(TC=130K) as the one studied here. They deduced a damping factor of the order of α=0.025 \nwith only a slight temperature dependence betw een 4K and 80K and an increase close to T C. \nHowever, these measurements were done at one  (X-band) microwave frequency only and the \nnumerical value of α was obtained by assuming a neglig ible inhomogeneous linewidth. As \nexplained above, the value  of α can be expected to be overestimated in this case.In the \nphotovoltage measurements of ref.7 the dampi ng factor of a low (x=0.02) doped GaMnAs \nlayer has been determined with microwave fr equencies from 2 to 19.6GHz but for one field \norientation H//[001] (h ard axis) and one temperature (T =9K) only. Interestingly, their \nmeasurements show a linear behavior even in  the low frequency range down to 4GHz which \ndemonstrates the negligible contribution from two magnon scattering in this case. \n The intrinsic damping factor α plays also an important role in the critical currents \nrequired to switch the magnetization in FM/NM/ FM trilayers [5]. However, in trilayer \nstructures interface and spin pumping effects wi ll add to the intrinsic damping factor of the \nferromagnetic material and give rise to an incr eased effective damping fa ctor. Sinova et al [5] \nhave estimated the critical current for real istic GaMnAs layers: based on a value of α= 0.02, \nthey estimated the critica l current density to J C=105A/cm2. It should be noted that the damping \nfactor involved in the domain wall motion [ 17, 18] is by definition different form the FMR \ndamping factor. Both are however linearly related with αFMR<αDW [17, 18]. First observations \nof current induced magne tization switching in Ga 0.956Mn 0.044As/GaAs/Ga 0.967Mn 0.033As \ntrilayers confirm these theoretical predictions [19]. These authors observed a critical current \ndensity of ≈  105A/cm2 which would have been predicted from Slonczewski’s formula [20] for \na  domain wall damping factor of αDW=0.002. \n \nConclusion:  \n  We have determined the intrinsic FM R Gilbert damping factor for annealed \nGa0.93Mn 0.07As thin films with high critical temperatures. To evaluate the influence of the \nstrain the two prototype cases of  compressive and tensile strained  layers were studied.   In \nboth cases we find an average damping factor of the order of 0.01. We thus see that the sign \nof the strain does not seem to  influence the damping factor strongly.  The homogeneity of the \nfilms as judged from the inhomogeneous linewid th is much higher in the case of GaAs \nsubstrates than for GaInAs substrates. This must  be attributed to the high dislocation density \nin the GaInAs layer [13]. In the case of the GaMnAs/GaAs layers, where the small linewidth \nallows a finer analysis of the data, we observe  an anisotropy of the da mping factor, which has \nthe lowest value for the easy axis orientation. This value of α[1−10]=0.003 is an order of \nmagnitude lower than the previous reported va lues. The few experiment al results available \nseem to indicate that the damping factor decr eases with increasing Mn concentration. This \ncorresponds well to the th eoretical predictions by Sinova et al [5] for the case of small quasi-\nparticle lifetime broadening. It wi ll be interesting to test this behavior further in more highly \ndoped layers (x ≈0.15), which become now available.  \n \nAcknowledgment: We thank Alain Mauger of the IMPMC laboratory of the University Paris \n6 and Bret Heinrich from the Simon Frazer University in Vancouver for many helpful \ndiscussions. \n \nReferences : \n \n[1] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A.H. MacDonald, Rev.Mod.Phys. 78, \n809 (2006). \n[2] X. Liu, and J. K. Furdyna , J. Phys.: Condens. Matter 18, R245 (2006). \n[3] X. Liu, Y. Sasaki and J. K. Furdyna, Phys.Rev.B 67 , 205204 (2003). \n[4] K.Khazen, H.J.von Bardeleben, J.L.Cantin, L.  Thevenard, L. Largeau, O. Mauguin, and A. \nLemaître,  Phys.Rev. B77, 165204 (2008) . \n[5] J.Sinova, T.Jungwirth, X.Liu, Y.Sa saki, J.K.Furdyna, W.A.Atkinson, and \nA.H.MacDonald, Phys.Rev. B69, 085209 (2004). \n [6] Y.H.Matsuda, A.Oiwa, K.Ta naka, and H.Munekata, Physica B 376-377,668 (2006) \n[7] A.Wirthmann, X.Hui, N.Mecking, Y.S.Gu i, T.Chakraborty, C.M.Hu, M.Reinwald, \nC.Schüller, and W.Wegschneider, Appl.Phys.Lett.92, 232106 (2008) \n[8] M. Farle, Rep.Prog.Phys. 61, 755 (1998). [9] B.Heinrich, « Spin relaxation in magnetic me tallic layers and multilayers », in Ultrathin \nMagnetic Structures III, ed. by J.A.C.Bland, and B.Heinrich, Springe r Verlag (Berlin 2005), \np.143 \n[10] C.Chappert,K.LeDang,P.Beauvilain,H .Hurdequint, and D.Renard, Phys.Rev.B34 , 3192 \n(1986) \n[11] F. Glas, G. Patriarche, L. Largeau, A. Lemaître, Phys. Rev. Lett., 93, 086107 (2004) \n[12] L.Thevenard, L.Largeau,, O.Mauguin, A.Lemaitre, K.Khazen and H.J.von Bardeleben, \nPhys. Rev.B75 ,195218 (2006) \n[13] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaître, N. Vernier and J. Ferré, Phys. Rev. B 73, 195331 (2006)  \n[14] B.Heinrich, D.Fraitova, V.Kambersky , Phys. Stat. Solidi 23, 501 (1967). \n[15] T.Y.Tserkovnyak, G.A.Fiete, B.I.Halperin, Appl.Phys.Lett. 84, 5234 (2004) \n[16] A.Mauger, D.L. Mills, Phys.Rev. B29, 3815 (1984) \n[17] S.C.Chen and HL.Huang, I EEE Transactions on Magnetics 33, 3978 (1997) \n[18] H.L.Huang, V.L.Sobolev, and S.C.Chen,  J.Appl.Phys. 81, 4066(1997) \n[19] D.Chiba, Y.Sato, T.Kita, F. Matsukura, and H.Ohno, Phys.Rev.Lett. 93, 216602(2004) \n[20] J.C.Slonczewski, J.Magn.Magn.Mater. 159, L1(1996) \n \n \nFigure Captions: \n \nFigure 1a: X-band FMR spectrum of the GaMnAs /GaAs film taken at T=20K and for H// \n[001]; the peak-to-peak linewidth is 60Oe. The experimental  spectrum is shown by circles and \nthe Lorentzian lineshape simulation by a line. \n \n \nFigure 1b: Q-band FMR spectrum of the GaMnAs /GaInAs film taken at T=80K and for H// \n[100]; the peak to peak  linewidth is 120Oe. The experimental spectrum is shown by circles \nand the Lorentzian lineshape simulation by a line \n \nFigure 2 (color on line): X-band peak-to-peak li ne widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles.   \nFigure 3 color on line): Q-band peak-to-peak line widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles. \n \nFigure 4 (color on line): Peak-t o-peak linewidth at 9GHz and 35GHz for the GaMnAs/GaAs \nfilm; T=80K and H//[001] black squares, H//[1-10] olive lowe r triangles, H//[110] red circles, \nH//[100] blue upper triangles \n \nFigure 5 (color on line): GaMnAs/GaAs inhomogeneous linewidth as a function of temperature for four orientations of the applie d field: H//[001] black squares, H//[1-10] olive \nlower triangles, H//[110]  red circles, H//[100] blue upper triangles \n \nFigure 6 (color on line): damping factor α as a function of temperature and magnetic field \norientation; H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] blue upper \ntriangles, H//[100] red circles; the maximum erro r in the determination of the linewidth is \nestimated to 10G which co rresponds to an error in α of  0.001 \n \nFigure 7a: X-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: \nexperimental points, line: simulation with Gaussian lineshape \n \nFigure 7b: Q-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape \n \n \nFigure 8 (color on line): GaMnAs/GaInAs:  X-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field: H//[001] black squares, \nH//[1-10] olive lower triangles, H//[110] red circ les, H//[100] blue upper triangles; the easy \naxis FMR spectrum is not observable below T=100K ; a typical hysteresis  curve as measured \nby SQUID is shown in the inset. \n Figure 9 (color on line): GaMnAs/GaInAs : Q-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field : H//[001] black squares, \nH//[1-10] olive lower triangl es, H//[110] red circles, H//[100] blue upper triangles \n \nFigure 10 (color on line): GaMnAs/GaInAs : FM R linewidth as a function of microwave \nfrequency for H//[1-10] at different temper atures; T=10K (black squares), T=25K (red \ncircles), T=55K(black spades), T=80K( blue stars), T=115K (red triangles) \n \nFigure 11 (color on line): GaMnAs/GaInAs: FMR linewidth as a function of microwave \nfrequency for H//[001] (easy ax is)  at different temperatur es; T=100K (red circles), T=120K \n(blue spades),T=130K(black squares) \n \nFigure 12: GaMnAs/GaInAs inhomogeneous linewi dth as a function of temperature for two \norientations of the applied field: H//[001]  squares, H//[1-10]  lower triangles,  \n \nFigure 13 : GaMnAs/GaInAs damping factor  α as a function of temperature and magnetic \nfield orientation; H//[001]  squa res, H//[1-10]  lower triangles.  \n \nTables \n \n \nGa 0.93Mn 0.07As/GaAs G a0.93Mn 0.07As/Ga 0.902In0.098As \nMs(T=4K)= 47e mu/cm3 Ms(T=4K)= 38 e mu/cm3 \nTC =157K  (SQUID) TC=130K  (SQUID) \nAnisotropy constants (T=80K) Anisotropy constants (T=55K) \nK2⊥ = - 55000 e rg/cm3 K 2⊥ = +91070 e rg/cm3 \nK2// =2617 er g/cm3 K 2// = -2464 erg/cm3 \nK4⊥ = 8483  erg/cm3 K 4⊥ = -34050  erg/cm3 \nK2// =2590 er g/cm3 K 2// = -1873er g/cm3 \nEasy axis of magnetization Easy axis of magnet ization \n[100] 4K<T<T C [001] 4K<T< T C \n \n \nTable I \n \nTable caption \n \nTable I: M icrom agnetic  param eters of the two sam ples studied in this work: s aturation \nmagnetization M s at T=4K, critical tem perature T C, magneto crystalline anisotropy constants \nof second and fourth order K 2⊥ , K 2//, K 4⊥, K 4// at T=55K and T=80K respectively and the \norientation of the easy axis for m agnetization. \n \n \n \n \n \n \n \n            \n \n7500 7600 7700 7800 7900 8000SW\n  FMR Signal (arb.u.)\nMagnetic Field H ( Oe)\n \n \nFigure 1a \n \n10000 1020 0 10400 1060 0 10800 1100 0\n  FMR Signal (arb.u.)\nMagnetic Field H (Oe)\n \nFigure 1b \n 0 20 40 60 80 100 120140 160050100150200250300Linew idth ∆H (Oe)\nTempera ture (K) 001\n 110\n 100 1-10\n \nFigure 2 \n \n0 20 40 60 80 100 120 140160050100150200250300Linew idth ∆H (Oe)\nTemperature ( K) 001\n 110\n 100 1-10\n \nFigure 3 \n \n 0 5 10 15 20 25 30 35 4004080120160200Linewi dth ( G)\nFreq uency  (GHz)\n  \n \n \nFigure 4 \n \n20 40 60 80 100 120 140 160020406080100120∆Hinhom (Oe)\nTemperature (K) 001\n 110\n 100\n 1-10\n \n \nFigure 5 \n \n 0 20 40 60 80 100 120140 1600.0040.0060.0080.0100.0120.0140.0160.0180.020Damping factor α\nTemperature (K ) 001\n 110\n 100\n 1-10\n \nFigure 6 \n \n4000 5000 6000 7000 8000SW\n  FMR S ignal (arb. u.)\nMagnetic Field (Oe )\n \nFigure 7a \n 13000 14000 15000 16000 17000SW\n  FMR Signal (arb. u.)\nMagnet ic Field  (Oe)\n \n \nFigure 7b \n \n0 20 406 08 0 100120140300400500600700800\n  Linewidth ∆H (Oe)\nTemperature (K) [110]\n [100]\n [1-10]\n [001]-500 0 500-40040 M (emu/cm2)\n Magnetic F ield (O e)\n \nFigure 8 \n 0 204 0 608 0 100 120 140300400500600700800\n  Linewidth ∆H (Oe)\nTemperat ure (K ) [110]\n [100]\n [1-10]\n [001]\n \n \nFigure 9 \n \n0 5 10 15 20 25 30 35 40350400450500550600650700750\n115K80K55K25K10K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 10 \n 0 5 10 15 20 25 30 35 40200250300350400450500\n100K120K130K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 11 \n \n0 204 0 608 0 100120140200300400500600\n  ∆Hinhom (Oe)\nTemperat ure (K ) [001]\n [1-10]\n \n \n \nFigure 12 \n 0 204 06 0 80 100 120 1400.0040.0060.0080.0100.0120.0140.0160.0180.020Dam ping factor α\n  \nTemperature (K)\n \n \nFigure 13 \n \n \n \n "    },    {        "title": "0810.1340v2.Transverse_spin_diffusion_in_ferromagnets.pdf",        "content": "Transverse spin di\u000busion in ferromagnets\nYaroslav Tserkovnyak,1E. M. Hankiewicz,2and Giovanni Vignale3\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Institut f ur Theoretische Physik und Astrophysik,\nUniversit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n(Dated: October 29, 2018)\nWe discuss the dissipative di\u000busion-type term of the form m\u0002r2@tmin the phenomenological\nLandau-Lifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of\n\fnite-momentum spin waves. This term arises physically from itinerant-electron spin \rows through\na perturbed ferromagnetic con\fguration and can be understood to originate in the ferromagnetic\nspin pumping in the continuum limit. We develop a general phenomenology as well as provide\nmicroscopic theory for the Stoner and s-dmodels of ferromagnetism, taking into account disorder\nand electron-electron scattering. The latter is manifested in our problem through the Coulomb drag\nbetween the spin bands. The spin di\u000busion is identi\fed in terms of the transverse spin conductivity,\nin analogy with the Einstein relation in the kinetic theory.\nPACS numbers: 75.30.Ds,72.25.-b,76.50.+g,75.45.+j\nI. INTRODUCTION\nThe problem of spin di\u000busion through conducting fer-\nromagnetic medium attracted much attention over sev-\neral decades.1,2,3,4,5,6Semiclassical spin transport in the\npresence of a weak magnetic \feld Hcan be captured by\nthe conventional di\u000busion equation (neglecting spin re-\nlaxation):\n@tS=H\u0002S+Dr2S; (1)\nwhereDis the di\u000busion coe\u000ecient and His the total\ne\u000bective \feld (omitting the gyromagnetic ratio), includ-\ning the applied and exchange contributions. The \frst\nterm on the right-hand side describes spin precession in\nthe local \feld while the second term stands for the ordi-\nnary di\u000busion of spin density S. Equation (1) is, how-\never, not applicable to most realistic ferromagnets, whose\nspin interactions are characterized by a large exchange\nenergy \u0001 xc. In particular, when \u0001 xcis comparable to\nthe Fermi energy (which is the case in transition met-\nals), the spin precession in the exchange \feld cannot be\ntreated in the di\u000busive transport framework. Further-\nmore, the time-dependent exchange \feld induces spin-\npumping currents7,8inside the ferromagnet with spatially\ninhomogeneous magnetization dynamics, which can con-\nsiderably modify the self-consistent magnetic equation of\nmotion. Here, we wish to elucidate the central role of\nsuch self-consistent dissipative spin currents, which gov-\nern the di\u000busion-like terms in the magnetic equation of\nmotion in the limit of strong ferromagnetic exchange cor-\nrelations.\nThis paper is a follow up to our previous work,9pro-\nviding additional technical details and o\u000bering a broader\nphenomenological base. Apart from assuming strong\nexchange correlations limit, our phenomenological ap-\nproach and the main results of the paper should not be\nsensitive to the microscopic details and do not rely on\nthe speci\fc model of the ferromagnetic material (suchas the Stoner or an s-dmodel, for example). The main\ngoals of this paper are as follows: (i) to put the results of\nRef. 9 into a broader phenomenological perspective, (ii)\nto explicitly show that two quite di\u000berent models|the\nspin-polarized itinerant electron liquid (treated in Ref. 9)\nand thes-dmodel|lead to the same phenomenology and\ncan be treated in parallel, and (iii) to make direct contact\nwith the spin-pumping theory.7,8\nTo be speci\fc, let us consider a continuous ferromag-\nnetic medium, with the e\u000bective \feld and spin density\ninitially pointing along the zaxis. For weak excitations\nclose to this state, we may try expanding the ensuing\ntransverse spin-current density as2,6\nji=\u0000D0z\u0002@iS\u0000D00@iS; (2)\nwhich enters in the continuity equation:\n@tS=H\u0002S\u0000X\ni=x;y;z@iji: (3)\nIn the limit of vanishing ferromagnetic correlations, we\nrecover Eq. (1) by setting D0!0 andD00!Din Eq. (2).\nHereafter, we are focusing exclusively on the transverse\nspin dynamics and spin currents. The longitudinal spin\n\rows are conventionally described in terms of the ordi-\nnary di\u000busion for spin-up and spin-down electrons with\nspin-dependent di\u000busion coe\u000ecients and spin-\rip scatter-\ning between the up and down spin bands.10Understand-\ning the transverse spin \rows and dynamics requires more\ncare, in part due to the inherently quantum-mechanical\nbehavior in the case of a strong exchange \feld. When\nthe magnetic excitation is driven by the self-consistent\ntransverse \feld h=z\u0002H\u0002z, there should also be \feld-\ndriven contributions to the transverse spin current (2),\nsuch as ji/@ih.\nThe problem in fact simpli\fes in the limit of strong\nexchange correlations. We will in the following employ\na mean-\feld view of ferromagnetism, where the collec-\ntive spin dynamics are driven by the exchange \feld, H=arXiv:0810.1340v2  [cond-mat.mes-hall]  17 Mar 20092\n\u0000\u0001xcm(r;t) (setting \u0016h= 1 throughout), parametrized\nby the local and instantaneous spin-density orientation,\nm=S=S, which has to be solved for self-consistently.\nSince we are only interested in the transverse spin dy-\nnamics, we set the magnitude of the spin density Sto be\nspatially and time independent. In the limit of large \u0001 xc,\nthe spin currents can be parametrized by m(r;t). We can\nthus proceed phenomenologically and expand jiin spa-\ntial and time derivatives of m(r;t). For a static magnetic\npro\fle m(r), we have the familiar exchange spin \row\nj0\ni=\u0000Am\u0002@im (4)\n(whereAis the material-dependent exchange-sti\u000bness\nconstant), which is the only \frst-order form allowed by\nspin-rotational and time-reversal symmetries. To avoid\nunnecessary complications, we will assume isotropic fer-\nromagnet throughout this paper. Dynamics allow for\ndissipative spin-current contributions that break time-\nreversal symmetry:\nj00\ni=\u0000\u0011m\u0002@i@tm: (5)\nFocusing on linear deviations of mfrom the equilibrium,\nm(0)=z, we omit terms such as @im\u0002@tm.\nAccording to the time-reversal property, the spin-\ncurrent density (4) corresponds to the D0term in Eq. (2),\nwhile the spin-current density (5) is analogous to the D00\nterm, although the latter two are certainly not identi-\ncal. In fact, we wish to emphasize the striking di\u000ber-\nence between the di\u000busive picture for the spin currents,\nEq. (2), on one side and Eqs. (4) and (5) on the other\nside, where we expand spin currents phenomenologically\nin terms of the time-dependent magnetic texture m(r;t).\nThe latter approximation is speci\fc to the limit of strong\nexchange correlations, where the nondissipative spin cur-\nrent, Eq. (4), is determined by the instantaneous mag-\nnetic pro\fle, while the dissipative spin current, Eq. (5),\ncan be interpreted as quasiparticle spin pumping by the\ncollective magnetic dynamics,8rather than ordinary spin\ndi\u000busion. It is also instructive to draw analogy between\ncoe\u000ecients Aand\u0011in Eqs. (4), (5) and the shear modu-\nlus and shear viscosity, respectively, in elasticity theory.\nIn the next section, we develop further the phenomeno-\nlogical grounds for Eqs. (4) and (5), before proceeding\nwith microscopic calculations for the dissipative coe\u000e-\ncient\u0011in Secs. III and IV. In Sec. V, we discuss a spin-\npumping interpretation of dissipative spin current (5),\nbefore summarizing our work in Sec. VI.\nII. PHENOMENOLOGY\nA. Landau-Lifshitz theory\nThe conventional starting point for studying ferro-\nmagnetic precession is the nondissipative Landau-Lifshitz\n(LL) equation11\n@tmjLL=H\u0003\u0002m; (6)where we de\fne the e\u000bective \feld H\u0003as the functional\nderivative of the free energy:\nH\u0003\u0011@mF[m]=S: (7)\nIn this Landau-Lifshitz phenomenology, which is applica-\nble well below the Curie temperature, only the position-\ndependent direction of the magnetization is taken to\nbe a dynamic variable, parametrizing the Free energy\nF[m(r)]. The angular-momentum density S=Smis\nassumed to be related to the magnetization by a con-\nstant conversion factor, the e\u000bective gyromagnetic ratio.\n(Abusing terminology, we say spin density synonymously\nwith angular-momentum density .) Since in the common\ntransition-metal ferromagnets the gyromagnetic ratio is\nnegative, we wrote Eq. (7) with an extra minus sign in\ncomparison to the standard de\fnition, where mis taken\nto be the direction of the magnetization rather than the\nspin density. The right-hand side of Eq. (6) is the phe-\nnomenological reactive torque on the spatially-resolved\nmagnetic precession, which generalizes the simple Larmor\nprecession of Eq. (1). Note that the dissipation power\nP[m(r;t)]\u0011\u0000SZ\nd3rH\u0003\u0001@tm (8)\nclearly vanishes according to Eq. (6). We also eas-\nily verify that the time reversal (under which t!\u0000t,\nm!\u0000m, and H\u0003!\u0000H\u0003) leaves Eq. (6) unchanged,\nas it should in the absence of dissipation. The only dis-\nsipative term we can write in the quasistationary limit\n(i.e., up to the \frst order in @t), assuming spatially uni-\nform and isotropic ferromagnet, is the so-called Gilbert\ndamping:12\n@tmjLLG=H\u0003\u0002m\u0000\u000bm\u0002@tm; (9)\nwhere\u000bis a material-dependent dimensionless (Gilbert)\nconstant. A typical experimental value for \u000bturns out\nto be often of the order of 10\u00002in various metallic ferro-\nmagnets, which means that it takes roughly 2 \u0019=\u000b\u001810\nprecession cycles for an out-of-equilibrium magnetiza-\ntion to relax to a static equilibrium direction along H\u0003.\nThe Gilbert damping breaks time-reversal symmetry and\ncauses a \fnite dissipation power:\nP[m(r;t)] =\u000bSZ\nd3r(@tm)2: (10)\nAs a side comment, we note that an alternative, so-called\nLandau-Lifshitz damping term m\u0002H\u0003\u0002mis mathemati-\ncally identical to the Gilbert damping m\u0002@tmin Eq. (9),\nup to an extra factor of (1 + \u000b2) on the left-hand side of\nthe equation.\nThe e\u000bective \feld H\u0003is in practice dominated by the\napplied magnetic \feld, magnetic crystal anisotropies, and\nmagnetostatic (dipole-dipole) interactions. In the pres-\nence of spatial inhomogeneities, there is also exchange\ncontribution to the free energy, which to the leading3\n(quadratic) order in magnetic inhomogeneities can be\nwritten as11\nFxc=A\n2Z\nd3r\u0002\n(@xm)2+ (@ym)2+ (@zm)2\u0003\n:(11)\nThe corresponding e\u000bective \feld is\nHxc=\u0000(A=S)r2m; (12)\nand the associated term in LL Eq. (6) is\n@tmjxc= (A=S)m\u0002r2m: (13)\nThis equation can also be formally written as\nS@tmjxc=\u0000X\ni=x;y;z@ij0\ni;\nj0\ni=\u0000Am\u0002@im; (14)\nwhich simply recovers our equilibrium spin current (4).\nWe emphasize that this spin current does not depend on\nmagnetic dynamics.\nTo summarize these preliminary considerations, the\nphenomenological LL equation describes collective mag-\nnetic precession driven by local e\u000bective \felds as well as\nequilibrium spin currents. At this point, there is, how-\never, a conspicuous asymmetry in the treatment of the\ndissipative correction to the LL equation, i.e., Gilbert\ndamping (9), which depends only on the local magnetic\ndynamics and thus does not involve spin currents. To\novercome this \\discrepancy,\" we expand the dissipative\nterms to second order in spatial derivatives, generalizing\nGilbert term to\n@tmjdiss=\u0000\u000bm\u0002@tm+ (\u0011=S)m\u0002r2@tm;(15)\nwhere\u0011is a new phenomenological parameter, charac-\nterizing spin-wave damping. Assuming spatial-inversion\nsymmetry (under which @i!\u0000@iandm!m), pre-\nvents us from writing any phenomenological terms linear\nin spatial derivatives. Recall also that we are always as-\nsuming small perturbations with respect to a uniform\nequilibrium magnetization, so that all spatial and time\nderivatives must hit a single m(for example, a dissipa-\ntive term of the formP\ni[@im\u0001(m\u0002@tm)]@imis dis-\nregarded since it is higher order in small deviations of\nm). Additional quadratic terms would be allowed phe-\nnomenologically if, e.g., we developed our linearized the-\nory with respect to an equilibrium magnetic texture, such\nas a domain wall or magnetic spiral. Some of such terms\nwere discussed in Ref. 13, which is beyond our present\nscope. Finally, we note that we wrote Eq. (15) with no\ndirect coupling to the e\u000bective \feld H\u0003. We justify this\nby assuming that the ferromagnetic correlations are char-\nacterized by a very large energy scale \u0001 xc, so that mi-\ncroscopic processes responsible for dissipation are driven\nby the collective variable m, rather than directly by H\u0003.\nIn transition-metal ferromagnets, the internal exchange\nenergy is of the order of eV, while the e\u000bective \feld H\u0003corresponds to microwave frequencies (i.e., at least three\norders of magnitude smaller than the exchange energy).\nThis means that when we excite magnetic dynamics by\nan external \feld, the microscopic degrees of freedom re-\nspond not to the small driving \feld but rather the much\nlarger self-consistent exchange \feld parametrized by the\ntime-dependent m. For the same reason, the spin current\nin Eq. (14) depends only on the magnetic pro\fle m(r),\nirrespective of how it is created by applied \felds.\nThe total dissipation power corresponding to Eq. (15)\nnow becomes\nP[m(r;t)] =Z\nd3r\u0002\n\u000bS(@tm)2+\u0011(@i@tm)2\u0003\n:(16)\nSimilarly to Eq. (14), we can also write the \u0011term in\nEq. (15) in the form of the divergence of the spin-current\ndensity\nj00\ni=\u0000\u0011@i(m\u0002@tm); (17)\nreproducing Eq. (5). We thus identi\fed two contribu-\ntions to the spin-current density: usual exchange spin\ncurrent (14) and dissipative spin current (17), which\nwe will later interpret as the dynamically-driven spin\npumping.7,8Spin current (17) can thus damp down\nspin-wave excitations even in the absence of any spin-\nrelaxation scattering.23The latter is, however, believed\nto be the culprit for a \fnite Gilbert damping \u000b,14which\nrelaxes uniform magnetic precession by transferring its\nangular momentum to the atomic lattice.\nIn the presence of dissipative currents (17), the relative\nlinewidth of the spin-wave resonance15is proportional to\n\u000b+ (\u0011=S)q2, for the wave vector q. In the absence of\nthe Gilbert damping \u000b, thus, the spectral width of the\nspin-wave excitation would vanish in the long-wavelength\nlimit.1\nB. Mermin ansatz for spin current\nWe now wish to establish a microscopic procedure for\nevaluating the dissipative component of the spin current,\nEq. (17). Ref. 9 adapted Mermin ansatz16for this pur-\npose, which we will reproduce below. Microscopically,\nthe spin-current density jiis carried by conducting elec-\ntrons responding to the mean-\feld exchange interaction\n^Hxc=\u0000\u0001xcm(r;t)\u0001^\u001b=2 (18)\nin the self-consistent single-electron Hamiltonian (which\ncould stem, e.g., either from the coupling to the localized\ndelectrons in the s\u0000dmodel or the itinerant electron\nStoner/LDA exchange). ^\u001bis the vector of Pauli matrices,\nwhich de\fnes the electron spin operator.\nLet us for the moment view exchange interaction (18)\nas an external parametric driving \feld, not concerning\nwith a self-consistent determination of m(r;t). In par-\nticular, we may allow for an instantaneous deviation of4\nthe electron spin density sfrom the exchange-\feld direc-\ntionm. This will allow us for a trick to \fnd the ensuing\nspin \rows, which is what we are after. The spin-density\ncontinuity equation corresponding to Hamiltonian (18) is\n@ts= \u0001 xcz\u0002(sm\u0000s)\u0000@iji: (19)\nThe equilibrium orientation of mis taken to be along the\nzaxis and we assume small-angle excitations, which do\nnot modulate the magnitude of the spin density, s=jsj.\nshere is the spin density of the conducting electrons,\nwhich in, e.g., the s\u0000dmodel has to be distinguished\nfrom the total spin density Sthat enters Eq. (3).\nWe next use the Mermin ansatz to relate the spin-\ncurrent density jito the spin density s:\nji=\u001b?\u0001xc@i(m\u0000s=s); (20)\nwhere\u001b?is the transverse spin conductivity, to be evalu-\nated later by the Kubo formula. Eq. (20) is analogous to\nOhm's law for electric current density, with the expres-\nsion on the right-hand side reminiscent of the gradient\nof the electrochemical potential. The physical reasoning\nbehind ansatz (20) is simple: there should be no dis-\nsipative spin currents in the static con\fguration, which\ncorresponds to s(r) =sm(r). The advantage in writing\nthe spin current in this form is that \u001b?will now have\nto be evaluated in the limit of ( q;!)!0. Combining\nEqs. (19) and (20) will then give us the spin current to\nthe linear order in qand!: exactly what we need to re-\nlate it to Eq. (17) and read out \u0011. In fact, it is su\u000ecient\nto \fnd \u0001 xc(m\u0000s=s)\u0019\u0000z\u0002@tmfrom Eq. (19), which\nis valid to the linear order in !and zeroth order in q,\nbefore putting it into Eq. (20) to \fnally \fnd\nji=\u0000\u001b?@i(z\u0002@tm): (21)\nComparing this with Eq. (17), we immediately identify \u0011\nwith the transverse spin conductivity:\n\u0011=\u001b?: (22)\nEquation (22) can be interpreted as an analog of the Ein-\nstein relation for transverse spin di\u000busion in strong fer-\nromagnets.\nC. Transverse spin conductivity\nAs is the case with the charge conductivity, it is con-\nvenient to evaluate the transverse spin conductivity in\nthe velocity gauge. Namely, we eliminate the spin \\po-\ntential,\" corresponding to small magnetization deviations\n\u000em=m\u0000zin Eq. (18), by the SU(2) gauge transfor-\nmation\n^ (r;t)!ei\u0001xcRt\n\u00001dt0\u000em(r;t0)\u0001^\u001b=2^ 0(r;t); (23)\nat the expense of introducing the SU(2) vector potential\n^Ai=\u0000\u0001xcZt\n\u00001dt0@im(r;t0)\u0001^\u001b=2; (24)which enters the kinetic part of the single-particle Hamil-\ntonian as\n^Hk=X\ni(pi\u0000^Ai)2=2m\u0003; (25)\nwherepi=\u0000i@iandm\u0003is the electron's e\u000bective mass\n(assuming exchange-split parabolic bands). It is easy to\nverify that the e\u000bective \feld driving the spin current in\nvelocity gauge (25), ^Ei=\u0000@t^Ai, is the same as the \fc-\ntitious \feld ^Ei=\u0000@i^Vin original length gauge (18).\nOne caveat is in order: Eqs. (23)-(25) are only valid for\nan Abelian exchange potential, which would be the case\nif only one vector component of \u000em(r;t) was modulated\n(e.g.,\u000emxor\u000emy) in space and time. Such scenario is\nsu\u000ecient for our purpose, in order to establish the trans-\nverse spin conductivity entering Eq. (20).\nFourier transforming the electric \feld ^Eiin time,R\ndtei!t, the usual relationship is obtained: ^Ei(!) =\ni!^Ai(!). We now proceed to construct the semiclassi-\ncal transport equation for the spin current driven by a\nspatially homogeneous \fctitious \feld Ei= Tr[ ^Ei^\u001b] =\n\u0001xc@im, to deduce the long-wavelength conductivity de-\n\fned by Ohm's law24\nji=\u001b?Ei: (26)\nThe semiclassical spin-current response, in the presence\nof the exchange splitting \u0001 xc, with disorder and electron-\nelectron scattering is given by17\n@tji+ \u0001 xcz\u0002ji=nEi\n4m\u0003\u0000ji\u00121\n\u001cdis\n?+1\n\u001cee\n?\u0013\n;(27)\nwherenis the total equilibrium (conducting) electron\ndensity. The second term on the right-hand side of\nEq. (27) describes spin-current relaxation, due to dis-\norder and electron-electron scattering. Note that even\nin Galilean-invariant systems, spin-independent Coulomb\ninteraction between electrons causes relaxation of a ho-\nmogeneous spin current, in contrast to the ordinary cur-\nrent. Solving Eq. (27) at low frequencies, we recover\nEq. (26) for the current component along Ei, with3,9\n\u001b?=n\n4m\u0003\u001c?\n1 + (\u001c?\u0001xc)2; (28)\nwhere the total transverse spin scattering rate is de\fned\nby\n1\n\u001c?=1\n\u001cdis\n?+1\n\u001cee\n?: (29)\nIn particular, in the limit of weak spin polarization and\nno electron-electron interactions, \u001c?should reduce to the\nordinary momentum scattering time \u001c, and\u001b?to the\nquarter of the Drude conductivity n\u001c=m\u0003.5\nIII. MICROSCOPIC CALCULATION\nA. Spin-current autocorrelator\nIn order to substantiate the preceding phenomenology,\nwe need to establish the microscopic expressions for the\ninvolved scattering times, \u001cdis\n?and\u001cee\n?. In the velocity\ngauge discussed in the previous section, the transverse\nspin conductivity is given, according to the Kubo for-\nmula, by the spin-current autocorrelation function:9\n\u001b?=\u00001\n4m\u00032Vlim\n!!0=mhhP\nl^\u001bxlpxl;P\nl^\u001bxlpxlii!\n!;(30)\nwhere the summation is over all electrons in volume V\nand\nhh^A;^Bii!=\u0000iZ1\n0dtei(!+i0+)th[^A(t);^B(0)]i(31)\nrepresents the Fourier-transformed retarded (Kubo)\nlinear-response function for the expectation value of the\nobservable ^Aunder the action of a classical \feld that\ncouples linearly to the observable ^B.=min Eq. (30) is\ninserted out of convenience, since the linear in !response\nfunction is guaranteed to be imaginary. (The zeroth-\norder in!correlator includes also the omitted \\diamag-\nnetic piece\" of the spin current in the velocity gauge.)\nAssuming isotropic disorder (and for the moment no\nelectron-electron interactions), the ladder vertex correc-\ntions to the conductivity vanish and we only need to eval-\nuate the bubble diagram de\fned by the (single-particle)\nspin-dependent Green's functions\nGR;A\n\u001b(p;!) =1\n!\u0000p2=2m\u0003+\u0016\u001b\u0006i=2\u001c\u001b; (32)\nwhere\u001b=\";#(=\u0006) is the spin index along the zaxis,\n\u0016\u001b=\u0016+\u001b\u0001xc=2 is the spin- \u001belectron Fermi energy, \u0016is\nthe chemical potential, and \u001c\u001bis the spin-dependent dis-\norder scattering time. In the Born approximation for\ndilute white-noise disorder, the scattering rate is pro-\nportional to the electron density of states, and we can\nwrite\u001c\u001b=\u001c\u0017=\u0017\u001b, where\u001cparametrizes the strength of\nthe scattering potential, \u0017\u001bis the spin-\u001bband density of\nstates, and \u0017= (\u0017\"+\u0017#)=2. A straightforward calcula-\ntion then leads to9\n\u001b?=n\n4m\u00031\n\u001cdis\n?\u00012xc; (33)\nin the strong exchange coupling limit, where\n1\n\u001cdis\n?=4\n3\u0016\"+\u0016#\nn\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)(34)\nidenti\fes the disorder contribution to e\u000bective transverse\nspin scattering rate (29).B. Spin-force autocorrelator\nIn the presence of electron-electron interactions, it\nis convenient to express the spin- current autocorrelator\n(30) in terms of the spin- force autocorrelator. To this\nend, we use the equation of motion for the operators\nde\fning Kubo formula (30) to \fnd\n\u001b?=\u00001\n4m\u00032\u00012xcVlim\n!!0=mhhP\nl^\u001bxlFxl;P\nl^\u001bxlFxlii!\n!;\n(35)\nwhereFxl= _pxl=\u0000i[pxl;^H] is the force operator along\nthexaxis for the lth electron. Evaluated with respect to\na uniform magnetization, m=z, the force operator Fxl\nconsists of two pieces: the disorder force and the electron-\nelectron interaction force. Evaluating correlator (35) in\nthe clean limit to second order in Coulomb interactions,\none \fnds for the transverse spin scattering rate:5,9\n1\n\u001cee\n?= \u0007(p)m\u0003a2\nBr4\ns(kBT)2; (36)\nwhereaBis the Bohr radius, Ttemperature, kBBoltz-\nmann constant, rsthe dimensionless Wigner-Seitz radius,\nand \u0007(p) is a dimensionless function of the spin polariza-\ntionp= (n\"\u0000n\")=n(nsbeing spin-selectron density),\nwhich was discussed in Refs. 5,9. Notice that scattering\nrate (36) has the Landau quasiparticle scaling with tem-\nperature. The \fnite-frequency modi\fcation of scatter-\ning rate (36) is, furthermore, accomplished by replacing\n(2\u0019kBT)2!(2\u0019kBT)2+!2.\nC. Spin-density autocorrelator\nIt is also possible to calculate the transverse spin dif-\nfusion directly, as a linear spin-density response to the\ntransverse magnetic \feld. We will carry that out in\nSec. IV for two popular mean-\feld models of ferromag-\nnetism in metals: the Stoner and the s\u0000dmodels. In ad-\ndition to o\u000bering an alternative approach to the problem,\nthis derivation provides a justi\fcation for the preceding\nheuristic utilization of the Mermin ansatz.\nStarting with the mean-\feld Hamiltonian for itinerant\nelectrons\n^H=p2\n2m\u0003+U(r)\u0000\u0016\u0000\u0001xc^\u001bz=2; (37)\nand directly solving for the self-consistent spin-density\nresponse to a small driving magnetic \feld, we will derive\nin the next section the following general relation:\n\u0011=\u00012\nxc\nq2lim\n!!0=m~\u001f+\u0000(q;!)\n!; (38)\nvalid at long wavelengths, q!0. The axially-symmetric\n(Kubo) spin-response function is de\fned by\n~\u001f+\u0000(q;!) =\u00001\n2hhs+(r;t);s\u0000(r0;0)iiq;!; (39)6\nwheres\u0006=sx\u0006isyis the transverse spin density of\nitinerant electrons. The disorder potential U(r) entering\nEq. (37) is, as before, taken to obey the Gaussian white-\nnoise correlations:\nhU(r)U(r0)i=1\n2\u0019\u0017\u001c\u000e(r\u0000r0); (40)where\u0017= (\u0017\"+\u0017#)=2 is the spin-averaged density of\nstates at the Fermi level and \u001cis the characteristic scat-\ntering time.\nWriting the spin density s(r) = Tr [ ^\u001b^\u001a(r)]=2 in terms of the electron density matrix \u001a\u000b\f(r) = \ty\n\f(r)\t\u000b(r) in spin\nspace, we proceed to evaluate ~ \u001f+\u0000in the standard imaginary-time formalism. At temperature T, we have:\n~\u001f+\u0000(q;i\nn) =\u0000T\n2VX\npp0;mG#(p+q;p0+q;i!m+i\nn)G\"(p0;p;i!m); (41)\nwhere\nG\u001b(p;p0;i!m) =\u00001\nVZ\nd3rd3r0Z1=T\n0d\u001ce\u0000ip\u0001r+ip0\u0001r0+i!m\u001c\n\t\u001b(r;\u001c)\ty\n\u001b(r0;0)\u000b\n(42)\nis the \fnite-temperature single-particle Matsubara Green's function. \n n= 2n\u0019T is the bosonic and !m= (2m+1)\u0019T\nfermionic Matsubara frequencies, where nandmare integer indices.\nThe disorder-averaged Green's function is given by\nhG\u001b(p;p0;i!m)i=\u000epp0\ni!m\u0000\"p\u001b+isign(!m)=2\u001c\u001b; (43)\nwhere\"p\u001b=p2=2m\u0003\u0000\u0016\u0000\u001b\u0001xc=2. The analytic continuation of the Matsubara Green's functions into the retarded\n(advanced) Green's functions is accomplished by replacing i!m!!\u0006i0+and sign(!m)!\u0006 . According to our\nconvention (40), \u001c\u001b=\u001c\u0017=\u0017\u001b. Taking into account the vertex ladder corrections (as shown in Fig. 1), we obtain for\nthe disorder-averaged response function:\n~\u001f+\u0000(q;i\nn) =\u0000T\n2VX\nmP\npG#(p+q;i!m+i\nn)G\"(p;i!m)\n1\u0000(\u0018=V)P\npG#(p+q;i!m+i\nn)G\"(p;i!m); (44)\nwhere\u0018= 1=2\u0019\u0017\u001c and by the Green's functions with a single wave-vector argument here we understand disorder-\naveraged propagators (43). Inserting Eq. (43) into Eq. (44) and performing an analytic continuation onto the real\nfrequencies, it is straightforward to calculate ~ \u001f+\u0000(q;!). Setting the temperature to zero and taking the !!0 limit,\nwe \fnd:\n=m~\u001f+\u0000(q;!) =!\n4\u0019<e~\u001fRA\u0000~\u001fAA\n(1\u0000\u0018~\u001fRA)(1\u0000\u0018~\u001fAA); (45)\nwhere ~\u001fXY(q) =R\ndpGX\n#(p)GY\n\"(p\u0000q) andR\ndp\u0011R\nd3p=(2\u0019)3in three dimensions. All energies entering these\nGreen's functions are set at the Fermi level. To the lowest order in 1 =\u001c\u0001, we now obtain:\n=m~\u001f+\u0000(q;!) =!\n8\u0019\u0014Z\ndpA#(p)A\"(p\u0000q) + 4\u0018=mZ\ndpGA\n\"(p\u0000q)A#(p)Z\ndpGA\n\"(p\u0000q)<eGR\n#(p)\u0015\n; (46)\nwhereA\u001b=\u00002=mGR\n\u001bis the spectral function.\nThe second term in Eq. (46) is the vertex ladder correc-\ntion, which is necessary for Eq. (46) to give a meaningful\nresult. In particular, the vertex correction cancels the\nspuriousq= 0 contribution of the \frst term, which would\ngive\u000b\u00181=\u001c\u0001xc. Finally, in the limit of q\u001c\u0019\u0001xc=vF,we arrive at:\n=m~\u001f+\u0000(q;!) =\u0016\"+\u0016#\n3m\u0003\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)\u00014xc!q2: (47)\nUsing Eq. (38), this \fnally gives:\n\u0011=\u0016\"+\u0016#\n3m\u0003\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)\u00012xc; (48)7\nwhich agrees with Eqs. (22), (28), and (34) in the relevant\nhere limit of \u001c\u00001\n?\u001c\u0001xc.\nIV. MEAN-FIELD FERROMAGNETISM\nA. Time-dependent LDA\nIn a spin-density-functional theory (s-DFT),5,18the\nmany-body problem of itinerant ferromagnetism is re-\nduced to the single-electron Hamiltonian\n^H(t) =p2\n2m\u0003+U(r)\u0000\u0016\n\u0000[\u0001xcm(r;t) +!0z+h(r;t)]\u0001^\u001b=2:(49)\n!0\u001c\u0001xcis the ferromagnetic Larmor precession fre-\nquency in the presence of a uniform magnetic \feld ap-\nplied along the zaxis. \u0001 xcm(r;t) is the self-consistent\nexchange \feld, such that Hamiltonian (49) produces the\ncorrect spin-density response. Since we are ultimately in-\nterested in the equation of motion for the collective ferro-\nmagnetic dynamics, the spin-density response is all that\nis needed. In the local-density approximation (LDA) of\nthe s-DFT, the exchange \feld follows the local and in-\nstantaneous magnetization direction m(r;t).h(r;t) is\nthe external rf driving \feld, which we will treat pertur-\nbatively.\nThe time-dependent portion of the Hamiltonian is thus\ngiven by\n^H0(t) =\u0000Z\nd3r[\u0001xc\u000em(r;t) +h(r;t)]\u0001^\u001b=2:(50)\nSince\u000em=\u000es=S(where\u000edenotes small deviations from\nequilibrium), we have for the transverse spin component\ns+=sx+isy:\ns+(q;!) = ~\u001f+\u0000(q;!)\u0014\nh+(q;!) +\u0001xc\nSs+(q;!)\u0015\n:(51)\nThe self-consistent response function to the rf \feld,\n\u001f+\u0000=s+=h+, is thus\n\u001f\u00001\n+\u0000(q;!) = ~\u001f\u00001\n+\u0000(q;!)\u0000\u0001xc\nS: (52)\nIn the LDA approximation, the problem thus trivially\nreduces to calculating the spin-spin response function for\na noninteracting Hamiltonian with a \fxed exchange \feld.\nLet us in general write\n~\u001f+\u0000(q;!) =S\n[!r(q;!) + \u0001 xc\u0000!]\u0000i\u000b(q;!)!;(53)\nin terms of functions !rand\u000bthat are to be determined.\nThe self-consistent response function then becomes:\n\u001f+\u0000(q;!) =S\n[!r(q;!)\u0000!]\u0000i\u000b(q;!)!: (54)Atq= 0, obviously !r(!)\u0011!0and\u000b(!)\u00110. This fol-\nlows in general from the spin conservation in the presence\nof Coulomb interactions and arbitrary spin-independent\npotentialU(r). In this paper, we are most interested in\ntheq-dependent damping function \u000b(q;!), which can be\nidenti\fed by a microscopic evaluation of ~ \u001f+\u0000(q;!). In\nthe limit of strong exchange correlations, \u0001 xc\u001d!r, we\nimmediately obtain from Eq. (53):\n\u000b(q;!!0)\u0019\u00012\nxc\nSlim\n!!0=m~\u001f+\u0000(q;!)\n!: (55)\nIn inversion-symmetric systems, the leading in qspin-\nwave contribution to Gilbert damping is \u000b(q;!!0) =\n(\u0011=S)q2, so that self-consistent response function (54)\ncorresponds to the dissipative term\n@tmjdiss= (\u0011=S)m\u0002r2@tm (56)\nin Landau-Lifshitz Eq. (6) of motion for the magnetic\nspin direction m(r;t). This is the desirable result and,\naccording to Eq. (55), the microscopic expression for \u0011\ngives Eq. (38) of the previous section. In the next section,\nwe will demonstrate that Eq. (55) is generic to mean-\feld\ntreatment of conducting ferromagnets.\nB.s\u0000dmodel in RPA\nIt is also instructive to pursue a more basic descrip-\ntion starting with a ferromagnetic lattice of localized d\nelectrons exchange-coupled to itinerant selectrons. The\ncorresponding Hamiltonian is\n^H(t) =^H0\u0000X\ni[JSi\u0001s(ri;t) +Si\u0001h(ri;t)];(57)\nwhere Siare localdspins and ^H0consists of the de-\ncoupled Hamiltonian for itinerant electrons, dc Zeeman\nHamiltonian of the delectrons, as well as the d\u0000dex-\nchange and possible dipolar interactions. his the applied\nrf \feld, which we take for simplicity to couple to the lo-\ncalized spin only. As long as the average exchange \feld\nexperienced by the selectrons is su\u000eciently strong and\nthe magnetization is dominated by the delectrons, we\ncan disregard their direct rf coupling for our purpose.\nIf also the Fermi wavelength is long in comparison to\nthedlattice spacing, we will treat the electronic band\nstructure in the e\u000bective-mass approximation, and also\ncoarse grain the local spins:P\niSi!R\nd3rS(r) and\nSi!(V=N)S(r), whereN=V is the density of dsites.\nLet us compute the spin-density response function for\nthedlattice:\n\u001f+\u0000(r;r0;t) =\u00001\n2hhS+(r;t);S\u0000(r0;0)ii: (58)\nFor this purpose, it is convenient to de\fne bosonic\nmagnon operators:\nap=1p\n2DNX\nie\u0000ip\u0001riSi+; (59)8\n8\nF:\nq,iΩn=\nq,iΩn+\np+q p/prime+qp/primep\n↓i(ωm+Ωn)↑iωm\n+\n +···\nFIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron\nspin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green’s functions and dashed lines\ndescribe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron\nspin propagators.\nEq. (61) with ˆH0for the selectrons, resulting in the\nlongitudinal mean-field exchange field ∆ xc=JS, where\nS=DN/V is the averaged d-electron spin density. The\nspin-flop term of Eq. (61) can be expanded in terms of\nthe electronic field operators Ψ σas:\nˆH/prime\nxc=−J\nV/radicalbigg\nDN\n2/summationdisplay\npp/prime/bracketleftBig\na†\npΨ†\n↑(p/prime)Ψ↓(p/prime+p)\n+apΨ†\n↓(p/prime)Ψ↑(p/prime−p)/bracketrightBig\n. (62)\nIn the imaginary-time (Matsubara) formalism, the re-\nsponse function (58) can be written as:\nχ+−(q,τ)=−SF(q,τ), (63)\nin terms of the magnon Green’s function\nF(q,τ)=−T/angbracketleftbig\naq(τ)a†\nq(0)/angbracketrightbig\n, (64)\nusing the definition (59). Here, Tis the time-ordering\nsymbol. For the noninteracting Hamiltonian (60),\nF0(q,iΩn)=1\niΩn−εq, (65)\nwhere F(q,iΩn)=/integraltext1/T\n0dτeiΩnτF(q,τ) and Ω n=2nπT,\nas previously, is the bosonic Matsubara frequency. In or-\nder to calculate Fin the presence of the s−dexchange\n(62), we will sum up the bubble diagrams (which con-\nstitute an RPA approximation) shown in Fig. 1. It is\neasy to recognize that each bubble (which is the magnon\nself-energy in this approximation) is just the itinerant\nelectron spin-density response function that was already\ncalculated in Sec. III C.\nSumming up the diagrams in Fig. 1, we thus obtain\nF−1(q,iΩn)=F−1\n0(q,iΩn)−Σ(q,iΩn), (66)\nwhere the self-energy (the s-electron spin-response func-\ntion)\nΣ(q,iΩn)=−J2S˜χ+−(q,iΩn) (67)\nfollows from Eq. (44). Combining Eqs. (63), (66), and\n(67), we finally obtain (after analytic continuation onto\nreal frequencies):\nχ−1\n+−(q,ω)=εq−ω\nS−J2˜χ+−(q,ω). (68)Eq. (68) is actually quite trivial: It can be also obtained\nby treating the d-orbital magnetization dynamics and\nthe associated s−dtorque in a mean-field approxima-\ntion analogous to the preceding discussion of the Stoner\nmodel. Inverting Eq. (68), we can write it in the form\n(54), after identifying\nωr(q,ω)=εq−SJ2/Rfracture˜χ+−(q,ω) (69)\nand\nα(q,ω)=SJ2/Ifracturm˜χ+−(q,ω)\nω, (70)\nwhich is, in fact, exactly the same as Eq. (55) in the\nlow-frequency limit, using ∆ xc=JS. We should em-\nphasize that although χ+−is defined in this section for\na different physical system than χ+−in Sec. IV A (and\nthus, not surprisingly, is found to be somewhat different),\nthe itinerant-electron response ˜ χ+−is the same through-\nout the paper. The reason why the q2magnetic damping\nα(q,ω) is identified in terms of the same quantity ˜ χ+−in\nthe two different models of ferromagnetism can be traced\nto our phenomenological identification of this damping in\nterms of the conducting-electron transverse conductivity,\nEq. (22). The latter is governed by the mean-field struc-\nture of the exchange field, irrespective of the microscopic\norigin of the ferromagnetic order.\nLet us also note in the passing that, unlike the idealized\nStoner model considered in the previous section, the s−d\nmagnetic damping may have a finite q= 0 value even in\nthe absence of any additional spin-dependent terms in the\nHamiltonian. When the gyromagnetic ratios of the two\nelectron species differ (ultimately stemming from some\nform of spin-orbit interaction), the total spin does no\nlonger precess undamped in the uniform field, and the\nuniform transverse spin component can decohere in the\npresence of ordinary scalar disorder. Since Eq. (70) cor-\nresponds to the magnetic field coupled to the delectrons\nonly, we implicitly set the selectron gfactor to zero.\nV. SPIN-PUMPING INTERPRETATION\nIt is illuminating to interpret the key result of this\npaper for the transverse spin diffusion of the form (17)\nFIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron\nspin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green's functions and dashed lines\ndescribe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron\nspin propagators.\nwhich obey the canonical commutation relations,\n[ap;ay\np0] =\u000epp0, close to the fully-magnetized ground\nstate. To be speci\fc, let us take the Heisenberg model for\nexchange coupling, so that in the ground state, Siz=D,\nthed-orbital spin [assuming the applied dc magnetic \feld\nto point along the \u0000zdirection, as in Eq. (49)]. The d-\norbital Hamiltonian for magnon excitations close to the\nground state can thus be written as:\n^H0=X\np\"pay\npap: (60)\nIn terms of the magnon operators, we, furthermore,\nrewrite the s\u0000dexchange interaction as\n^Hxc=\u0000J\nVr\nDN\n2X\np\u0002\nay\nps+(p) +aps\u0000(\u0000p)\u0003\n\u0000JX\niSizsz(ri); (61)\nwheres\u0006(p) =R\nd3re\u0000ip\u0001rs\u0006(r) is the Fourier-\ntransformed transverse s-electron spin density. Approx-\nimatingSiz\u0019D, we can combine the second term in\nEq. (61) with ^H0for theselectrons, resulting in the\nlongitudinal mean-\feld exchange \feld \u0001 xc=JS, where\nS=DN=V is the averaged d-electron spin density. The\nspin-\rop term of Eq. (61) can be expanded in terms of\nthe electronic \feld operators \t \u001bas:\n^H0\nxc=\u0000J\nVr\nDN\n2X\npp0h\nay\np\ty\n\"(p0)\t#(p0+p)\n+ap\ty\n#(p0)\t\"(p0\u0000p)i\n: (62)\nIn the imaginary-time (Matsubara) formalism, re-\nsponse function (58) can be written as:\n\u001f+\u0000(q;\u001c) =\u0000SF(q;\u001c); (63)\nin terms of the magnon Green's function\nF(q;\u001c) =\u0000T\naq(\u001c)ay\nq(0)\u000b\n; (64)\nusing de\fnition (59). Here, Tis the time-ordering sym-\nbol. For noninteracting Hamiltonian (60),\nF0(q;i\nn) =1\ni\nn\u0000\"q; (65)whereF(q;i\nn) =R1=T\n0d\u001cei\nn\u001cF(q;\u001c) and \nn= 2n\u0019T,\nas previously, is the bosonic Matsubara frequency. In or-\nder to calculateFin the presence of s\u0000dexchange (62),\nwe will sum up the bubble diagrams (which constitute an\nRPA approximation) shown in Fig. 1. It is easy to recog-\nnize that each bubble (which is the magnon self-energy\nin this approximation) is just the itinerant electron spin-\ndensity response function that was already calculated in\nSec. III C.\nSumming up the diagrams in Fig. 1, we thus obtain\nF\u00001(q;i\nn) =F\u00001\n0(q;i\nn)\u0000\u0006(q;i\nn); (66)\nwhere the self-energy (the s-electron spin-response func-\ntion)\n\u0006(q;i\nn) =\u0000J2S~\u001f+\u0000(q;i\nn) (67)\nfollows from Eq. (44). Combining Eqs. (63), (66), and\n(67), we \fnally obtain (after analytic continuation onto\nreal frequencies):\n\u001f\u00001\n+\u0000(q;!) =\"q\u0000!\nS\u0000J2~\u001f+\u0000(q;!): (68)\nEquation (68) is actually quite trivial: it can be also ob-\ntained by treating the d-orbital magnetization dynamics\nand the associated s\u0000dtorque in a mean-\feld approxima-\ntion analogous to the preceding discussion of the Stoner\nmodel. Inverting Eq. (68), we can write it in form (54),\nafter identifying\n!r(q;!) =\"q\u0000SJ2<e~\u001f+\u0000(q;!) (69)\nand\n\u000b(q;!) =SJ2=m~\u001f+\u0000(q;!)\n!; (70)\nwhich is, in fact, exactly the same as Eq. (55) in the\nlow-frequency limit, using \u0001 xc=JS. We should em-\nphasize that, although \u001f+\u0000is de\fned in this section for\na di\u000berent physical system than \u001f+\u0000in Sec. IV A (and\nthus, not surprisingly, is found to be somewhat di\u000berent),\nthe itinerant-electron response ~ \u001f+\u0000is the same through-\nout the paper. The reason why the q2magnetic damping\n\u000b(q;!) is identi\fed in terms of the same quantity ~ \u001f+\u0000in9\nthe two di\u000berent models of ferromagnetism can be traced\nto our phenomenological identi\fcation of this damping in\nterms of the conducting-electron transverse conductivity,\nEq. (22). The latter is governed by the mean-\feld struc-\nture of the exchange \feld, irrespective of the microscopic\norigin of the ferromagnetic order.\nLet us also note in the passing that, unlike the idealized\nStoner model considered in the previous section, the s\u0000d\nmagnetic damping may have a \fnite q= 0 value even in\nthe absence of any additional spin-dependent terms in the\nHamiltonian. When the gyromagnetic ratios of the two\nelectron species di\u000ber (ultimately stemming from some\nform of spin-orbit interaction), the total spin no longer\nprecesses undamped in the uniform \feld, and the uniform\ntransverse spin component can decohere in the presence\nof ordinary scalar disorder. Since Eq. (70) corresponds\nto the magnetic \feld coupled to the delectrons only, we\nimplicitly set the selectrongfactor to zero.\nV. SPIN-PUMPING INTERPRETATION\nIt is illuminating to interpret the key result of this\npaper for the transverse spin di\u000busion of form (17) in\nterms of the spin pumping associated with a nonuni-\nform magnetic dynamics in ferromagnetic bulk.8The\nferromagnetic spin pumping was originally proposed in\nthe context of magnetic multilayers with sharp normal-\nmetaljferromagnetic interfaces. This paper shows that\nanalogous processes also take place in the continuous fer-\nromagnetic medium.\nTo illustrate the direct connection between the trans-\nverse spin di\u000busion and the spin pumping, we consider\na periodic stack of alternating F and N layers forming a\ntwo-component superlattice in the xdirection.8We treat\nthe model depicted in Fig. 2, in which an F jN bilayer\nforms the unit cell with thickness b=L+d, where the\nnormal-metal spacer of width Lseparates the magnetic\n\flms of thickness\nd\u001d\u0015sc=vF=\u0019\u0001xc: (71)\nThe latter approximation allows us to neglect the trans-\nverse spin-current coherence between two interfaces of\nthe same magnetic layer.8Translational invariance is as-\nsumed for simplicity in the lateral directions. We con-\nsider here collective spin-wave excitations, taking both\nthe static and dynamic exchange couplings into account.7\nThe static (RKKY-like) exchange interaction between\nneighboring ferromagnetic layers is mediated by the dis-\nsipationless spin currents \rowing through the normal-\nmetal spacer.19We will parametrize the strength of this\ncoupling by the corresponding precession frequency !xc\nof a single ferromagnetic \flm that is exchange-coupled to\na pinned \flm. In the presence of the magnetic dynam-\nics, additional dissipative spin currents set in. Their ori-\ngin lies in the spin pumping by the individual magnetic\nlayers into the adjacent normal spacers, which at low\nfrequencies is given by8Ipump\ns = (\u0016h=4\u0019)~g\"#\nNjFm\u0002@tm.\nLb=L+d\nd\nxFIG. 2: A schematic view of the superlattice considered in the\ntext: an FjN bilayer is repeated along the xaxis, with either\nferromagnetic or antiferromagnetic alignment of the consecu-\ntive magnetic layers. The system is translationally invariant\nalong the two remaining axes.\n~g\"#\nNjFis the dimensionless spin-mixing conductance per\nunit area of the F jN interface (which is assumed to be\nreal-valued, for simplicity). This interfacial spin pump-\ning induces nonlocal spin transfer in magnetoelectronic\ncircuits, which can in general be treated as a source term\nentering spin transport equations in normal and mag-\nnetic layers. In a collinear superlattice of Fig. 2, the\nproblem simpli\fes considerably, because the spin-current\nvector Ipump\ns/m\u0002@tmis transverse with respect to\nthe magnetic alignment (in both ferromagnetic and anti-\nferromagnetic cases), within the linear-response regime.\nThis means that the spin current pumped by one fer-\nromagnetic layer is either scattered back by the normal\nspacer and reabsorbed, or transmitted and absorbed by a\nneighboring layer, with no possibility to reach more dis-\ntant neighbors, subject to condition (71). Spin relaxation\nin normal spacers would only cause an overall increase\nin the e\u000bective Gilbert damping parameter of a uniform\nmagnetic precession, and will thus be omitted, since our\nprimary interest here is nonlocal damping e\u000bects. The\nproblem of dynamic exchange between two adjacent fer-\nromagnetic layers thus e\u000bectively reduces to the analo-\ngous e\u000bect in magnetic bilayers, which was studied in de-\ntail in Ref. 7. In particular, the net spin pumping through\na given normal spacer is /m1\u0002@tm1\u0000m2\u0002@tm2, which\nre\rects the dynamic spin injection in the opposite direc-\ntions by the adjacent magnetic layers m1andm2. Notice\nthat the total pumping vanishes in the case of a perfectly\nsynchronous precession, m1(t) =m2(t).\nLet us now put the static and dynamic exchange inter-\nactions into the equation of motion for small-angle spin\ndynamics of a multilayer with respect to an all-parallel\ncon\fguration, ui(t) =mi(t)\u0000z. For long-wavelength ex-\ncitations, it may be approximated as a continuous func-\ntionu(x;t) of the coordinate xnormal to the interfaces.\nFor the uniaxial e\u000bective \feld H\u0003=\u0000!0z, the spin-wave\ndynamics obey the di\u000berential equation\n@tu=\u0002\n!0u\u0000!xcb2@2\nxu+\u000b@tu\u0000\u000b0b2@2\nx@tu\u0003\n\u0002z;(72)\nwhere we made the following de\fnitions: !xc=Jxc=Sd\nand\u000b0=<eA\"#\nFjNjF=4\u0019Sd. Here,Jxcis the exchange10\ncoupling between two consecutive magnetic layers and\n1=A\"#\nFjNjF= 2=~g\"#\nNjF+e2L\u001a=\u0019 (73)\nis the e\u000bective pumping resistance of the spacer. The \frst\nterm on the right-hand side of Eq. (73) parametrizes the\npumping strength of the individual interfaces, as was dis-\ncussed above, and the second term is the ordinary ohmic\nresistance of the normal spacer (neglecting any spin relax-\nation), which backscatters the pumped currents and thus\nsuppresses the dynamic exchange. The second spatial\nderivatives in Eq. (72) re\rect simply the di\u000berence of the\nstatic and dynamic exchange spin currents through two\nconsecutive normal spacers (which themselves require a\n\fnite misalignment of the adjacent magnetic layers) in\nthe continuum limit. The static Heisenberg coupling can\nbe interpreted as the superlattice equivalent of the bulk\nexchange-sti\u000bness parameter Aof Eq. (11), which for the\nsuperlattice becomes A=Jxcb. Both!xcand\u000b0are sen-\nsitive to the normal-interlayer thickness L, vanishing in\nthe limitL! 1 . It follows from Eq. (72) that the\nsmall-momentum, q\u001cb\u00001, spin-wave excitations of the\nsuperlattice, propagating perpendicular to the interfaces,\nu/exp[i(qx\u0000!t)], obey the dispersion relation\n!(q) =!0+ (bq)2!xc\n1 +i[\u000b+ (bq)2\u000b0]: (74)\nWhenq!0,!(q) reduces to the Larmor frequency !0\nof the individual magnetic layers because the static and\ndynamic exchange couplings vanish when the consecutive\nmagnetic layers move coherently in phase. Equation (74)\nholds up to momenta comparable to b\u00001, whenbqhas to\nbe replaced by 2 sin( bq=2).\nThe matters are quite di\u000berent for an\nantiferromagnetically-aligned superlattice, which is\nthe ground state when, for example, Jxc<0 and\nH\u0003= 0. In this case, we have a more complex\ndispersion:\n!(q) =\u0000!xc\u0006p\n(bq)2(1 +\u000b2)\u00004\u000b2+i[2\u000b+ (bq)2\u000b0]\n1 +\u000b2+ 4\u000b\u000b0+ (bq)2\u000b02;\n(75)\nwhere plus and minus signs refer, respectively, to the\nmodes with antisymmetric and symmetric dynamics in\nthe adjacent layers for overdamped motion, and to the\nright- and left-propagating modes when the real part\nof!(q) is signi\fcant. Note that now !xc<0, so that\n=m! > 0, as required for a stable con\fguration. In the\nabsence of bulk magnetization damping, \u000b= 0, Eq. (75)\nreduces to\n!(q) =\u0006(bq)!xc\n1\u0006i(bq)\u000b0; (76)\nwith linear dispersion and damping at small q. Equa-\ntions (75) and (76) can also be generalized to large mo-\nmenta by replacing bqwith 2 sin(bq=2). Notice that in\nEqs. (72), (74), and (76), the dynamic coupling modi-\n\fes the damping similarly to the way the static couplinga\u000bects the excitation frequency of the magnetic superlat-\ntice. Crystal and shape anisotropies on top of the simple\ne\u000bective \felds assumed above might become important\nin real structures, and their inclusion is straightforward.\nLet us now compare the damping ( bq)2\u000b0in Eq. (74)\nwith\u000b(q) = (\u001b?=S)q2corresponding to Eq. (33), which\nis the analogous quantity for the bulk. Keeping only\nthe mixing conductance contribution to Eq. (73) and\napproximating8~g\"#\u0019p2\nF=2\u0019in terms of the character-\nistic Fermi momentum pFin the normal metal, we have\nfor theq-dependent part of the damping:\n\u000b(q) = (bq)2\u000b0\u0018(b=\u0015F)2\nSdq2; (77)\nup to a numerical constant. At the same time, the bulk\n\u000b(q), corresponding to Eq. (33), can be written as\n\u000b(q)\u0018(\u0015sc=\u0015F)2\nSlq2; (78)\nwhich establishes a loose formal correspondence between\nthe two results. Here, l=vF\u001cis the mean free path, \u0015F\nthe Fermi wavelength, and the ferromagnetic coherence\nlength\u0015scwas de\fned in Eq. (71).\nComparing Eqs. (77) and (78), we interpret the length\nscaleb$\u0015scto describe the longest distance over which\nferromagnetic regions can communicate via spin trans-\nfer. The length scale d$\u0015characterizes momentum\nscattering relevant for spin transfer, which in the case\nof the superlattice with sharp interfaces corresponds to\nthe magnetic \flm width d: Approximating ~ g\"#\u0019p2\nF=2\u0019\nabove, we e\u000bectively took the normal spacers to be bal-\nlistic and, because of Eq. (71), the spin transfer does\nnot penetrate deep into the ferromagnetic layers, mak-\ning possible disorder scattering there irrelevant for our\nproblem.\nVI. DISCUSSION AND OUTLOOK\nEstimating the numerical value of the dimensionless q2\ndamping, according to Eq. (28),\n\u000b(q) =\u001b?q2\nS\u0018\u0012\u0016F=\u0001xc\npF=q\u00132\u001c?\u0001xc\n1 + (\u001c?\u0001xc)2; (79)\nwe can see that it will most likely be at most compa-\nrable or smaller than the typical q= 0 Gilbert damp-\ning\u000b\u001810\u00002, in metallic ferromagnets. Damping (79)\nmay, however, become dominant in weak ferromagnets,\nsuch as diluted magnetic semiconductors. We are not\naware of systematic experimental investigations of the\nq2damping in metallic ferromagnets. q2scaling of rel-\native linewidth was reported in Ref. 20 for the iron-rich\namorphous Fe 90\u0000xNixZr10alloys. However, we are not\ncertain whether the strong damping observed there can\nbe attributed to the mechanism discussed in our paper.\nAnother intriguing context where the physics discussed\nhere can play out to be important is the current-driven11\nnonlinear ferromagnetic dynamics in mesoscopic as well\nas bulk magnetic systems. The q2magnetic damping\ndescribed by Eq. (15) can be physically thought of the\nviscous-like spin transfer between magnetic regions pre-\ncessing slightly out-of-phase. The obvious consequence\nof this is the enhanced damping of the inhomogeneous\ndynamics and thus the synchronization of collective mag-\nnetic precession. This phenomenon was predicted in\nRef. 7 and unambiguously observed in Ref. 21, in the case\nof the coupled dynamics of a magnetic bilayer: When the\ntwo layers are tuned to similar resonance conditions, only\nthe symmetric mode corresponding to the synchronized\ndynamics produces a strong response, while the antisym-\nmetric mode is strongly suppressed. It is thus natural to\nsuggest that the q2viscous magnetic damping in the con-\ntinuum limit may have far-reaching consequences for the\ncurrent-driven nonlinear power spectrum as that mea-\nsured in Ref. 22. This needs a further investigation.\nThe role of electron-electron interactions was mani-\nfested in our theory through the spin Coulomb drag,\nwhich enhances the e\u000bective transverse spin scattering\nrate (29). This becomes particularly important, in com-\nparison to the disorder contribution to the transverse spin\nscattering, in the limit of weak magnetic polarization.9We \fnally emphasize that the study in this paper was\nlimited exclusively to weak linearized perturbations of\nthe magnetic order with respect to a uniform equilib-\nrium state. When the equilibrium or out-of-equilibrium\nmagnetic state is macroscopically nonuniform, as is the\ncase with, e.g., the magnetic spin spirals, domain walls,\nvortices, and other topological states, the longitudinal as\nwell as transverse spin currents become relevant for the\nmagnetic dynamics. The longitudinal spin currents lead\nto additional contributions to the spin-transfer torques,\nmodifying the magnetic equation of motion. Such spin\ntorques leading to the dissipative q2damping terms were\ndiscussed in Ref. 13. These latter contributions to the\nmagnetic damping are likely to dominate in strongly-\ntextured magnetic systems.\nAcknowledgments\nWe are grateful to Gerrit E. W. Bauer and Arne\nBrataas for stimulating discussions. This work was sup-\nported in part by the Alfred P. Sloan Foundation (YT)\nand NSF Grant No. DMR-0705460 (GV).\n1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898\n(1969).\n2A. J. Leggett, J. Phys. C: Sol. State Phys. 3, 448 (1970).\n3A. Singh, Phys. Rev. B 39, 505 (1989); A. Singh and\nZ. Te\u0014 sanovi\u0013 c, ibid.39, 7284 (1989).\n4V. L. Sobolev, I. Klik, C. R. Chang, and H. L. Huang, J.\nAppl. Phys. 75, 5794 (1994); A. E. Meyerovich and K. A.\nMusaelian, Phys. Rev. Lett. 72, 1710 (1994); D. I. Golosov\nand A. E. Ruckenstein, ibid.74, 1613 (1995); Y. Takahashi,\nK. Shizume, and N. Masuhara, Phys. Rev. B 60, 4856\n(1999).\n5Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002).\n6V. P. Mineev, Phys. Rev. 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Lett.\n87, 206403 (2001).\n19J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n20J. A. Fernandez-Baca, J. W. Lynn, J. J. Rhyne, and G. E.\nFish, J. Appl. Phys. 61, 3406 (1987).\n21B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n22I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley,\nJ. C. Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev.\nB76, 024418 (2007).\n23It is natural to also wonder about a possible additional\nspin current of the form ji/@i@tm, which does not\nbreak time-reversal symmetry. Such spin current leads to a\nwave-vector-dependent correction to the e\u000bective gyromag-\nnetic ratio, which is very small in practice. It parallels the\nstructure of the spin pumping in magnetic nanostructures,\nwhich consists of the dominant dissipative piece of the form\nm\u0002@tmand a smaller piece of the form @tm. The latter\nmerely causes a slight rescaling of the gyromagnetic ratio.\nWhile the dissipative piece of the spin pumping has been\nunambiguously established in a number of experiments,8\nthe small correction to the gyromagnetic ratio is yet to be\nobserved.\n24We need to remark here that the above gauge transfor-\nmation does not a\u000bect the transverse spin current in the\nlinearized theory."    },    {        "title": "0810.2870v1.Interaction_of_reed_and_acoustic_resonator_in_clarinetlike_systems.pdf",        "content": "arXiv:0810.2870v1  [physics.class-ph]  16 Oct 2008\n/C1/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/CU /D6/CT/CT/CS /CP/D2/CS /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CX/D2 \r/D0/CP/D6/CX/D2/CT/D8/B9/D0/CX/CZ /CT /D7/DD/D7/D8/CT/D1/D7/BY /CP/CQ /D6/CX\r/CT /CB/CX/D0/DA/CP/B8 /C2/CT/CP/D2 /C3/CT/D6/CV/D3/D1/CP /D6/CS/B8 /CP/D2/CS /BV/CW/D6/CX/D7/D8/D3/D4/CW/CT /CE /CT/D6/CV/CT/DE\n/CP/B5/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/CT /C5/GH \r /CP/D2/CX/D5/D9/CT 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/BW/CX/B9/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9\r/CT/CS /CW/CT/D6/CT/BMθ /B8Ye\n/B8 /CP/D2/CSD/CP/D6/CT /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /CX/D2/D4/D9/D8 /CP/CS/D1/CX/D8/D8/CP/D2\r/CT /CP/D2/CS/D8/CW/CT /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2\r/D8/CX/D3/D2/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA\nθ=ω\nωr,Ye(θ) =Zc\nZe(θ)\n/CP/D2/CSD(θ) =1\n1+jqrθ−θ2. /B4/BH/B5/CC/CW/CT/D6/CT /CP/D6/CT /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 \r/D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM\nγ=Pm\nKy0\n/CP/D2/CSζ=ZcW/radicalbigg2y0\nKρ. /B4/BI/B5\nγ /CX/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /CQ /CT/D8 /DB /CT/CT/D2 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT/D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD \r/D0/D3/D7/CT /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D2 /D7/D8/CP/D8/CX\r/D6/CT/CV/CX/D1/CT/B8 /DB/CW/CX/D0/CTζ /D1/CP/CX/D2/D0/DD /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D1/D3/D9/D8/CW/D4/CX/CT\r/CT \r/D3/D2/D7/D8/D6/D9\r/B9/D8/CX/D3/D2 /CP/D2/CS /D0/CX/D4 /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS /CP/D2/CS /CX/D7 /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT /D1/CP/DC/B9/CX/D1 /D9/D1 /AT/D3 /DB /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /B4ζ /CT/D5/D9/CP/D0/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD\n2β /CX/D2 /CA/CT/CU/BA /BE/B5/BA/C4/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BG/B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D7/D3/B9\r/CP/D0/D0/CT/CS \r /CW/CP/D6/CP\r/B9/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2/BM\nYe(θ) =ζ√γ/braceleftbigg\nD(θ)−1−γ\n2γ/bracerightbigg\n, /B4/BJ/B5/DB/CW/CX\r /CW \r/CP/D2 /CQ /CT /D7/D4/D0/CX/D8 /CX/D2 /D8/D3 /D6/CT/CP/D0 /CP/D2/CS /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D7/BM/C1/D1(Ye(θ)) =ζ√γ /C1/D1(D(θ)), /B4/BK/B5/CA/CT(Ye(θ)) =ζ√γ/parenleftbigg/CA/CT(D(θ))−1−γ\n2γ/parenrightbigg\n. /B4/BL/B5/BT /D8 /D0/CP/D7/D8/B8 /CP 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/CU/D6/D3/D1 /CA/CT/CU/BA /BE/BN /D3/D9/D6 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/B5/BA\n/BY/C1/BZ/BA /BE/BA /BW/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D2/CS /D4/D6/CT/D7/D7/D9/D6/CT/CU/D3/D6 /CP /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BMqr= 0.008 /B8fr= 700 Hz /B8β=\n0.05 /BA /CA/CT/D7/D9/D0/D8/D7 /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5 /CP/D2/CS /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /B4/D7/D5/D9/CP/D6/CT/D7/B5 /CU/D6/D3/D1/CA/CT/CU/BA /BE/BN /D3/D9/D6 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/B5/BA\r/CP/D2 /CQ /CT /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8/BA /C1/D2 /CP /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /D4/CW/CP/D7/CT /D3/CU /D3/D9/D6 /D2 /D9/D1/CT/D6/B9/CX\r/CP/D0 /CP/D0/CV/D3/D6/CX/D8/CW/D1/D7/B8 /D3/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/CT/CT /BY/CX/CV/BA /BD /CP/D2/CS /BE/B5 /DB /CT/D6/CT \r/D3/D1/B9/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT/CX/D6 /D3/D2/CT/D7 /B4/D8/CW/CT /CS/CP/D8/CP /CQ /CT/CX/D2/CV /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT/CX/D6/CP/D6/D8/CX\r/D0/CT/B5/B8 /D7/D3/D0/DA/CX/D2/CV /CT/DC/CP\r/D8/D0/DD /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/BA /BW/CX/AR/CT/D6/CT/D2\r/CT/D7/CP/D4/D4 /CT/CP/D6 /CQ /CT/D8 /DB /CT/CT/D2 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 \r/D3/D2\r/CT/D6/D2/CX/D2/CV /D8/CW/D6/CT/D7/CW/D3/D0/CS/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7/B8 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /DB/CW/CT/D2 /D8/CW/CT /D0/CT/D2/CV/D8/CW/CS/CT\r/D6/CT/CP/D7/CT/D7/BA /C1/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8 /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7 /D8/CW/CT/CP/D9/D8/CW/D3/D6/D7 /CS/D3 /D2/D3/D8 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8/BM /D0/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /AT/D3 /DB/D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CX/D7 /DA /CP/D0/CX/CS /D3/D2/D0/DD /DB/CW/CX/D0/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D7 /D2/D3/D8 \r/D0/D3/D7/CT/CS/CP/D8 /D6/CT/D7/D8/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2Pm< Ky 0\n/BA /CD/D7/CX/D2/CV /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CU/D3/D6/D1/CU/D3/D6 /CW/CX/CV/CW/CT/D6 /DA /CP/D0/D9/CT/D7 /D3/CUPm\n/DB /D3/D9/D0/CS /CQ /CT /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7/B8 /CT/DA /CT/D2 /CU/D3/D6/CU/D6/CT/CT /D6/CT/CT/CS /CP/CT/D6/D3/D4/CW/D3/D2/CT/D7/BM /D8/CW/CT /D3/D4 /CT/D2/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /B4/D0/CX/D2/CZ /CT/CS /D8/D3/D8/CW/CT /D6/CT/CT/CS /CS/CX/D7/D4/D0/CP\r/CT/D1/CT/D2 /D8/B5 /D8/CP/CZ/CX/D2/CV /D4/CP/D6/D8 /CX/D2 /AT/D3 /DB \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2\r/CP/D2 /D2/CT/DA /CT/D6 /CQ /CT /D2/CT/CV/CP/D8/CX/DA /CT/BA /BY /D3/D6 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /CU/D3/D6 /DB/CW/CX\r /CW /D6/CT/CT/CS/CQ /CT/CP/D8/D7 /CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D1/D3/D9/D8/CW/D4/CX/CT\r/CT/B8 /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D7 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD\r/D0/D3/D7/CT/CS /CP/D2/CS /D8/CW/CT/D2 /D7/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 \r/CP/D2/D2/D3/D8 /D3 \r\r/D9/D6 /CU/D3/D6/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /CF/BU /D8/CW/CT/D3/D6/DD /D4/D6/CT/CS/CX\r/D8/D7 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS /CP/CQ /D3 /DA /CT /D7/D8/CP/D8/CX\r /CQ /CT/CP/D8/CX/D2/CV /D6/CT/CT/CS /D4/D6/CT/D7/D7/D9/D6/CT /CQ /DD /CT/DC/D8/CT/D2/CS/CX/D2/CV /D0/CX/D2/B9\n/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /CQ /CT/DD /D3/D2/CS /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7/BA/CC/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /D6/CT/B9/D1/CP/CX/D2/D7 /D0/D3 /DB /CT/D6 /D8/CW/CP/D2 /BD /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /CP /D1/CP/DC/CX/D1 /D9/D1/D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/BA /CF /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8/D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /DA /CP/D0/D9/CTθ/D1/CP/DC\n/CX/D7 /CP/D7/D7/D3 \r/CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1/DA /CP/D0/D9/CT /D3/CUγ /B4/CX/BA/CT/BA/B8/BD/B5/BA /C6/D3 /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D4/D6/D3 /D3/CU /CX/D7 /CV/CX/DA /CT/D2 /CW/CT/D6/CT/B8 /CQ/D9/D8/D8/CW/CX/D7 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /CP/D0/D0 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/B9/D8/CP/CX/D2/CT/CS/BA /CC/CW/CT/D2 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 θ/D1/CP/DC\n/CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nYe(θ/D1/CP/DC) =ζD(θ/D1/CP/DC), /B4/BD/BC/B5/DB/CW/CX\r /CW /D0/CT/CP/CS/D7 /D8/D3 /D6/CT/D0/CP/D8/CX/D3/D2\n1−θ2/D1/CP/DC=ζ /CA/CT(Ze(θ/D1/CP/DC))>0 /B4/BD/BD/B5/CP/D7 /D8/CW/CT /CQ /D3/D6/CT /CX/D7 /CP /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1/BA /BT/D7 /CP \r/D3/D2\r/D0/D9/D7/CX/D3/D2/B8 /CX/D8 /CX/D7 /D2/D3/D8/D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D4/D0/CP /DD /D7/CW/CP/D6/D4 /CT/D6 /D8/CW/CP/D2 /CP /CU/D6/CT/D5/D9/CT/D2\r/DD /D7/D0/CX/CV/CW /D8/D0/DD /AT/CP/D8/B9/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D6/CT/CT/CS/B9/D0/CX/D4/B9/D1/D3/D9/D8/CW/D4/CX/CT\r/CT /D7/DD/D7/D8/CT/D1 /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/B9/D5/D9/CT/D2\r/DD /B8 /D8/CW/CT /CS/CT/DA/CX/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /CQ /CT/CX/D2/CV /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT/D0/D3/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /CQ /D3/D6/CT/BA/BY/BA /C5/CX/D2/CX/D1/D9/D1 /D4 /D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /C1/D1/D4 /D6/D3/DA/CT/CS /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CU/D3 /D6/CX/D2/D8/CT/D6/CP\r/D8/CX/D2/CV /D6/CT/D7/D3/D2/CP/D2\r/CT/D7/BY /D3/D6 /CT/CP\r /CW /DA /CP/D0/D9/CT /D3/CUqr\n/B8 /D8/CW/CT/D6/CT /CT/DC/CX/D7/D8/D7 /D3/D2/CT /D3/D6 /D1/D3/D6/CT /D6/CP/D2/CV/CT/D7 /D3/CU/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CX/D7 /CV/D6/CT/CP/D8/D0/DD /CX/D1/D4/D6/D3 /DA /CT/CS/BA /C1/D2/B9/CS/CT/CT/CS/B8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/D9/D6/DA /CT/D7 /D7/CW/D3 /DB /CP /D1/CX/D2/CX/D1 /D9/D1 /CU/D3/D6 /CP\r/CT/D6/D8/CP/CX/D2 /DA /CP/D0/D9/CT /D3/CUkrL /B8 /CS/CT/D2/D3/D8/CX/D2/CV /CP/D2 /CX/D2\r/D6/CT/CP/D7/CT/CS /CT/CP/D7/CX/D2/CT/D7/D7 /D8/D3/D4/D6/D3 /CS/D9\r/CT /D8/CW/CT /D2/D3/D8/CT \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CX/D7 /D0/CT/D2/CV/D8/CW/BA /BT/D7/D7/D3 \r/CX/B9/CP/D8/CX/D2/CV /CP \r/D0/CP/D6/CX/D2/CT/D8 /D1/D3/D9/D8/CW/D4/CX/CT\r/CT /DB/CX/D8/CW /CP /D8/D6/D3/D1 /CQ /D3/D2/CT /D7/D0/CX/CS/CT/B8 /CX/D2/B9/CU/D3/D6/D1/CP/D0 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 \r/D3/D2/AS/D6/D1 /D8/CW/CP/D8 /CX/D8 /CX/D7 /CT/CP/D7/CX/CT/D6 /D8/D3 /D4/D6/D3 /CS/D9\r/CT/D7/D3/D1/CT /D2/D3/D8/CT/D7 /D8/CW/CP/D2 /D3/D8/CW/CT/D6 /D3/D2/CT/D7/BA /BT/D2/CP/D0/DD/D8/CX\r/CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /CT/DC/B9/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CW/CP /DA /CT /CQ /CT/CT/D2 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA /CD/D2/B9/CS/CT/D6 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CX/D7 /D1/CX/D2/CX/D1/CP/D0 /DA /CP/D0/D9/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/CU/D3/D6 /CP/D2 /CT/D1/CT/D6/CV/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2\r/DD /D0/D3 \r/CP/D8/CT/CS \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/B9/D2/CP/D2\r/CT/B8 /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /CX/D7 /D1/CP/CX/D2/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV/B8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 \r/CP/D2 /CQ /CT /CX/CV/D2/D3/D6/CT/CS /CA/CT(Ye(ω)) = 0 /B8 /BX/D5/BA /B4/BL/B5/D0/CT/CP/CS/CX/D2/CV /D8/CW /D9/D7 /D8/D3/BM\nγ=1\n1+2 /CA/CT(D(θ)). /B4/BD/BE/B5/C1/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS /D1/D3 /CS/CT/D0 /B4D(θ) = 1 /B5/B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/B9/D7/D9/D6/CT /CX/D7 /CT/D5/D9/CP/D0 /D8/D31/3 /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CU/D6/CT/B9/D5/D9/CT/D2\r/CX/CT/D7 /CU/D3/D6 /DB/CW/CX\r /CW /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D2/D4/D9/D8/CX/D1/D4 /CT/CS/CP/D2\r/CT /DA /CP/D2/CX/D7/CW/CT/D7/B8 /DB/CW/CX\r /CW /CX/D7 \r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D6/CT/D7/D9/D0/D8/D7 /CP/D0/B9/D6/CT/CP/CS/DD /D4/D9/CQ/D0/CX/D7/CW/CT/CS\n/BF/BG/BA /CC/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3 \r/B9\r/D9/D6/D7 /CP/D8 /CP /D1/CP/DC/CX/D1 /D9/D1 /D3/CU /CA/CT(D(θ)) /BM/CA/CT(D(θ)) =1−θ2\n(1−θ2)2+(qrθ)2, /B4/BD/BF/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6θ=√1−qr\n/B4/DB/CW/CX\r /CW /CX/D7 \r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 θ≃1 /B5/B8 /D8/CW /D9/D7/B8\nγ0=qr(2−qr)\n2+qr(2−qr), /B4/BD/BG/B5/D2/CT/CP/D6/D0/DD /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3qr\n/CU/D3/D6 /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /CT/DA /CP/D0/D9/CP/D8/CT /D8/CW/CT /CT/AR/CT\r/D8 /D3/CU /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /D3/D2 /D8/CW/CX/D7/D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CP /D3/D2/CT/B9/D1/D3 /CS/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /DB/CX/D8/CW/D0/D3/D7/D7/CT/D7 /CX/D7 /D2/D3 /DB \r/D3/D2/D7/CX/CS/CT/D6/CT/CS/BM\nYe(ω) =Yn/parenleftbigg\n1+jQn/parenleftbiggω\nωn−ωn\nω/parenrightbigg/parenrightbigg\n, /B4/BD/BH/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BG/DB/CW/CT/D6/CTYn\n/CX/D7 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CT /CP/CS/D1/CX/D8/D8/CP/D2\r/CT/CP/D2/CSQn\n/CX/D7 /D8/CW/CT /D5/D9/CP/D0/CX/D8 /DD /CU/CP\r/D8/D3/D6/BN /BX/D5/D7/BA /B4/BK/B5 /CP/D2/CS /B4/BL/B5 /CQ /CT\r/D3/D1/CT\nYn+ζ1−γ\n2√γ=ζ√γ1−θ2\n(1−θ2)2+(qrθ)2, /B4/BD/BI/B5\nYnQn/parenleftbiggθ\nθn−θn\nθ/parenrightbigg\n=−ζ√γqrθ\n(1−θ2)2+(qrθ)2. /B4/BD/BJ/B5/C1/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CUYn\n/DB/CX/D8/CW /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CQ /D3/D6/CT/B8/D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BD/BI /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3krL /D0/CT/CP/CS/D7 /D8/D3/CP /D1/CX/D2/CX/D1 /D9/D1 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CU/D9/D2\r/D8/CX/D3/D2 γ=f(krL) /CU/D3/D6θ2/D1/CX/D2=\n1−qr\n/CP/D2/CSγ/D1/CX/D2\n/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2\nYn\nζ√γ/D1/CX/D2+1−γ/D1/CX/D2\n2γ/D1/CX/D2=1\nqr(2−qr), /B4/BD/BK/B5/DB/CW/CX\r /CW /D8/CW/CT /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nγmin≃γ0/parenleftbigg\n1+2Yn\nζ√γ0/parenrightbigg/B4/BD/BL/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6\nωn=ωr/parenleftbigg\n1−qr\n2+1\n2Qn+ζ\n2YnQn√qr/parenrightbigg\n. /B4/BE/BC/B5/BY /D3/D6 /CP/D2 /D3/D4 /CT/D2/BB\r/D0/D3/D7/CT/CS \r/DD/D0/CX/D2/CS/CT/D6 ωn= (2n−1)πc/2L /B8 /D8/CW/CT/D6/CT/D7/D9/D0/D8 /CX/D7\n(krL)min≃(2n−1)π\n2/parenleftbigg\n1+qr\n2−1\n2Qn−ζ\n2YnQn√qr/parenrightbigg\n./B4/BE/BD/B5/CC /DD/D4/CX\r/CP/D0 /DA /CP/D0/D9/CT/D7Yn= 1/25 /B8ζ= 0.4 /B8 /CP/D2/CSqr= 0.4 /D0/CT/CP/CS /D8/D3/CP/D2 /CX/D2\r/D6/CT/CP/D7/CT /CX/D2γmin\n/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3γ0\n/D3/CU /CP/CQ /D3/D9/D88% /B8 \r/D3/D2/AS/D6/D1/CX/D2/CV/D8/CW/CT /D4/D6/CT/D4 /D3/D2/CS/CT/D6/CP/D2 /D8 /CT/AR/CT\r/D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1/D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /CW/D3 /DB \r/D3/D9/D4/D0/CX/D2/CV /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /CP/D2/CS/D1/CT\r /CW/CP/D2/CX\r/CP/D0 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 \r/D3/D9/D0/CS /D6/CT/CS/D9\r/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D3 /DA /CT/D6 /CP /DB/CX/CS/CT/D6 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /D6/CP/D2/CV/CT/B8 /D8/CW/CT /D2/CT/CX/CV/CW /CQ /D3/D6/CW/D3 /D3 /CS /D3/CU /D8/CW/CT/D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D1/CX/D2/CX/D1 /D9/D1 /CW/CP/D7 /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS/BA /CD/D7/CX/D2/CV/CP/CV/CP/CX/D2 /CP /D7/CX/D2/CV/D0/CT /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /D1/D3 /CS/CT /CU/D3/D6 /D8/CW/CT \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/B8 /CS/CT/D6/CX/DA /CP/B9/D8/CX/D3/D2 /D3/CU /CP /D4/CP/D6/CP/CQ /D3/D0/CX\r /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /DB /CP/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CF /D6/CX/D8/CX/D2/CV\nγ=γ/D1/CX/D2(1+ε2) /B8θ2=θ2/D1/CX/D2+δ /CP/D2/CSωn= (ωn)/D1/CX/D2(1+ν) /B8/DB/CX/D8/CWε /B8δ /B8 /CP/D2/CSν /D7/D1/CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D8/CW/CT /CC /CP /DD/D0/D3/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D3/CU /BX/D5/D7/BA /B4/BD/BI /B5 /CP/D2/CS /B4/BD/BJ /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7 /D0/CT/CP/CS/D7/D8/D3 /D8/CW/CT /D2/CT/DC/D8 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7/BM\nε2∼δ2/(2q2\nr), /B4/BE/BE/B5\n2q2\nrYnQn(δ−2ν) =−δζ√qr. /B4/BE/BF/B5/BY/CX/D2/CP/D0/D0/DD /B8 /D2/CT/CP/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CS/CT/D4 /CT/D2/B9/CS/CT/D2\r/CT /D3/D2 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nγ=γ/D1/CX/D2\n1+2qr/parenleftbigg\nq3/2\nr+ζ\n2YnQn/parenrightbigg2/parenleftbiggkrL−krL/D1/CX/D2\nkrL/D1/CX/D2/parenrightbigg2\n./B4/BE/BG/B5/C1/D2 /CP /AS/D6/D7/D8 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CP/D4 /CT/D6/D8/D9/D6/CT /D3/CU /D8/CW/CT /CP/D4/D4/D6/D3 /DC/CX/B9/D1/CP/D8/CT/CS /D4/CP/D6/CP/CQ /D3/D0/CP/B8 /D8/CW /D9/D7 /D8/CW/CT /DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /D6/CP/D2/CV/CT /CU/D3/D6 /DB/CW/CX\r /CW/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D0/D3 /DB /CT/D6/CT/CS/B8 /CX/D7 /D1/CP/CX/D2/D0/DD \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD\n/D8/CW/CT /D1 /D9/D7/CX\r/CX/CP/D2 /CT/D1 /CQ /D3/D9\r /CW /D9/D6/CT/B8 /CX/BA/CT/BA/B8 /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /D0/CX/D4/D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8/B8 /D8/CW/CP/D2/CZ/D7 /D8/D3 /CX/D8/D7 /CT/D1/B9/CQ /D3/D9\r /CW /D9/D6/CT/B8 /D8/CW/CT /D4/D0/CP /DD /CT/D6 \r/CP/D2 /CT/DC/D4 /CT\r/D8 /CP/D2 /CT/CP/D7/CX/CT/D6 /D4/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU/D8/D3/D2/CT/D7 /CU/D3/D6 \r/CT/D6/D8/CP/CX/D2 /D2/D3/D8/CT/D7/BA/BY /D3/D6 /CP /D0/D3/D7/D7/DD \r/DD/D0/CX/D2/CS/D6/CX\r/CP/D0 /D3/D4 /CT/D2/BB\r/D0/D3/D7/CT/CS /CQ /D3/D6/CT/B8 /D1/D3 /CS/CP/D0 /CT/DC/B9/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2\r/CT /CV/CX/DA /CT/D7YnQn=ωnL/2c=\n(2n−1)π/4 /D7/D3 /D8/CW/CP/D8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /CS/D3 /D2/D3/D8 /D7/CT/CT/D1 /D8/D3 /CW/CP /DA /CT /CP/CV/D6/CT/CP/D8 /CX/D2/AT/D9/CT/D2\r/CT /D3/D2 /D4/D0/CP /DD/CX/D2/CV /CU/CP\r/CX/D0/CX/D8 /DD /B8 /CP/D8 /D0/CT/CP/D7/D8 /DB/CW/CT/D2 \r/D3/D2/D7/CX/CS/B9/CT/D6/CX/D2/CV /D1/CX/D2/CX/D1/CP/D0 /CQ/D0/D3 /DB/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT γmin\n/B4Qn\n/CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6/CP/D0/D3/D2/CT /CX/D2 /AS/D6/D7/D8 /D3/D6/CS/CT/D6 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/B5/BA /C7/D2 /D8/CW/CT \r/D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/CT/DD/CP/D6/CT /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CU/D3/D6 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D8/CW/CT /CT/DC/D8/CX/D2\r/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2\n/BF/BH/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 /D8/CW/CT /D6/CT/CT/CS /CX/D7 /CW/CT/D0/CS /D1/D3/D8/CX/D3/D2/D0/CT/D7/D7/D0/DD/CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D0/CP /DD /BA/C1 /C1 /C1/BA /C5/C7/BW/BX/C4 /C1/C5/C8/CA/C7 /CE/BX/C5/BX/C6/CC/CB/C4/CP/D7/D8 /CU/D3/D9/D6 /CS/CT\r/CP/CS/CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /CU/D6/D9/CX/D8/CU/D9/D0 /CX/D2 /D4/CW /DD/D7/CX\r/CP/D0 /D1/D3 /CS/B9/CT/D0/CX/D2/CV /D3/CU /D1 /D9/D7/CX\r/CP/D0 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/B8 /CT/D7/D4 /CT\r/CX/CP/D0/D0/DD /CU/D3/D6 /D7/CX/D2/CV/D0/CT /D6/CT/CT/CS/CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/BA /C8/CX/D4 /CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D8/CW/CT /CU/D3 \r/D9/D7 /D3/CU /CP /CV/D6/CT/CP/D8 /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D7/D8/D9/CS/CX/CT/D7 /D7/CX/D2\r/CT /BU/CT/D2/CP/CS/CT\n/BF/BI/B8 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU/D4 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/D0/CX/D1/CX/D8/CT/CS /D8/D3 /CX/D8/D7 /D6/CT/D7/D3/D2/CP/D2\r/CT/BA/CB/CX/D2\r/CT /C6/CT/CS/CT/D6/DA /CT/CT/D2\n/BG/BD/CP/D2/CS /CC/CW/D3/D1/D4/D7/D3/D2\n/BG/BE/B8 /CX/D8 /CX/D7 /D4/D6/D3 /DA /CT/CS /D8/CW/CP/D8 /D8/CW/CT/AT/D3 /DB /CT/D2 /D8/CT/D6/CX/D2/CV /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D6/CT/CT/CS /D3/D4 /CT/D2/CX/D2/CV /CX/D7 /CS/CX/DA/CX/CS/CT/CS /CX/D2 /D3/D2/CT/D4/CP/D6/D8 /CT/DC\r/CX/D8/CX/D2/CV /D8/CW/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2/CS /CP/D2/D3/D8/CW/CT/D6 /D4/CP/D6/D8 /CX/D2/CS/D9\r/CT/CS /CQ /DD/D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2/BA /C1/D2 /CU/CP\r/D8/B8 /D8/CW/CT /DA/CX/CQ/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /D3/CU/D8/CW/CT /D6/CT/CT/CS /D4/D6/D3 /CS/D9\r/CT/D7 /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D7\r/CX/D0/D0/CP/D8/CX/D2/CV /AT/D3 /DB/BA /CC/CW /D9/D7/D8/CW/CT /CT/D2 /D8/CT/D6/CX/D2/CV /AT/D3 /DBU \r/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7\nU(ω) =Ye(ω)P(ω)+Sr(jωY(ω)), /B4/BE/BJ/B5/DB/CW/CT/D6/CTSr\n/CX/D7 /D8/CW/CT /CT/AR/CT\r/D8/CX/DA /CT /CP/D6/CT/CP /D3/CU /D8/CW/CT /DA/CX/CQ/D6/CP/D8/CX/D2/CV /D6/CT/CT/CS /D6/CT/D0/CP/D8/CT/CS/D8/D3 /D8/CW/CT /D8/CX/D4 /CS/CX/D7/D4/D0/CP\r/CT/D1/CT/D2 /D8 y(t) /BA /BT/D0/D8/CT/D6/D2/CP/D8/CT/D0/DD /B8 /CP /D0/CT/D2/CV/D8/CW∆l\r/CP/D2 /CQ /CT /CP/D7/D7/D3 \r/CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /AS\r/D8/CX/D8/CX/D3/D9/D7 /DA /D3/D0/D9/D1/CT /DB/CW/CT/D6/CT /D8/CW/CT/D6/CT/CT/CS /D7/DB/CX/D2/CV/D7/BA /BW/CP/D0/D1/D3/D2 /D8 /CT/D8 /CP/D0/BA\n/BD/BF/D6/CT/D4 /D3/D6/D8/CT/CS /D8 /DD/D4/CX\r/CP/D0 /DA /CP/D0/D9/CT/D7/D3/CU10mm /CU/D3/D6 /CP \r/D0/CP/D6/CX/D2/CT/D8/BA /C6/CT/CS/CT/D6/DA /CT/CT/D2\n/BG/BD/D0/CX/D2/CZ /CT/CS∆l /D8/D3 /D6/CT/CT/CS/D7/D8/D6/CT/D2/CV/D8/CW /B4/D3/D6 /CW/CP/D6 /CS/D2/CT/D7/D7 /B5/BM∆l /D1/CP /DD /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/D0/DD /DA /CP/D6/DD /CU/D6/D3/D1\n6mm /B4/D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/D7/B5 /D8/D39mm /B4/D7/D3/CU/D8/CT/D6 /D6/CT/CT/CS/D7/B5/BA /CC/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7/CQ /CT/CX/D2/CV /D7/D1/CP/D0/D0 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 \r/D0/CP/D6/CX/D2/CT/D8 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/B8 /D8/CW/CT /D6/CT/CT/CS/D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB \r/CP/D2 /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/CW/D6/D3/D9/CV/CW /CP /D1/CT/D6/CT/D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 /CX/D2 \r/D3/D1/D1/D3/D2 /DB /D3/D6/CZ/B8 /CQ/D9/D8 /CX/D8/D7 /CX/D2/AT/D9/CT/D2\r/CT /D3/D2/D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2/CS /D6/CT/CT/CS /CX/D7/D2/D3/D8 /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /D3/D2 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /CP/D7 /CX/D8 /CX/D7 /D7/D8/D9/CS/CX/CT/CS/D2/D3 /DB/BA/C1/D2 /CP /AS/D6/D7/D8 /D7/D8/CT/D4/B8 /D8/CW/CX/D7 /CT/AR/CT\r/D8 /CX/D7 \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /B8 /CP/D0/D0/D0/D3/D7/D7/CT/D7 /CQ /CT/CX/D2/CV /CX/CV/D2/D3/D6/CT/CS /B4qr= 0 /CP/D2/CSη= 0 /B5/BA /BX/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BJ /B5\r/D3/D9/D4/D0/CT/CS /D8/D3 /BX/D5/BA /B4/BJ /B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D7/DD/D7/D8/CT/D1/BM\n− /C1/D1(Ye) =kr∆lθ\n1−θ2, /B4/BE/BK/B5\n1\n1−θ2−1−γ\n2γ= 0⇔γ=1−θ2\n3−θ2. /B4/BE/BL/B5/BT/D7 /D7/CT/CT/D2 /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /B8 /D7/CT/DA /CT/D6/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/DD /D7/D3/D0/D9/D8/CX/D3/D2/D7 θ /CT/DC/CX/D7/D8 /CU/D3/D6/CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D7 γ /BA /BX/DC/CP/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BL /B5/CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 γ=f(θ) /B4/CU/D3/D6θ <1 /B5 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/D3/B9/D0/D9/D8/CX/D3/D2 /CW/CP /DA/CX/D2/CV /D8/CW/CT /D0/D3 /DB /CT/D7/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D8/CW/CT /D3/D2/CT /CQ /CT/CX/D2/CV /D8/CW/CT\r/D0/D3/D7/CT/D7/D8 /D8/D3 /D8/CW/CT /D6/CT/CT/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /BA/BT/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CX/D2 /D7/D3/D1/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7/BA/CF/CW/CT/D2 /D4/D0/CP /DD/CX/D2/CV \r/D0/D3/D7/CT /D8/D3 /CP /CQ /D3/D6/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/D5/D9/CT/D2\r/DD θn≪/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BI/BY/C1/BZ/BA /BG/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /CP/D2/CS/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8 /B4∆l= 12mm /CP/D2/CS /D3/D8/CW/CT/D6 \r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7/CX/D2 /BY/CX/CV/BA /BD/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5/B8/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/B9/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT\r/D8/D7 /B4/D4/D0/CP/CX/D2/D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT\r/CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA\n1 /B8 /D8/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BE/BK /B5 /CX/D7 /D7/D1/CP/D0/D0/B8 /D7/D3 /D8/CW/CP/D8/B8 /DB/CX/D8/CW\nYe=−jcot(θkrL) /B8 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CP\r/D8/D7/D1/CT/D6/CT/D0/DD /CP/D7 /CP /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2/BM\n∆l1\n1−θ2n\n/DB/CW/CT/D6/CTθn=(2n−1)π\n2krL. /B4/BF/BC/B5/CC/CW/CX/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CX/D7 /DA /CP/D0/CX/CS /DB/CW/CT/D2 /D8/CW/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /CQ /D3/D6/CT/CU/D6/CT/D5/D9/CT/D2\r/DD θn\n/D6/CT/D1/CP/CX/D2/D7 /D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /D9/D2/CX/D8 /DD /BA /BT/D2 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CU/D3/D6 /D8/CW/CT/CT/AR/CT\r/D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CU/D6/D3/D1 /BX/D5/BA /B4/BK/B5/BM\n∆lq≃ζqr√\n3kr. /B4/BF/BD/B5/BT /D8/D6/CX/CP/D0 /CP/D2/CS /CT/D6/D6/D3/D6 /D4/D6/D3 \r/CT/CS/D9/D6/CT /CW/CP/D7 /CQ /CT/CT/D2 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /D8/D3 /CP/CS/CY/D9/D7/D8/D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /CE /CT/D6/DD /CU/CT/DB /CX/D8/CT/D6/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /D2/CT/CT/CS/CT/CS/D8/D3 /CT/DC/CW/CX/CQ/CX/D8 /CP /DA /CP/D0/D9/CT /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /B4∆l≃\n12mm /B5 /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /B4∆lq≃2mm /B5/CX/D2 /D8/CW/CT \r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/CU /BY/CX/CV/D7/BA /BD /CP/D2/CS /BG/BA /CF/CW/CT/D2 /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /CP/D2/CS/D1/CT\r /CW/CP/D2/CX\r/CP/D0 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /CP/D6/CT /DA /CT/D6/DD \r/D0/D3/D7/CT /B4θ= 1−ε /CP/D2/CSθn=\n1−εn\n/B5/B8 /CP /D7/CT\r/D3/D2/CS/B9/D3/D6/CS/CT/D6 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 \r/CP/D2 /CQ /CT /CS/CT/CS/D9\r/CT/CS/BM\nε=εn\n2/parenleftigg\n1+/radicaligg\n1+2∆l\nLε2n/parenrightigg\n, /B4/BF/BE/B5/D8/CW/CT /CP/D4/D4/CP/D6/CX/D8/CX/D3/D2 /D3/CU /CP /D7/D5/D9/CP/D6/CT /D6/D3 /D3/D8 /CQ /CT/CX/D2/CV /D8 /DD/D4/CX\r/CP/D0 /D3/CU /D1/D3 /CS/CT \r/D3/D9/B9/D4/D0/CX/D2/CV/B8 /D1/CP/CZ/CX/D2/CV /D8/CW/CT /CP\r /CW/CX/CT/DA /CT/D1/CT/D2 /D8 /D3/CU /CP/D2/CP/D0/DD/D8/CX\r/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/CS/CXꜶ\r/D9/D0/D8/BA /CC/CW/CT/D2/B8 /DB/CW/CT/D2 /D8/CW/CT /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CS/CT\r/D6/CT/CP/D7/CT/D7 /CT/D2/D3/D9/CV/CW/D7/D3 /D8/CW/CP/D8 /D3/D2/CT /D3/CU /CX/D8/D7 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /CX/D2\r/D6/CT/CP/D7/CT/D7 /CP/CQ /D3 /DA /CT /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT/B4θn>1 /B5/B8 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/D4/D6/D3/CP\r /CW/CT/D7 /D8/CW/CT /D6/CT/CT/CS/D3/D2/CT /D9/D2 /D8/CX/D0 /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT\r/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CS/CX/D7/CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6krL=nπ/B4/D7/CT/CT /BY/CX/CV/BA /BH /B5/BA /C6/CT/CP/D6 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/B8 /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /CP/D4/D4/D6/D3 /DC/B9/CX/D1/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS/BM\nθ≃1−1\n2kr∆ltan(krL), /B4/BF/BF/B5\nγ≃1\n2kr∆ltan(krL). /B4/BF/BG/B51rL k2/ π=θ1rL k2/ π2 =θ13rL k2/ π3 =θ\nθ0r)2θ−1 (/ l ∆ k θ r)L k θ(t o c/BY/C1/BZ/BA /BH/BA /BZ/D6/CP/D4/CW/CX\r/CP/D0 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BK /B5 /CV/CX/DA/CX/D2/CV /D3/D7\r/CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /D0/CT/CU/D8/B9/CW/CP/D2/CS /D8/CT/D6/D1− /C1/D1(Y) /B4/D8/CW/CX\r /CZ/D0/CX/D2/CT/D7/B5/B8 /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1kr∆lθ/(1−θ2) /B4/D8/CW/CX/D2 /D0/CX/D2/CT/B5/B8 /CP/D2/CS /D7/D3/D0/D9/B9/D8/CX/D3/D2/D7 /B4/D1/CP/D6/CZ /CT/D6/D7/B5/BA/CC/CW/CT/D7/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CP/D6/CT /DA /CP/D0/CX/CS /CX/CUtan(krL)/greaterorsimilar0 /B8 /CX/BA/CT/BA/B8krL/greaterorsimilar\nnπ /BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /BX/D5/BA /B4/BE/BL /B5/B8 /DB/CW/CT/D2 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/B9/D4/D6/D3/CP\r /CW/CT/D7 fr\n/B8 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CS/CT\r/D6/CT/CP/D7/CT/D7 /D8/D3 /DE/CT/D6/D3 \r/D3/D2/B9/D8/D6/CP/D6/DD /D8/D3 /DB/CW/CP/D8 /CW/CP/D4/D4 /CT/D2/D7 /DB/CW/CT/D2 \r/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV /CT/AR/CT\r/D8/BA/BY/CX/CV/D9/D6/CT/D7 /BG /CP/D2/CS /BI /D7/CW/D3 /DB /CP /D2 /D9/D1/CT/D6/CX\r/CP/D0 \r/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT /CT/AR/CT\r/D8/D7 /D3/CU /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV/BA /CC/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /DB/CX/D8/CW∆l=\n12mm /CX/D2 /BY/CX/CV/BA /BG /CP/D2/CS5mm /CX/D2 /BY/CX/CV/BA /BI /CP/CS/CY/D9/D7/D8/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CU/D3/D6 /CQ /D3/D8/CW /CW/CT/CP /DA/CX/D0/DD /CP/D2/CS /D7/D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/B8/CT/DA /CT/D2 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/B8 /CP/D2/CS /CX/D7 /D4/D6/CT/D4 /D3/D2/B9/CS/CT/D6/CP/D2 /D8 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU/CP /D1/CX/D7/D8/D9/D2/CT/CS /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4∆l= 12mm /B5/CX/D7 /CT/DC/CW/CX/CQ/CX/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BI /B8 /D6/CT/DA /CT/CP/D0/CX/D2/CV /CP /D1/CX/D7/D1/CP/D8\r /CW /D3/CU /D3/D7\r/CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /CC/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D1/CP/CX/D2/D0/DD \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS/CQ /DD /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP\r /CW/B9/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /BV/D0/CP/D7/D7/CX\r/CP/D0 /CP/D4/D4/D6/D3/CP\r /CW/CT/D7/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BG/BD /B5 /CU/D3/D6 /D8/CW/CT \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /D4/D0/CP /DD/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2/B9\r/CX/CT/D7/B8 /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /AT/D3 /DB /CS/D9/CT /D8/D3 /D4/D6/CT/D7/D7/D9/D6/CT /CS/D6/D3/D4/B8 /CP/D2/CS /D7/CT/CP/D6\r /CW/B9/CX/D2/CV /CU/D3/D6 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1 /CX/D2\r/D0/D9/CS/CX/D2/CV/D8/CW/CT /CQ /D3/D6/CT /CP/D2/CS /D8/CW/CT /D6/CT/CT/CS /D3/D2/D0/DD /CP/D6/CT /D8/CW /D9/D7 /CY/D9/D7/D8/CX/AS/CT/CS/BA /C1/D8 \r/CP/D2 /CQ /CT/D2/D3/D8/CX\r/CT/CS /D8/CW/CP/D8 /BX/D5/BA /B4/BE/BK/B5 /DB /CP/D7 /CP/D0/D6/CT/CP/CS/DD /CV/CX/DA /CT/D2 /CQ /DD /CF /CT/CQ /CT/D6\n/BG/BF/CX/D2/D8/CW/CT /CT/CP/D6/D0/DD /BD/BL/D8/CW \r/CT/D2 /D8/D9/D6/DD /B4/D7/CT/CT /D4/CP/CV/CT /BE/BD/BI/B5/B8 /CP/D7/D7/D9/D1/CX/D2/CV /CP /D6/CT/CT/CS/CP/D6/CT/CP /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT \r/DD/D0/CX/D2/CS/D6/CX\r/CP/D0 /D8/D9/CQ /CT /D7/CT\r/D8/CX/D3/D2/BA /CC/CW/CX/D7 /D8/CW/CT/D3/D6/DD /B8/D9/D7/CT/CS /CQ /DD /D7/CT/DA /CT/D6/CP/D0 /CP/D9/D8/CW/D3/D6/D7 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BF/BE /B5/B8 /DB /CP/D7 /CS/CX/D7\r/D9/D7/D7/CT/CS/CQ /DD /DA /D3/D2 /C0/CT/D0/D1/CW/D3/D0/D8/DE\n/BF/BF/CP/D2/CS /BU/D3/D9/CP/D7/D7/CT\n/BG/BG/B8 /CT/D7/D4 /CT\r/CX/CP/D0/D0/DD \r/D3/D2\r/CT/D6/D2/CX/D2/CV/D8/CW/CT /D0/CP\r /CZ /D3/CU /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D4/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /D7/CT/D0/CU/B9/D7/D9/D7/D8/CP/CX/D2/CT/CS/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7/BA /C7/D2 /D8/CW/CT \r/D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/D7/D9/D6/CT \r/D9/D6/DA /CT/D7/CT/DC/CW/CX/CQ/CX/D8 /D8/CW/CP/D8 /CQ /D3/D8/CW /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /AT/D3 /DB /CW/CP /DA /CT /CX/D2/B9/AT/D9/CT/D2\r/CT /D3/D2 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D8/D3 /D3/D7\r/CX/D0/D0/CP/D8/CT/BA/CB/D3 /CP \r/D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/CW/CT/D2/D3/D1/CT/D2/CP /CW/CP/D7 /D8/D3 /CQ /CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3/CP\r\r/D3/D9/D2 /D8/B8 /D2/D3/D2/CT /D3/CU /D8/CW/CT/D1 /CQ /CT/CX/D2/CV /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /CX/D2 /D8/CW/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS/CS/D3/D1/CP/CX/D2/BA/C1/CE/BA /BZ/C7/C1/C6/BZ /BU/BX/CH/C7/C6/BW /C1/C6/CB/CC /BT/BU/C1/C4/C1/CC/CH /CC/C0/CA/BX/CB/C0/C7/C4/BW/BT/BA /C4/CX/D2/CT/CP /D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /DB/CX/D8/CW /D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/BT/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/CP/D2 /CQ /CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS/D9/D7/CX/D2/CV \r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6/D1/CP/D0/CX/D7/D1/BA /BY /D3/D6 /CP /CV/CX/DA /CT/D2 \r/D3/D2/AS/CV/B9/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /D3/D6 /CT/B9/D6 /CT /CT /CS/B9/D1/D9/D7/CX\r/CX/CP/D2 /B4L /B8S /B8/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BJ/BY/C1/BZ/BA /BI/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8 /B4\r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7 /CX/D2 /BY/CX/CV/BA /BE/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS/AT/D3 /DB /D3/D2/D0/DD /B4∆l= 5mm /BM /D4/D0/CP/CX/D2 /D8/CW/CX/D2 /D0/CX/D2/CT/D7/B8∆l= 12mm /BM /CS/D3/D8/D8/CT/CS/D0/CX/D2/CT /CX/D2 /D9/D4/D4 /CT/D6 /CV/D6/CP/D4/CW /D3/D2/D0/DD/B5/B8 /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D8/CW/CX/D2/D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT\r/D8/D7 /B4/D4/D0/CP/CX/D2 /D8/CW/CX\r /CZ /D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT/CP\r/CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA\nωr\n/B8qr\n/B8γ /B8 /CP/D2/CSζ /CQ /CT/CX/D2/CV /D7/CT/D8/B5/B8 /CX/D8/D7 \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7\nsn=jωn−αn\n\r/CP/D2 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/BA /CC/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D3/CUsn\n\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /D8/CW/CT /D6/CT/CP/D0 /D4/CP/D6/D8αn\n/CQ /CT/B9/CX/D2/CV /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CX/D7 /D1/D3 /CS/CT/B8 /CU/D3/D6 /D8/CW/CT \r/D3/D9/D4/D0/CT/CS /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/D7/DD/D7/D8/CT/D1 \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D7/D8/CP/D8/CX\r /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/BA /BV/D0/CP/D7/D7/CX\r/CP/D0/D0/DD /B8/DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /CQ /CT/D0/D3 /DB /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS/B8 /CP/D0/D0 /CT/CX/CV/CT/D2/CS/CP/D1/D4/CX/D2/CV/D7 αn\n/CP/D6/CT /D4 /D3/D7/CX/D8/CX/DA /CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT/CQ /CT/CX/D2/CV /D7/D8/CP/CQ/D0/CT/BA /BT/D2 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D1/CP /DD /CP/D4/D4 /CT/CP/D6 /DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8/D3/D2/CT /D1/D3 /CS/CT /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /CT\r/D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT/B8 /CX/BA/CT/BA/B8/DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8 /D3/D2/CT /D3/CU /D8/CW/CTαn\n/CQ /CT\r/D3/D1/CT/D7 /D2/CT/CV/CP/D8/CX/DA /CT/BA /C4/D3 /D3/CZ/CX/D2/CV/CU/D3/D6 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/CP/D2 /CQ /CT /CS/D3/D2/CT /CQ /DD /DA /CP/D6/DD/CX/D2/CV /CP /CQ/CX/CU/D9/D6\r/CP/B9/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4/CT/CX/D8/CW/CT/D6L /B8γ /B8 /D3/D6ζ /B5 /CP/D2/CS /CT/DC/CP/D1/CX/D2/CX/D2/CV /D8/CW/CT /D6/CT/CP/D0/D4/CP/D6/D8 /D3/CU \r/D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7/BA /CC /DB /D3 /CT/DC/CP/D1/D4/D0/CT/D7 /CP/D6/CT/D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/D7/BA /BJ /CP/D2/CS /BK /BA /C1/D8 /CX/D7 /D2/D3/D8/CX\r/CT/CP/CQ/D0/CT /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2/B9\r/CX/CT/D7 /CT/DA /D3/D0/DA /CT /D3/D2/D0/DD /D7/D0/CX/CV/CW /D8/D0/DD /DB/CX/D8/CW /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 γ/CP/D2/CS /CP/D6/CT \r/D0/D3/D7/CT /D8/D3 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /CT/CX/D8/CW/CT/D6 /D8/CW/CT /CQ /D3/D6/CT /D3/D6 /D8/CW/CT/D6/CT/CT/CS /B4 /C1/D1(jω/ωr)≃1 /B5/BA /BY /D3/D6 /D8/CW/CT /AS/D6/D7/D8 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX\r/D6/CT/CV/CX/D1/CT /CQ /CT\r/D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6γ≃0.28 /CP/D2/CS /CP /CU/D6/CT/D5/D9/CT/D2\r/DD/D2/CT/CP/D6 /D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2\r/CT /D3/CU /D8/CW/CT /CQ /D3/D6/CT /B4/CS/D3/D8/B9/CS/CP/D7/CW/CT/CS \r/D9/D6/DA /CT/D7/B5/BA/C7/D8/CW/CT/D6 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /B4/D8/CW/CT/CX/D6 /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CQ /CT/CX/D2/CV /D3 /CS/CS/D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU /D8/CW/CT /AS/D6/D7/D8 /D3/D2/CT/B5 /CP/D2/CS /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /CW/CP /DA /CT /CW/CX/CV/CW/CT/D6/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D6/CT/D1/CP/CX/D2 /CS/CP/D1/D4 /CT/CS /CU/D3/D6 /D8/CW/CX/D7 \r/D3/D2/AS/CV/D9/B9/D6/CP/D8/CX/D3/D2/BA /BY /D3/D6 /CP /D0/D3/D2/CV/CT/D6 /D8/D9/CQ /CT /B4/BY/CX/CV/BA /BK/B5/B8 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6\nγ≃0.3 /CP/D8 /CP /CU/D6/CT/D5/D9/CT/D2\r/DD /D0/D3 \r/CP/D8/CT/CS /D2/CT/CP/D6 /D8/CW/CT /D8/CW/CX/D6/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/D3/CU /D8/CW/CT /CQ /D3/D6/CT /CJ/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT /CP/D8 /C1/D1(jω/ωr)≃0.8≃3×0.27 ℄/B8/D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2\r/CT /CQ /CT\r/D3/D1/CX/D2/CV /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6 /CP /D0/CP/D6/CV/CT/D6 /D1/D3/D9/D8/CW/D4/D6/CT/D7/D7/D9/D6/CT/BA /CC/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /B4/D7/D3/D0/CX/CS\r/D9/D6/DA /CT/D7/B5 /D7/D8/CX/D0/D0 /D6/CT/D1/CP/CX/D2/D7 /CS/CP/D1/D4 /CT/CS/BA/BV/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /D1/CP /DD /CQ /CT /D7/CX/D1/D4/D0/CX/AS/CT/CS /CP/D2/CS /CP\r\r/CT/D0/CT/D6/CP/D8/CT/CS /CQ /DD /D9/D7/B9/CX/D2/CV /CP /D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D1/D4 /CT/CS/CP/D2\r/CT Ze(ω) /BM/D8/CW/CX/D7 /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /D8/D3 /CQ /CT /DB/D6/CX/D8/D8/CT/D2/CP/D7 /CP /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CUjω /B8 /CP/D2/CS /D3/D4/D8/CX/D1/CX/DE/CT/CS /CP/D0/CV/D3/B9/D6/CX/D8/CW/D1/D7 /CU/D3/D6 /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /D6/D3 /D3/D8 /AS/D2/CS/CX/D2/CV \r/CP/D2 /CQ /CT /D9/D7/CT/CS/BA /C5/D3 /CS/CP/D0/CT/DC/D4/CP/D2/D7/CX/D3/D2 \r/D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CTN /AS/D6/D7/D8 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /D3/CU /D8/CW/CT\n/BY/C1/BZ/BA /BJ/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D7 /CP /CU/D9/D2\r/B9/D8/CX/D3/D2 /D3/CU /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8\nqr= 0.3 /B8L= 16cm /B4θ1= 0.53 /CP/D2/CSkrL= 2.96 /B5/B8 /CP/D2/CSζ= 0.2 /BA\n/BY/C1/BZ/BA /BK/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D7 /CP /CU/D9/D2\r/B9/D8/CX/D3/D2 /D3/CUγ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8qr= 0.3 /B8\nL= 32cm /B4θ1= 0.26 /CP/D2/CSkrL= 5.91 /B5/B8 /CP/D2/CSζ= 0.2 /BA/CQ /D3/D6/CT/BM\nZe(ω)\nZc=jtan/parenleftbiggωL\nc−jα(ω)L/parenrightbigg/B4/BF/BH/B5\nZe(ω)\nZc≃2c\nLN/summationdisplay\nn=1jω\nω2n+jqnωωn+(jω)2, /B4/BF/BI/B5/DB/CW/CT/D6/CT /D1/D3 /CS/CP/D0 \r/D3 /CTꜶ\r/CX/CT/D2 /D8/D7 ωn\n/CP/D2/CSqn\n\r/CP/D2 /CQ /CT /CS/CT/CS/D9\r/CT/CS /CT/CX/B9/D8/CW/CT/D6 /CU/D6/D3/D1 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2\r/CT /D3/D6 /CU/D6/D3/D1 /CP/D2/CP/D0/DD/D8/CX\r/CP/D0/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CJ/BX/D5/BA /B4/BF/BH /B5℄/B8 /CP/D7/D7/D9/D1/CX/D2/CV α(ω) /D8/D3 /CQ /CT /CP /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/B9/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /BA/BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CS/CX/D6/CT\r/D8 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D8/CW/D6/CT/D7/CW/D3/D0/CS γ/D8/CW\n/D9/D7/CX/D2/CV /CF/BU /D1/CT/D8/CW/D3 /CS /CP/D2/CS /CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D9/D7/CX/D2/CV/D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /CP/D2/CS \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 \r/D3/D1/B9/D4/D9/D8/CX/D2/CV /CW/CP/D7 /CQ /CT/CT/D2 /CS/D3/D2/CT/BA /CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT /CS/CXꜶ\r/D9/D0/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT/AS/D6/D7/D8 /D1/CT/D8/CW/D3 /CS /CP/D6/CX/D7/CT /CS/D9/CT /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7\r/CT/D2/CS/CT/D2 /D8/CP/D0 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/B9/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/CW/CT /D7/CT\r/D3/D2/CS /D3/D2/CT /D6/CT/D5/D9/CX/D6/CT/D7 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/CX/CV/CT/D2/B9/DA /CP/D0/D9/CT/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /B8 /D9/D7/CX/D2/CV/CP/D2 /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6\r /CW /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /CC/CW/CT /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D1/D3 /CS/CT/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /CW/CP/D7 /CQ /CT/CT/D2 \r /CW/D3/D7/CT/D2 /D7/D9\r /CW/D8/CW/CP/D8 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CUγ/D8/CW\n/CX/D7 /D0/CT/D7/D7 /D8/CW/CP/D20.01 /B8 /DB/CW/CX\r /CW /CX/D7 /CP/D0/D7/D3 /D8/CW/CT/D8/D3/D0/CT/D6/CP/D2\r/CT /D9/D7/CT/CS /CU/D3/D6 /D8/CW/CT /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6\r /CW/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /CX/D7/CV/CX/DA /CT/D2 /CX/D2 /CC /CP/CQ/D0/CT /C1 /CU/D3/D6 /CP /CW/CT/CP /DA/CX/D0/DD /CS/CP/D1/D4 /CT/CS /CP/D2/CS /D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/BA/C1/D8 /D7/CW/D3 /DB/D7 /DA /CT/D6/DD /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /D1/CT/D8/CW/D3 /CS/D7/CU/D3/D6 /CQ /D3/D8/CWγ /CP/D2/CSθ /BA /CC/CW/CX/D7 /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /CP/D0/D0/D3 /DB/D7 /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT\r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/D4/D6/D3/CP\r /CW/B8 /DB/CW/CX\r /CW /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP/D2 /CTꜶ\r/CX/CT/D2 /D8/CP/D0/CV/D3/D6/CX/D8/CW/D1 /D8/CW/CP/D8 \r/CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D1/D3/D6/CT \r/D3/D1/D4/D0/CT/DC /D6/CT/D7/B9/D3/D2/CP/D8/D3/D6/D7 /DB/CW/CT/D2/CT/DA /CT/D6 /CP /D1/D3 /CS/CP/D0 /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /CX/D7 /CP /DA /CP/CX/D0/CP/CQ/D0/CT/BA/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BKkrLθCFθWB∆θγCFγWB∆γ/BK/BA/BH /BC/BA/BD/BK/BH /BC/BA/BD/BK/BI /BC/BA/BH/B1 /BC/BA/BG/BF /BC/BA/BG/BF /BC/BA/BD/B1/BE /BC/BA/BJ/BG/BJ /BC/BA/BJ/BG/BJ /BC/BA/BD/B1 /BC/BA/BF/BC /BC/BA/BF/BC /BC/BA/BI/B1/BD /BD/BA/BC/BE/BE /BD/BA/BC/BE/BG /BC/BA/BE/B1 /BF/BA/BK/BI /BF/BA/BK/BE /BD/BA/BC/B1/BC/BA/BK/BD /BD/BA/BC/BF/BF /BD/BA/BC/BF/BG /BC/BA/BD/B1 /BK/BA/BJ/BJ /BK/BA/BJ/BD /BC/BA/BJ/B1/CC /BT/BU/C4/BX /C1/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/CU/D6/CT/D5/D9/CT/D2\r/DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV \r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6/D1/CP/D0/CX/D7/D1 /B4/CX/D2/B9/CS/CT/DC/CT/CS /CQ /DDCF /B5 /CP/D2/CS /CF/CX/D0/D7/D3/D2 /B2 /BU/CT/CP /DA /CT/D6/D7 /D1/CT/D8/CW/D3 /CS /B4WB /B5 /CU/D3/D6\nr= 7mm /B8ωr= 2π×750rad/s /B8qr= 0.4 /CP/D2/CSζ= 0.13 /BA/CC/CW/CT /DB/D6/CX/D8/CX/D2/CV /D3/CU /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CP /D7/CX/D2/B9/CV/D0/CT /CP\r/D3/D9/D7/D8/CX\r /D1/D3 /CS/CT /CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT \r/D3/D9/D4/D0/CT/CS/D3/D7\r/CX/D0/D0/CP/D8/D3/D6/D7/BM\n/bracketleftbigg\nω2\nn+jω/parenleftbigg\nqnωn+c\nLζ1−γ√γ/parenrightbigg\n−ω2/bracketrightbigg\n×/bracketleftbig\nω2\nr+jqrωωr−ω2/bracketrightbig\n=jω2c\nLω2\nr/parenleftbigg\nζ√γ+jω∆l\nc/parenrightbigg\n./B4/BF/BJ/B5/BV/D3/D9/D4/D0/CX/D2/CV /D6/CT/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT /AT/D3 /DB /CX/D2 /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /D1/D3 /CS/B9/CX/AS/CT/D7 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CT /CP\r/D3/D9/D7/D8/CX\r /D1/D3 /CS/CT/BM /CX/D2 /CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3/D8/CW/CT /D9/D7/D9/CP/D0 /D8/CT/D6/D1 /B4\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /DA/CX/D7\r/D3/D8/CW/CT/D6/D1/CP/D0 /D0/D3/D7/D7/CT/D7 /CP/D2/CS/CT/DA /CT/D2 /D8/D9/CP/D0/D0/DD /D6/CP/CS/CX/CP/D8/CX/D3/D2/B5/B8 /CS/CP/D1/D4/CX/D2/CV /CX/D7 /CX/D2\r/D6/CT/CP/D7/CT/CS /CQ /DD /CP /D5/D9/CP/D2 /D8/CX/D8 /DD/D6/CT/D0/CP/D8/CT/CS /D8/D3 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7/D1/CP /DD /CQ /CT /D6/CT/CV/CP/D6/CS/CT/CS /D8/D3 /CP/D7 /CP /D6/CT/D7/CX/D7/D8/CX/DA /CT /CP\r/D3/D9/D7/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /CP/D8 /D8/CW/CT/CQ /D3/D6/CT /CT/D2 /D8/D6/CP/D2\r/CT/BA/BT/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /CP /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS /D1/D3 /CS/CT/D0 /CX/D7 /D7/D8/CX/D0/D0 /D6/CT/D0/CT/DA /CP/D2 /D8 /CS/D9/D6/B9/CX/D2/CV /D8/CW/CT /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /B4/CQ /CT/CU/D3/D6/CT /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D7/CP/D8/B9/D9/D6/CP/D8/CX/D3/D2 /D1/CT\r /CW/CP/D2/CX/D7/D1 /CP/D4/D4 /CT/CP/D6/D7/B5/B8 /D8/CW/CX/D7 /CP/D4/D4/D6/D3/CP\r /CW \r/CP/D2 /CQ /CT /CT/DC/B9/D8/CT/D2/CS/CT/CS /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT \r/D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1/BA /BV/CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CX/D2/CV /D8/CW/CT /CS/CT/CV/D6/CT/CT /D3/CU /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /DDσ= min nαn\n/B8 /D8/CW/CT /D7/D0/D3/D4 /CT /D3/CU /D8/CW/CT \r/D9/D6/DA /CT\nσ=f(γ) /CV/CX/DA /CT/D7 /CP/D2 /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/B9/CV/D6/CT/CT /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D7/D0/CX/CV/CW /D8/D0/DD/CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /BT /CV/D6/CT/CP/D8 /D7/D0/D3/D4 /CT/DB /D3/D9/D0/CS \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CP /DA /CT/D6/DD /D9/D2/D7/D8/CP/CQ/D0/CT \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS/CP /D5/D9/CX\r /CZ /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT/CP/D7 /D2/CT/CP/D6/D0/DD \r/D3/D2/D7/D8/CP/D2 /D8\r/D9/D6/DA /CT /DB /D3/D9/D0/CS /D0/CT/CP/CS /D8/D3 /CP /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/AS\r/CP/D8/CX/D3/D2 \r/D3 /CTꜶ\r/CX/CT/D2 /D8 /CP/D2/CS/D7/D0/D3 /DB/D0/DD /D6/CX/D7/CX/D2/CV /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT/D2 /D8/D3 /D0/D3/D2/CV/CT/D6 /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CP/D8/B9/D8/CP\r /CZ /CQ /CT/CU/D3/D6/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7/BA/C4/CX/D2/CZ/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT \r/D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /D8/CW/CT \r/D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CW/CT/D6/CT /CP/D2/CS /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6/CW/CP /DA /CT /D7/D8/CX/D0/D0 /D8/D3 /CQ /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA/BU/BA /C5/D3/D9/D8/CW /D4 /D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP /CV/CX/DA/CT/D2 /B4/D7/D1/CP/D0/D0/B5/CP/D1/D4/D0/CX/D8/D9/CS/CT/CC/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4/CP/D4 /CT/D6 /CS/CT/CP/D0 /DB/CX/D8/CW /D8/CW/CT /D7/D8/CP/CQ/CX/D0/B9/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT/B8 /D0/D3 /D3/CZ/CX/D2/CV /CU/D3/D6 /D8/CW/CT \r/D3/D2/CS/CX/D8/CX/D3/D2 /D8/D3 /D1/CP/CZ /CT/CP /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CB/D3/D1/CT /CS/CT/DA /CT/D0/D3/D4/D1/CT/D2 /D8/D7 \r/D3/D2\r/CT/D6/D2/CX/D2/CV/D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D2/CV /D6/CT/CV/CX/D1/CT /CP/CQ /D3 /DA /CT /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /D2/D3 /DB/BA /C6/CT/CX/D8/CW/CT/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D2/D3/D6 /D8/D3/D2/CT/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CX/D7/D7/D9/CT /DB/CX/D0/D0 /CQ /CT /CS/CX/D7\r/D9/D7/D7/CT/CS /CW/CT/D6/CT/BA/BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA\n/BE/BC/D7/D9/CV/CV/CT/D7/D8/CT/CS /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CX/D1/B9/CX/D8/CT/CS /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /CX/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS \r/CP/D7/CT/BA\n/CC/CW/CT /D8/CT\r /CW/D2/CX/D5/D9/CT /CX/D7 /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX\r /CQ/CP/D0/CP/D2\r/CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D3/D7\r/CX/D0/B9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7/BA /BV/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /CW/CT/D6/CT/B9/CP/CU/D8/CT/D6 /CQ /DD /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /D8/CW/CT /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /CX/D2 /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/B8 /DB/CW/CX\r /CW /CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6 /CX/D2 /D8/CW/CT/D1/CT/D2 /D8/CX/D3/D2/CT/CS /D4/CP/D4 /CT/D6/BA /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /CS/CT/B9/D4 /CT/D2/CS/D7 /D3/D2 /BY /D3/D9/D6/CX/CT/D6 \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D7/CX/CV/D2/CP/D0Pm−p(t) /CP/D2/CS\ny(t) /BA /BT/D7/D7/D9/D1/CX/D2/CV /D7/D8/CT/CP/CS/DD /D7/D8/CP/D8/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/B9/D5/D9/CT/D2\r/DDω /B8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D7/CX/CV/D2/CP/D0/D7 /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7\np(t) =/summationdisplay\nn/negationslash=0pnenjωt, u(t) =u0+/summationdisplay\nn/negationslash=0Ynpnenjωt, /B4/BF/BK/B5\ny(t)\ny0= (1−γ)+/summationdisplay\nn/negationslash=0Dnpnenjωt, /B4/BF/BL/B5/DB/CW/CT/D6/CTYn=Ye(nω) /CP/D2/CSDn=D(nω) /CP/D6/CT /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CQ /D3/D6/CT /CP/CS/D1/CX/D8/D8/CP/D2\r/CT /CP/D2/CS /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2\r/D8/CX/D3/D2/CU/D3/D6 /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/D5/D9/CT/D2\r/DD nω /BA /CC/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/CX/D7 /D6/CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7\nu2(t) =ζ2(y(t)/y0)2(γ−p(t)). /B4/BG/BC/B5/CB/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /D3/CU /DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D6/CT /D7/D8/D9/CS/B9/CX/CT/CS/B8 /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8p1\n/CX/D7 /CP /D2/D3/D2 /DA /CP/D2/CX/D7/CW/CX/D2/CV \r/D3 /CTꜶ\r/CX/CT/D2 /D8 \r/D3/D2/B9/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CP /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /C6/D3/D8/CP/D8/CX/D3/D2/D7 CEn\n/CP/D2/CS\nF(n\nm)\n/CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9\r/CT/CS/BM\nCEn=Yn/(ζ√γ)+1−γ\n2γ−Dn, /B4/BG/BD/B5\nF(m\nn)=F(n\nm)=DnDm−1−γ\nγ(Dn+Dm)−YnYm\nζ2γ/B4/BG/BE/B5/BV/CP/D2\r/CT/D0/D0/CP/D8/CX/D3/D2 /D3/CUCEn\n/CU/D3/D6 /CV/CX/DA /CT/D2ω /CP/D2/CSγ /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BJ/B5 /CX/D7 /D7/D3/D0/DA /CT/CS /CU/D3/D6nω /CP/D2/CSγ /BA /BX/DC/B9/D4/CP/D2/CS/CX/D2/CV /BX/D5/BA /B4/BG/BC /B5 /D0/CT/CP/CS/D7 /D8/D3\n0 =/bracketleftbiggu2\n0\nζ2γ−(1−γ)2/bracketrightbigg\n+2(1−γ)/summationdisplay\nn/negationslash=0/bracketleftbiggu0Yn\nζ2γ(1−γ)−Dn+1−γ\n2γ/bracketrightbigg\npnenjωt\n−/summationdisplay\nn,m/negationslash=0F(n\nm)pnpme(n+m)jωt\n+1\nγ/summationdisplay\nn,m,q/negationslash=0DnDmpnpmpqe(n+m+q)jωt./B4/BG/BF/B5/C1/D8 /CX/D7 /CW/CT/D6/CT /CP/D7/D7/D9/D1/CT/CS /D8/CW/CP/D8pn\n/CX/D7 /D3/CU /D3/D6/CS/CT/D6|n| /B4/DB/CX/D8/CWp−n=p∗\nn\n/B5/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BE/BC /B5/BA /CC/CW/CT \r/D3/D2 /D8/CX/D2 /D9/D3/D9/D7 \r/D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /CX/D7 \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BE/BM\nu2\n0\nζ2γ=(1−γ)2+/summationdisplay\nn/negationslash=0F“+n\n−n”|pn|2\n−1\nγ/summationdisplay\nn,m,n+m/negationslash=0DnDmpnpmp∗\nn+m\n≃(1−γ)2+2F“+1\n−1”|p1|2+o(p2\n1)\n/B4/BG/BG/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BL/D7/D3 /D8/CW/CP/D8/B8 \r/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV u0\n/D8/D3 /CQ /CT /D6/CT/CP/D0\nu0≃ζ√γ(1−γ)\n1+|p1|2F“+1\n−1”\n(1−γ)2\n+o(p2\n1). /B4/BG/BH/B5/BY /D6/D3/D1 /BX/D5/BA /B4/BG/BF/B5/B8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD (Nω) /CX/D7 /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D3/D6\nN≥1 /BM\n0 = 2(1−γ)/bracketleftbigg\nDN−1−γ\n2γ−u0Yn\nζ2γ(1−γ)/bracketrightbigg\npN\n+/summationdisplay\nn/negationslash=0F(n\nN−n)pnpN−n\n−1\nγ/summationdisplay\nn,m/negationslash=0DnDmpnpmpN−n−m. /B4/BG/BI/B5/BY /D3/D6N≥2 /B8 /CC /CP /DD/D0/D3/D6 /D7/CT/D6/CX/CT/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D9/D4 /D8/D3 /D3/D6/CS/CT/D6N /CX/D7/CP/D4/D4/D0/CX/CT/CS/BM /CX/D2 /D8/CW/CT /AS/D6/D7/D8 /D7/D9/D1/B8 /D3/D2/D0/DD /D8/CT/D6/D1/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3\n0≤n≤N \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CT /CP/D8 /D3/D6/CS/CT/D6N /B8 /DB/CW/CX/D0/CT/B8 /CX/D2 /D8/CW/CT /D7/CT\r/D3/D2/CS/D3/D2/CT/B8 /D8/CW/CT /D8/CT/D6/D1/D7 /D8/D3 \r/D3/D2/D7/CX/CS/CT/D6 /CP/D6/CT /D8/CW/CT /D3/D2/CT/D7 /CU/D3/D6 /DB/CW/CX\r /CW0<\nn < N /CP/D2/CS0< m < N −n /BA /CC/CW/CT \r/D3/D1/D4 /D3/D2/CT/D2 /D8 pN\n\r/CP/D2 /CQ /CT/CS/CT/CS/D9\r/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D7/CT/D5/D9/CT/D2\r/CT (pn)0<n<N\n/BM\npN=1\n2(1−γ)CEN/bracketleftigg/summationdisplay\n0<n<NF(n\nN−n)pnpN−n\n−1\nγ/summationdisplay\n0<n<N\n0<m<N −nDnDmpnpmpN−n−m/bracketrightigg\n+o(pN\n1). /B4/BG/BJ/B5/BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /CU/D3/D6N= 2 /B8 /D8/CW/CT /D7/CT\r/D3/D2/CS /D7/D9/D1 /CQ /CT/CX/D2/CV /CT/D1/D4/D8 /DD/BM\np2≃F(1\n1)p2\n1\n2(1−γ)CE2+o(p2\n1). /B4/BG/BK/B5/BT/D7 /CT/DC/D4 /CT\r/D8/CT/CS /CP/D2/CS /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D7/D3/B9\r/CP/D0/D0/CT/CS /AG/CF /D3/D6/B9/D1/CP/D2 /D6/D9/D0/CT/AH\n/BD/BK/B8 /CW/CX/CV/CW/CT/D6 \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D4/D4 /CT/CP/D6 /D8/D3 /CQ /CT /CW/CX/CV/CW/CT/D6 /D3/D6/B9/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BM /D3/D6/CS/CT/D6 /BE /CU/D3/D6p2\n/B8 /BF /CU/D3/D6p3\n/B8 /BG /CU/D3/D6p4\n/B8 /CT/D8\r/BA/BY /D3 \r/D9/D7 /CX/D7 /D2/D3 /DB /CV/CX/DA /CT/D2 /D8/D3 /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /BV/CP/D0\r/D9/B9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BF/BM\n0 =2(1−γ)/bracketleftbigg\nD1−1−γ\n2γ−u0Y1\nζ2γ(1−γ)/bracketrightbigg\np1\n+/summationdisplay\nn/negationslash=0,1F(n\n1−n)pnp1−n\n−1\nγ/summationdisplay\n1−n−m,n,m/negationslash=0DnDmpnpmp1−n−m\n≃2(1−γ)/bracketleftbigg\nD1−1−γ\n2γ−u0Y1\nζ2γ(1−γ)/bracketrightbigg\np1\n+2F(2\n−1)p2p∗\n1−1\nγD1p1|p1|2(D1+2D∗\n1)\n/B4/BG/BL/B5/CA/CT/D4/D0/CP\r/CX/D2/CV u0\n/CP/D2/CSp2\n/CV/CX/DA /CT/D7\n|p1|2≃2(1−γ)2CE1\nF(1\n1)F“+2\n−1”\nCE2−2Y1\nζ√γF“+1\n−1”−1−γ\nγ(D2\n1+2|D1|2)./B4/BH/BC/B5\n/CC/CW/CT /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AS/D6/D7/D8 /CW/CP/D6/D1/D3/D2/CX\r /D1/CT/D8/CW/D3 /CS /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT/CX/D2/AT/D9/CT/D2\r/CT /D3/CUpn≥2\n/D3/D2 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT |p1| /D0/CT/CP/CS/D7 /D8/D3\n|p1|2≃2(1−γ)2CE1\n−2Y1\nζ√γF“+1\n−1”−1−γ\nγ(D2\n1+2|D1|2). /B4/BH/BD/B5/C1/D2 /D8/CW/CT/CX/D6 /D4/CP/D4 /CT/D6/B8 /BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA\n/BE/BC/D7/D8/CP/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D4/D6/D3 \r/CT/CS/D9/D6/CT/D9/D7/CT/CS /CX/D2 /CP /AS/D6/D7/D8 /D7/CX/D1/D4/D0/CX/AS/CT/CS \r/CP/D7/CT \r/CP/D2 /CQ /CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /CP /D1/D3 /CS/CT/D0/CX/D2\r/D0/D9/CS/CX/D2/CV /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8Ze(ω) /CX/D7 /D6/CT/D4/D0/CP\r/CT/CS/CQ /DDZe(ω)D(ω) /BA /BV/D3/D2\r/D0/D9/D7/CX/D3/D2 /CX/D7 /D2/D3/D8 /D7/D3 /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS /CP/D7/D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /CP/D0/D7/D3 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D7 /DB/CX/D8/CW /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/B9/D8/CX/D3/D2/D7/CW/CX/D4 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D2/CV /D8/D3 /CP /D1/D3/D6/CT \r/D3/D1/D4/D0/CT/DC /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6/D8/CW/CT /AS/D6/D7/D8 \r/D3/D1/D4 /D3/D2/CT/D2 /D8 /CP/D1/D4/D0/CX/D8/D9/CS/CT/BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT/CX/D6 /DB /D3/D6/CZ/B8\nDn\n/D8/CT/D6/D1/D7 /DB /D3/D9/D0/CS /D3/D2/D0/DD /D3 \r\r/D9/D6 /DB/CX/D8/CWYn\n/CX/D2 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU\n|p1|2/DB/CW/CX\r /CW /CX/D7 /D2/D3/D8 /D8/CW/CT \r/CP/D7/CT/BA\n/BY/C1/BZ/BA /BL/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU|p1| \r/D3/D1/D4/D9/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX\r /CQ/CP/D0/B9/CP/D2\r/CT /CU/D3/D6 /D7/D1/CP/D0/D0 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CUkrL /BM /CU/D6/D3/D1/D8/CW/CX/D2 /D8/D3 /D8/CW/CX\r /CZ /D0/CX/D2/CT/D7/B8krL= 9,5,4,3,2,1.8,1.5,1.25 /B4∆l= 0 /B8\nqr= 0.4 /B8 /CP/D2/CSfr= 1050Hz /B5/BA/C1/D2 /BY/CX/CV/BA /BL /CP/D6/CT /D7/CW/D3 /DB/D2 /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CS/CX/CP/CV/D6/CP/D1/D7 /CU/D3/D6 /CP /CW/CT/CP /DA/B9/CX/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BA /BY /D6/D3/D1 /D8/CW/CT /AT/CP/D8/D8/CT/D7/D8 /D8/D3/D2/CT /B4krL= 9.00 /B5 /D8/D3/D8/CW/CT /D7/CW/CP/D6/D4 /CT/D7/D8 \r/D3/D1/D4/D9/D8/CT/CS /B4krL= 1.25 /B5/B8 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW/DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7 /CP/D6/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /CP/D7 /D8/CW/CT /CS/CX/CP/CV/D6/CP/D1/D7 /D7/CW/D3 /DB/C0/D3/D4/CU /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /CP/D6/CT /D7/D9/D4 /CT/D6\r/D6/CX/D8/CX\r/CP/D0/B8 /CX/BA/CT/BA/B8 /CS/CX/D6/CT\r/D8/B8 /CU/D3/D6 /CP/D0/D0/D8/CW/CT \r/D3/D1/D4/D9/D8/CT/CS \r/CP/D7/CT/D7/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CU/D3/D6 \r/CP/D7/CT/D7 /DB/CW/CT/D6/CT/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CX/D7 \r/D0/D3/D7/CT/CS /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT /B4/BY/CX/CV/BA /BD/BC /B5/B8/CP /D7/D9/CQ \r/D6/CX/D8/CX\r/CP/D0 /C0/D3/D4/CU /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D3 \r\r/D9/D6/D7 /CU/D3/D6 /CP /D6/CP/D2/CV/CT /D3/CU /CQ /D3/D6/CT/D0/CT/D2/CV/D8/CW/B8 /DB/CX/D8/CW /CP /DA /CT/D6/DD /D7/D1/CP/D0/D0 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D0/CX/D2/CZ 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product\nin dissipative phase-space quantum mechanics\nB. Belchev, M.A. Walton\nDepartment of Physics, University of Lethbridge\nLethbridge, Alberta, Canada T1K 3M4\nborislav.belchev@uleth.ca, walton@uleth.ca\nMarch 1, 2022\nAbstract\nDito and Turrubiates recently introduced an interesting mo del of the\ndissipative quantum mechanics of a damped harmonic oscilla tor in phase\nspace. Its key ingredient is a non-Hermitian deformation of the Moyal star\nproduct with the damping constant as deformation parameter . We com-\npare the Dito-Turrubiates scheme with phase-space quantum mechanics\n(or deformation quantization) based on other star products , and extend\nit to incorporate Wigner functions. The deformed (or damped ) star prod-\nuct is related to a complex Hamiltonian, and so necessitates a modified\nequation of motion involving complex conjugation. We find th at with this\nchange the Wigner function satisfies the classical equation of motion. This\nseems appropriate since non-dissipative systems with quad ratic Hamilto-\nnians share this property.\n11 Introduction\nThe quantum mechanics of dissipative systems has been studied inte nsively; for\nreviews, see [1, 2, 3]. Work started soon after the birth of quantu m mechanics\nand continues today.\nPerhaps the most fundamental approachis to consider the syste m of interest\nas interacting with an appropriate reservoir. Then the well-known q uantization\nmethods that are valid for non-dissipative, closed systems can be a pplied to the\nsystem plus reservoir as a whole. Effective equations of motion for t he system\ncan then be found by integrating out the reservoir degrees of fre edom, while\nmaking appropriate physical assumptions.\nAnother technique is to work backwards, and derive the effective e quations\nby adapting quantization procedures to non-dissipative systems. The adapted\nquantization procedures should, in the end, agree with more funda mental treat-\nments. Provided they do, they would be helpful, as shortcuts for t he more\nfundamental derivations, and possibly more.\nCanonical (operator) quantization has been adapted to dissipativ e systems\neither by using time-dependent Hamiltonians, complex Hamiltonians, o r by\nmodifying the canonical Poisson brackets, and the corresponding operator com-\nmutators.1\nWe concern ourselves here, however, with a different quantization method:\nquantum mechanics in phase space, or deformation quantization (s ee Sect. 2).\nSpecifically, we study Dito and Turrubiates [8] recent adaptation o f deforma-\ntion quantization to the paradigmatic dissipative system, the dampe d harmonic\noscillator.\nThe important innovation they introduce is a non-Hermitian γ-deformation\nof the Moyal star product of non-dissipative deformation quantiz ation, where\nγdenotes the damping constant. Their damped star product is built in turn\non aγ-deformation of the classical Poisson bracket, and so recalls the m odified\nbrackets of canonical operator quantum mechanics adapted to d issipation that\nwerejust mentioned. Perhapsthis is not surprising, since canonica lclassicalme-\nchanics is a key ingredient of deformation quantization; the algebra of quantum\nobservables is described as an ¯ h-deformation of the classical Poisson algebra on\nphasespace. Unlike in other schemes, the one proposedby Dito and Turrubiates\nonlymodifies the classical bracket and thereby the corresponding qua ntum star\nproduct–itusestheundamped Hamiltonian, forexample. TheDito-T urribiates\n1Some relatively recent works with the latter approach are [4 , 5, 6, 7].\n2proposal is at least economical, since in other formulations involving m odified\nbrackets, extra structure must be input.\nThis paper is organized as follows. The next section is a brief review of\nordinary (undamped) quantum mechanics in phase space. Section 3 studies the\nDito-Turrubiates scheme and extends it to include Wigner functions and their\nevolution. The final section is a discussion, consisting of a concise su mmary,\nand a list of some of the many questions that remain.\n2 Phase-space quantum mechanics\nWe now give a brief review of phase-space quantum mechanics, to es tablish\nour notation and provide the essential background for next sect ion’s discussion\nof the Dito-Turrubiates scheme. Throughout this paper our trea tment will be\nlimited to one non-relativistic particle travelling on the real line with coo rdinate\nq, and conjugate momentum p.\nFor pedagogical introductions to phase-space quantum mechanic s (or defor-\nmation quantization), see [9, 10], for examples; and for more advan ced reviews,\nsee the book [11], and references therein, including the pioneering work [12].\nIn our context, operators will be expressible entirely in terms of th e operator\nposition ˆqandmomentum ˆ p. Thesemoregeneraloperatorswillalsobeindicated\nby carets. Phase-space quantum mechanics has a direct relations hip with the\nmore commonly used operator quantum mechanics. The Wigner tran sformW\nmaps operators to functions and distributions on phase space:\nW/parenleftBig\nˆf/parenrightBig\n=f(p,q). (1)\nHereˆfis an operator, and f(p,q) is the corresponding phase-space distribution,\nsometimes called the symbolofˆf. The most important property of the Wigner\ntransform is encoded in\nW(ˆfˆg) = (Wˆf)∗(Wˆg). (2)\nConsequently, operators can be Wigner-mapped to symbols whose algebra is\nhomomorphic to the original operator algebra, as long as the symbo ls are mul-\ntiplied using the so-called star-product ( ∗-product)\n∗= exp/bracketleftBigi¯h\n2(←\n∂q→\n∂p−←\n∂p→\n∂q)/bracketrightBig\n. (3)\nOperators can therefore be replaced by functions/distributions on phase space,\nand quantum mechanics can be done in phase space, without refere nce to oper-\nators.\n3The Moyal star product ∗inherits associativity from the operator product,\nby (2). Since ( ˆfˆg)†= ˆg†ˆf†, we also have\n(f∗g) =g∗f . (4)\nHerefindicates the complex conjugate of f. We say that ∗is Hermitian.\nThe inverse of the Wigner transform is familiar in operator quantizat ion.\nThe so-called Weyl map W−1, satisfies\nW−1(f∗g) = (W−1f)(W−1g). (5)\nThe image of a polynomial in qandpof this Weyl map is the corresponding\nWeyl-ordered operator. The famous Moyal star product ∗of (3) and (5) is\ntherefore intimately related to Weyl ordering. With other operato r orderings,\nother∗-products arise.\nThe operator encoding information about the quantum state is the density\nmatrix, and its Wigner transform is the central object of phase-s pace quantum\nmechanics. Up to normalization, it is the celebrated Wigner function.\nTo consider evolution, start with the Liouville Theorem for the classic al\nphase-space density ρc:\ndρc\ndt=∂ρc\n∂t+{ρc,H}= 0. (6)\nThe Dirac correspondence then yields the equation of motion of the density\noperator (matrix) ˆ ρ\ni¯h∂ˆρ\n∂t+ [ ˆρ,ˆH] = 0. (7)\nThe Wigner transform of this is the evolution equation for the Wigner function\nρ=ρ(p,q;t)\ni¯h∂ρ\n∂t+ [ρ, H]∗= 0, (8)\nwhere the ∗-commutator is\n[a, b]∗=a∗b−b∗a . (9)\nEquivalently, the phase-space version of the Dirac quantization ru le,\n{a,b} →[a, b]∗\ni¯h, (10)\nleads directly from (6) to (8).\nDefining\nadf[∗]g:= [f , g]∗, (11)\n4(8) can be rewritten as\ni¯h∂ρ\n∂t= adH[∗]ρ=:i¯hLρ . (12)\nThe solution is just\nρ(p,q;t) = exp/braceleftbigg−it\n¯hadH[∗]/bracerightbigg\nρ(p,q;0) = exp {tL}ρ(p,q;0).(13)\nUsing the associativity of ∗-multiplication, this reduces to\nρ(p,q;t) =U(p,q;t)∗ρ(p,q;0)∗U(p,q;−t), (14)\nwith\nU(p,q;t) =∞/summationdisplay\nn=01\nn!/parenleftbigg−itH\n¯h/parenrightbigg∗n\n= Exp[∗]/parenleftbigg−itH\n¯h/parenrightbigg\n.(15)\nHere\nExp[∗](a) :=∞/summationdisplay\nn=01\nn!a∗n(16)\ndenotes the so-called ∗-exponential. Note that U(p,q;t) =U(p,q;−t), and\nU(p,q;t)∗U(p,q;t) =U(p,q;t)∗U(p,q;t) = 1. (17)\nAlternatively, substituting (14) into (8) yields the dynamical equat ion\ni¯h∂tU(p,q;t) =H∗U(p,q;t), (18)\nand its complex conjugate. Its solution, the symbol of the propag ator, is (15).\nThe spectrum ofenergies and the Wigner functions ofstationarys tates can then\nbe found by the spectral decomposition (or Fourier-Dirichlet expa nsion)\nU(p,q;t) =/summationdisplay\nEρE(p,q)e−iEt/¯h, (19)\nwhere\nH∗ρE=ρE∗H=EρE. (20)\nFor the simple harmonic oscillator, with Hamiltonian\nH=p2\n2m+1\n2mω2q2, (21)\nthe well-known results are\nU(p,q;t) =1\ncos(ωt/2)exp/braceleftbigg2Htan(ωt/2)\ni¯hω/bracerightbigg\n, (22)\n5and\nρEn= 2(−1)nLn/parenleftbigg4H\n¯hω/parenrightbigg\nexp/parenleftbigg\n−2H\n¯hω/parenrightbigg\n, (23)\nwhereEn= ¯hω(n+1/2), andLnis then-th Laguerre polynomial.\nThe time evolution invokes more than the pure-state, or diagonal W igner\nfunctions, however. We need the off-diagonal\nρE,E′=1\n2π¯hW(|E/an}bracketri}ht/an}bracketle{tE′|), (24)\nsatisfying the ∗-eigen equations\nH∗ρE,E′=EρE,E′, ρE,E′∗H=E′ρE,E′; (25)\nso thatρE,E=ρE. Assuming that the ∗-eigen functions (25) are complete, we\nexpand\nρ(p,q;t) =/summationdisplay\nE,E′RE,E′(t)ρE,E′(p,q). (26)\nSubstituting into the equation of motion then gives the time-evolved Wigner\nfunction\nρ(p,q;t) =/summationdisplay\nE,E′RE,E′(0) exp/bracketleftbigg−i(E−E′)t\n¯h/bracketrightbigg\nρE,E′(p,q).(27)\nTo close this section, we will discuss alternatives to the Moyal star p roduct\n(see [10] and [12], e.g.). First, consider operator-ordering ambiguit ies. Since\nthe Moyal product is intimately related to Weyl ordering, a different operator\nordering will lead to a different star product. For example, if we use t he so-\ncalled standard ordering, where every ˆ qis placed to the left of every ˆ p, we find\ninstead the standard star product\n∗S=ei¯h←\n∂q→\n∂p=∗ei¯h\n2(←\n∂q→\n∂p+←\n∂p→\n∂q). (28)\nSince the two orderings can be simply related, there is a nice relation b etween\nthe two star products:\nTS(f∗Sg) = (TSf)∗(TSg), (29)\ninvolving the invertible transition operator\nTS= exp/braceleftbigg\n−i¯h\n2∂q∂p/bracerightbigg\n. (30)\nAn ordering change preserves a bi-grading in the powers of pandq. By the\nHeisenberg commutation relation, ¯ hhas bi-grade (1,1) and so that of TSis\n(0,0).\n6This kind of connection between different star products can be ext ended. In\ngeneral, a star product ˜∗is considered equivalent to the Moyal one ∗, if we have\n˜T(f∗g) = (˜Tf)˜∗(˜Tg). (31)\nHere˜Tstands for an invertible transition operator. If ˜∗, like∗, is an ¯h-\ndeformation of the point-wise product of functions, then\n˜T= 1 + O(¯h). (32)\nAssociativity of ˜∗is guaranteed by the relation (31), if ˜Tis invertible:\nf˜∗g˜∗h=˜T/parenleftBig\n(˜T−1f)∗(˜T−1g)∗(˜T−1h)/parenrightBig\n. (33)\nIf˜Tisreal,then ˜∗willbeHermitian, as ∗is.∗Sisanexampleofanon-Hermitian\nproduct, since TSis not real.\nBy the Liebniz product rule, the equivalence relation (31) can be rew ritten\nas\n˜∗=∗˜T−1[←\n∂]˜T[←\n∂+→\n∂]˜T−1[→\n∂]. (34)\nHere∂stands for either ∂qor∂p,\n˜T[←\n∂+→\n∂] :=˜T|\n∂→←\n∂+→\n∂, (35)\nand similarly for ˜T−1[←\n∂] and˜T−1[→\n∂]. For example,\nTS[←\n∂+→\n∂] = exp/braceleftbigg\n−i¯h\n2(←\n∂q+→\n∂q)(←\n∂p+→\n∂p)/bracerightbigg\n, (36)\nby (30).\nEqn. (34) makes it clear that equivalent star products ∗and˜∗only differ by\na factor that is left-right symmetric. Consequently, it can be show n that\nlim\n¯h→0[f,g]˜∗\ni¯h= lim\n¯h→0[f,g]∗\ni¯h={f,g}, (37)\nthe Poisson bracket.\nThe equivalence between ˜∗and∗defined by (31) is known as c-equivalence.\nIt is important because the Moyal ∗-product (3) is the unique, associative de-\nformation of the point-wise multiplication of functions on R2(andR2n), up to\nc-equivalence.\nThe c in c-equivalence stands for cohomological. As such, it is a mathe mat-\nical, rather than a physical equivalence. For example, consider the Husimi star\nproduct∗H(see [13], e.g.)\n∗H=∗exp/braceleftbigg¯h\n2(s2←\n∂q→\n∂q+1\ns2←\n∂p→\n∂p)/bracerightbigg\n, (38)\n7related to ∗by the transition operator\nTH= exp/braceleftbigg¯h\n4/parenleftbig\ns2∂2\nq+1\ns2∂2\np/parenrightbig/bracerightbigg\n. (39)\nIt is also easy to see that ∗Hsatisfies the condition (37). But ∗His used with\nthe Husimi phase space distribution\nρH(p,q;t) :=THρ(p,q;t) (40)\nwhichdoesnotencodepreciselythesamephysicsastheWignerfunc tionρ(p,q;t).\nEqn. (40) can be rewritten as\nρH(p,q;t) =1\nπ¯h/integraldisplay\ndp′dq′ρ(p′,q′;t)\n×exp/braceleftbigg\n−1\n¯h/bracketleftbigg(q−q′)2\ns2+s2(p−p′)2/bracketrightbigg/bracerightbigg\n,(41)\nindicating that the Husimi distribution is a smoothed version of the Wig ner\nfunction, coarse-grained by a squeezed2Gaussian weighting in phase space. So,\nalthough ∗His c-equivalent to the Moyal product, it describes different physics .\n3 The damped harmonic oscillator in phase-\nspace quantum mechanics\nWe start in this section with a review of the Dito and Turrubiates [8] mo del\nof damping in a quantum harmonic oscillator in phase space, with comme nts\nadded. A new contribution will then be described: our incorporation of Wigner\nfunctions and their evolution. First, a na¨ ıve proposal will be examin ed, and\nrejected, since it leads to an unacceptable evolution equation. The equation\nof motion will then be modified, and explained. Finally, its consequence s are\noutlined. One such consequence is that the Wigner function of the d amped\nharmonic oscillator follows the canonical flow. This property is also co mmon to\nall non-dissipative systems having quadratic Hamiltonians, including t he simple\n(undamped) harmonic oscillator. Since the damped and undamped ha rmonic\noscillator and such systems are treated similarly in some other appro aches, this\nseems physically reasonable.\nDito and Turrubiates [8] describe the damped harmonic oscillator usin g the\nHamiltonian of the undamped simple harmonic oscillator. In a sense the n, they\n2More precisely, the distribution functions for general swere introduced in [14, 15], while\nthe Husimi distribution has s= 1. We call the Gaussian weighting of (41) squeezed because\nit is proportional to the Wigner transform of a squeezed stat e.\n8describe its dissipation as kinematics. More precisely, the dissipation is encoded\nin a deformation of the classical Poisson bracket of the harmonic os cillator and\nits consequent quantum ∗-product.\nTheinitialobservationisthattheclassicalequationsofmotionofth edamped\nharmonic oscillator can be written as\n˙q={q,H}γ=p/m; ˙p={p,H}γ=−mω2q−2γp , (42)\nusing the undamped Hamiltonian (21). The price paid is that we must use a\ndeformed Poisson bracket\n{f,g}γ:={f,g} −2γm∂pf ∂pg\n=f/parenleftBig←\n∂q→\n∂p−←\n∂p→\n∂q−2γm←\n∂p→\n∂p/parenrightBig\ng , (43)\nwith the damping constant γas deformation parameter. Notice that the de-\nformed bracket is no longer skew,3and doesn’t obey the Jacobi identity.\nThis result applies to a general class of classical systems: if we use a ny\nHamiltonian of the form\n˜H=p2\n2m+V(q), (44)\nwith an arbitrary potential V(q), the same damping term 2 mγ˙qis still the only\nmodification of the equation of motion for q(t). We will restrict attention here\nto the simple harmonic oscillator and its damped version, however.\nJust as the Moyal product is related to the Poisson bracket, the D ito-\nTurrubiates damped ∗-product is obtained by exponentiation of the deformed\nclassical bracket (43):\n∗γ:= exp/bracketleftBigi¯h\n2(←\n∂q→\n∂p−←\n∂p→\n∂q−2γm←\n∂p→\n∂p)/bracketrightBig\n=∗e−i¯hγm←\n∂p→\n∂p.(45)\nWhyexponentiate? Intheundampedcase,itisnecessaryfortheh omomorphism\nof the∗-algebra with the operator algebra. Also, the Moyal ∗-product is the\nunique associative deformation of the point-wise multiplication of fun ctions on\nR2n, up to isomorphism (by transition operators – see below). No similar r esult\nforthe dampedproductisknown; neitheristhe operatoralgebraf orthe damped\ncase. Exponentiation is therefore an assumption.\n3The deformed bracket can be obtained from the Poisson bracke t by the substitutions\n←\n∂q→←\n∂q−2αmγ←\n∂pand→\n∂q→→\n∂q+2βmγ→\n∂p, for any real αandβsuch that α+β= 1. The\nsign difference in front of γbetween the substitutions for the left- and right-acting de rivatives\nforeshadows our proposal (70).\n9On the other hand, suppose that we posit\n∗γ:=F/parenleftbiggi¯h\n2(←\n∂q→\n∂p−←\n∂p→\n∂q−2γm←\n∂p→\n∂p)/parenrightbigg\n, (46)\nfor some function F. Since{·,·}γ→ {·,·}asγ→0, requiring that lim γ→0∗γ=\n∗selects the exponential function. The possibility that the form (46 ) is too\nrestrictive remains, however.\nOne key result of [8] is\nT(f∗g) = (Tf)∗γ(Tg), (47)\nwith\nT= exp/parenleftBig−i¯hmγ\n2∂2\np/parenrightBig\n. (48)\nThis can be provedeasily using (34). That is, the Dito-Turrubiates d amped star\nproduct is c-equivalent to the Moyal star product. As discussed in the previous\nsection, c-equivalence does not imply physical equivalence, and so t he damped\nstar product has the potential to describe the dissipation of a dam ped harmonic\noscillator.\nSo, what effect does the Dito-Turrubiates transition operator Thave? First,\nit is clear that Tdoes more than change the ordering, since it has no fixed\nbi-grade in the powers of pandq. Its effect is therefore more profound. An\nimportant example is\nT/parenleftbiggp2\n2m+V(q)/parenrightbigg\n=p2\n2m+V(q)−i¯hγ\n2; (49)\n(see also (58)).4Ttransforms a real Hamiltonian into a complex one.\nFor such a conversion to be possible, it is necessary, but not sufficie nt, that\nT/ne}ationslash=T. Another consequence of a non-real transition operator is that ∗γis not\nHermitian:\na∗γb=b∗−γa/ne}ationslash=b∗γa . (50)\nIts c-equivalence with the Moyal star product ensures that the d amped star\nproduct∗γis associative, for any value of the damping constant γ. For example,\n∗−γwill be useful later because of (50), and it is also associative. Howev er,\nproblems exist if any two different damping parameters γandγ′are used. Care\nmust be taken with regard to the order of multiplications. Clearly,\nf∗γ(1∗γ′g)/ne}ationslash= (f∗γ1)∗γ′g , (51)\n4This last result is also interesting in its own right, since i t recalls a different approach to\ndamped systems that uses complex Hamiltonians (see [1, 2], a nd also [16]).\n10ifγ/ne}ationslash=γ′, for example. A unique product such as a∗γb∗−γcdoes not (automat-\nically) exist.\nUse of a na¨ ıve substitution\n{a,b}γ?→[a, b]∗γ\ni¯h(52)\nleads to\n0 =∂ργ\n∂t+1\ni¯h[ργ,H]∗γ, (53)\nwhereργindicates the Wigner function of the dampedharmonic oscillator. This\nequation of motion integrates to\nργ(p,q;t) =Uγ(p,q;t)∗γργ(p,q;0)∗γUγ(p,q;−t). (54)\nHere\nUγ(p,q;t) :=∞/summationdisplay\nn=01\nn!/parenleftbigg−itH\n¯h/parenrightbigg∗γn\n= Exp[∗γ]/parenleftbigg−itH\n¯h/parenrightbigg\n(55)\nsatisfies the damped analogue of (18), i.e.\ni¯h∂tUγ(p,q;t) =H∗γUγ(p,q;t). (56)\nDito-Turrubiatessolve(56), withoutdiscussingtheevolutionofWig nerfunc-\ntions. They then Fourier-Dirichlet expand the solution to find the sp ectrum of\neigenvalues and eigenfunctions of\nH∗γργ,E=Eγργ,E. (57)\nUsing (47), one finds\nUγ(p,q;t) = Exp[ ∗γ]/parenleftbigg−itH\n¯h/parenrightbigg\n=T/parenleftBig\nExp[∗]/parenleftbigg−itT−1(H)\n¯h/parenrightbigg/parenrightBig\n=T/parenleftBig\nExp[∗]/parenleftbigg−it[H+i¯hγ/2]\n¯h/parenrightbigg/parenrightBig\n=eγt/2T/parenleftBig\nU(p,q;t)/parenrightBig\n.(58)\nUsing (22) and (48), this gives [8]\nUγ(p,q;t) =exp(γt/2)\ncos(ωt/2)/bracketleftbig\n1+2γ\nωtan(ωt/2)/bracketrightbig\n×exp/braceleftBigg\n2tan(ωt/2)\ni¯hω/parenleftBigg\np2\n2m[1+2γ\nωtan(ωt/2)]+1\n2mω2q2/parenrightBigg/bracerightBigg\n.(59)\nThe expansion of Uγgives the solutions of (57) in terms of the simple harmonic\noscillator analogues:\nργ,n=T/parenleftBig\nρEn/parenrightBig\n, (60)\n11with\nEγ,n=En+i¯hγ\n2=¯h\n2[(2n+1)ω+iγ]. (61)\nFor example, for n= 0 Dito and Turrubiates [8] find\nρEγ,0=2/radicalbig\n1−2iγ/ω\n×exp/braceleftbigg\n−2\n¯hω/parenleftbiggp2\n2m[1−2iγ/ω]+1\n2mω2q2/parenrightbigg/bracerightbigg\n. (62)\nHowever, there is a fundamental problem with the basic equation of motion,\neqn. (53). For the damped harmonic oscillator it yields\n0 =∂ργ\n∂t+1\ni¯h[ργ,H]∗γ=∂ργ\n∂t+1\ni¯h[ργ,H]∗+iγ¯h∂p∂qργ. (63)\nNotice that the term proportional to the damping constant γis not real. But\nthe eigenvaluesof the density matrix arethe populations, and they must be real.\nTime evolution must preserve the reality of the density matrix. The im aginary\ndamping term in (63) means that a real density matrix does not rema in real as\nit evolves.\nAn equally important flaw is revealed by the limit\nlim\n¯h→01\ni¯h[H,ργ]∗γ={H,ργ} /ne}ationslash={H,ργ}γ. (64)\nThis is a direct consequence of the c-equivalence of ∗γand∗. But this shows\nthat there is no damping in the classical limit. The classical limit of (53) is not\ncorrect!\nTo try to understand better the origin of the difficulties with the pur ported\nequation of motion (53), let us consider how it might be “derived”. No tice that\n(53) would result from the undamped evolution equation (8) by the s ubstitution\nργ?=T(ρ), (65)\nageneralizationof(60). Thissimpleidentificationisappealinginpartbe causeit\nis similar to (40), that defining the Husimi phase-space distribution ρH(p,q;t).\nThe Husimi equation of motion is found directly from that for the Wign er\nfunction, by substituting (40). Additional terms arise in the Husimi equation\nof motion compared to that for the Wigner function, since the coar se-graining\nevolves in time [13]. Just as the “twisting” by THmodifies the equation of\nmotion, so does application of the Dito-Turrubiates transition oper atorT. The\nphysical damping is meant to be introduced that way.\n12Following (41), one might hope for an interpretation of T(ρ) as the Wigner\ndistribution coarse-grained in momentum space. That point of view is not\nsensible, however, since the weighting would have to be Gaussian-like with an\nimaginaryexponent. Thisdoespointtotheoriginofthemainproblem, however:\nunlike the Husimi transition operator TH, the Dito-Turrubiates Tis not real.\nAs a consequence, eqn. (49) can hold: Ttransforms a real Hamiltonian into\na complex one. That is the crucial point here, we believe. Compare to a similar\nsituation – if a non-self-adjoint Hamiltonian is used, the equation of m otion of\nthe density operator is modified,5to\n0 =i¯h∂ˆρ\n∂t+ ˆρˆH−ˆH†ˆρ . (66)\nBy analogy, we should consider6\n−i¯h∂ργ\n∂t=ργ∗γH−ργ∗γH . (67)\nNotice that the right-hand side of this equation is purely imaginary, s o that\n∂ργ\n∂t=−2\n¯hIm/parenleftbig\nργ∗γH/parenrightbig\n. (68)\nThis implies that the reality of the Wigner function\nργ=ργ (69)\nis preserved in evolution.7We can therefore also write\n−i¯h∂ργ\n∂t=ργ∗γH−H∗−γργ. (70)\nWith this prescription it is easy to show that the classical limit makes se nse for\nthe simple harmonic oscillator Hamiltonian:\nlim\n¯h→0ργ∗γH−H∗−γργ\ni¯h={ργ,H}γ− {H,ργ}−γ\n2={ργ,H}γ.(71)\nWe emphasize that the relation (65) is then notobeyed.\nLet us now attempt to argue for (70) from other grounds. As a st arting\npoint, let us assume that the Liouville Theorem still holds, and try to m odify\nthe argument that led to (8) to justify the damped equation of mot ion (70). An\n5See [17], for example, where the analogous modified equation of motion for a Heisenberg\noperator was shown to lead to a quantum anomaly.\n6This form may possibly be related to the bi-orthogonal quant um mechanics discussed by\nCurtright and Mezincescu [18]. For related work in phase spa ce, see [19] and [20].\n7According to (68), if ργhad a non-zero imaginary part, it would not evolve.\n13important advantageofthe Dito-Turrubiatesmethod is that it only modifies the\nclassical brackets. That advantage would be lost if we need to input something\nto replace the Liouville Theorem. Therefore, we assume the equatio n of motion\nisdργ,c\ndt=∂ργ,c\n∂t+{ργ,c,H}γ= 0. (72)\nThe phase-space version of the Dirac quantization rule (10) above should there-\nfore be deformed to\n{a,b}γ→a∗γb−a∗γb\ni¯h=a∗γb−b∗−γa\ni¯h(73)\nfor real observables aandb, in order to recover the equation of motion (67).\nOne might be tempted to write\nργ(p,q;t)?=Uγ(p,q;t)∗γργ(p,q;0)∗−γUγ(p,q;t) (74)\nas a real solution to the equation of motion (70). But this expressio n is ambigu-\nous at best, as shown by the discussion around eqn. (51).\nLuckily, however, a formal solution to (70) canbe written, by deforming the\nundamped solution of (11-13). If we define\nad(γ)\n∗[f]g:=f∗−γg−g∗γf , (75)\nthen (70) is\ni¯h∂ργ\n∂t= ad(γ)\n∗[H]ργ=:i¯hLγργ. (76)\nThe solution is just\nργ(p,q;t) = exp/braceleftbigg−it\n¯had(γ)\n∗[H]/bracerightbigg\nργ(p,q;0) = exp/braceleftbig\ntLγ/bracerightbig\nργ(p,q;0).(77)\nThere is no associative ambiguity in this explicit solution.\nA more explicit result can be given immediately for the damped harmonic\noscillator, having\nLγ=mω2q∂\n∂p−p\nm/parenleftbigg∂\n∂q−2mγ∂\n∂p/parenrightbigg\n. (78)\nThe quantum evolution is (70), but that reduces to (72), with ργ,c→ργ. The\nquantum Wigner function satisfiesthe classical equation ofmotion. The Wigner\nfunction at time tis therefore simply\nργ(p,q;t) =f(pc(−t),qc(−t)), (79)\n14wherepc(t) andqc(t) characterize the classical (damped) trajectories in phase\nspace. Thus the quantum damped harmonic oscillatorfollows the clas sicalback-\nward flow of the phase-space coordinates. Perhaps this is not sur prising, since\nfor any quadratic Hamiltonian, such as that of the undamped simple h armonic\noscillator, the same results holds (see [11], e.g.).\n4 Discussion\nWe first summarize. We have shown how to describe the dynamics of W igner\nfunctions in the Dito-Turrubiates scheme [8]. The non-Hermitian da mped star\nproduct ∗γis c-equivalent to the Moyal product ∗. Therefore, if the evolution\nequation of the damped Wigner function only involves ∗γ-commutators, only\nPoisson brackets survive in the classical limit, rather than the damp ed bracket\n{·,·}γ. The classical limit would then be damping-free, and therefore incor rect.\nHowever, the damped transition operator Ttransforms the simple harmonic\noscillator Hamiltonian into a complex one, according to (49). Consequ ently, the\nevolution equation does not involve ∗γ-commutation, but must instead be (70).\nThis ensures that a real Wigner function remains real as it evolves, and the\nclassical limit is correct. Not only is the classical limit correct, it is exac t. The\ndamped harmonicoscillatorin this formalismthereforefollowsthe clas sicalflow,\na property shared with non-dissipative systems in phase space hav ing quadratic\nHamiltonians.\nMost helpful to us were (i) comparisons of the damped star produc t∗γ\nwith other physical star products that are also c-equivalent to th e Moyal∗, viz.\nthe standard product ∗S, and the Husimi product ∗H; and (ii) the Heisenberg\nequation of motion for an operator observable modified for a non-H ermitian\nHamiltonian (see [17], e.g.).\nLet us now conclude with a few of the many questions that remain. We hope\nprogress can be made toward their answers.\nShould the damped ∗γ-product be obtained by exponentiating i¯h{·,·}γ/2?In\nthe case at hand, much of the structure of the γ-deformed star product is irrel-\nevant. Since we use the quadratic Hamiltonian (21), the only terms t hat enter\nare those that are up to quadratic in←\n∂and→\n∂. In this sense, our main result has\na certain robustness. More work on this question would be helpful, h owever.\nIs there a deformed structure analogous to the Heisenberg-W eyl group that\nis relevant to the damped case? In the undamped case, the Heisenberg-Weyl\ngroup is the structure of paramount importance. The Moyal ∗-product simply\n15provides a ∗-realization of that group. As discussed around (51), associativit y\ncan be a problem in the damped case. But if (77) is an appropriate guid e,\nperhaps\nexp/braceleftBig\nad(γ)\n∗[ap+bq]/bracerightBig\necp+dq=ecp+dqe−i¯h(ad−bc+2mγac)(80)\ncan servethe purpose. Here a,b,canddareconstants, so that eap+bqandecp+dq\n∗-representelements ofthe Heisenberg-Weylgroup. When γ→0 in (80), a form\nof the defining ∗-relations of the Heisenberg-Weyl group is recovered.\nCan a solution to the damped equation of motion (70) analogou s to the un-\ndamped formula (27) be written? It is clear from (77) that the ∗-eigen equation\nof ad(γ)\n∗[H] is relevant, so one can start there. One possible ansatz follows,\nalthough it may only have relevance for very small γ. Suppose we find the\ntime-independent off-diagonal Wigner matrix elements ργ;E,E′=ργ;E,E′(p,q)\nsatisfying\nH∗−γργ;E,E′=Eργ;E,E′, ργ;E,E′∗γH=E′ργ;E,E′,(81)\nwhere the complex eigenvalues8are\nE=E+iλ ,E′=E′+iλ′, (82)\nwithE,E′∈R, andλ,λ′∈R+. If the real eigenfunctions of (81) exist and are\ncomplete at all times t, then we can expand\nργ(p,q;t) =/summationdisplay\nE,E′RE,E′(t)ργ;E,E′(p,q). (83)\nThe equation of motion then yields\nργ(p,q;t) =/summationdisplay\nE,E′RE,E′(0) exp/bracketleftbigg−i(E −E′)t\n¯h/bracketrightbigg\nργ;E,E′(p,q)\n=/summationdisplay\nE,E′RE,E′(0) exp/bracketleftbigg−i(E−E′)t\n¯h/bracketrightbigg\nexp/bracketleftbigg−(λ+λ′)t\n¯h/bracketrightbigg\nργ;E,E′(p,q).(84)\nIf this conjecture is correct, then the physically important ∗-eigen equations are\nthose given in (81). It would not be clear then that the phase-spac e functions\nof (60) are directly relevant.\nThe ansatz of (81-84) reduces to the undamped system when ¯ h→0. It is\nalsodevoidofunphysicalstationarystates–the imaginarypartso feigenvalues λ\n8For the rolesuch complex eigenvalues play in another descri ption ofthe damped harmonic\noscillator, see [21]. Wigner functions involving such eige nvalues are considered in [22].\n16andλ′bothcontributetothedampingofthedynamics. Furthermore,thesys tem\nis consistent with (79) if ( E −E′)/¯his independent of ¯ h. Such eigenvalues were\nobtained in [8], as indicated in (61). However, the corresponding solu tions (60)\nare not real – see (62), e.g.\nCan the Dito-Turrubiates scheme be related to other quantiz ation methods\nfor dissipative systems (see [1, 2, 3])? Certainly, hints of connections with other\nmodels are apparent. For example, Dekker’s use of a complex Hamilto nian is\nrecalled by T(H) =H−i¯hγ/2. Bateman’s doubled system of a damped and\nanti-damped oscillator comes to mind from ad(γ)\n∗[H]ρ=H∗γρ−ρ∗−γH; in\nthis last expression, however, the anti-damped ( γ→ −γ) system is dual rather\nthan extra/auxiliary. The importance of resonances has been emp hasized by\n[22, 21] and others, and seems relevant to the ∗γ- and∗−γ-eigen equations of\nthe previous paragraph. Finally, if the correspondence (65) were correct, then\nwe would be able to define a damped Wigner transform\nWγ:=T−1W=WˆT−1, (85)\nwhere\nˆT−1= exp/braceleftbigg\n−im\n2¯h/parenleftBig\nad[ˆx]/parenrightBig2/bracerightbigg\n. (86)\nThe form of this ˆTappears to be consistent with Tarasov’s [23] proposal to\ninclude superoperators in the operator formulation of quantum me chanics in\norder to adapt it to dissipative systems.\nCan other dissipative physical systems be treated in a simil ar way? One very\nuseful example might be spin systems, with relaxation.\nInstead of dissipation, might other physical effects, such a s decoherence, be\ndescribable in the Dito-Turrubiates manner?\nAcknowledgements\nWe thank our colleagues Saurya Das, Arundhati Dasgupta, and So urav Sur for\nuseful discussion. M.A.W. also thanks M. Razavy for a helpful conve rsation.\nThisworkwassupportedinpartbyaDiscoveryGrantfromtheNatu ralSciences\nand Engineering Research Council (NSERC) of Canada.\n17References\n[1] H. Dekker, Phys. Rep. 80 (1981) 1.\n[2] M. Razavy, Classical and Quantum Dissipative Systems, Imperial College\nPress, London, 2005.\n[3] U. Weiss, Quantum Dissipative Systems, 2nd ed., World Scientific, S inga-\npore, 1999.\n[4] A.N. Kaufmann, Phys. Lett. A 100 (1984) 419.\n[5] M. Grmela, Phys. Lett. A 111 (1985) 36.\n[6] C.P. Enz, Found. Phys. 24 (1994) 1281.\n[7] G. Bimonte, G. Esposito, G. Marmo, C. Stornaiolo, Phys. Lett. A 318\n(2003) 313.\n[8] G. Dito, F. Turrubiates, Phys. Lett. A 352 (2006) 309.\n[9] J. Hancock, M.A. Walton, B. Wynder, Eur. J. Phys. 25 (2004) 52 5.\n[10] A.C. Hirshfeld, P. Henselder, Am. J. Phys. 70 (2002) 537.\n[11] C. Zachos, D. Fairlie, T. Curtright, Quantum Mechanics in Phase Space,\nWorld Scientific, Singapore, 2005.\n[12] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheim er, Ann.\nPhys. 111 (1978) 61 and 111.\n[13] K. Takahashi, Prog. Theor. Phys. Suppl. 98 (1989) 109.\n[14] N.D. Cartwright, Physica 83 A (1976) 210.\n[15] A.K. Rajagopal, Phys. Rev. A 27 (1983) 558.\n[16] S.G. Rajeev, Ann. Phys. 322 (2007) 1541.\n[17] J.G. Esteve, Phys. Rev. D 66 (2002) 125013.\n[18] T. Curtright, L. Mezincescu, J. Math. Phys. 48 (2007) 09210 6.\n[19] T. Curtright, A. Veitia, J. Math. Phys. 48 (2007) 102112.\n[20] F.G. Scholtz, H.B. Geyer, J. Phys. A: Math. Gen. 39 (2006) 101 89.\n[21] D. Chruscinski, Ann. Phys. 321 (2006) 854.\n18[22] D. Chruscinski, preprint arXiv:math-ph/0209008 (2002).\n[23] V.E. Tarasov Phys. Lett. A 288 (2001) 173.\n19"    },    {        "title": "0810.4633v1.The_domain_wall_spin_torque_meter.pdf",        "content": "The domain wall spin torque-meter\nI.M. Miron, P.-J. Zermatten, G. Gaudin, S. Au\u000bret, B. Rodmacq, and A. Schuhl\nSPINTEC, CEA/CNRS/UJF/GINP,\nINAC, 38054 Grenoble Cedex 9, France\n(Dated: September 15, 2021)\nAbstract\nWe report the direct measurement of the non-adiabatic component of the spin-torque in domain\nwalls. Our method is independent of both the pinning of the domain wall in the wire as well as\nof the Gilbert damping parameter. We demonstrate that the ratio between the non-adiabatic and\nthe adiabatic components can be as high as 1, and explain this high value by the importance of\nthe spin-\rip rate to the non-adiabatic torque. Besides their fundamental signi\fcance these results\nopen the way for applications by demonstrating a signi\fcant increase of the spin torque e\u000eciency.\nPACS numbers: 72.25.Rb,75.60.Ch,75.70.Ak,85.75.-d\n1arXiv:0810.4633v1  [cond-mat.other]  25 Oct 2008The possibility of manipulating a magnetic domain wall via spin torque e\u000bects when\npassing an electrical current through it opens the way for conceptually new devices such as\ndomain wall shift register memories[1]. Early spin-torque theories[2, 3, 4] were based on a\nso called adiabatic approximation which assumed that the incoming electron's spin follows\nexactly the magnetization as it changes direction within the domain wall. Nevertheless, the\nobserved critical currents needed to trigger the domain wall motion were lower than the\nvalue predicted within this framework[5]. As \frst predicted by Zhang[6], the existence of a\nnon-adiabatic term in the extended Landau-Lifshitz-Gilbert equation leads to the vanishing\nof the intrinsic critical current. The action of this non-adiabatic torque on a DW is expected\nto be identical to that of an easy axis magnetic \feld. Micromagnetic simulations have been\nused to predict the velocity dependence on current for a DW submitted to the action of the\ntwo components of the spin-torque[7]. The quantitative measurement of this non-adiabatic\ntorque can be achieved either by demonstrating the equivalence of \feld and current in\na static regime, or by observing the complex dynamic behavior[7]. The main di\u000eculty\nof these measurements comes from the pinning of the DW by material imperfections. It\nmasks the existence of the intrinsic critical current, and in addition, above the depinning\ncurrent, obscures the DW velocity dependence on current. Moreover, most of the DW\nvelocity measurements were done using materials with in-plane magnetization[5, 8, 9, 10,\n11], where the velocity can also depend on the micromagnetic structure of the wall[12]\n(transverse wall or vortex wall). Despite the simpler micromagnetic structure of the DWs,\nvery few results were reported[13] for Perpendicular Magnetic Anisotropy (PMA) materials.\nIn this case the intrinsic pinning is much stronger, probably due to a local variation of\nthe perpendicular anisotropy. Up to now, none of the measurements were able to clearly\nevidence the equivalence between \feld and current, nor to reproduce the predicted dynamic\nbehavior; hence the value of the non-adiabatic torque is still under debate.\nIn this letter we use a novel approach for the measurement of the non-adiabatic component\nof spin-torque. Instead of measuring the DW velocity, we perform a quasistatic measurement\nof its displacement under current and magnetic \feld. In principle this method is similar to\nany quasi-static force measurement: a small displacement is created, \frst with the unknown\nforce and then with a known reference force. In our case the unknown force is caused by\nthe electric current passing through the DW while the reference force is due to an applied\nmagnetic \feld. By comparing the two displacements one directly compares the applied\n2FIG. 1: Schematic representation of the experimental setup. The inset shows an SEM picture of a\nsample.\nforces. Due to the high sensitivity of our method (able to detect DW motion down to\n\u001810\u00002nm[14]) we can study the displacement of the DW inside its pinning center. Since the\nmeasurement relies on the comparison to a reference force, the method is independent of\nthe strength of the pinning. Moreover, as the \feld and current are applied quasi-statically,\nthe damping parameter does not play any role.\nAccording to recent theories[6, 15, 16] that derived the value of the spin torque, \f(the\nratio between the non-adibatic and adiabatic torques) is given by the ratio between the\nrate of the spin-\rip of the conduction electrons and that of the s-d exchange interaction.\nGenerally, two conditions must be ful\flled to obtain a high spin-\rip rate. First it is necessary\nto have a strong crystalline \feld inside the material. The electric \felds will yield a magnetic\n\feld in the rest frame of the moving electrons. Second, a breaking of the inversion symmetry\nis needed. Otherwise the total torque of the magnetic \feld on the electron spin averages\nout, and the spin-\rip may only occur during momentum scattering[17].\nIn order to highlight these e\u000bects we have patterned samples from a Pt 3nm/\nCo0.6nm/(AlO x)2nmlayer[18]. In this case the symmetry is broken by the presence of the\n3AlO xon one side of the Co layer, and of the heavy Pt atoms on the other[19, 20]. We\nwill emphasize the importance of the spin-\rip interaction to spin torque by comparing\nresults from these samples with those for samples fabricated from a symmetric Pt 3nm/\nCo0.6nm/Pt 3nmlayer[21], where a much smaller spin-\rip rate is expected. As the only di\u000ber-\nence between the two structures is the upper layer, we expect similar growth properties for\nthe Co layer. Both samples exhibit PMA and a strong Anomalous Hall E\u000bect (AHE)[22].\nThe \flms are patterned into the shape depicted in Figure 1. This shape is well suited for a\nquasi static measurement as a constriction is created by the presence of the four wires used\nfor the AHE measurement (\fgure 1 inset). This way a DW can be pinned in a position\nwhere changes in the out of plane component of the magnetization (i.e. DW motion) can\nbe detected by electrical measurements. A current is passed through the central wire. This\ncurrent will serve to push the domain wall as well as to probe the eventual displacement.\nIn the case where the DW does not move under the action of the current, the transverse\nresistance remains unchanged and the voltage measured across the side wires (AHE) will be\nlinear with the current. If the DW moves due to the electric current, the exciting force will\ncreate resistance variations, causing a nonlinear relationship between the measured voltage\nand the applied current. A simple way to detect such nonlinearities is to apply a perfectly\nharmonic low frequency (10 Hz) ac current, and look at the \frst harmonic in the Fast Fourier\nTransform (FFT) of the measured voltage. Its value is a measure of the amplitude of the\nDW displacement at the frequency of the applied current. To quantitatively compare the\naction of a magnetic \feld to that of an electric current, the magnetic \feld is applied at\nthe same frequency and in phase (or opposition of phase) with the electric current. By\napplying current and \feld simultaneously, we ensure that their corresponding torques act on\nthe same DW con\fguration. In addition to the displacement provoked by the current, the\n\feld induced displacement will add to the value of the \frst harmonic, which can be either\nincreased if the \feld and current push the wall in the same direction, or decreased if they\nact in opposite directions.\nFigure 2 shows the dependence of the resistance variation at the frequency of the current\n(\u0001R f) on the current amplitude for di\u000berent values of the \feld amplitude. First, at low\ncurrent and \feld amplitudes the displacement is almost linear ( \u0018107A/cm2), but for higher\nvalues, the \u0001R fvaries more rapidly. A simple estimation based on the value of the resistance\nvariation compared to the total Hall resistance of a cross (1 \n) yields \u00181 nm for the\n4FIG. 2: (a) Dependence of the resistance variation on the current amplitude for several \feld\namplitudes (Pt/Co/AlO xsample). The inset shows a possible nonlinear and asymmetric\npotential well. The energy landscape can be modeled by an e\u000bective out of plane magnetic\n\feld that has negative values on one side of the equilibrium position and positive values\non the other. (b) A zoom on the small amplitude regime. The inset shows the perfect\nsuperposition obtained by shifting the curves horizontaly with 1.25 \u0001105Acm\u00002Oe\u00001.5maximum amplitude of the DW motion in the \frst regime and \u00187 nm for the second regime.\nThis behavior can be explained by the anatomy of the local pinning. The local potential\nwell trapping the DW can be considered as a superposition of the geometric pinning [23]\nand intrinsic pinning caused by defects randomly distributed inside the material[13, 21].\nBecause the potential well for the small scale displacements (below 10nm) is dominated by\nthe random intrinsic pinning rather than geometric pinning (the increase of the length of\nthe DW is small \u00181%) in the general case it should be asymmetric. We have veri\fed the\nsupposed asymmetry of the e\u000bective potential well by applying alongside the ac current and\n\feld, a dc bias \feld that changes the local potential well (inset of \fgure 3). By varying\nthis \feld we observed a reduction of the current amplitude needed to access this strongly\nnon-linear regime (\fgure 3). When the magnetic bias \feld was reversed this second regime\nwas no longer attained with the available current densities (not shown).\nThe observed dependence of \u0001R fon current and \feld (\fgures 2 and 3) is in perfect\nagreement with the characteristic features of the non-adiabatic component of the spin-torque.\nFirst, we do not observe any critical current down to the lowest current value (106A/cm2-\n\fgure 4 in [14]). Futhermore, by extrapolating the amplitude of the DW displacement (\fgure\n2), when the current is reduced, the displacement goes to zero as the current goes to zero,\nin agreement with the absence of the critical current.\nHowever, the most important feature of the \u0001R fbehavior is that the curves obtained\nfor any \feld amplitude can be obtained from the curve corresponding to zero \feld just by\nshifting it horizontally (in current): towards the lower current values when the \feld and\ncurrent act in the same direction on the DW and towards higher values when their actions\nare opposed. This means that any displacement of the DW can also be achieved with a\ndi\u000berent current if a magnetic \feld is added. The di\u000berence in current is compensated by\nthe magnetic \feld. The value of this horizontal shift gives the \feld to current correspondence.\nThe inset of \fgure 1b shows that all the curves corresponding to di\u000berent \feld amplitudes\nhave the same shape; by shifting them horizontally (using the \feld-current correspondence),\nthey all collapse on the zero \feld amplitude curve. This shows that independently of the\ndirection or strength of the applied current and \feld, as predicted by the theories, their e\u000bect\non the DW is fundamentally similar. Moreover, further evidence that this correspondence is\nintrinsic and not in\ruenced by pinning is that its value remains the same within the di\u000berent\namplitude regimes as well as when the local potential well is tuned by a constant bias \feld.\n6FIG. 3: The nonlinear regime (Pt/Co/AlO xsample). When an external bias \feld is added,\nthe e\u000bective pinning \feld changes (inset) and the nonlinear regime is reached for di\u000berent\ncurrent and \feld amplitudes. However, this does not cause any change in the \feld to current\ncorrespondence: the horizontal distance between the curves remains the same.\n7Since the motion of the DW is quasi-static the magnetization can be considered to be\nat equilibrium during motion. In this case the sum of all torques must be zero. In order\nfor the DW to remain at rest, the torque from the applied current must be compensated\nby the torque generated by the magnetic \feld. The upturn observed on the -60 Oe curve\n(\fgure 2b) determines the position of the zero amplitude point. Note that the position of\nthis point is in perfect agreement with the \feld to current correspondence obtained from the\nhorizontal shifting of the curves. By taking into account the micromagnetic structure of the\nDW (very thin 5nm Bloch wall) the two torques are integrated over the width of the wall,\nand by comparing their values (the \feld torque is easily calculated; [14]) the non-adiabatic\nterm of the spin-torque is determined. In the case of Pt/Co/AlO xstacks the current-\feld\ncorrespondence is approximately 1.25 105A/cm2to 1Oe, corresponding to a value of \f= 1.\nSimilar measurements (\fgure 1 in [14]) were also performed in the saturated state (with-\nout the DW). They con\frm that there is no contribution to the signal from the ordinary\nHall e\u000bect, but indicate a small contribution from thermoelectric e\u000bects - the Nernst-\nEttingshausen E\u000bect(NEE)[24]. The contribution from DW motion to \u0001R fis much higher\nthan the NEE for the Pt/Co/AlO xstack. In the case of Pt/Co/Pt layers we \fnd that the\namplitude of the current induced DW motion is much smaller and entirely masked by the\nNEE. When a DW is moving inside the perfectly harmonic region at the bottom of the po-\ntential well, its displacement depends linearly on the applied force. In such a scenario, the\ncurrent induced DW motion and the NEE are indistinguishable. They both lead to a linear\ndependence of the \u0001R fresponse on current. The only possibility to separate these e\u000bects,\nfor the Pt/Co/Pt layer, is to attain the high amplitude nonlinear regime of DW motion.\nThis is done by keeping the current amplitude constant and varying the \feld amplitude.\nWhen the current and \feld push the wall in the same direction, the nonlinear regime should\nbe reached for smaller \feld amplitudes, than if their actions were opposed.\nIn the presence of current induced displacements, the nonlinearities observed in the\n\u0001R fversus \feld amplitude curve should be asymmetric. Moreover the asymmetry should\ndepend on the current value. Such an asymmetry is observed (inset of \fgure 4) in the case\nof Pt/Co/AlO xsamples. In contrast to this behavior, a fully symmetric dependence that\ndoes not depend on the current amplitude is measured for the Pt/Co/Pt samples (\fgure 4).\nWe conclude that in this case the spin torque induces DW displacements smaller than the\nresolution limit of this method. This limit value leads to (supplementary notes) \f\u00140.02.\n8FIG. 4: The nonlinear response of a DW to magnetic \feld. (a) \u0001R fvs. the amplitude of the \feld\nfor three di\u000berent current densities in the case of Pt/Co/Pt layers (inset Pt/Co/AlO x).(b)\nDerivative of \u0001R fvs. the \feld amplitude for a Pt/Co/Pt sample (inset Pt/Co/AlO x).\nTheoretical estimations[6] based on a spin-\rip frequency of 1012Hz yield a value \f=0.01.\nTo clarify the di\u000berence of the spin-torque e\u000eciency in the two samples, the symmetry\nbreaking due to the presence of the AlO xsurface must be taken into account. As a metallic\n\flm gets thinner, the conduction electron's behavior resembles more and more to that of a\ntwo-dimensional electron gas. When such a gas is trapped in an asymmetric potential well,\n9the spin-orbit coupling is much stronger than in the case of a symmetric potential due to\nthe Rashba interaction[25]. This e\u000bect was \frst evidenced in nonmagnetic materials where\nthis interaction leads to a band splitting (0.15 eV for the surface states of Au (111)[26]).\nIn the case of ferromagnetic metals this e\u000bect was already proposed to contribute as an\ne\u000bective magnetic \feld[27] for certain DW micromagnetic structures, but should not have\nany e\u000bect for Bloch walls in PMA materials. The simple 1D representation used in this\ncase[27] to model the DW accounts for the coherent rotation of the spins of the incoming\nelectrons around the e\u000bective \feld, but excludes any de-coherence between electrons having\ndi\u000berent k-vector directions on the Fermi sphere (di\u000berent directions of the Rashba e\u000bective\n\feld) as well as possible spatial inhomogeneities of this \feld (surface roughness). Since the\nspin-torque is caused by the cumulative action of all conduction electrons [6], the relevant\nparameter is not the spin-\rip rate of a single electron but the relaxation rate of the out\nof equilibrium spin-density [6]. In a more realistic 2D case, in the presence of the above\nmentioned strong decoherence e\u000bects, the relaxation rate of the out of equilibrium spin-\ndensity approaches the rate of spin precession around the Rashba e\u000bective \feld. The above\nvalue of the measured spin-orbit splitting (0.15 eV) will yield in this case an e\u000bective spin-\n\rip rate of 30 \u00011012Hz, which is in excellent agreement with the order of magnitude of the\nmeasured non-adiabatic parameter, supporting this scenario.\nIn summary, a technique that allows the direct measurement of the torque from an\nelectric current on a DW was developed. We have pointed out the importance of spin-\n\rip interactions to spin torque by comparing its e\u000eciency between two di\u000berent systems.\nWe show that the Pt/Co/AlO xsample with the required symmetry properties to increase\nthe spin-\rip frequency (breaking of the inversion symmetry) shows an enhanced spin torque\ne\u000bect. A value of the order of 1 was measured for the \fparameter approaching the maximum\nvalue predicted by existing theories. This value can be explained by order of magnitude\nconsiderations on the Rashba e\u000bect observed on surface states of metals. Obtaining a high\ne\u000eciency spin torque in a low coercivity material would make possible the development\nof nanoscale devices whose magnetization could be switched at low current densities. The\norder of magnitude of the current densities would be similar to the one observed for magnetic\n10semiconductors[28], but, as the resistance is smaller, the supplied power will be lower.\n[1] S. Parkin, US Patent 309,6,834,005 (2004).\n[2] L. Berger, J. Appl. Phys. 55, 1954 (1984).\n[3] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, J. Appl. Phys. 95, 70497051 (2004).\n[4] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[5] M. Klui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman,\nF. Nolting, and U. Rudiger, Phys. Rev. Lett. 94, 106601 (2005).\n[6] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[7] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett 69, 990996 (2005).\n[8] G. S. D. Beach, C. Knutson, C. 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Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. E. Prieto, K. Dbrich, S. Blgel, and\nG. Kaindl, Phys. Rev. B 71, 201403(R) (2005).\n[20] H. Cercellier, C. Didiot, Y. Fagot-Revurat, B. Kierren, L. Moreau, and D. Malterre, Phys.\nRev. B 73, 195413 (2006).\n11[21] P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferr, V. Baltz, B. Rodmacq, B. Di-\neny,and R. L. Stamps, Phys. Rev. Lett. 99, 217208 (2007).\n[22] C. L. Canedy, X. W. Li, and G. Xiao, Appl. Phys. Lett. 81, 5367 (1997).\n[23] J. Wunderlich, D. Ravelosona, C. Chappert, F. Cayssol, V. Mathet, J. Ferr, J.-P. Jamet, and\nA. Thiaville, IEEE Transactions on Magnetics 37, 2104 (2001).\n[24] B. J. Thaler, R. Fletcher, and J. Bass, J. Phys. F: Met. Phys. 8, 131 (1978).\n[25] L. Petersen and P. Hedegard, Surface science 459, 49 (2000).\n[26] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett. 77, 3419 (1996).\n[27] K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008).\n[28] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature 428, 539 (2004).\n12"    },    {        "title": "0811.0425v1.Amplitude_Phase_Coupling_in_a_Spin_Torque_Nano_Oscillator.pdf",        "content": "arXiv:0811.0425v1  [cond-mat.mtrl-sci]  4 Nov 2008Amplitude-Phase Coupling in a Spin-Torque Nano-Oscillato r\nKiwamu Kudo,∗Tazumi Nagasawa, Rie Sato, and Koichi Mizushima\nCorporate Research and Development Center, Toshiba Corpor ation, Kawasaki, 212-8582, Japan\n(Dated: October 31, 2018)\nThe spin-torque nano-oscillator in the presence of thermal fluctuation is described by the normal\nform of the Hopf bifurcation with an additive white noise. By the application of the reduction\nmethod, the amplitude-phase coupling factor, which has a si gnificant effect on the power spectrum of\nthe spin-torque nano-oscillator, is calculated from the La ndau-Lifshitz-Gilbert-Slonczewski equation\nwith the nonlinear Gilbert damping. The amplitude-phase co upling factor exhibits a large variation\ndepending on in-plane anisotropy under the practical exter nal fields.\nWhen a direct current Iflows into a magnetoresistive\n(MR) device, a stationary magnetic state becomes un-\nstable and a steady magnetic oscillation is excited by the\nspin-transfer torque. The oscillation is expected to be\napplicableto ananoscalemicrowavesource, i.e., the spin-\ntorque nano-oscillator (STNO).1,2According to the the-\nory based on the spin-wave Hamiltonian formalism,3,4,5,6\nthe frequency nonlinearity plays a key role in determin-\ning the behavior of the oscillator. It has been shown that\nthe strong frequency nonlinearity leads to significant ef-\nfects on the power spectrum of STNO in the presence\nof thermal fluctuation: a linewidth enhancement5and\nnon-Lorentzian lineshapes6. In this paper, the impor-\ntantnonlinearityisexamined. FromtheLandau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation as the model of\nSTNO,wecalculateexplicitlythemagnitudeofthequan-\ntity corresponding to the normalized frequency nonlin-\nearityN/Γeff(see, e.g., Eq. (4) in Ref. 6) of the spin-\nwaveapproach. Inparticular,wetakeaccountofin-plane\nanisotropy of a magnetic film which has been neglected\nin the early studies3,4,5,6, finding the large effect of the\nanisotropy on the nonlinearity.\nWe describe STNO by a generic oscillator model. It\nis known that small-amplitude oscillations near the Hopf\nbifurcation point are generally governed by the simple\nevolution equation for a complex variable W(t) known\nas the Stuart-Landau (SL) equation.7The SL equation\nis derived as a normal form of the supercritical Hopf bi-\nfurcation from the general system of ordinary differen-\ntial equations. Accordingly, the LLGS equation similarly\nreduces to the SL equation in the case where the Hopf\nbifurcation,whichrepresentsagenerationofmagneticos-\ncillations in STNO, occurs. The reduction of the LLGS\nequation can be executed by the reduction method based\non the center-manifold theorem. At finite temperature,\nthere exists inevitable thermal magnetization fluctuation\nin STNO.8,9We include the thermal effect into the mag-\nnetization dynamics by just adding white noise term to\nthe SL equation, i.e., STNO in the presence of thermal\nfluctuation is described by the ‘noisy’ Hopf normal form:\nd˜W\nd˜t=i˜Ω˜W+(1+iδ)(p−|˜W|2)˜W+η(˜t),(1)\nwhere˜Wis the normalized complex variable represent-\ning the amplitude and phase of a magnetization vec-torM(see Eq. (7) below). In Eq. (1), ˜Ω represents a\nfundamental frequency, ˜tis a normalized dimensionless\ntime, andη(˜t) is the zero-mean, white Gaussian noise\nwith the only non-vanishing second moment given by\n∝an}bracketle{tη(˜t)¯η(˜t′)∝an}bracketri}ht= 4δ(˜t−˜t′).pis the bifurcation parame-\nter. An oscillation is generated when pbecomes positive.\nIn the context of STNO, p∝(I−Ic) whereIcis the\nthreshold current. The parameter δquantifies the cou-\npling between the amplitude and phase fluctuations and\nis called the amplitude-phase coupling factor . It isδthat\nwe calculate numerically in this paper and that corre-\nsponds to the normalized frequency nonlinearity N/Γeff\nof the spin-wave approach. The amplitude-phase cou-\npling factor δaffects the power spectrum of an oscillator\nand leads to linewidth enhancement and non-Lorentzian\nlineshapes.10,11Due toits effect, the factor δisalsocalled\nthelinewidth enhancement factor .12Eq. (1) is often used\nas the simplest model of a noisy auto-oscillator in many\nfields, for example, electrical engineering, chemical reac-\ntions, optics, biology, and so on.10,13Therefore, we can\neasily compare STNO with conventional oscillators and\nclarify its features.\nThe amplitude-phase coupling factor δis obtained in\nthe procedure of the reduction of the LLGS equation. In\nthe following, we first explain the LLGS equation. Then,\nfollowing Kuramoto’s monograph7, we consider an insta-\nbility of a steady solution and execute the reduction of\nthe LLGS equation.\nThe magnetic energy density of the free layer of STNO\nis assumed to have the form\nE=−M·Hext−Ku\nM2s(M·ˆx)2+1\n24πM·N ·M,(2)\nwhereMsisthesaturationmagnetization, Hext=Hxˆx+\nHyˆy+Hzˆzis an external field, Kuis uniaxial anisotropy\nalong thexdirection, and Nis the demagnetizing ten-\nsor;N= diag(Nx,Ny,Nz). Using the spherical coordi-\nnate system (see Fig. 1), we describe the magnetization\ndynamics of STNO by the LLGS equation\n/braceleftBigg\ncosψ˙φ=−α(ξ)˙ψ−F1(φ,ψ,ω J)\n˙ψ=α(ξ)cosψ˙φ+F2(φ,ψ,ω J),(3)\nwhereF1(φ,ψ,ω J)≡(γ/Ms)∂E/∂ψ−a(φ,ωJ) and\nF2(φ,ψ,ω J)≡(γ/(Mscosψ))∂E/∂φ+b(φ,ψ,ω J).γis2\nφHH\nM\nxHxyyz\nψz\nuK\npI\nFIG. 1: The spherical coordinate system ( φ,ψ) for the direc-\ntion of the free layer magnetization m=M/Msof STNO.\npdenotes the direction of the pinned layer magnetization;\np= (cosψpcosφp,cosψpsinφp,sinψp).\nthe gyromagnetic ratio. The second terms of Fire-\nsult from the Slonczewski term TJ= (γaJ/Ms)M×\n(M×p) in which aJis proportional to the cur-\nrent density Jthrough the free layer14. Therefore,\na(φ,ωJ)≡ωJcosψpsin(φ−φp) andb(φ,ψ,ω J)≡\nωJ[cosψpsinψcos(φ−φp)−sinψpcosψ], whereωJ=\nγaJ.α(ξ)-terms of Eqs. (3) are the generalized Gilbert\ndamping terms proposed by Tiberkevich and Slavin.15\nWe take into account only the first non-trivial term of\nthe Taylor series expansion for α(ξ) by the magneti-\nzation change rate ξ≡(∂m/∂t)2/(γ4πMs)2;α(ξ) =\nαG(1+q1ξ). According to Ref. 15, the nonlinear LLGS\nmodel with q1= 3 gives a good agreement with the ex-\nperimental results of Ref. 1 and Ref. 16.\nAn instability of a steady solution of Eq. (3) is consid-\nered. A steady solution ( φ0(ωJ),ψ0(ωJ)) is derived from\nFi(φ0,ψ0,ωJ) = 0. Shifting the variables as u1≡φ−φ0\nandu2≡ψ−ψ0, we have the Taylor series of Eq. (3) as\nfollows,\n˙u=Lu+N2uu+N3uuu+··· (4)\nwhereu= (u1,u2)T. Here, the diadic and triadic nota-\ntions7have been used. The stability of a steady solution\nis determined by the eigenvalues of the linear coefficient\nmatrixL:λ±= Γ±(Γ2−detL)1/2. Γ is defined as\nΓ = Γ(ωJ)≡(1/2)trLand plays the role as a control pa-\nrameter since it depends on ωJ. We confine ourselves to\nthecasewheretheHopfbifurcationoccurs. Then, λ±isa\npair of complex-conjugate eigenvalues. The point, Γ = 0,\nis the Hopf bifurcation point; while a steady solution re-\nmains stable for Γ <0, it becomes unstable for Γ >0.\nThe bifurcation point corresponds to the threshold ωc\nJ\nwhich is determined by tr L= 0 andFi(φ0,ψ0,ωc\nJ) = 0.\nNear the bifurcation point, we divide Linto the two\nparts;L=L0+ ΓL1, whereL0is the critical part and\nΓL1istheremainingpart. Correspondingto L,λ+isalso\ndivided into the two parts; λ+=λ0+Γλ1. Although L1\nandλ1generally depend on Γ further, we neglect their\ndependence and evaluate them by the values at Γ = 0.\nAccordingly, λ0=iω0and\nλ1= 1−1\n2iω0d\ndΓdetL/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΓ=0, (5)\nwhereω0≡√detL0. The right and left eigenvector ofFIG.2: (Color online)(a)Power PdividedbyR0I2withR0=\n13.6 Ω and (b) linewidth (FWHM) of the signal of STNO as\na function of applied current I. Dots are experimental data\natT= 150 K taken from Ref. 16. Red lines are theoretical\nfitting curves based on the model of Eq. (1).\nL0corresponding to the eigenvalue λ0are denoted as U\nandU∗, respectively. These are normalized as U∗U=\n¯U∗¯U= 1 where ¯Umeans a complex conjugate of U.\nLet us apply the reduction method to Eq. (4). The SL\nequation for a complex amplitude W(t),\n˙W= Γλ1W−g|W|2W (6)\nand the neutral solution for the magnetization dynamics,\n/parenleftbigg\nφ\nψ/parenrightbigg\n=/parenleftbigg\nφ0\nψ0/parenrightbigg\n+W(t)eiω0tU+¯W(t)e−iω0t¯U(7)\nareobtainedwithinthelowestorderapproximation.7Un-\nder the approximation, only the Taylor expansion coef-\nficients up to the third order are needed. The complex\nconstantgin Eq. (6) is given by\ng≡ν1+iν2=−3(U∗,N3¯UUU)\n+4(U∗,N2UV0)+2(U∗,N2¯UV+),(8)\nwhereV0=L−1\n0N2U¯UandV+= (L0−2iω0)−1N2UU.\nThe amplitude-phase coupling factor δis obtained from\nthe complex constant gand is given by\nδ=ν2/ν1. (9)\nIn this way, the factor δfor STNO can be calculated\nnumerically from the parameters of the LLGS equation.\nThe noisy Hopf normal form given by Eq. (1) is de-\nrived when we add the noise term f(t) with∝an}bracketle{tf(t)¯f(t′)∝an}bracketri}ht=\n4Dγ2δ(t−t′) to the SL Eq. (6). f(t) has the di-\nmension of frequency. The components in Eq. (1) are\ndefined as ˜W(t) = (Dγ2/ν1)−1/4W(t)ei(ω0+Γδ−ΓImλ1)t,\n˜t≡/radicalbig\nDγ2ν1t,p≡Γ//radicalbig\nDγ2ν1, and˜Ω≡ω0//radicalbig\nDγ2ν1.\nTherefore, we can make the most of many well-known\nproperties of Eq. (1)10,11to examine the behavior of\nSTNO. It is known, for example, that the spectrum\nlinewidth ∆ ωFWHMfar abovethe threshold ( p≫0) is in-\ncreased by a factor of (1+ δ2).10In the context of STNO,\nwhen Γ≫0, the linewidth can be expressed as\n∆ωFWHM= ∆ωres×kBT\nEosci×1\n2(1+δ2),(10)3\nFIG.3: (Color online)(a)Dependenceof δonthenonlinearity\nof the damping q1for various values of an external magnetic\nfieldH. An uniaxial anisotropy field is taken as Hk/4πMs=\n0.04. (b) Dependence of δon an external magnetic field H\nfor various values of an uniaxial anisotropy field Hk.\nwhich corresponds to Eq. (11) in Ref. 5. Here, kBTis\nthe thermal energy. ∆ ωresis the linewidth at thermal\nequilibrium ( ωJ= 0) given by ∆ ωres= 2Γeq, where\nΓeq≡ −Γ(ωJ= 0). Moreover, Eosciis the magneti-\nzation oscillating energy and can be written as Eosci≃\n2U†[∂(∂u1E,∂u2E)\n∂(u1,u2)]u=0UPWVfree=1\n2ΓeqkBT\nDγ2PWwhen it is\nassumed that Eosci≃kBTnear thermal equilibrium ( en-\nergy equipartition ). Here,Vfreeis the volume of the free\nlayer andPWis the total power of W(t) given byPW=/radicalbig\nDγ2/ν1{p+2/F(p)}withF(p)≡√πep2/4[1+ erf(p/\n2)]. From the expression of Eq. (10), it is found that\nthe MR device in STNO itself is nothing but a resonator\non the analogy of electrical circuits. The other one of\nwell-known properties of Eq. (1) is that the amplitude-\nphasecouplingfactordistortsthepowerspectrumtonon-\nLorentzian lineshapes especially near the threshold (see,\ne.g., FIG. 5 of Ref. 11). The degree of the lineshape\ndistortion is determined by the magnitude of δandp,\ncorresponding to the calculation in Ref. 6. We com-\nment on the validity of Eq. (1) for large-amplitude os-\ncillations. In Fig. 2, the theoretical fitting curves based\non the model Eq. (1) are compared with the experimen-\ntal data of Ref. 16 and give a good agreement with them\nup toI≃5.6 mA (p≃8.2) beyond the threshold cur-\nrentIc= 4.8 mA (p= 0) estimated by the fitting.17\nTherefore, although the derivation of Eq. (1) is based on\na perturbation expansion around the bifurcation point,\nit is considered to be valid for rather large-amplitude os-\ncillations with p∼10.\nWe briefly mention the oscillating frequency ωosci.From Eqs. (1) and (7), the oscillating frequency of a\nfree layer magnetization far above threshold is written as\nωosci=ω0−Γδ+ΓImλ1. Although the calculationresults\nfor Imλ1of Eq. (5) are not shown here, we have found\nthat this quantity has a small value with Im λ1∼αG\nfor wide range of parameters of the LLGS equation. Ac-\ncordingly,ωosciisapproximatelygivenby ωosci≃ω0−Γδ.\nSince Γ∝(I−Ic), while the frequency ωoscidecreases as\nthe current I(>Ic) increaseswhen δ>0 (red shift), ωosci\nincreases when δ<0 (blue shift) in accordance with the\nspin-wave models3,4,5,6.\nAsillustratedabove,theamplitude-phasecouplingfac-\ntorδplays a key role to determine the behavior of an\noscillator. Therefore, the features of STNO can be found\nout by the calculation of δ.\nSome calculation examples of δare shown in Fig. 3.\nIt is considered the case where a free layer is an in-\nplane magnetic film with an in-plane external field ap-\nplied along the xdirection, Hext=Hˆx. It is assumed\nthatN= diag(0,0,1),αG= 0.02, and (φp,ψp) = (0,0).\nIn Fig. 3(a), the dependence of δon the nonlinearity\nof the damping q1is shown. It is found that δmono-\ntonically decreases for q1and the variation of δis very\nlarge. This result suggests that a nonlinear damping sig-\nnificantly changes the LLG dynamics.15In Fig. 3(b), the\ndependence of δon an external magnetic field Hfor var-\nious values of an uniaxial anisotropy field Hk(= 2Ku/\nMs) is shown. The nonlinearity of the damping is taken\nasq1= 3.15In the practical external field region, δis\nvery sensitive to an uniaxial anisotropy field and varies\nlargely. Therefore, when the dynamics of STNO is con-\nsidered, it is necessary to take the effect of an uniaxial\nanisotropy field into account seriously. This is the main\nresult of the present paper.\nIn summary, we have considered the dynamics of\nSTNO by reducing the LLGS equation to a generic oscil-\nlatormodelandcalculatedexplicitlytheamplitude-phase\ncoupling factor which is the key factor for the power\nspectrum. The amplitude-phase coupling factor δis very\nsensitive to magnetic fields, in-plane anisotropy, and the\nnonlinearity of damping. The large variation of δis the\nremarkable feature of STNO in comparison with conven-\ntional oscillators. The calculation way for δshown is ap-\nplicable for an arbitrarymagnetization configurationand\nis useful for finding a stable STNO with small ∆ ωFWHM\n(Eq. (10)), which is preferable for applications.\n∗Electronic address: kiwamu.kudo@toshiba.co.jp\n1S. I. Kiselev et al, Nature 425, 380 (2003).\n2W. H. Rippard et al, Phys. Rev. Lett. 92, 027201 (2004).\n3A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264\n(2005).\n4V. Tiberkevich, A. N. Slavin, and J.-V. Kim, Appl. Phys.\nLett.91, 192506 (2007).\n5J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev.\nLett.100, 017207 (2008).6J.-V. Kim et al., Phys. Rev. Lett. 100, 167201 (2008).\n7Y. Kuramoto, Chap. 2 of Chemical Oscillations, Waves,\nand Turbulence (Springer-Verlag, Berlin, 1984).\n8J.-V. Kim, Phys. Rev. B 73, 174412 (2006).\n9K. Mizushima, K. Kudo, and R. Sato, J. Appl. Phys. 101,\n113903 (2007).\n10H. Risken, Chap. 12 of Fokker-Planck Equation (2nd Ed.\nSpringer-Verlag, Berlin, 1989).\n11J. P. Gleeson and F. O’Doherty, SIAM J. Appl. Math. 66,4\n1669 (2006).\n12C. H. Henry, IEEE Journal of Quantum Electronics, QE-\n18, 259 (1982).\n13H. Haken, Advanced Synergetics (Springer-Verlag, New\nYork, 1993).\n14J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n15V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440\n(2007).\n16Q. Mistral et al., Appl. Phys. Lett. 88, 192507 (2006).17The dimensionless power in Fig. 2(a) is given by P/R0I2=\na{p+2/F(p)}witha≃2.3063×10−9andp≃10.202(I−\nIc). To obtain the linewidth in Fig. 2(b), we have used the\nparameters ofp\nDγ2ν1/2π= 11.24 MHz and δ= 0.5, and\nhave solved the eigenvalue problem of the Fokker-Planck\nequation corresponding to Eq. (1) as done in Ref. 10 or\nRef. 6."    },    {        "title": "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. Zhang, Z. Li,Phys. Rev. Lett. 93, 127204 (2004); S. E.\nBarnes, S. Maekawa,Phys. Rev. Lett. 95, 107204 (2005);\nA. Thiaville, Y.Nakatani, J. Miltat, Y.Suzuki,Europhys.\nLett.69, 990 (2005); Y. Tserkovnyak, H. J. Skadsem,\nA. Brataas, G. E. W. Bauer, Phys. Rev. B 74, 144405\n(2006); A. K. Nguyen, H. J. Skadsem, A. Brataas, Phys.\nRev. Lett. 98, 146602 (2007).\n[3] For reviews see e.g.D.C. Ralph, M.D. Stiles, J. Magn.\nMagn. Mater., 3201190 (2008).\n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater., 3201282 (2008).\n[5] A. K. Nguyen, R. V. Shchelushkin and A. Brataas, Phys.\nRev. Lett. 97, 136603 (2006); R. Oszwaldowski, J. A.\nMajewski and T. Dietl,Phys. Rev. B 74, 153310 (2006).\n[6] G. E. Volovik,J. Phys.C: Sol. State Phys. 20, L83(1987);\nS.E. BarnesandS.Maekawa,Phys.Rev.Lett. 98, 246601\n(2007); J.I. Ohe, A. Takeuchi, and G. Tatara, Phys. Rev.\nLett.99, 266603 (2007); S. A.Yang, D. Xiao, andQ. Niu,\ncond-mat/0709.1117.\n[7] R. A. Duine, Phys. Rev. B 77, 014409 (2008);\narXiv:0809.2201v1; Y. Tserkovnyak and M. Mecklen-\nburg,Phys. Rev. B 77, 134407 (2008).\n[8] S. A. Yang et al.,Phys. Rev. Lett. 102, 067201 (2009).\n[9] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[10] A. Brataas, Y. Tserkovnyak and G.E.W Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[11] S. R. de Groot, Thermodynamics of Irreversible Pro-\ncesses(North-Holland, Amsterdam, 1952).\n[12] The magnetic texture is inverted in this equation, i.e.\nLcw[m] =Lwc[−m].\n[13] We define the spin-polarization P=/angbracketleftJz/angbracketright=P\nj(ψ†\njJzψj)/N(m=−ˆ z), where the sum is over N\npropagating modes, ψjare the corresponding spinor-\nvalued wavefunctions, and Jzthe dimensionless angular\nmomentum operator.\n[14] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[15] A. K. Nguyen, and A. Brataas, Phys. Rev. Lett. 101,\n016801 (2008).\n[16] J. 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": "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": "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": "0812.0832v1.Observation_of_ferromagnetic_resonance_in_strontium_ruthenate__SrRuO3_.pdf",        "content": "arXiv:0812.0832v1  [cond-mat.mtrl-sci]  3 Dec 2008Observationofferromagnetic resonance instrontium ruthe nate (SrRuO 3)\nM.C. Langner,1,2C.L.S. Kantner,1,2Y.H. Chu,3L.M.\nMartin,2P. Yu,1R. Ramesh,1,4and J. Orenstein1,2\n1Department of Physics, University of California, Berkeley , CA 94720\n2Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, CA 94720\n3Department of Materials Science and Engineering,\nNational Chiao Tung University, HsinChu, Taiwan, 30010\n4Department of Materials Science and Engineering,\nUniversity of California, Berkeley, CA 94720\n(Dated: October 26, 2018)\nAbstract\nWe report the observation of ferromagnetic resonance (FMR) in SrRuO 3using the time-resolved\nmagneto-optical Kerr effect. The FMR oscillations in the ti me-domain appear in response to a sudden,\noptically induced change in the direction of easy-axis anis tropy. The high FMR frequency, 250 GHz, and\nlarge Gilbert damping parameter, α≈1, are consistent with strong spin-orbit coupling. We find th at the\nparameters associated with the magnetization dynamics, in cludingα, have a non-monotonic temperature\ndependence, suggestive of alink to theanomalous Hall effec t.\nPACS numbers: 76.50.+g,78.47.-p,75.30.-m\n1Understanding and eventually manipulating the electron’s spin degree of freedom is a major\ngoal of contemporary condensed matter physics. As a means to this end, considerable attention\nis focused on the spin-orbit (SO) interaction, which provid esa mechanism for control of spin po-\nlarization by applied currents or electric fields [1]. Despi te this attention, many aspects of SO\ncoupling are not fully understood, particularly in itinera nt ferromagnets where the same elec-\ntrons are linked to both rapid current fluctuations and slow s pin dynamics. In these materials,\nSO coupling is responsible for spin-wave damping [2, 3], spi n-current torque [4, 5], the anoma-\nlous Hall effect (AHE) [6], and magnetocrystalline anisotr opy (MCA) [7]. Ongoing research is\naimed toward a quantitative understanding of how bandstruc ture, disorder, and electron-electron\ninteractionsinteracttodeterminethesizeandtemperatur edependenceoftheseSO-driveneffects.\nSrRuO 3(SRO) is a material well known for its dual role as a highly cor related metal and\nan itinerant ferromagnet with properties that reflect stron g SO interaction [8, 9, 10]. Despite\nits importance as a model SO-coupled system, there are no pre vious reports of ferromagnetic\nresonance (FMR) in SRO. FMR is a powerful probe of SO coupling in ferromagnets, providing\na means to measure both MCA and the damping of spin waves in the small wavevector regime\n[11]. HerewedescribedetectionofFMRbytime-resolvedmag netoopticmeasurementsperformed\non high-quality SRO thin films. We observe a well-defined reso nance at a frequency ΩFMR=\n250 GHz. This resonant frequency is an order of magnitude hig her than in the transition metal\nferromagnets,which accountsforthenonobservationbycon ventionalmicrowavetechniques.\n10-200nmthickSROthinfilmsweregrownviapulsedlaserdepo sitionbetween680-700◦Cin\n100 mTorr oxygen. High-pressure reflection high-energy ele ctron diffraction (RHEED) was used\nto monitor the growth of the SRO film in-situ. By monitoring RH EED oscillations, SRO growth\nwas determined to proceed initially in a layer-by-layer mod e before transitioning to a step-flow\nmode. RHEED patterns and atomic force microscopy imaging co nfirmed the presence of pristine\nsurfaces consisting of atomically flat terraces separated b y a single unit cell step ( 3.93 ˚A). X-ray\ndiffractionindicatedfullyepitaxialfilmsandx-rayreflec tometrywasusedtoverifyfilmthickness.\nBulk magnetization measurements using a SQUID magnetomete r indicated a Curie temperature,\nTC, of∼150K.\nSensitive detection of FMR by the time-resolved magnetoopt ic Kerr effect (TRMOKE) has\nbeen demonstrated previously [12, 13, 14]. TRMOKE is an all o ptical pump-probe technique in\nwhichtheabsorptionofan ultrashortlaserpulseperturbst hemagnetization, M, ofaferromagnet.\nThe subsequent time-evolutionof Mis determined from the polarization state of a normally inci -\n2dent, time-delayed probe beam that is reflected from the phot oexcited region. The rotation angle\nof the probe polarization caused by absorption of the pump, ∆ΘK(t), is proportional to ∆Mz(t),\nwherezisthedirectionperpendiculartotheplaneofthefilm[15].\nFigs. 1a and 1b show ∆ΘK(t)obtained on an SRO film of thickness 200 nm. Very similar\nresults are obtained in films with thickness down to 10 nm. Two distinct types of dynamics are\nobserved,dependingonthetemperatureregime. Thecurvesi nFig. 1aweremeasuredattempera-\nturesnearT C. Therelativelyslowdynamicsagreewithpreviousreportsf orthisTregime[16]and\nare consistent with critical slowing down in the neighborho od of the transition [17]. The ampli-\ntudeofthephotoinducedchangeinmagnetizationhasalocal maximumnearT=115K.Distinctly\ndifferentmagnetizationdynamicsareobservedasTisreduc edbelowabout80K,asshowninFig.\n1b. The TRMOKE signal increases again, and damped oscillati ons with a period of about 4 ps\nbecomeclearly resolved.\nFIG. 1: Change in Kerr rotation as a function of time delay fol lowing pulsed photoexcitation, for several\ntemperatures below the Curie temperature of 150 K. Top Panel : Temperature range 100 K <T<150 K.\nBottom panel: Temperature range 5K <T<80 K.Signal amplitude and oscillations grow with decreasin g\nT.\nIn order to test if these oscillations are in fact the signatu re of FMR, as opposed to another\n3photoinduced periodic phenomenon such as strain waves, we m easured the effect of magnetic\nfield on theTRMOKE signals. Fig. 2a shows ∆ΘK(t)for several fields up to 6 T applied normal\nto the film plane. The frequency clearly increases with incre asing magnetic field, confirming that\ntheoscillationsare associatedwithFMR.\nThemechanismfortheappearanceofFMRinTRMOKEexperiment siswellunderstood[14].\nBeforephotoexcitation, Misorientedparallelto hA. Perturbationofthesystembythepumppulse\n(by local heating for example) generates a sudden change in t he direction of the easy axis. In the\nresulting nonequilibrium state, MandhAare no longer parallel, generating a torque that induces\nMto precess at the FMR frequency. In the presence of Gilbert da mping,Mspirals towards the\nnewhA, resultingin thedamped oscillationsof Mzthat appearin theTRMOKE signal.\nTo analyze the FMR further we Fourier transform (FT) the time -domain data and attempt to\nextract thereal and imaginary parts of the transversesusce ptibility,χij(ω). Themagnetization in\nthetime-domainisgivenby therelation,\n∆Mi(t) =/integraldisplay∞\n0χij(τ)∆hj\nA(t−τ)dτ, (1)\nwhereχij(τ)is the impulse response function and ∆hA(t)is the change in anisotropy field.\nIf∆hA(t)is proportional to the δ-function, ∆Mi(t)is proportional χij(τ)and the FT of the\nTRMOKE signal yields χij(ω)directly. However, for laser-induced precession one expec ts that\n∆hA(t)will be more like the step function than the impulse function , as photoinduced local\nheating can be quite rapid compared with cooling via thermal conduction from the laser-excited\nregion. When ∆hA(t)is proportional to the step function, χij(ω)is proportionalto the FT of the\ntimederivativeof ∆Mi(t), ratherthan ∆Mi(t)itself. Inthiscase, theobservable ωIm{∆ΘK(ω)}\nshouldbecloselyrelated tothereal, ordissipativepart of χij(ω).\nInFig. 2bweplot ωIm{∆ΘK(ω)}foreachofthecurvesshowninFig. 2a. Thespectrashown\nin Fig. 2b do indeed exhibit features that are expected for Re χij(ω)near the FMR frequency. A\nwell-definedresonancepeakisevident,whosefrequencyinc reaseswithmagneticfieldasexpected\nforFMR.TheinsettoFig. 2bshows ΩFMRasafunctionofappliedmagneticfield. Thesolidline\nthroughthedatapointsisafitobtainedwithparameters |hA|=7.2Tandeasyaxisdirectionequal\nto 22 degrees from the film normal. These parameter values agr ee well with previous estimates\nbased onequilibriummagnetizationmeasurements[8, 10].\nAlthough the spectra in Fig. 2b are clearly associated with F MR, the sign change at low fre-\nquency is not consistent with Re χij(ω), which is positive definite. We have verified that the\n4FIG. 2: Top panel: Change in Kerr rotation as a function of tim e delay following pulsed photoexcitation at\nT=5K,forseveral valuesofappliedmagneticfieldranging up to6Tesla. Bottompanel: Fourier transforms\nof signals shown intop panel. Inset: FMRfrequency vs. appli ed field.\nnegativecomponentis always present in the spectra and is no t associated with errors in assigning\nthet=0pointinthetime-domaindata. Theoriginofnegativecomp onentoftheFTismadeclearer\nby referring back to the time domain. In Fig. 3a we show typica l time-series data measured in\nzero field at 5 K. Forcomparison we show theresponseto a step f unction changein theeasy axis\ndirection predicted by the Landau-Lifschitz-Gilbert (LLG ) equations [18]. It is clear that, if the\nmeasured and simulated responses are constrained to be equa l at large delay times, the observed\n∆ΘK(t)ismuchlarger thantheLLGpredictionat smalldelay.\nWe have found that ∆ΘK(t)can be readily fit by LLG dynamics if we relax the assumption\nthat∆hA(t)is a step-function, in particular by allowing the change in e asy axis direction to\n”overshoot” at short times. The overshoot suggests that the easy-axis direction changes rapidly\nas the photoexcited electrons approach quasiequilibriumw ith the phonon and magnon degrees of\nfreedom. The red line in Fig. 3a shows the best fit obtained by m odeling∆hA(t)byH(t)(φ0+\nφ1e−t/τ), whereH(t)is the step function, φ0+φ1is the change in easy-axis direction at t= 0,\nandτis the time constant determining the rate of approach to the a symptotic value φ0. The fit is\n5FIG. 3: Components of TRMOKE response in time (top panel) and frequency (bottom panel) domain.\nBlacklinesaretheobserved signals. Greenlineinthetoppa nel isthesimulated response toastepfunction\nchange in easy-axis direction. Best fits to the ”overshoot” m odel described in the text are shown in red.\nBluelines are thedifference between the measured and best- fit response.\nclearly much better when the possibility of overshoot dynam ics in∆hA(t)is included. The blue\nline shows the difference between measured and simulated re sponse. With the exception of this\nveryshortpulsecenterednear t=0,theobservedresponseisnowwelldescribedbyLLGdynami cs.\nInprinciple,analternateexplanationforthediscrepancy withthestep-functionassumptionwould\nbe to consider possible changes in the magnitude as well as di rection of M. However, we have\nfound that fitting the data then requires |M(t)|to be larger at t >20ps than|M(t <0)|, a\nphotoinducedincrease thatisunphysicalfora systemin ast ableFM phase.\nIn Fig. 3b we compare data and simulated response in the frequ ency domain. With the al-\nlowance for an overshoot in ∆hA(t)the spectrum clearly resolves into two components. The\npeak at 250 GHz and the sign change at low frequency are the bot h part of the LLG response to\n∆hA(t). The broad peak or shoulder centered near 600 GHz is the FT of t he short pulse compo-\nnentshowninFig. 3a. Wehavefoundthiscomponentisessenti allylinearinpumppulseintensity,\n6and independent of magnetic field and temperature - observat ions that clearly distinguish it from\nthe FMR response. Its properties are consistent with a photo induced change in reflectivity due to\nband-filling,whichiswell-knowntocross-coupleintotheT RMOKEsignalofferromagnets [19].\nByincludingovershootdynamicsin ∆hA(t),weareabletodistinguishstimulusfromresponse\nin the observed TRMOKE signals. Assuming LLG dynamics, we ca n extract the two parameters\nthatdescribetheresponse: ΩFMRandα;andthetwoparametersthatdescribethestimulus: φ1/φ0\nandτ. In Fig. 4 we plot all four parameters as a function of tempera ture from 5 to 80 K. The\nT-dependence of the FMR frequency is very weak, with ΩFMRdeviating from 250 GHz by only\nabout 5%overthe range ofthe measurement. TheGilbert damping param eterαis of order unity\nat all temperatures, avaluethatis approximatelyafactor 102largerthan found intransitionmetal\nferromagnets. Over the same T range the decay of the easy axis overshoot varies from about 2\nto 4 ps. We note that the dynamical processes that characteri ze the response all occur in strongly\noverlapping time scales, that is the period and damping time of the FMR, and the decay time of\nthehAovershoot,areeach inthe2-5ps range.\nWhileΩFMRisessentiallyindependentofT,theparameters α,φ1/φ0andτexhibitstructurein\ntheirT-dependencenear40K.Thisstructureisreminiscent oftheT-dependenceoftheanomalous\nHallcoefficient σxythathasbeenobservedinthinfilmsofSRO[20,21,22]. Forcom parison,Fig.\n4dreproduces σxy(T)reportedinRef. [20]Thesimilaritybetween theT-dependen ceofAHEand\nparameters related to FMR suggests a correlation between th e two types of response functions.\nRecently Nagaosa and Ono [23] have discussed the possibilit y of a close connection between\ncollective spin dynamics at zero wavevector (FMR) and the of f-diagonal conductivity (AHE). At\na basic level,both effects are nonzero only in the presence o f both SO couplingand time-reversal\nbreaking. However, the possibilityof a more quantitativec onnection is suggested by comparison\nof the Kubo formulas for the two corresponding functions. Th e off-diagonal conductivity can be\nwrittenin theform [24],\nσxy(ω) =i/summationdisplay\nm,n,kJx\nmn(k)Jy\nnm(k)fmn(k)\nǫmn(k)[ǫmn(k)−ω−iγ], (2)\nwhereJi\nmn(k)is current matrix element between quasiparticle states wit h band indices n,mand\nwavevector k. The functions ǫmn(k)andfmn(k)are the energy and occupation difference, re-\nspectively,between such states, and γis a phenomenologicalquasiparticledamping rate. FMR is\nrelated to theimaginary part of theuniformtranverse susce ptibility,with thecorresponding Kubo\n7FIG. 4: Temperature dependence of (a) FMR frequency (triang les) and damping parameter (circles), (b)\novershoot decay time, (c) ratio of overshoot amplitude to st ep-response amplitude ( φ1/φ0), and (d) σxy\n(adapted from [20]).\nform,\nImχxy(ω) =γ/summationdisplay\nm,n,kSx\nmn(k)Sy\nnm(k)fmn(k)\n[ǫmn(k)−ω]2+γ2, (3)\nwhereSi\nmnisthematrixelementofthespinoperator. Ingeneral, σxy(ω)andχxy(ω)areunrelated,\nas they involvecurrent and spin matrix elements respective ly. However, it has been proposed that\nin several ferromagnets, including SRO, the k-space sums in Eqs. 2 and 3 are dominated by a\nsmall number of band-crossings near the Fermi surface [22, 2 5]. If the matrix elements Si\nmnand\nJi\nmnvary sufficiently smoothly with k, thenσxy(ω)andχxy(ω)may be closely related, with both\nproperties determined by thepositionofthechemical poten tialrelativeto theenergy at which the\n8bandscross. Furthermore,asGilbertdampingisrelatedtot hezero-frequencylimitof χxy(ω),i.e.,\nα=ΩFMR\nχxy(0)∂\n∂ωlim\nω→∞Imχxy(ω), (4)\nand AHE is the zero-frequency limit of σxy(ω), the band-crossing picture implies a strong corre-\nlationbetween α(T)andσxy(T).\nIn conclusion,we havereported the observationof FMR in the metallictransition-metaloxide\nSrRuO 3. Both the frequency and damping coefficient are significantl y larger than observed in\ntransition metal ferromagnets. Correlations between FMR d ynamics and the AHE coefficient\nsuggest that both may be linked to near Fermi surface band-cr ossings. Further study of these\ncorrelations, as Sr is replaced by Ca, or with systematic var iation in residual resistance, could be\na fruitful approach to understanding the dynamics of magnet ization in the presence of strong SO\ninteraction.\nAcknowledgments\nThis research is supported by the US Department of Energy, Of fice of Science, under contract\nNo. DE-AC02-05CH1123. Y.H.C. would also like to acknowledg e the support of the National\nScience Council,R.O.C., underContract No. NSC97-3114-M- 009-001.\n[1] I.ˆZuti´ c, J. Fabian, and S.DasSarma, Rev.Mod. Phys. 76, 323 (2004).\n[2] V. Korenman and R.E.Prange, Phys. Rev.B 6, 2769 (1972).\n[3] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[4] J. C.Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[5] L.Berger, Phys. Rev.B 54, 9353 (1996).\n[6] J. M.Luttinger and R. Karplus, Phys. Rev. 94, 782 (1954).\n[7] H.Brooks, Phys. Rev. 58, 909 (1940).\n[8] L. Klein, J. S. Dodge, C. H. Ahn, J. W. Reiner, L. Mieville, T. H. Geballe, M. R.Beasley, and A. Ka-\npitulnik, J.Phys. Cond.-Matt. 8, 10111 (1996).\n[9] P.Kostic,Y.Okada,N.C.Collins,Z.Schlesinger, J.W.R einer,L.Klein,A.Kapitulnik, T.H.Geballe,\nand M. R.Beasley, Phys.Rev. Lett. 81, 2498 (1998).\n9[10] A.F.Marshall, L.Klein, J.S.Dodge, C.H.Ahn,J.W.Rein er, L.Mieville, L.Antagonazza, A.Kapit-\nulnik, T.H.Geballe, and M.R. Beasley, J.Appl. Phys. 85, 4131 (1999).\n[11] B.Heinrich and J. F.Cochran, Adv. Phys. 42, 523 (1993).\n[12] W.K.Hiebert, A.Stankiewicz, and M.R.Freeman, Phys. R ev.Lett.79, 1134 (1997).\n[13] Y. Acremann, C. H.Back, M. Buess, O.Portmann, A. Vaterl aus, D.Pescia, and H.Melchior, Science\n290, 492 (2000).\n[14] M.vanKampen,C.Jozsa, J.T.Kohlhepp, P.LeClair, L.La gae,W.J.M.deJonge, andB.Koopmans,\nPhys. Rev. Lett. 88, 227201 (2002).\n[15] K. Shinagawa, in Magneto-optics , edited by S. Sugano and N. Kojima (Springer-Verlag, Berlin , Ger-\nmany, 2000).\n[16] T. Ogasawara, K. Ohgushi, Y. Tomioka, K. S. Takahashi, H . Okamoto, M. Kawasaki, and Y. Tokura,\nPhys. Rev. Lett. 94, 087202 (2005).\n[17] T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev.\nLett.85, 1986 (2000).\n[18] W.F.Brown, Micromagnetics (Krieger, 1963).\n[19] B.Koopmans,M.vanKampen,J.T.Kohlhepp,andW.J.M.de Jonge,Phys.Rev.Lett. 85,844(2000).\n[20] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. K awasaki, Z. Fang, N. Nagaosa, and\nY. Tokura, Phys. Rev. Lett. 93, 016602 (2004).\n[21] L. Klein, J. R. Reiner, T. H. Geballe, M. R. Beasley, and A . Kapitulnik, Phys. Rev. B 61, R7842\n(2000).\n[22] Z. Fang, N. Nagaosa, K. Takahashi, A. Asamitsu, R. Mathi eu, T. Ogasawara, H. Yamada,\nM. Kawasaki, Y. Tokura, and K.Terakura, Science 302, 92(2003).\n[23] M. Onoda, A.S.Mishchenko, and N. Nagaosa, J.Phys. Soc. Jap.77, 013702 (2008).\n[24] M. Onoda and N.Nagaosa, J. Phys.Soc. Jap. 71, 19 (2002).\n[25] X.Wang, J.R. Yates, I. Souza, and D.Vanderbilt, Phys.R ev. B.74, 195118 (2006).\n10"    },    {        "title": "0812.2209v1.Frequency_dependent_Drude_damping_in_Casimir_force_calculations.pdf",        "content": "arXiv:0812.2209v1  [quant-ph]  11 Dec 2008Frequency-dependent Drude damping in Casimir force\ncalculations\nR. Esquivel-Sirvent1,∗\n1Instituto de F´ ısica, Universidad Nacional Aut’onoma de M´ exico,\nApdo. Postal 20-364, M´ exico D.F. 01000, M´ exico\nAbstract\nThe Casimir force is calculated between Au thin films that are described by a Drude model\nwith a frequency dependent damping function. The model para meters are obtained from available\nexperimental data for Au thin films. Two cases are considered ; annealed and nonannealed films\nthat have a different damping function. Compared with the calc ulations using a Drude model with\na constant damping parameter, we observe changes in the Casi mir force of a few percent. This\nbehavior is only observed in films of no more than 300 ˚Athick.\n∗Corresponding author. Email:raul@fisica.unam.mx\n1I. INTRODUCTION\nThe advent of precise and systematic Casimir force experiments sin ce the late 90 [1, 2,\n3, 4, 5, 6, 7, 8] has prompted an intense research on the role of th e dielectric properties of\nthe involved materials. Although, the Lifshitz theory [9] explicitly req uires the dielectric\nfunction of the materials, an important issue is which one is the corre ct dielectric function\nthat is consistent in describing the optical properties of the mater ials and the measurements\nof the Casimir force.\nThe first approachis to assume a plasma model for the dielectric fun ction[10] or the more\nrealistic Drude model, that has been extensively used when extrapo lating to low frequencies\ntabulated data. Although it may be thought that the problem of usin g a dielectric function\nis straight forward, controversial results have been reported, in particular in relation to the\nuse of the Drude model in finite temperature calculations of the Cas imir force .\nThe use of of the Drude model in Lifshitz theory seems to violate Ner nst heat theorem,\nwhile the plasma model presents no problem at all, but is not realistic in t he representation\nof the dielectric properties of metals. The plethora of papers and c omments shows that the\nissue is far from settled [11, 12, 13, 14, 15, 16, 17].\nEven without considering finite-temperature effects, the choice o f the dielectric function\ncan change the calculations of the Casimir force. For example, for A u samples the Drude\nparameters extracted from tabulated data vary depending on th e sample. The variations on\nthe Drude parameters have important implications in the Casimir forc e calculations since\ndifference of up to 5% are obtained[18]. A similar result was obtained by Svetovoy et al.\nwhere [19] different Au samples were prepared under similar condition s with thicknesses\nranging from 120 nm to 400 nm. From measured ellipsometry data it wa s verified that the\nplasma frequency varies from 6.8 eV to 8.4 eV for this set of particula r samples, changing\nthe Casimir force a few percent. The conclusions of these works sh ow that there is not a\nstandard plasma frequency or damping parameter for Au, it is samp le dependent and in situ\nmeasurements are needed. Experimentally, the effect of thin films o n the Casimir force was\nshown experimentally by Iannuzzi [21, 22] who demonstrated tha t the Casimir attraction\nbetween a metallic plateanda metal coatedsphere depended onthe thickness of thecoating.\nThe reduction of size can significantly change the physical paramet ers of a system. In the\ncaseofthinfilms, asthethickness ofthefilmapproachesthemeanf reepath, theconductivity\n2show a sharp decrease in its values. This was shown experimentally by Kastle [23] with Au\nfilms whose thickness varied from 2 nmto 70nm. Indeed, a conductor-insulator transition\nis observed as a function of film thickness in Au [24]. As a function of film thickness, the\nCasimir force decreases with decreasing film thickness until a critica l thickness is reached\nafter which the Casimir force increases even with decreasing film thic kness[25].\nTo further the discussion about the possible factors that influenc e Casimir force calcula-\ntions, in this paper we introduce a frequency dependent damping γ(ω) in the Drude model.\nThis model describes the dielectric properties of thin films and chang es if the film has been\nannealed.\nII. FREQUENCY DEPENDENT DAMPING\nThe classical Drude model the local dielectric function is given by\nǫ(ω) = 1−ω2\np\nω(ω+iγ), (1)\nwhereωis the frequency, ωpthe plasma frequency and γthe damping parameter that is\nconstant for a fixed temperature.\nMeasurements of the dielectric properties of Au thin films by M. L. Th eye [26] from\nreflectance and transmittance data showed a deviation from the b ulk Drude behavior of Au.\nThe deviations from the Drude model were explained by introducing a frequency dependent\nrelaxation time due to electron-phonon and electron-ion interactio ns of the form:\nγ=γ0+Aω2. (2)\nHowever, a more precise correction to the Drude model was introd uced by Nagel [27] to\ninclude the frequency dependence of the damping parameter. It w as observed that sample\npreparationwas relevant intheoptical behavior of thematerial, sin ce themeasured datawas\ndifferent for annealed and nonannealed samples. An explanation of t he frequency dependent\nrelaxation time with the sample and the role that annealing plays in the o ptical properties\nwas given by Nagel [27] using a classical two carrier model. The model assumes that in a\nthin film sample there are two regions labeled aandb. One where the electrons see a perfect\nlattice, inside crystallitesandasecondhighlydisorderedregionbetw een thecrystallites. The\nelectrons respond differently in each of these regions and have a diff erent damping rate, say\n3γaandγb, and a different plasma frequency ωpaandωpb. Ignoring local field corrections in\nthe determination of the optical response, the two carrier model yields an effective damping\nparameter given by\nγeff=γa/bracketleftBigg\n1+ωpb\nωpa/parenleftBiggω2+γ2\na\nω2+γ2\nb/parenrightBigg/bracketrightBigg−1\n+γb/bracketleftBigg\n1+ωpa\nωpb/parenleftBiggω2+γ2\nb\nω2+γ2\na/parenrightBigg/bracketrightBigg−1\n, (3)\nwhereωpa,b= 4πNa,be2/m∗a,b. This last expression is general and the behavior observed\nby They´ e Eq. (2) is obtained if ωτa>>1 andωτb<<1. Equation (3) assumes that the\neffective masses in both regions are the same, thus Nb/Na=ωpb/ωpa. Thus the thin film\ncan be modeled by a Drude dielectric function with an effective damping constant.\nInthispaper wewill usetheparameters considered by Nagel [27] t hat fittheexperimental\ndata of They´ e [26] for an annealed and a nonanneald Au film. The par ameters are shown\nin Table 1.\nTABLE I: Parameters for the two films used in our calculations , after [27].\nfilm Nb/Naγa×1014s−1γb×1014s−1\nannealed 0.0077 0.93 25\nnonannealed 0.058 1.18 25\nIn Figure 1, we plot the effective damping γeffEq. (3) as a function of frequency for\nthe samples considered by Nagel. As expected, annealing the film will r educe the number\nof impurities and the damping constant should be smaller.\nTo study the effect that a frequency dependent damping has on th e Casimir force, we\nconsider two plates separated a distance Lwith a thin film deposited on their surface. The\nfilms are described by the parameters given in Table. 1 We calculate th e reduction factor\ndefine as the Casimir force calculated using Lifshitz formula Fdivided by the Casimir force\nbetween perfect conductors F0; this isη=F/F0. This is,\nη=120L4\nπ4/integraldisplay∞\n0QdQ/integraldisplay\nq>0dkk2\nq(Gs+Gp), (4)\nwhereGs= (r−1\n1sr−1\n2sexp(2kL)−1)−1andGp= (r−1\n1pr−1\n2pexp(2kL)−1)−1. In these expres-\nsions, the factors rp,sare the reflectivities for porspolarized light , Qis the wavevector\ncomponent along the plates, q=ω/candk=√q2+Q2.\n4In Figure (2) we show the reduction factor for Au samples describe d by a classical Drude\n(ηc) model where the damping is constant, and the Drude model with a f requency dependent\ndamping parameter for an annealed ( ηa) and nonannealed samples ( ηn). For the classical\nDrude model the parameters are ωp= 9eVandγ= 0.02eV. The difference between the\nannealed and nonannealed sample is small. The reduction factor incre ases as compared to\nthe annealed and nonannealed samples. The difference in reduction f actor can better be seen\nby computing the percent difference ∆ = 100 |ηa,n−ηc|/ηcas a function of the separation\nbetween the plates. At large separations, the percent difference is at most of ∼2.2% for the\nnonannealed sample and ∼1.6% for the annealed film.\nIII. CONCLUSIONS\nThe goal of high-precission measurements at small separations of the Casimir force, re-\nquires that the dielectric function of the materials are well known. I n this paper we have\nconsidered the effects of the frequency dependent damping when Au films of a few hundreds\nof Angstroms are considered. Besides, the effect of sample prepa ration such as annealing\nthe film also changes the Casimir force. Although the changes are of less than 3% at large\nseparations, we wish to point out that this is an example where using a simple Drude model,\nfor example, to extrapolate to low frequencies tabulated data, is n ot granted unless thick\nfilms are used. This is important if high precision measurements are se ek. Furthermore,\npreparing a sample in ultra high vacuum can change the optical prope rties of the Au films\n[28], showing the need for in-situ determination of the dielectric prop erties of the samples\nused in Casimir experiments.\nFinally, the relevance of the frequency dependent damping parame ter is in the finite-\ntemperature effects, since now the damping will also depend on the M atsubara frequencies.\nThis will be consider in a future paper.\nAcknowledgements: Partial support from DGAPA-UNAM project N o. 113208 .\n50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.060.070.080.090.10.110.120.130.140.150.16\nω(eV)γeff(eV)\n  \nannealed\nnonannealed\nFIG. 1: Effective damping as a function of frequency using Eq. ( 3) for an annealed sample (dashed\nline) and a nonannealed sample (solid line).\n100 200 300 400 500 600 7000.20.30.40.50.60.70.8\nSeparation (nm)η\n  \nDRUDE\nnonannealed\nannealed\nFIG. 2: Reduction factor as a function of separation for the c lassical Drude model with a constant\ndamping parameter (solid line), the annealed Au (*) sample a nd the nonannealed Au sample (+).\n6100 200 300 400 500 600 7000.811.21.41.61.822.22.42.6\nSeparation (nm)∆(%)\n  \nannealed\nnonannealed\nFIG. 3: Percent difference ∆ between the reduction factors cal culated in Fig.(2).\nReferences\n[1] Lamoreaux S K 1997 Phys. Rev. Lett. 785; 1998Phys. Rev. Lett. 815475\n[2] Mohideen U and Roy A 1998 Phys. Rev. Lett. 814549 ; Roy A , Lin C Y and Mohideen U\n1999Phys. Rev. D60, 111101(R) ; Harris B W , Chen F and Mohideen U 2000 Phys. Rev. A\n62, 052109\n[3] Ederth T (2000) Phys. Rev. A62062104\n[4] Chan H B, Aksyuk V A , Kleiman R N, Bishop D J and F. Capasso 20 01Science2911941;\n2001Phys. Rev. Lett. 87211801\n[5] Bressi G, Carugno G , Onofrio R, and Ruoso G, Phys. Rev. Lett. 88041804 (2002)\n[6] Decca R S, L´ opez D, Fischbach E, and Krause D E 2003 Phys. Rev. Lett. 91050402 ; Decca\nR S, Fischbach E, Klimchitskaya G L, Krause D E, L´ opez D, and M ostepanenko V M 2003\nPhys. Rev. D68116003\n[7] van Zwol P J, Palasantzas G, De Hosson J Th M 2008 Phys. Rev. B77075412\n[8] Chan H B, Bao Y, Zou J Cirelli R A Klemens F Mansfield W M and Pa i S C 2008 Phys. Rev.\nLett101030401\n[9] E. M. Lifshitz 1956 Zh. Eksp. Teor. Fiz. 2994 [1956 Sov. Phys. JETP 273 ].\n7[10] Lambrecht A, Reynaud S 2000 Eur. Phys. J. D8309\n[11] J. S Hoye, I. Brevik, S. E. Ellingsen and J. B. Aarseth 200 7 Phys. Rev. E 75, 051127\n[12] Bimonte G 2007 New J. Phys. 9281\n[13] Ellingsen S A 2008 Phys. Rev. E78021120\n[14] Mostepanenko V M and Geyer B 2008 J. Phys. A: Math. Thor. 4116014\n[15] Lamoreaux S K 2008 Possible resolution of the Casimir fo rce finite temperature correction\n”controversies” Preprint arXiv:0801.1283v1 [quant-ph]\n[16] Decca R S, Fischbach E, Geyer B, Klimchitskaya G L, Kraus e D E, Lopez D, Mohideen\nU, Mostepanenko V M 2008 Comment on ” Possible resolution of t he Casimir force finite\ntemperature correction ”controversies”” Preprint arXiv:0803.4247v1 [quant-ph]\n[17] Ellingsen A., Brevik I, Hoye J S , Milton K A (2008) Low tem perature Casimir-Lifshitz free\nenergy and entropy: the case of poor conductors Preprint arXiv:0809.0763v1 [quant-ph]\n[18] Pirozhenko I, Lambrecht A and Svetovoy V B 2006 New Journal of Physics 8238\n[19] Svetovoy V B, van Zwol P J, Palasantzas G, De Hosson J Th M 2 008Phys. Rev. B77035439\n[20] Pirozhenko I and Lambrecht A 2008 Phys. Rev. A77, 013811\n[21] Iannuzzi D, Lisanti M, Munday J N and Capasso F 2006 J. Phys. A: Math. Gen. 396445\n[22] Lisanti M, Iannuzzi D and Capasso F 2005 Proc. Nat. Acad. Sci. 10211989\n[23] Kastle G, Boyen H G , Schroder A , Plettl A and Ziemann P 200 470, 165414\n[24] Walther M , Cooke D G , Sherstan C, Hajar M, Freeman M R and H egmann F A 2007 Phys.\nRev.B76125408\n[25] Esquivel-Sirvent R. 2008 Phys. Rev A77042107\n[26] Theye M L 1970 Phys. Rev. B23060\n[27] Nagel S R and Schnatterly E 1974 Phys. Rev. B91299\n[28] Bennett H E, Bennett J M 1966 Optical Properties and Electronic Structure of Metals and\nAlloysed F Abel´ es and L Erwin (Amsterdam: North Holland) p 175\n8"    },    {        "title": "0812.2570v1.Non_Adiabatic_Spin_Transfer_Torque_in_Real_Materials.pdf",        "content": "arXiv:0812.2570v1  [cond-mat.mtrl-sci]  13 Dec 2008Non-Adiabatic Spin Transfer Torque in Real Materials\nIon Garate1, K. Gilmore2,3, M. D. Stiles2, and A.H. MacDonald1\n1Department of Physics, The University of Texas at Austin, Au stin, TX 78712\n2Center for Nanoscale Science and Technology, National Inst itute\nof Standards and Technology, Gaithersburg, MD 20899-8412 a nd\n3Maryland NanoCenter, University of Maryland, College Park , MD, 20742\n(Dated: October 30, 2018)\nThe motion of simple domain walls and of more complex magneti c textures in the presence of a\ntransport current is described by the Landau-Lifshitz-Slo nczewski (LLS) equations. 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": "0812.3184v2.Origin_of_intrinsic_Gilbert_damping.pdf",        "content": "Origin of Intrinsic Gilbert Damping\nM. C. Hickey\u0003and J. S. Moodera\nFrancis Bitter Magnet Laboratory, Massachusetts Institute of Technology,\n150 Albany Street, Cambridge, Massachusetts 02139 USA.\nThe damping of magnetization, represented by the rate at which it relaxes to equilibrium, is\nsuccessfully modeled as a phenomenological extension in the Landau-Lifschitz-Gilbert equation.\nThis is the damping torque term known as Gilbert damping and its direction is given by the vector\nproduct of the magnetization and its time derivative. Here we derive the Gilbert term from \frst\nprinciples by a non-relativistic expansion of the Dirac equation. We \fnd that this term arises when\none calculates the time evolution of the spin observable in the presence of the full spin-orbital\ncoupling terms, while recognizing the relationship between the curl of the electric \feld and the time\nvarying magnetic induction.\nPACS numbers: 76.20.-m, 75.30.-m and 75.45.+j\nThe Gilbert damping torque in magnetic systems de-\nscribes the relaxation of magnetization and it was intro-\nduced into the Laudau-Lifschitz equation [1, 2] for de-\nscribing spin dynamics. Gilbert damping is understood\nto be a non-linear spin relaxation phenomenon and it con-\ntrols the rate at which magnetization spins reach equilib-\nrium. The introduction of this term is phenomenological\nin nature [3] and the question of whether it has an in-\ntrinsic physical origin has not been fully addressed, in\nthe face of rather successful modeling of the relaxation\ndynamics of measured systems. Correlating ferromag-\nnetic resonance spectral line-widths [4, 5] in magnetic\nthin \flms with the change in damping has been success-\nful for con\frming the form of the damping term in the\nunderlying dynamical equations. The intrinsic origin of\nthe damping itself is still an open question. The damping\nconstant,\u000bis often reformulated in terms of a relaxation\ntime, and the dominant relaxation processes are invoked\nto calculate this, but this approach presupposes preces-\nsional damping torque.\nIt has been long thought that intrinsic Gilbert damp-\ning had its origin in spin-orbital coupling because this\nmechanism does not conserve spin, but it has never been\nderived from a coherent framework. Non-local spin re-\nlaxation processes [6] and disorder broadening couple to\nthe spin dynamics and can enhance the Gilbert damp-\ning extrinsically in thin \flms and heterostructures. This\ntype of spin relaxation, which is equivalent to ensemble\ndephasing [7], is modeled as the (S-S 0)/T\u0003\n2decay term\nin the dynamical Bloch equation, where T\u0003\n2is the decay\ntime of the ensemble of spins. Crudely speaking, during\nspin relaxation, some spins lag behind the mean mag-\nnetization vector and the exchange and magnetostatic\n\felds then exert a time dependent torque. Calculations\non relaxation driven damping of this kind presuppose the\nGilbert damping term itself which begs the question.\nThe inhomogeneous damping term can be written as\nM\u0002dr2M=dtwhich gives rise to non-local e\u000bects such\n\u0003Electronic mail : hickey@mit.eduas spin wave dissipation [6, 8]. These non-local theo-\nries are successful in quantifying the enhancement of the\nGilbert damping, but do not derive the intrinsic Gilbert\nterm itself. There are models [9, 10] which deal with\nthe scattering of electron spins from thermal equilibrium\nin the presence of phonon and spin-orbital interactions\nwhich is a dynamic interaction and this allows us to de-\ntermine the strength of the Gilbert damping for itiner-\nant ferromagnetic metals, generalizing the Gilbert damp-\ning response to a tensorial description. Both the s-d\nexchange relaxation models [11, 12] and the Fermi sur-\nface breathing models of Kambersky [9, 13] either pre-\nsuppose a Gilbert damping term in the dynamical equa-\ntion or specify a phenomenological Hamiltonian H = -\n1/(\rMs)^\u000b.dM/dt. While this method is ab initio from\nthe point of view of electronic structure, it already as-\nsumes the Gilbert term ansatz. Hankiewicz et al. [14]\nconstruct the inhomogeneous Gilbert damping by con-\nnecting the spin density-spin current conservation law\nwith the imaginary part of magnetic susceptibility ten-\nsor and show that both electron-electron and impurity\nscattering can enhance the damping through the trans-\nverse spin conductivity for \fnite wavelength excitations\n(q6= 0). In previous work [15], there are derivations\nof the Gilbert constant by comparing the macroscopic\ndamping term with the torque-torque correlations in ho-\nmogeneously magnetized electron gases possessing spin\norbital coupling. For the case of intrinsic, homogeneous\nGilbert damping, it is thought that in the absence of\nspin-orbital scattering, the damping vanishes. We aim to\nfocus on intrinsic, homogeneous damping and its physical\norigin in a \frst-principles framework and the question as\nto whether spin in a homogeneous time-varying magne-\ntization can undergo Gilbert damping is addressed.\nIn this work, we show that Gilbert damping does indeed\narise from spin-orbital coupling, in the sense that it is\ndue to relativistic corrections to the Hamiltonian which\ncouple the spin to the electric \feld and we arrive at the\nGilbert damping term by \frst writing down the Dirac\nequation for electrons in magnetic and electric potentials.\nWe transform the Hamiltonian in such a way as to write\nit in a basis in which the canonical momentum terms arearXiv:0812.3184v2  [cond-mat.other]  1 Apr 20092\neven powers. This is a standard approach in relativistic\nquantum mechanics and we do this in order to calculate\nthe terms which couple the linear momentum to the spin\nin a basis which is diagonal in spin space. This is often\nreferred to as a non-relativistic expansion of the Dirac\nequation. This allows us to formulate the contributions\nas a perturbation to an otherwise non-relativistic parti-\ncle. We then wish to calculate the rate equation for the\nspin observable with all of the spin-orbital corrections in\nmind.\nNow, we start with a purely relativistic particle, a Dirac\nparticle and we write the Dirac-Pauli Hamiltonian, as\nfollows :\nH=c\u000b:(p\u0000eA\nc) +\fm 0c2+e\u001e (1)\n=O+\fm 0c2+\" (2)\nwhere Aand\u001eare the magnetic vector potential and the\nelectrostatic potential, respectively and\n\u000b=\u0012\n0\u001bi\n\u001bi0\u0013\nwhile\n\f=\u0012\n1 0\n0\u00001\u0013\n:\nWe observe immediately that \fO=\u0000O\f.Ois the Dirac\ncanonical momentum , c and e are the speed of light in\na vacuum and the electronic charge, respectively.\nWe now need to rewrite the Hamiltonian in a basis where\nthe odd operators (whose generators are o\u000b diagonal in\nthe Pauli-Dirac basis : \u000bi,\ri,\r5..) and even operators\n(whose generators are diagonal in the Pauli-Dirac basis :\n(1,\f, \u0006,.. ) are decoupled from one another.\nIf we are to \fnd S so that H0does not contain odd powers\nof spin operators, we must chose the operator S, in such\na way as to satisfy the following constraint :\n[S;\f] =\u0000O\nim0c2(3)\nIn order to satisfy cancelation of the odd terms of O\nto \frst order, we require S=\u0000iO\f\n2m0c2and this is known\nas the Foldy-Wouthuysen transformation in relativistic\nquantum mechanics and it is treated in some detail in,\nfor example, reference [16]. We now would like to collect\nall of the terms into the transformed Hamiltonian, and\nthis is written as\nH0=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\"\u00001\n8m2\n0c4[O;[O;\"]] +\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\nThe expression above contains odd powers of the canon-\nical momentumO, so we rede\fne the canonical momen-\ntum to encapsulate all of these odd power terms. So wenow apply the procedure of eliminating odd powers once\nagain :\nS0=\u0000i\f\n2m0c2O0=\u0000i\f\n2m0c2\u0012\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\u0013\n(4)\nH00=eiS0\nH0e\u0000iS0\n=\fm 0c2+\"0+O00; (5)\nwhereO00is now O(1\nm2\n0c4), which can be further elimi-\nnated by applying a third transformation (S00=\u0000i\fO00\n2m0c4),\nwe arrive at the following Hamiltonian :\nH000=eiS00\u0010\nH00\u0011\ne\u0000iS00\n=\fm 0c2+\"0\n=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\n\"\u00001\n8m2\n0c4[O;[O;\"]]\nThus we have the fully Foldy-Wouthuysen transformed\nHamiltonian :\nH000=\f\u0012\nm0c2+(p\u0000eA=c)2\n2m0\u0000p4\n8m3\n0c6\u0013\n+e\b\n\u0000e~\n2m0c2\f\u0006:B\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E)\n\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000e~2\n8m2\n0c2(r:E)\nThe terms which are present in the above Hamiltonian,\nshow us that we have a p4kinetic part which is the rela-\ntivistic expansion of the mass of the particle. The terms\nwhich couple to the spin \u0006 are of importance and we see\nthat these terms correspond to the Zeeman, spin-orbital\n(comprising momentum and electric \feld curl terms) and\nthe Darwin term, respectively. Strictly speaking, the\npresence of the scalar potential \u001ebreaks the gauge invari-\nance in the Pauli-Dirac Hamiltonian and a fully gauge in-\nvariant theory would require that this contain the gauge-\nfree electromagnetic \feld energy. We omit the term\ne2~\n4m2c3\u0006:(A\u0002E) (which establishes gauge invariance in\nthe momentum terms) in this rotated Hamiltonian, as it\nis O(1=m2c3) and we are only interested in calculating\nsemiclassical rate equations for \felds, which are mani-\nfestly gauge-invariant, and not wavefunctions or energy\neigenvalues. We can now de\fne the spin dependent cor-\nrections to a non-relativistic Hamiltonian :\nH\u0006=\u0000e~\n2m0c2\f\u0006:B\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E):\n(6)\nwhere\n\u0006=\u0012\n\u001bi0\n0\u001bi\u0013\n\u0011^Si:3\nand\u001biare the Pauli matrices. Note that the last\ntwo terms in Equation 6 encapsulate the entire spin\norbital coupling in the sense that these terms couple\nthe particle's linear momentum to the spin ^Si. The\n\frst spin-orbital term in the Hamiltonian is well known\nand give rise to momentum dependent magnetic \felds.\nWhen the ensuing dynamics are calculated for this\ncase, it gives rise to spin relaxation terms which are\nlinear in spin [17]. Note that, while neither spin-orbital\nterm is Hermitian, the two terms taken together are\nHermitian and so the particles angular momentum\nis a conserved quantity and the total energy lost in\ngoing from collective spin excitations (spin waves) to\nsingle particles states via spin-orbital coupling is gained\nby the electromagnetic \feld. Recognizing the curl of\nthe electric \feld in the last term, we now rewrite this\nthe time varying magnetic \feld as given by Maxwells\nequations asr\u0002E=\u0000@B\n@t. We now have an explicitly\ntime-dependent perturbation on the non-relativistic\nHamiltonian. We can write the time-varying magnetic\n\feld seen by the spin (in, for example a magnetic\nmaterial) as@B\n@t=@B\n@M\u0001@M\n@t=\u00160(1 +\u001f\u00001\nm)@M\n@t. We now\nhave the spin dependent Hamiltonian :\nHS=\u0000e~\n2m0c2\fS:B\u0000e~\n4m2\n0c2S:E\u0002p\n+ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt=\nHS=HS\n0+HS(t):\nWe focus our attention on the explicitly time-dependent\npart of the Hamiltonian HS(t) ;\nHS(t) =ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt: (7)\nIn this perturbation scheme, we allow the Hermitian\ncomponents of the Hamiltonian to de\fne the ground sate\nof the system and we treat the explicitly time-dependent\nHamiltonian (containing the spin orbital terms) as a time\ndependent perturbation. In this way, the rate equation is\nestablished from a time dependent perturbation expan-\nsion in the quantum Liouville description. We now de\fne\nthe magnetization observable as ^M=X\n\u000bg\u0016B\nVTr\u001a^S\u000b(t)\nwhere the summation is taken over the site of the magne-\ntization spin \u000b. We now examine the time dependence of\nthis observable by calculating the rate equation according\nto the quantum-Liouville rate equation ;\nd\u001a(t)\ndt+1\ni~[^\u001a;H] = 0 (8)\nThis rate equation governs the time-evolution of the\nmagnetization observable as de\fned above, in the non-\nequilibrium regime. We can write the time derivative ofthe magnetization [18], as follows ;\ndM\ndt=X\nn;\u000bg\u0016b\nVh\tn(t)j1\ni~[\u001aS\u000b;H] +@\u001a\n@tS\u000b+\u001a@S\u000b\n@tj\tn(t)i;\nand we can use the quantum Liouville rate equation as\nde\fned by Equation 8 to simplify this expression and we\narrive at the following rate equation :\ndM\ndt=X\n\u000bg\u0016b\nV1\ni~Trf\u001a[S\u000b;HS(t)]g (9)\nIn the case of the time dependent Hamiltonian derived\nin equation 7, we can assume a \frst order dynamical\nequation of motion given bydM\ndt=\rM\u0002Hand calculate\nthe time evolution for the magnetization observable :\ndM\ndt=X\n\u000b;\fg\u0016B\nV1\ni~Tr\u001a[Si\n\u000b;ie~2\u00160\n8m2c2Sj\n\f]:(1 +\u001f\u00001\nm) !@M\ndt\n=X\n\u000bg\u0016B\nVie~2\u00160\n8m2c21\ni~Tr\u001ai ~\u000fijkSk\n\u000b\u000e\u000b\f(1 +\u001f\u00001\nm)\u000ejl !@ Ml\ndt\n=\u0000ie~\u00160\n8m2c2(1 +\u001f\u00001\nm)M\u0002 !@M\ndt;\nwhere, in the last two steps, we have used the fol-\nlowing commutation relations for magnetization spins :\n[Si\n\u000b;Sj\n\f] =i~\u000fijkSk\n\u000b\u000e\u000b\fwhich implies that the theory pre-\nsented here is that which relates to local dynamics and\nthat the origin of the damping is intrinsic. We now rec-\nognize the last equation as the which describes Gilbert\ndamping, as follows :\ndM\ndt=\u0000\u000b\nMs:M\u0002 !@M\n@t(10)\nwhereby the constant \u000bis de\fned as follows :\n\u000b=ie~\u00160Ms\n8m2\n0c2\u0000\n1 +\u001f\u00001\nm\u0001\n(11)\nThe\u000bde\fned above corresponds with the Gilbert\ndamping found in the phenomenological term in the\nLandau-Lifschitz-Gilbert equation and \u001fmis the mag-\nnetic susceptibility. In general, the inverse of the suscep-\ntibility can be written in the form [19],\n\u001f\u00001\nij(q;!) = ~\u001f\u00001\n?(q;!)\u0000!ex\n\r\u00160M0\u000eij; (12)\nwhere the equilibrium magnetization points along the z-\naxis and!exis the excitation frequency associated with\nthe internal exchange \feld. The \u000eijterm in the in-\nverse susceptibility does not contribute to damping mech-\nanisms as it corresponds to the equilibrium response.4\nIn the basis (M x\u0006iMy,Mz), we have the dimensionless\ntransverse magnetic susceptibility, as follows :\n~\u001fm?(q;!) =\r\u00160M0\u0000i\r\u001b?q2\n!0\u0000!\u0000i\r\u001b?q2!0=M0\nThe \frst term in the dimensionless Gilbert coe\u000ecient\n(Equation 11) is small ( \u001810\u000011) and the higher damp-\ning rate is controlled by the the inverse of the suscep-\ntibility tensor. For uniformly saturated magnetization,\nthe damping is critical and so the system is already at\nequilibrium as far as the Gilbert mechanism is concerned\n(dM/dt = 0 in this scenario). The expression for the\ndimensionless damping constant \u000bin the dc limit ( !=0\n) is :\n\u000b=e~\u00160Ms\n8m2\n0c2Im0\n@!0\n\r\u00160M0\u0000i\u001b?q2!0\n\u00160M2\n0\n1\u0000i\r\u001b?q2=M01\nA; (13)\nand we have the transverse spin conductivity from the\nfollowing relation (in units whereby ~=1) :\n\u001b?=n\n4m\u0003!2\n0\u00121\n\u001cdis\n?+1\n\u001cee\n?\u0013\n;\nwhere\u001cdis\n?and\u001cee\n?are the impurity disorder and electron\nelectron-electron scattering times as de\fned and param-\neterized in Reference [14]. We calculate the extrinsically\nenhanced Gilbert damping using the following set of pa-\nrameters as de\fned in the same reference ; number den-\nsity of the electron gas, n=1.4 \u00021027m\u00003, polarization p,\nequilibrium magnetization M 0=\rpn/2, equilibrium ex-\ncitation frequency !0=EF[(1 +p)2=3\u0000(1\u0000p)2=3] and\nwave-number de\fned as q = 0.1 k F, where E Fand kF\nare the Fermi energy and Fermi wave number, respec-\ntively. m\u0003is taken to be the electronic mass. Using these\nquantities, we evaluate \u000bvalues and these are plotted as\na function of both polarization and disorder scattering\nrate in Figure 1.\nIn general, the inverse susceptibility \u001f\u00001\nmwill deter-\nmine the strength of the damping in real inhomogeneous\nmagnetic systems where spin relaxation takes place, sub-\nbands are populated by spin orbit scattering and spin\nwaves and spin currents are emitted. The susceptibil-\nity term gives the Gilbert damping a tensorial quality,\nagreeing with the analysis in Reference [10]. Further, the\nconnection between the magnetization dynamics and the\nelectric \feld curl provides the mechanism for the energy\nloss to the electromagnetic \feld. The generation of radi-\nation is caused by the rotational spin motion analog of\nelectric charge acceleration and the radiation spin inter-\naction term has the form :\nHS(t) =ie~2\u00160\n8m2\n0c2X\n\u000b\u0000\n1 +\u001f\u00001\nm\u0001\nS\u000b:dM\ndt: (14)In conclusion, we have shown that the Gilbert term,\nheretofore phenomenologically used to describe damping\nFIG. 1: (Color Online) Plot of the dimensionless Gilbert\ndamping constant \u000bin the dc limit ( !=0), as a function of\nelectron spin polarization and disorder scattering rate.\nin magnetization dynamics, is derivable from \frst prin-\nciples and its origin lies in spin-orbital coupling. By a\nnon-relativistic expansion of the Dirac equation, we show\nthat there is a term which contains the curl of the elec-\ntric \feld. By connecting this term with Maxwells equa-\ntion to give the total time-varying magnetic induction,\nwe have found that this damping term can be deduced\nfrom the rate equation for the spin observable, giving the\ncorrect vector product form and sign of Gilberts' origi-\nnal phenomenological model. Crucially, the connection\nof the time-varying magnetic induction and the curl of\nthe electric \feld via the Maxwell relation shows that\nthe damping of magnetization dynamics is commensu-\nrate with the emission of electromagnetic radiation and\nthe radiation-spin interaction is speci\fed from \frst prin-\nciples arguments.\nAcknowledgments\nM. C. Hickey is grateful to the Trinity and the uni-\nformity of nature. We thank the U.S.-U.K. Fulbright\nCommission for \fnancial support. The work was sup-\nported by the ONR (grant no. N00014-09-1-0177), the\nNSF (grant no. DMR 0504158) and the KIST-MIT pro-\ngram. The authors thank David Cory, Marius Costache\nand Carlos Egues for helpful discussions.5\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowiet. Un. 8, 153\n(1935).\n[2] E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics\nPart 2 (Pergamon Press, Oxford, United Kingdom,\n1980).\n[3] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004).\n[4] C. E. Patton, C. H. Wilts, and F. B. Humphrey, J. Appl.\nPhys. 38, 1358 (1967).\n[5] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Japanese Journal of Ap-\nplied Physics 45, 3889 (2006).\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[7] Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[8] G. Eilers, M. L uttich, and M. M unzenberg, Phys. Rev.\nB74, 054411 (2006).\n[9] V. Kambersk\u0012 y, Canadian Journal of Physics 48, 2906\n(1970).[10] D. Steiauf and M. F ahnle, Phys. Rev. B 72064450\n(2005).\n[11] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[12] M. F ahnle, R. Singer, D. Steiauf, and V. P. Antropov,\nPhys. Rev. B 73, 172408 (2006).\n[13] J. Kune\u0014 s and V. Kambersk\u0013 y, Phys. Rev. B 65, 212411\n(2002).\n[14] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 78, 020404 (2008).\n[15] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 75, 174434 (2007).\n[16] W. Greiner, Relativistic Quantum Mechanics (Springer-\nVerlag, Berlin, Germany, 1987).\n[17] H.-A. Engel, E. I. Rashba, and B. I. Halperin, Phys. Rev.\nLett. 98, 036602 (2007).\n[18] J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett.\n92, 097601 (2004).\n[19] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002)."    },    {        "title": "0901.2191v1.The_sound_damping_constant_for_generalized_theories_of_gravity.pdf",        "content": "arXiv:0901.2191v1  [hep-th]  15 Jan 2009The sound damping constant\nfor generalized theories of gravity\nRam Brustein\nDepartment of Physics, Ben-Gurion University,\nBeer-Sheva, 84105 Israel, E-mail: ramyb@bgu.ac.il\nA.J.M. Medved\nPhysics Department, University of Seoul,\nSeoul 130-743 Korea, E-mail: allan@physics.uos.ac.kr\nAbstract\nThe near-horizon metric for a black brane in Anti-de Sitter ( AdS)\nspace and the metric near the AdS boundary both exhibit hydro dy-\nnamic behavior. We demonstrate the equivalence of this pair of hy-\ndrodynamic systems for the sound mode of a conformal theory. This\nis first established for Einstein’s gravity, but we then show how the\nsound damping constant will be modified, from its Einstein fo rm, for\na generalized theory. The modified damping constant is expre ssible as\nthe ratio of a pair of gravitational couplings that are indic ative of the\nsound-channel class of gravitons. This ratio of couplings d iffers from\n1both that of the shear diffusion coefficient and the shear viscos ity to\nentropy ratio. Our analysis is mostly limited to conformal t heories\nbut suggestions are made as to how this restriction might eve ntually\nbe lifted.\n1 Introduction\nThenear-horizon geometry of ablack branein an Anti-de Sitt er (AdS)\nspacetime provides a translationally invariant and therma lly equili-\nbrated background; two of the characteristic features of an y hydrody-\nnamic theory. Indeed, the long-wavelength fluctuations of t he near-\nhorizon metric are known to satisfy equations of motion that are com-\npletely analogous to the hydrodynamic equations of a viscou s fluid\n[1]. The very same statements can be made about the metric nea r the\nAdS boundary. It is, however, quite remarkable that this pai r of ef-\nfective theories appears to be described by an equivalently defined set\nof hydrodynamic parameters [1, 2, 3, 4]; this, in spite of the ir obvious\nlack of proximity.\nThe relevant thermodynamic and hydrodynamic parameters —\nsuch as the entropy density or the various transport coefficie nts —\nare intrinsic properties of the black brane horizon. Conseq uently,\nsuch parameters should be determined by the near-horizon me tric.\nThis makes it all the more phenomenal that the very same param -\neters are employed in the AdS boundary theory. In the context of\nthe gauge–gravity duality, this apparent non-locality bec omes much\nmore sensible. The duality relates the AdS boundary hydrody namics\nto the hydrodynamics of strongly coupled gauge theories [5, 3]. Mean-\n2while, thedual gauge theory is supposedto have its thermal p roperties\nascribed to it, holographically, by the thermodynamic natu re of the\nblack brane.\nMost calculations inthis genretake place at theouter bound ary, as\nthis is the most convenient surface for relating the bulk gra vitational\ntheory to its gauge-theory dual. In many ways, however, the m ost\nnatural setting is at the black brane horizon, where the vari ous hydro-\ndynamicparameters are actually defined. Thegraviton hydro dynamic\n“fluid” can be interpreted as “living” on the stretched horiz on, and\nso it would be rather disturbing if the actual calculations c ould only\nbe performed on a surface that is displaced a spacetime away. Our\nresults will make it clear that there is really nothing parti cularly spe-\ncial about either the stretched horizon or the AdS outer boun dary.\nRather, all calculations might as well be done on any radial s hell that\nis external to the horizon.\nStrongly coupled gauge theories provide an intriguing theo retical\nlaboratory to investigate the field of relativistic hydrody namics. It\nis hoped that, by applying the duality to certain calculatio ns on the\ngravity side, one would be able explain the experimental res ults of\n— for instance — heavy-ion collisions [6]. However, advance ments in\nthis direction has been somewhat impeded for the following r eason:\nStudieson thehydrodynamicsof theAdS boundaryhave, for th emost\npart, been limited to Einstein’s theory of gravity, which — f rom the\ngauge-theory perspective — corresponds to infinitely stron g ’t Hooft\ncoupling. Insofar as the objective is to apply what can be lea rnt from\nthe duality to physically real systems, one actually requir es knowledge\nabout gauge theories at finitevalues of ’t Hooft coupling.\n3As it so happens, a strong-coupling expansion on the gauge-t heory\nside corresponds to an expansion in the number of derivative s on the\ngravity side. Since Einstein’s gravity is only a two-deriva tive theory, it\nshould be clear that describing a finitely coupled gauge theo ry neces-\nsitates some sort of extension of Einstein’s theory. To put i t another\nway, any discernible progress will depend upon our understa nding of\nthe boundary hydrodynamics for theories of generalized gravity.\nIn a previous paper [7], we were able to establish two relevan t\npoints. First, with the focus on the shear channel of fluctuat ing grav-\nitational modes, it was shown that the AdS boundary hydrodyn amics\ncan be translated to and localized on any radial shell in the a ccessi-\nble spacetime; including at the (stretched) horizon of the b lack brane.\nThen, by following [8], we explicitly demonstrated how this formalism\ncan be extended to any generalized (or Einstein-corrected) gravita-\ntional theory.\nIn [8, 7], we used insight from [9] to make a pertinent observa tion:\nVarious hydrodynamic parameters of an AdS brane theory can b e\nidentified with the different components of a (generally) pola rization-\ndependent gravitational coupling κµν. Meaning that, for a generic\ntheory, differently polarized gravitons will effectively have differing\nNewton’s constants. As shown in [9], this distinction can be quanti-\nfied at a remarkably rigorous level. With this prescription, the shear\nviscosity to entropy density ratio η/sis generalized from its “stan-\ndard” (Einstein) value of 1 /4πaccording to [8]η\ns=1\n4π(κrt)2\n(κxy)2, with\nthe precise meaning of the subscripts to be clarified below. M oreover,\nthe central finding of [7] was that the shear diffusion coefficien tDis\nmodified from its usual expression 1 /4πTinto the form D=κ2\nzx\nκ2\ntx1\n4πT,\n4withTbeing the temperature.\nOneshouldtake notethat thecouplingratios for η/sandDinvolve\ndifferent polarization directions. This is a natural consequ ence of the\nclass of gravitons that is implicated by each of the hydrodyn amic\nparameters. In the so-called radial gauge, the non-vanishi ng gravitons\nseparate into three decoupled classes or “channels”: scala r, shear and\nsound [10]. The shear viscosity ηis most directly associated with\nthe first of these classes, whereas the shear diffusion coefficie ntD\nis a characteristic of the second. As for the third class, one would\nanalogously associate with it the sound damping constant Γ, as well\nas the sound velocity (squared) c2\ns.\nThe purpose of the current paper is to analyze the case of soun d-\nmode fluctuations. A straightforward extension of previous analyses\nis inhibited by two technical issues that are intrinsic to al most any\nrigorous study of the sound channel. First, for a non-confor mal gauge\ntheory, the sound-channel analysis is highly model specific . Second,\nthe same non-conformality induces would-be radial invaria nts to vary\nwith radial position in the bulk. (See [11] for a discussion. ) As a con-\nsequence, Γ and c2\nsare, even for Einstein’s gravity, model-dependent\nparameters that vary with radial position in a model-specifi c way.\nWe can still be quite definitive by restricting the immediate con-\nsiderations to conformal theories. When conformality is pr otected by\neffectively “switching off” all massive fields, the above compli cations\nwill no longer be of issue. At the same time, we will still be ab le\nto make statements about how deviations from conformality s hould\ninfluence the ensuing results. In this sense, the current stu dy can be\nviewed as a significant first step towards a fully generic anal ysis.\n5Similarly to [7], we will begin here by establishing a direct con-\nnection between sound-mode (conformal) hydrodynamics on t he AdS\nouter boundary and on any other radial shell up to the horizon of the\nblack brane. This will be accomplished by examining the corr elator\nof an appropriately defined graviton and verifying that its p ole struc-\nture, which determines the associated dispersion relation , is a radial\ninvariant.\nNext, we will determine how this correlator pole is explicit ly modi-\nfied for a generalized (although still conformal) theory of g ravity. This\nwill enable us to extract the Einstein-corrected form of the damping\nconstant Γ. Additionally, we will confirm that the sound velo cityc2\ns\nremains fixed at its conformal value. As also discussed, the v ery same\noutcomes can be deduced through an inspection of the conserv ation\nequation for the dissipative stress tensor.\nThe paper will conclude with a preliminary discussion of pos sible\nextensions of our analysis to the non-conformal case.\nNote that, to avoid needless repetition, some salient point s that\nare already covered thoroughly in [7] (also see [8]) will onl y be glossed\nover here.\n2 Soundmode conformalhydrodynam-\nics for Einstein’s gravity\nLet us first introduce some notation and conventions, as well as es-\ntablish the basic framework. We will be considering a black p–brane\nin ad+ 1-dimensional (asymptotically) AdS spacetime. (Note tha t\n6d=p+ 1≥4.) Given translational invariance and spatial isotropy\non the brane along with a static spacetime, the associated me tric can\nalways be expressed in the generic brane form\nds2=−gtt(r)dt2+grr(r)dr2+gxx(r)/parenleftBiggp/summationdisplay\ni=1dx2\ni/parenrightBigg\n,(1)\nwheregtt(r) has a simple zero and grr(r) has a simple pole at the\nhorizonr=rh, whilegxx(rh) is finite and positive. For any r >\nrh, these metric components are all well-defined and strictly p ositive\nfunctions that go asymptotically to their respective AdS va lues (L2/r2\nforgrr, otherwise r2/L2)asr→ ∞. (ListheAdSradiusofcurvature.)\nIf the background theory is conformal, then one can be much mo re\nexplicit. Assuming, for the sake of simplicity, that thebra neis electro-\nmagnetically neutral, we obtain the Schwarzschild-like fo rm such that\ngxx=r2/L2andgtt= 1/grr=gxxf(r), withf(r) = 1−(rh/r)p+1.\nIt is often convenient to re-express this conformal metric b y changing\nthe radial coordinate to u=r2\nh/r2; then\nds2=−r2\nh\nL2uf(u)dt2+L2\n4u2du2\nf(u)+r2\nh\nL2u/parenleftBiggp/summationdisplay\ni=1dx2\ni/parenrightBigg\n,(2)\nwithf(u) = 1−up+1\n2and the horizon (outer boundary) now located\natu= 1 (u= 0). When non-conformal theories are discussed, u\nwill refer to a radial coordinate that is appropriately defin ed so as to\nextend over the same range of values.\nBrane hydrodynamics entails expanding the metric: gµν→gµν+\nhµν, withhµνrepresenting the fluctuations or gravitons. Let us —\nwithout loss of generality — specify xpto be the direction of gravi-\nton propagation on the brane and re-label it as z. It follows that\n7hµν∼exp[−iΩt+iQz] (and, otherwise, depending only on u), where\n(Ω,0,...,0,Q) is thep+1–momentum of the graviton.\nThe choice of radial gauge, huα= 0 for any α, is known to separate\nthe non-vanishing fluctuations into three decoupled classe s [10]. Our\nclass of current interest — namely, the sound channel — inclu des the\nnon-vanishing diagonal gravitons hαα(α/negationslash=u) along with htz.\nLet ustake noteof thesound-modedispersionrelation Ω = ±csQ−\niΓQ2+O(Q3) or, equivalently (given that the hydrodynamic or long-\nwavelength limit is in effect),\nΩ2=c2\nsQ2+i2ΓΩQ2+O(Q4). (3)\nHere,c2\nsis the sound velocity (squared) and Γ is the sound damping\nconstant. For a p+ 1-dimensional conformal theory, c2\ns= 1/pand\nΓ is directly proportional to the shear viscosity to entropy density\nratio times the inverse temperature: Γ =p−1\np1\nTη\ns. So that, for a p-\nbrane theory of Einstein’s gravity, one can deduce that Γ =p−1\np1\n4πT\n[12, 13], where Tis the coordinate-invariant Hawking temperature of\nthe brane. For this conformal case, T= (p+1)rh/4πL2. Meanwhile,\nfor a non-conformal theory, both parameters can differ apprec iably\nfrom their conformal values. For Γ, this model-specific devi ation is\nexpressible in terms of the bulk viscosity ζ.\nFor a complete derivation of the sound-mode correlator (whi ch\nis not needed here), one can follow the by-now standard presc rip-\ntion as documented in, for instance, [14, 15, 16]. The first st ep is\nto identify a gauge-invariant combination of the sound-mod e fluctu-\nationsHtt= (1/gtt)htt,Hzz= (1/gxx)hzz,Htz= (1/gxx)htzand\n8HX= (1/gxx)1\np−1/parenleftBig/summationtextp−1\ni=1hxixi/parenrightBig\n:\nZ=q2gtt\ngxxHtt+2qωHtz+ω2[Hzz−HX]+q2gtt′\ngxx′HX.(4)\nHere, a prime indicates a differentiation with respect to u; while\nω= Ω/2πTandq=Q/2πTrepresent, respectively, a dimension-\nless frequency and wavenumber. In the hydrodynamic limit, ωandq\nare both vanishing although not a prioriat the same rate.\nTheconformalversionofthesolutionfor Zcanreadilybeextracted\nout of the existent literature — for instance, [15, 16, 17]. F or the\nappropriately chosen boundary conditions (as discussed be low), one\nfinds that\nZ=Cf(u)−iω\n2/bracketleftbigg\nY(u)−ω2\nq2p−iω(p−1)f(u)+O(q2,ω2)/bracketrightbigg\n,(5)\nwhereCis an integration constant (to be fixed by normalization con-\nsiderations) and we have defined Y(u)≡gtt′/gxx′. For this theory in\nparticular, Y= (f/u)′/(1/u)′=f−uf′, which is everywhere positive,\nnon-vanishing and O(1). Also note that Y= 1 on the outer boundary.\nA specified pair of boundaryconditions determines the solut ion for\nZ. At the horizon u= 1, the solution should be that of an incoming\nplane wave, which determined the form of Eq. (5). In addition , the so-\ncalled Dirichlet boundary condition still needs to be impos ed. It has\nbecome almost traditional to single out the AdS boundary and choose\nu∗= 0 as the radius at which this condition is enforced; however , one\ncan freely impose this condition at any fixed radius u∗within 0 ≤\nu∗<1. The Dirichlet boundary condition necessitates that Z(u) is,\npriorto its normalization (see below), vanishing as u→u∗. Applying\nthiscondition toEq. (5), we promptlyobtain theassociated dispersion\n9relation\nω2=q21\npY(u∗)+iωq2p−1\npf(u∗)+O(q4), (6)\nwhere it is now clear that qandωare of the same order in the hydro-\ndynamic limit. Let us choose, for instance, the “orthodox” b oundary\nlocation of u∗= 0, so that Eq. (6) leads to ω2=q2/p+iωq2(p−\n1)/p+O(q4). Comparing this to the standard dispersion relation in\nEq. (3), one can readily verify the expected identifications c2\ns= 1/p\nand Γ = ( p−1)/(4πpT) for a conformal theory.\nOne further normalization condition that complements the D irich-\nlet boundary condition is that Z, rather than vanishing at u∗, should\nultimately be normalized to unity there. This can be achieve d by the\nunique choice\nC−1=/bracketleftbigg\nY(u∗)−ω2\nq2p−iω(p−1)f(u∗)/bracketrightbigg\n. (7)\nLet us take notice that, given the associated dispersion rel ation,C−1\nis a vanishing quantity as it must be to obtain a finite value of Z(u∗).\nThe normalized value of the field mode is simply 1, and so one\nmight wonder as to the physical significance of the implied di sper-\nsion relation. However, we are simply using the standard “tr ick” of\nfield-theoretic calculations to obtain the pole in the corre lator. The\nproperly normalized correlator GZZfor this gauge-invariant variable\n(up to an inconsequential numerical factor) is given by the b oundary\nresidue of the canonical term in the bulk action or GZZ∼ZZ′|u=u∗.\nIt should not be difficult to convince oneself that, at the lead ing hy-\ndrodynamic order, this quantity goes as GZZ∼C· O(q0), which is\nnotably divergent and of finite hydrodynamic order. A proper ac-\ncounting of the metrical factors in the action — namely, the p roduct\n10√−gguu— reveals that there are no other hidden zeros or infinities\nin this calculation at any permissible value of u∗.\nWhat is really significant here is that — from the quasinormal -\nmode perspective of brane hydrodynamics [15] — the pole in th e cor-\nrelator assigns a clear physical credence to Eq. (6) as the sp ectrum\nfor the dissipative modes of the black brane. However, one co uld (and\nshould!) be rightfully concerned that this dispersion rela tion appears\nto vary as the Dirichlet-boundary surface is moved radially through\nthe spacetime. This is not only in conflict with intuitive exp ectations\nbut with the analysis of [11], where it is made evident that (i nasmuch\nas the theory is conformal) both the sound mode and its correl ator\nshould be radial invariants.\nWe can readily account for the undesirable factor of f(u∗) in the\nsecond term of Eq. (6): As detailed in [7], ωandqshould natu-\nrally be sensitive to the the effects of a gravitational redshi ft. It was\nthen argued — in the context of shear modes — that consistency of\nthe hydrodynamic expansion along with protection of the inc oming\nboundary condition necessitates that ωremains fixed while q2scales\nasf−1. That is, ω(u) =ωbandq2(u) =q2\nb/f(u) (with the subscript\nbindicating the outer-boundary value of a quantity). Then, s ince the\ngravitational redshift should not be able to discriminate b etween the\ndifferent channels being probed, it follows that these same re lations\nshould persist for the current case.\nThis brings us to the first term in Eq. (6), which has the awkwar d\nappearance of Y(u∗) to be dealt with. Clearly, this will require new\ninputs. Thekey hereis theassociation of this term with c2\ns;cf, Eq. (3).\nNormally, the sound velocity of a hydrodynamic fluid is prese nted as\n11the variation of the pressure with respect to the energy dens ity or\nc2\ns=δP/δǫ. This cannot, however, be a universally accurate account.\nA closer look at the derivation of the sound dispersion relat ion (see,\ne.g.,[13])revealsthattheactual variation whichentersunder theguise\nof the sound velocity comes packaged in the term/parenleftbig\nδTzz/δTtt/parenrightbig\n∂2\nzTtt,\nwhereTαβis the stress tensor for the brane theory. For a flat or\nan effectively flat brane, such as at the AdS outer boundary, thi s\ndistinction is of no consequence, but this is not a general tr uism. On\na “warped” brane, rather, Tzz=gzzPandTtt=−gttǫ. Hence, the\ncorrect statement about the relevant term is (by way of the ch ain rule)\nδTzz\nδTtt∂2\nzTtt=gzzδP\nδǫ∂2\nzǫ+gttP\nǫ∂ugzz\n∂ugtt∂2\nzǫ . (8)\nWe will now argue that the first term on the right-hand side of\nEq. (8) is parametrically smaller than the second and, thus, the for-\nmer can be disregarded for current purposes. It follows from the ther-\nmodynamic relation sT=ǫ+Pand the infinite transverse volume of\nthe brane that ∂P/∂ǫ= 0. So, to the leading non-vanishing order,\nδP=1\n2∂2P\n∂ǫ2(δǫ)2orδP\nδǫ∼δǫ\nǫ≪1 . On the other hand, P/ǫ=1\npis\nof the order of unity. Now, comparing gtt∂ugzz\n∂ugttwithgzz, one will find\nthat the ratio of these quantities is of O(1).\nHaving deemed the first term in Eq. (8) as inconsequential, we\nneed only to evaluate the second. Since the brane metric is di agonal\n(so that gαα=g−1\nαα), the right-hand side reduces to\nδTzz\nδTtt∂2\nzTtt=P\nǫgtt\ng2xxY−1∂2\nzǫ+···, (9)\nwhere we have returned to the brane notation of Eq. (1) (so tha t\ngtt>0) and recalled the definition of Y(u) beneath Eq. (5).\n12Actually, all other terms in thedispersionrelation contai n a spatial\ncomponent of the brane stress tensor, and so share a common fa ctor\nofg−1\nxx. Hence, we can strip off one factor of this from the right side o f\nEq. (9). Next, let us identify P/ǫas the sound velocity as measured\non the outer AdS boundary and everything to the left of ∂2\nzǫas the\nsound velocity as measured on a radial shell of arbitrary rad ius. Then\nit follows that the sound velocity scales relative to the out er boundary\nas\n/bracketleftbig\nc2\ns/bracketrightbig\nu=Y−1(u)gtt(u)\ngxx(u)[c2\ns]b, (10)\nwhere a subscript of udenotes the value of a parameter at that radius.\nCalling again on our conformal-theory notation, let us take note that,\nbydefinition, f=gtt/gxx, andsothesoundvelocity equivalently scales\nasf/Y.\nNext, let us re-express Eq. (6) in a way that makes the scaling\nproperties of the parameters explicit:\nω2\nu∗=q2\nu∗Y(u∗)[c2\ns]u∗+iωu∗q2\nu∗p−1\npf(u∗)+O(q4\nu∗).(11)\nHere, we have made the identification1\np→[c2\ns]u∗on the basis that\nthe Dirichlet-boundary surface is where the sound velocity should be\ncalibrated to its conformal value — just like it is the Dirich let surface\nthat defines where the field Zis exactly unity. We can now apply the\npreviously discussed scalings ( q2∼1/f,c2\ns∼f/Yand an invariant ω)\nto convert the above expression into one that involves only t he outer-\nboundary values of the parameters. Also recalling that f=Y= 1 at\nthe AdS boundary u= 0, we then have\nω2\nb=q2\nbY(0)[c2\ns]b+iωbq2\nbp−1\npf(0)+O(q4\nb). (12)\n13But this is precisely what would have been obtained had we mad e the\nchoice of u∗= 0 in the first place. Hence, the dispersion relation is\nindeed a radial invariant and, by direct implication, the co rrelator is\nas well.\n3 Soundmode conformalhydrodynam-\nics for generalized theories of gravity\nNext on the agenda, we will investigate as to how the scenario changes\nwhen the theory is extended from Einstein’s gravity. It will be shown\nthat, for a quite general (although still conformal) gravit y theory, the\ndamping constant is modified in a very precise way. Meanwhile , the\nsound velocity is shown to be unmodified, as must be the case fo r\na conformal theory. These tasks will be accomplished by exam ining\nthe (modified) pole of the just-discussed correlator. These general-\nizations will be further supported by a simple argument that is based\nupon inspecting the conservation equation that gives rise t o the sound\ndispersion relation.\nBy ageneralized gravity theory, wehave inmindaLagrangian that\ncan be expressed as Einstein’s form plus higher-derivative s terms. If\nEinstein’s gravity is “non-trivially” modified by these cor rections —\nmeaning that the general Lagrangian can notbe converted into Ein-\nstein’s form by a field redefinition — then the gravitational c oupling\nis no longer as simple as κ2\nE= constant. Rather, the coupling (or\neffective Newton’s constant) can be expected to depend on the p olar-\nization of the gravitons being probed. We will denote this de pendence\n14by expressing the general couplings as κµν.\nIt is now well understood as to how one should calculate these\ncouplings for a given theory [9, 8, 7]. These formalities nee d not\nconcern the present discussion, although a schematic under standing\nof how the couplings come about should prove useful. One begi ns by\nwriting the Lagrangian as a perturbative expansion in power s of the\nmetric fluctuations or h’s. Of particular significance are the terms\nthat are quadratic in hand contain exactly two derivatives. For such\nterms, the gravitational couplings are identified on the pre mise that\nhµν→κµνhµνleads to a canonical kinetic term for the µν-polarized\ngraviton.\nAs it turns out, the gravitational couplings are expressibl e strictly\nin terms of the metric at the horizon. Like the metric, they ar e\ntypically radial functions; however, at the level of a two-d erivative\nexpansion of the Lagrangian, the couplings can safely be tre ated as\n(polarization-dependent) constants. Moreover, sincethe horizon is the\ntrue arena for black brane hydrodynamics, this locality is q uite nat-\nural and falls in line with other parameters, such as the entr opy and\nshear viscosity, being intrinsic properties of this specia l surface.\nLetusre-emphasizethatanygivenhydrodynamicparameters hould\nbe modified according to the class of gravitons that it probes . By\nworking in the radial gauge and then restricting to the decou pled set\nof modes that defines the sound channel, we are limited to a sel ect\nclass. Namely, the zz,tt, andtz-polarized gravitons, as well as the\n“trace mode”, which can be identified with HXin Eq. (4).\nWhen the theory is conformal, we can anticipate a further lim i-\ntation. To elaborate, in obtaining the solution for Z(see,e.g., [15,\n1516, 17]), one finds that the Httmode makes no direct contribution to\nEq. (5). (This is not at all true when conformality is broken. ) Recall-\ning that the gauge–gravity duality identifies the tt-polarized gravitons\nwith fluctuations in the energy density, we suspect that this null con-\ntribution is another manifestation of the suppression of th e variation\nδP/δǫ(as discussed in the previous section). On this basis, it see ms\nreasonable to suggest that the ttfluctuations can be excluded from a\nconformal theory in the hydrodynamic limit.\nAs implied above, the modifications of interest can be extrac ted\nfromthepolestructureof the(generalized) correlator GZZ. Critical to\nthis procedureis the identification of the gravitational co uplinghµν→\nκµνhµν, which persuades us to adapt the gauge-invariant variable Z\nof Eq. (4) as follows:\nZ= 2qωκtzHtz+ω2κzzHZ+q2YκzzHX, (13)\nwhereHZ≡Hzz−HX, the non-contributing mode Htthas been\ndroppedand, as before, Y=gtt′/gxx′. Also, the spatial isotropy of the\nbranehasenabledustomaketheconvenientsubstitution1\np−1/summationtextp−1\ni=1κxixi→\nκzz.\nThe scaling properties of the damping constant can now be de-\ntermined with a methodology akin to dimensional analysis: First,\nredefine the wavenumber and the frequency (and other paramet ers as\nnecessary) with a scaling operation, second, re-express the solution in\nterms of these revised parameters and, third, interpret the modified\npole structure. With regard to the first step, it is actually n ecessary\nto fixω, otherwise the incoming boundary condition at the horizon\nwould be jeopardized. We are, however, free at this level of a nalysis\n16to change the normalization of Z. On this basis, we arrive at\nZ= 2qωκtz\nκzzHtz+ω2HZ+q2YHX (14)\nor\nZ= 2/tildewideqωHtz+ω2HZ+/tildewideq2/tildewideYHX, (15)\nwith\n/tildewideq≡qκtz\nκzz,\n/tildewideY≡Yκ2\nzz\nκ2\ntz. (16)\nBy invoking Y→/tildewideY, we do not mean to suggest that this function\nactually gets rescaled. Rather, the presence of Yin Eq. (6) for Z\nrepresents a direct contribution from q2HX, which — after rescaling\nq2— picks up the extra factor κ2\nzz/κ2\ntz.\nSince the couplings can be regarded as constants, the soluti on in\nEq. (5) is formally unchanged and need only be rewritten in te rms of\nthe rescaled parameter. By this logic, the same can be said ab out the\ndispersion relation in Eq. (6), which takes on the modified fo rm\nω2=/tildewideq21\np/tildewideY+iω/tildewideq2p−1\npfκ2\nzz\nκ2\ntz. (17)\nTakingu∗= 0andthencomparingdirectlytoEq.(3), wecanpromptly\nextract the damping constant for a generalized (but conform al) theory\nof gravity:\nΓ =κ2\nzz\nκ2\ntzp−1\np1\n4πT(18)\nand, as advertised, the sound velocity is clearly unmodified .\nLet us briefly comment upon the significance of this result. It is\ncommonplace, for a conformal theory, to relate the sound dam ping\n17constant directly to the shear diffusion coefficient or (equiva lently)\nthe shear viscosity to entropy ratio: Γ =p−1\npDand Γ =p−1\np1\nTη\ns\nrespectively. This is all indisputably true for an Einstein theory of\ngravity; however, as we have now shown, these relations can n ot be\ntaken verbatim for a generalized theory. To be clear, let us c ompare\nEq. (18) to our prior results from [7] D=κ2\nxz\nκ2\ntz1\n4πTand from [8] η/s=\n1\n4πκ2\nrt\nκ2xz(wherexandzcould beany pair of orthogonal directions on the\nbrane and note that, in general, κxz/negationslash=κzz). It should now be evident\nthat both of the above relations for Γ will generally be modifi ed for\nan Einstein-corrected theory.\nThe very same outcome as in Eq. (18) can be surmised from the\nz-component of the conservation equation for the dissipativ e stress\ntensor; with this being the equation that gives rise to the so und-mode\ndispersion relation (see, e.g., [13]). An inspection of this conservation\nequation ∂tTtz+∂zTzz= 0 and the steps leading up to the dispersion\nrelation (3) is quite revealing. It is the tzcomponent of the stress ten-\nsorthataccounts fortheΩ2terminEq.(3), whereasthe zzcomponent\ngives riseto the Q2Ω term. Now, given a gravitational pedigreefor the\nhydrodynamic modes, it is natural to associate a coupling of κ2\nµνwith\ntheµνcomponent of the stress tensor. Hence, we anticipate that, f or\na generalized gravity theory, the conservation equation sh ould really\nbeκ2\ntz∂tTtz+κ2\nzz∂zTzz= 0. Similarly, we can expect the dispersion\nrelation to take on the modified form Ω2=κ2\nzz\nκ2\ntz/bracketleftbig\ni2ΓΩQ2/bracketrightbig\n+.... (with\nthe dots referring to the sound-velocity and higher-order t erms). Ab-\nsorbing this ratio of couplings into the damping constant, w e have\nprecisely the same generalized form as obtained in Eq. (18).\nNaively, this latter argument would also suggest that c2\nsscales in\n18the same way as Γ, given that both are associated with the same\nzzcomponent of the stress tensor. However, this is not really c or-\nrect: The sound velocity is associated with the variation of the pres-\nsure; with the pressure having originated from the non-diss ipative\nbackground part of the stress tensor. Meanwhile, the other t erms in\nthe dispersion relation are strictly associated with the flu ctuations or\nleading-order dissipative part. On this basis, we would not anticipate\nthe sound velocity to be scaled for a generalized (conformal ) theory;\nagain in compliance with the previous analysis.\n4 Discussion: Some aspectsofthenon-\nconformal case\nTo summarize, we have demonstrated two important outcomes f or the\nsound-mode conformal hydrodynamics of an AdS brane theory. First,\nwe have confirmed, for Einstein’s theory, that the hydrodyna mics at\nthe outer boundary is equivalent to that of any other radial s hell up to\n(and including at) the stretched horizon. Second, we have sh own —\nquite precisely — how the sound velocity and damping coefficie nt will\nbe modified for a generalized (but conformal) theory of gravi ty. More\nspecifically, c2\nsis unaffected, whereas Γ is scaled by a particular ratio\nof (generalized) gravitational couplings. Further note th at, inasmuch\nas the couplings can be treated as constants, the former outc ome will\ncarry through unfettered for any Einstein-corrected confo rmal theory.\nIt is also of some interest to reflect upon how a non-conformal\ntheory would impact upon our findings. Let us first consider th e is-\n19sue of radial invariance for Einstein’s theory. Clearly, th is invariance\nfor the correlator depended, in large part, on being able to d isregard\nthe first term in Eq. (8). However, the introduction of a massi ve field\ninto the spacetime (a prerequisite for breaking conformali ty) would be\ntantamount totheinclusionofachemical potential intothe thermody-\nnamics. Such an inclusion would then negate our previous arg ument\nfor the suppression of the scrutinized term; in particular, sT=ǫ+P\ncould no longer be true. Hence, there could no longer be any re ason to\nexpect thatδP\nδǫis a parametrically small quantity for a non-conformal\ntheory — meaning that the radial scaling of the sound velocit y would\ncertainly be more complicated. However, that this deviatio n from the\nconformal calculation is seemingly encapsulated in the sin gle varia-\ntionδP\nδǫgives one hope of being able to describe even the fully genera l\nsituation by way of a radial “flow” equation. Although, it sho uld be\nkept in mind that a further breach of radial invariance is pos sible (if\nnot probable) from additional terms that would (almost inev itably)\nappear in the O(q1) solution for Z.\nFor the case of generalized gravity, the state of affairs can be come\nsignificantly more convoluted for a non-conformal theory. H ere, the\nfirst order of business is to re-incorporate the previously d isregarded\nttmode — but then what? Well, at a first glance, the situation doe s\nnot appear to look too bad. For the reason discussed at the end of\nthe prior section, we would not expect the sound velocity to b e modi-\nfied irrespective of the generalized gravitational couplin gs. As for the\ndamping coefficient, one can show that Httmakes no contribution to\nthisparticularterm, soitseemsreasonabletosuggestthat Γmaintains\nits modified form of Eq. (18).\n20It is, however, a nearly certain likelihood that the situati on can\nnot be as simple as so far discussed. For a non-conformal theo ry,\nthere is an inevitable mixing between HXand the massive bulk fields,\nand it is not yet clear as to how this mixing might effect the scal ing\nrelationsforeitherΓor c2\ns(withbothofthesebeingdirectlyimplicated\nwith the “polluted” HXmode). Certainly, a mode formed out of HX\nand some, for instance, massive scalar field, could no longer have an\neffective coupling as trivial as κzz.\nThe main issue of non-conformal treatments is that, due to th e\nhigh degree of model dependence in the formalism, very littl e can be\nsaid in a generic sense. There has, however, been some recent progress\nin such a direction [18, 19]. These papers indicate that a bet ter start-\ning point might be to look at certain classes of non-conforma l theories,\nas opposed to the “extreme limiting cases” of a specific model or com-\npletely generality. Work along this line is only at a prelimi nary stage.\nAcknowledgments: The research of RB was supported by The Is-\nrael Science Foundation grant no 470/06. The research of AJM M is\nsupported by the University of Seoul.\nReferences\n[1] P. Kovtun, D. T. Son and A. O. Starinets, “Holography and\nhydrodynamics: Diffusion on stretched horizons,” JHEP 0310,\n064 (2003) [arXiv:hep-th/0309213].\n21[2] A. Buchel and J. T. Liu, “Universality of the shear vis-\ncosity in supergravity,” Phys. Rev. Lett. 93, 090602 (2004)\n[arXiv:hep-th/0311175].\n[3] See, for a review and more references as well, D. T. Son and\nA.O.Starinets, “Viscosity, Black Holes, andQuantumField The-\nory,” Ann. Rev. Nucl. Part. Sci. 57, 95 (2007) [arXiv:0704.0240\n[hep-th]].\n[4] A. O. Starinets, “Quasinormal spectrumandthe black hol e mem-\nbrane paradigm,” arXiv:0806.3797 [hep-th].\n[5] G. Policastro, D. T.SonandA. O.Starinets, “Theshearvi scosity\nof strongly coupled N = 4 supersymmetric Yang-Mills plasma, ”\nPhys. Rev. Lett. 87, 081601 (2001) [arXiv:hep-th/0104066].\n[6] See, for recent reviews with many references,\nS. Mrowczynski and M. H. Thoma, “What do electromagnetic\nplasmas tell us about quark-gluon plasma?,” Ann. Rev. Nucl.\nPart. Sci. 57, 61 (2007) [arXiv:nucl-th/0701002];\nE. Shuryak, “Physics of Strongly coupled Quark-Gluon Plasm a,”\narXiv:0807.3033 [hep-ph].\n[7] R. Brustein and A. J. M. Medved, “Theshear diffusion coeffici ent\nfor generalized theories of gravity,” to appear in Phys. Let t. B\n[arXiv:0810.2193 [hep-th]].\n[8] R. Brustein and A. J. M. Medved, “The ratio of shear viscos ity\nto entropy density in generalized theories of gravity,” to a ppear\nin Phys. Rev. D [arXiv:0808.3498 [hep-th]].\n22[9] R. Brustein, D. Gorbonos and M. Hadad, “Wald’s entropy is\nequal to a quarter of the horizon area in units of the effective\ngravitational coupling,” arXiv:0712.3206 [hep-th].\n[10] G. Policastro, D. T. Son and A. O. Starinets, “From AdS/C FT\ncorrespondence to hydrodynamics,” JHEP 0209, 043 (2002)\n[arXiv:hep-th/0205052].\n[11] N. Iqbal and H. Liu, “Universality of the hydrodynamic l imit in\nAdS/CFT and the membrane paradigm,” arXiv:0809.3808 [hep-\nth].\n[12] G. Policastro, D. T. Son and A. O. Starinets, “From AdS/C FT\ncorrespondence to hydrodynamics. II: Sound waves,” JHEP\n0212, 054 (2002) [arXiv:hep-th/0210220].\n[13] C. P. Herzog, “The sound of M-theory,” Phys. Rev. D 68, 024013\n(2003) [arXiv:hep-th/0302086].\n[14] D. T. Son and A. O. Starinets, “Minkowski-space correla tors\nin AdS/CFT correspondence: Recipe and applications,” JHEP\n0209, 042 (2002) [arXiv:hep-th/0205051].\n[15] P. K. Kovtun and A. O. Starinets, “Quasinormal modes\nand holography,” Phys. Rev. D 72, 086009 (2005)\n[arXiv:hep-th/0506184].\n[16] J. Mas and J. Tarrio, “Hydrodynamics from the Dp-brane, ”\nJHEP0705, 036 (2007) [arXiv:hep-th/0703093].\n[17] M. Fujita, “Non-equilibrium thermodynamics near the h orizon\nand holography,” JHEP 0810, 031 (2008) [arXiv:0712.2289 [hep-\nth]].\n23[18] S. S. Gubser, S. S. Pufu and F. D. Rocha, “Bulk viscosity o f\nstrongly coupled plasmas with holographic duals,” JHEP 0808,\n085 (2008) [arXiv:0806.0407 [hep-th]].\n[19] T. Springer, “Sound Mode Hydrodynamics from Bulk Scala r\nFields,” arXiv:0810.4354 [hep-th].\n24"    },    {        "title": "0904.0813v2.Projective_Space_Codes_for_the_Injection_Metric.pdf",        "content": "arXiv:0904.0813v2  [cs.IT]  8 Apr 2009Projective Space Codes for the Injection Metric\nAzadeh Khaleghi, Frank R. Kschischang\nDepartment of Electrical and Computer Engineering, Univer sity of Toronto, Canada\nEmail:{azalea,frank }@comm.utoronto.ca\nAbstract—In the context of error control in random linear\nnetwork coding, it is useful to construct codes that compris e\nwell-separated collections of subspaces of a vector space o ver\na finite field. In this paper, the metric used is the so-called\n“injection distance,” introduced by Silva and Kschischang . A\nGilbert-Varshamov bound for such codes is derived. Using th e\ncode-construction framework of Etzion and Silberstein, ne w non-\nconstant-dimension codes are constructed; these codes con tain\nmore codewords than comparable codes designed for the sub-\nspace metric.\nI. INTRODUCTION\nThe problem of error-correction in random network coding\nhas recently become an active area of research [1], [2], [3],\n[4], [5], [6]. The main motivation for this problem is the\nphenomenon of error-propagation in the network. Since the\nreceived packets are random linear combinations of packets\ninserted at intermediate nodes, the system is very sensitiv e\nto transmission errors. Due to the vector-space preserving\nnature of random linear network coding, it has been shown\nthat codes constructed in the projective space are suitable for\nerror-correction for network coding.\nOur focus in this paper is on construction of codes in\nthe projective space for adversarial error-correction in r andom\nnetworkcoding.Asshownin[7]asuitablemeasureofdistanc e\nbetween subspaces for an adversarial error-control model i s\ngiven by\ndI(U,V) = max{dimU,dimV}−dim(U∩V),\na measure known as the “ injection metric ”. This choice of dis-\ntance metric is the main parameter that distinguishes our wo rk\nfrom the existing literature on (subspace) codes construct ed\nfor random linear network coding. All existing bounds and\nconstructions are based on a metric known as the subspace\ndistancedSoriginally introduced by K¨ otter and Kschischang\nin [5]. In the special case where codes are contained in the\nGrassmannian, codes designed for dIcoincide with those\ndesigned for dS. However, as shown in [7], in general non-\nconstant dimension codes designed for dImay have higher\nrates than those designed for dS.\nIn this paper we present a construction of a class of\ncodes in the projective space for the injection distance. Th is\nconstructionismotivatedbytheworkofEtzionandSilberst ein\nin [8], with the main difference that the construction in [8] is\nbased on dS.\nIn Section II, we present a brief overview of the projective\nspace and the Grassmannian,as well as rank-metriccodes. We\nalso briefly review Etzion and Silberstein’s “Ferrers diagr am\nlifted rank-metric codes”, as our work in Section V is relate dto this construction. In Section III, we present a Gilbert-\nVarshamov-type bound on the size of codes of a certain\nminimum injection distance in the projective space. As we\nare precluded by space in this paper, we present this theorem\nwithout proof. In Section IV we present a construction for th e\nFerrers diagram rank-metric codes as subcodes of linear MRD\ncodes. In Sections V and VI, we provide an algorithm for the\nconstruction of a class of non-constant-dimensioncodes in the\nprojective space designed for dI. Finally, in Section VII we\npresent our numerical results. As shown in this section our\nconstruction results in codes of slightly higher rates than the\ncodes of [8].\nII. PRELIMINARIES\nA. Notation\nLetq≥2be a power of a prime. In this paper, all\nvectorsand matrices are defined over the finite field Fq, unless\notherwisementioned.We denoteby Fm×n\nq, the set of all m×n\nmatrices over Fq. Ifvis a vector then the ithentry ofvis\ndenoted by vi. We denote the logical complement of a binary\nvectorv= (v1,v2,···,vn)by¯v= (¯v1,¯v2,···,¯vn). The\nnumber of non-zero elements of vis denoted by wt(v). We\ndefine the support set of a vector v, denoted supp(v)to be the\nset of indices corresponding to the non-zero entries of v. Let\nxandybe two binary vectors of the same length. We denote\nthe number of 1→0transitions from xtoybyN(x,y), their\nHammingdistanceby dH(x,y)andthelogicalANDoperation\nbetweenxandyby∧. IfXis a matrix then the rank of Xis\ndenoted by rankXand its row space is denoted by /an}bracketle{tX/an}bracketri}ht. Let\nn >0be an integer. We denote by [n]the set of all positive\nintegers less than or equal to n, i.e.[n] ={0,1,2,···,n}.\nB. Rank-Metric Codes\nLetXandYbe two matrices in Fm×n\nq. Therank dis-\ntancebetween XandY, denoted dR(X,Y)is defined as\ndR(X,Y)/definesrank(Y−X). As shown in [9] the rank\ndistance is indeed a metric. Let Fqbe a base field and Fqm\nwithm≥1be an extension of Fq. The rank of a vector\nv= (v1,v2,···,vn)∈(Fqm)nis the rank of the m×n\nmatrix obtained by expanding each entry of vto anm×1\ncolumn vector over Fq. A code CRis a rank-metric code\noverFqmof minimum distance d, ifCR⊆(Fqm)nand for\nallX, Y∈CRdR(X,Y)≥d. As shown in [9] CRmust\nsatisfylogq|CR| ≤max{m,n}(min{m,n}−d+1),and rank\nmetric codes achieving this bound with equality are said to\nbe Maximum Rank Distance (MRD) codes. Gabidulin codes,\npresented by Gabidulin [9] are an extensive class of MRD\ncodes,whichare the analogsof the generalizedReed-Solomo ncodes designed for the rank metric. Efficient polynomial-ti me\ndecoding algorithms exist that correct errors of rank up to/floorleftbiggd−1\n2/floorrightbigg\n. See for example [10], [11], [12].\nC. Projective Space\nLetVbe ann-dimensional vector space over the finite\nfieldFqof order q. For a non-negative integer k≤n\ndenote by G(n,k)the set of all k-dimensional subspaces\nofV. This set is known as a Grassmannian and its cardi-\nnality is given by the q-aryGaussian coefficient defined as/bracketleftbiggn\nk/bracketrightbigg\nq/defines(qn−1)(qn−1−1)···(qn−k+1−1)\n(qk−1)(qk−1−1)···(q−1). The set of all\nsubspacesof Vformaprojectivespace Pn\nqofordernoverFq.\nThusPn\nqcan be viewed as a union of the Grassmannians for\nallk≤n, i.e.Pn\nq=n/uniondisplay\nk=0G(n,k). A code Cis an(n,M,d)dS\ncode inPn\nqif|C|=Mand for all U, V∈ C, dS(U,V)≥d.\nSimilarly, a code C ⊆ Pn\nqis an(n,M,d)dIcode if|C|=M\nand for all U, V∈ C, dI(U,V)≥d. A code Cis an\n(n,M,d,k )constant-dimension code if C ⊆ G(n,k)for some\nk∈[n]. Since in this case dIanddSare equal up to scale,\nthere is no need to distinguish between (n,M,d,k )dIand\n(n,M,d,k )dScodes.\nD. Ferrers Diagram Lifted Rank-Metric Codes\nIn this section we review the code construction of [8] with\na slightly different notation. The key idea in this construc tion\nis the observation that every k-dimensional vector space V\ninPn\nqarises uniquely as the row space of a k×nmatrix in\nReduced Row Echelon Form (RREF). Let Vbe a vector space\ninPn\nqand letE(V)be its corresponding generator matrix in\nRREF. We define the profile vector ofVdenotedp(V), to be\na binary vector of length nwhose non-zero elements appear\nonlyin positions where E(V)has a leading 1. Consider an\nequivalence relation ∼onPn\nqwhere,\n∀V1, V2∈ Pn\nq, V1∼V2↔p(V1) =p(V2).(1)\nThis relation partitions Pn\nqinto equivalence classes, where\nV1andV2belong to the same class provided that they are\nidentified by the same profile vector. Let Γdenote the set\nof all equivalence classes generated in Pn\nqaccording to (1).\nConsider an equivalence class γ∈Γwith a profile vector v\nof length nand weight k. We define the profile matrix PM(v)\nto be ak×nmatrix in RREF where the leading coefficients\nof its rows appear in columns indexed by supp(v), and has\n•’s in all its entries which are not required to be terminal\nzeros or leading ones. For example if p= (0,1,0,1,1,0,0)\nthenPM(v) =\n0 1•0 0• •\n0 0 0 1 0 • •\n0 0 0 0 1 • •\n. Notice that the\ngenerator matrices in RREF of the elements of γdiffer only\nin entries of PM(v)marked as •’s. Letηdenote the number\nof columns of PM(v)which contain at least a single •. Let\nS(v)be thek×ηsub-matrix of PM(v)composed of all such\ncolumnsof PM(v).Acodeisan [S,κ,δ]Ferrersdiagramrank-\nmetric code if it forms a rank-metric code with dimension κand minimum rank-distance δ, all of whose codewords are\nm×ηmatrices with zeros in all their entries where S(v)has\nzeros.\nIn the construction presented in [8], a set Ω⊆Γis\nconstructed in such a way that for all γ1, γ2∈Ωwith\nγ1/ne}ationslash=γ2and for all V1∈γ1,V2∈γ2, dS(V1,V2)≥d.\nBy Lemma 2 in [8], this is possible by selecting the profile\nvectorsoftheequivalenceclassesaccordingtoabinarycod eof\nminimumHammingdistance d. Then within each class γ∈Ω,\na Ferrers diagram rank-metric code is used to ensure that for\nallV1, V2∈γ,dS(V1,V2)≥d. Finally C={V∈γ|γ∈Ω}.\nIII. A G ILBERT-VARSHAMOV -TYPEBOUND ON THE SIZE\nOFCODES IN THE PROJECTIVE SPACE\nLetVbe ak-dimensional vector space in Pn\nq. We define\nSt(V)to be the set of all vector spaces in Pn\nqat an injection\ndistance at most tfromV. i.e.\nSt(V) ={W∈ Pn|dI(V,W)≤t}\nWe may view St(V)as a hypothetical sphereof radius t\ncentered at V. In Theorem 1 we give the cardinality of\nSt(V)centered at some k-dimensional vector space with\nk≤n. Since the projective space is non-homogeneous, the\nsize ofSt(V)does not depend merely on its radius, but\nalso on the dimension of its center. In other words for two\nvector spaces V1andV2withdimV1/ne}ationslash= dimV2, we have\n|St(V1)| /ne}ationslash=|St(V2)|.\nTheorem 1. LetVbe ak-dimensional vector space in Pn\nq,\nwithk≤n, and let N(k,t)denote the cardinality of St(V).\nThen,\nN(k,t) =t/summationdisplay\nr=0qr2/bracketleftbiggk\nr/bracketrightbigg\nq/bracketleftbiggn−k\nr/bracketrightbigg\nq+\nr/summationdisplay\nj=1qr(r−j)/parenleftBigg/bracketleftbiggk\nr/bracketrightbigg\nq/bracketleftbiggn−k\nr−j/bracketrightbigg\nq+/bracketleftbiggn−k\nr/bracketrightbigg\nq/bracketleftbiggk\nr−j/bracketrightbigg\nq/parenrightBigg\nUsing Theorem 1, and following an approach similar to\nthat of Etzion and Vardy in [13], we obtain the following\ngeneralized Gilbert-Varshamov-type bound on the size of\ncodes in the projective space.\nTheorem 2. LetAq(n,d)denote the maximum number of\ncodewords in an (n,M,d)code inPn\nq. Then,\nAq(n,d)≥/vextendsingle/vextendsinglePn\nq/vextendsingle/vextendsingle2\nn/summationdisplay\nk=0/bracketleftbiggn\nk/bracketrightbigg\nN(k,d−1)\nIV. FERRERSDIAGRAM RANK-METRICCODE\nCONSTRUCTION\nLetvbe a binary vector of length nand weight m. LetCF\nbe an[S(v),κ,δ]Ferrers diagram rank-metric code that fits\nS(v), i.e. every codeword in CFhas zeros in all its entries\nwhereS(v)has zeros. We may view CFas a subcode of a\nlinearrank-metriccode Cofminimumrank-distance dR(C)≥\nδ, with a further set of linear constraints ensuring that CFfits\nS(v).In Theorem 3 we provide a lower bound on the dimension\nκof the largest [S(v),κ,δ]Ferrers diagram rank-metric code\nobtained as a subcode of a linear MRD code.\nTheorem 3. Letvbe a binary vector of weight mand let\nS(v)be them×ηsub-matrix of PM(v)composed of all the\ncolumns of PM(v)that contain at least a single •. Assume\nthatS(v)contains a total of w•’s. Consider the dimension κ\nof the largest [S,κ,δ]Ferrer’s diagram rank-metric code CF.\nWe have, κ≥w−max{m,η}(δ−1).\nProof:LetV=Fm×η\nq. Note that Fm×η\nqis anmη-\ndimensionalvectorspaceover Fq.LetCbealinearMRDcode\nwithdR(C)≥δ. This code is a k-dimensional subspace of\nFm×η\nqwithk= max{m,η}(min{m,η}−δ+1). There exists\na linear transformation Φ :V−→V/CwithkerΦ =C, and\nby the First Isomorphism Theorem dimV/C=mη−k. Let\nA={(i,j)|S(v)ij= 0}bethesetof (i,j)indiceswhere S(v)\nhas zeros, and note that |A|=mη−w. Letf:V−→Fmη−w\nq\nsuch that f(x) = (xij),(i,j)∈A. Now any subcode C′ofC\nsatisfying f(c) = 0∀c∈C′is an[S(v),κ,δ]Ferrers diagram\nrank-metric code. Let CFbethe largest such subcode of C.\nDefine a linear transformation Φ′:V−→V/C×Fmη−w\nq, by\nwhichx/mapsto→(Φ(x),f(x)). Now by construction kerΦ′=CF.\nNoting that Φ′(V)⊆V/C×Fmη−w\nqwe have dimΦ′(V)≤\n2mη−k−w, and by the rank-nullity theorem we obtain\ndimCF≥w+k−mη=w−max{m,η}(δ−1), and the\ntheorem follows.\nAs an example, given a profile vector vof length n, with\nwt(v) =mwe may construct an [S(v),κ,d]code by taking a\nGabidulin code over Fη\nqmwithdR≥d, expand the elements\nof its parity-check matrix Hover the base field Fq, and add\nappropriate parity-check equations to HinFqto ensure that\nthe resulting code fits S(v).\nV. FERRERSDIAGRAM LIFTEDRANK-METRICCODES\nFOR THE INJECTION METRIC\nInspiredbytheconstructionof[8],inthissectionweprese nt\na scheme for constructing (n,M,d)dIFerrers diagram lifted\nrank-metric codes in Pn\nq. The following theorem is key in our\nconstruction.\nTheorem 4. LetUandVbe two vector spaces in Pn\nq,\nwith profile vectors u, andvrespectively. Then we have,\ndI(U,V)≥max{N(u,v),N(v,u)}.\nProof:First note that the dimension of a vector space\nis equal to the Hamming weight of its profile vector, i.e.\ndimU=wt(u)anddimV=wt(v). Now let w=u∧v\nand observe that dimU∩V≤wt(w). Therefore we have\ndimU−dim(U∩V)≥wt(u)−wt(w). Similarly, dimV−\ndim(U∩V)≥wt(v)−wt(w). Taking the maxof both equa-\ntions we obtain, dI(U,V)≥max{wt(u),wt(v)} −wt(w) =\nmax{N(u,v),N(v,u)}.\nFor two binary vectors xandy, the quantity\nmax{N(x,y),N(y,x)}is a metric, known as the asymmetric\ndistance between xandy. The asymmetric distance was\nfirst introduced by Varshamov in [14] for construction of\ncodes for the Z channel. Constructions exist mainly for\nsingle-asymmetric error-correcting codes, and some multi -error correcting codes ([15] and references therein). Plea se\nrefer to [16] for a more recent work on general t-asymmetric\nerror-correcting codes.\nBy Theorem 4 two spaces are guaranteed to have an\ninjection distance of at least d, provided that the asymmetric\ndistance between their profile vectors is at least d. Thus to\nconstruct a code in Pn\nqwith minimum injection distance d,\nwe may select a set of subspaces according to an asymmetric\ncode in the Hamming space with minimum asymmetric dis-\ntancedand follow a procedure similar to that presented in\n[8]. Construction of our (n,M,d)dIcode can be described\nalgorithmically as follows:\n1) Take a binary asymmetric code Aof length nand\nminimum asymmetric distance d.\n2) For each codeword c∈ A, obtainS(c), (composed of\nthe columns of PM(c)with at least one •).\n3) Given each k×ηmatrixS(c), use the construction of\nSectionIVtoobtainan [S(c),κ,d]Ferrersdiagramrank-\nmetric code.\n4) Lift each matrix S(c)to its correspondingprofile matrix\nPM(c), to obtain a generator matrix Gc.\n5) Finally C={V∈ Pn\nq|V=/an}bracketle{tGc/an}bracketri}ht}.\nNote that a slight modification to Step 1 and Step 3 in the\nabove procedure allows for the construction of an (n,M,d)dS\ncode in the projective space. More specifically, in order to\nconstruct an (n,M,2δ)dScode inPn\nq, we may first take a\nbinarycode Hofminimum Hamming distancedH≥2δ.Then\nfor each codeword c∈ Hwe may construct an [S(c),κ,δ]\nFerrers diagram rank-metric code. Following the rest of the\nsteps are described above, we obtain an (n,M,2δ)dScode.\nVI. PROFILEVECTORSELECTION\nAs suggested by Theorem 3, the dimension of an [S,κ,δ]\nFerrers diagram rank-metric code depends not only on the\ndesired minimum distance δ, but also on the number of •’s in\nS. Since the number of •’s inSis directly related to its corre-\nsponding profile vector, the choice of the asymmetric code in\nthe first step is crucial to the size of our codes. For instance ,\nthe vector v= (1,1,0,0,0)results in a profile matrix with\na higher number of •’s than that of v= (1,1,0,1,1). Thus\na code of lower rate that contains vectors which potentially\nresult in larger number of •’s in their corresponding profile\nmatrices may be preferable over one with a higher rate, that\ninvolves vectors resulting in smaller number of •’s.\nWith this observation, given a minimum asymmetric dis-\ntancedwe define a scoring function score(v,d)on the set\nof all binary vectors, which calculates for every v∈ {0,1}n,\nthe lower bound κof the dimension of the largest [S(v),κ,d]\nFerrers diagram rank-metric code induced by v. It is easy to\nobserve that\nscore(v,d) =n/summationdisplay\ni=1i/summationdisplay\nj=1¯vivj−max{wt(v),η(v)}(d−1)\nwhereη(v) =n−(wt(v)+ min\nt∈supp(v)t)+1\nNow in order to select a set Pof profile vectors at aminimum asymmetric distance d, we use a standard greedy\nalgorithm that maintains a list of available profile vectors\nA⊆ {0,1}n, (withAinitialized to {0,1}n). At each step an\navailableprofilevector v∈Awiththelargestscore score(v,d)\nis added to P, and vectors within asymmetric distance dof\nvare made unavailable. The algorithm proceeds until A=∅.\nBy a slight modification to this algorithm we may allow for\nthe same greedy selection of a set of profile vectors at a\ncertain minimum Hamming distance as opposedto a minimum\nasymmetric distance.\nVII. N UMERICAL RESULTS\nAs constant-dimensioncodes designed for dScoincide with\nthose designedfor dI, we are interested in the analysis of non-\nconstant dimension (n,M,d)dIcodes.\nA. Our(n,M,d)dIvs.(n,M,d)dSCodes\nFor our(n,M,d)dIcodes we first used the selection algo-\nrithm presented in Section VI to obtain a set of binary profile\nvectors at a minimum asymmetric distance da≥d. Using this\nprocedure along with the bound of Theorem 3 we obtained\n|(n,M,d)dI|. As discussed previously, for every (n,M,d)dI\ncode constructed according to the procedure described in\nSection V, we may construct an (n,M,2d)dScounterpart\nthrough a similar procedure. In order to select a set of profil e\nvectors for our (n,M,2d)dScodes we used the algorithm of\nSection VI for dH≥2d. As shown in Table I our (n,M,d)dI\ncodes denoted by C2have a slightly higher rate than their\n(n,M,2d)dScounterparts, C1.\nB. Our(n,M,d)dICodes vs. Codes of [8]\nThe best(n,M,d,k )dSconstant-dimensioncodes of [8] are\nobtainedby using constant-weightlexicodesas profile vect ors.\nThese codes achieve maximum cardinality when k=/floorleftBign\n2/floorrightBig\n.\nThe column corresponding to C3in Table I shows rates of the\n(n,M,d S,/floorleftbign\n2/floorrightbig\n)dSconstant-dimension codes of [8].\nNon-constant-dimension (n,M,d)dScodes of [8] are con-\nstructed by means of a puncturing operation performed on\nconstant-dimension codes. As shown in [8] puncturing an\n(n,M,d,k )dScode results in an (n−1,M′,d−1)dScode\nwithM′≥M(qn−k+qk−2)\nqn−1. In Table I, logq|C4|denotes\nthe guaranteed minimum rate of (n,M′,dS)punctured codes\nobtained from the best (n+ 1,M,dS+ 1,/floorleftbign+1\n2/floorrightbig\n)dScodes\nof [8]. As shown in the table, our (n,M,d)dIcodes have\na slightly higher rate than both constant and non-constant-\ndimension codes of [8].\nVIII. C ONCLUSION\nWe presented a construction for the Ferrers diagram rank-\nmetric codes as subcodes of linear MRD codes, and provided\na lower bound on the dimension of the largest such codes.\nUsing a greedy profile vector selection algorithm along with\nour construction of Ferrers diagram rank-metric codes we\npresented a class of non-constant dimension lifted Ferrers\ndiagram rank-metric codes for the injection distance. We al so\npresented a similar construction for non-constant dimensi onTABLE I\nPARAMETERS OF CODES CONSTRUCTED WITH C1=OUR(n,M,d S)dS\nCODES,C2=OUR(n,M,d I)dICODES,C3= (n,M,d S,n/2)dSCODES\nOF[8],C4=PUNCTURED CODES OF [8]\nq dIdSnlogq|C1|logq|C2|logq|C3|logq|C4|\n2 2 4 9 15 .1732 15 .6245 15 .1731 10 .9588\n2 2 4 10 20 .1551 20 .3294 20 .1548 13 .5585\n2 2 4 12 30 .1561 30 .3346 30 .1559 13 .7676\n2 3 6 10 15 .0031 15 .0071 15 .0032 7 .5581\n2 3 6 13 28 .0032 28 .0263 28 .0032 21 .9888\n3 2 4 7 8 .0177 8 .1331 8 .0170 4 .6210\n3 2 4 8 12 .0138 12 .0311 12 .0138 6 .2567\n4 2 4 7 8 .0039 8 .0522 8 .0038 4 .4974\n4 2 4 8 12 .0031 12 .0068 12 .0031 6 .1599\ncodes designed for the subspace distance. We observed that\nour non-constant dimension codes designed for the injectio n\ndistance have a slightly higher rate than their counterpart s\ndesigned for the subspace distance. Moreover, comparing\nour codes designed for the injection distance, with the best\nsubspace codes of [8], we observed a minor improvement in\nrate. The Ferrers diagram lifted rank-metric codes introdu ced\nby [8], as well as those presented in our paper achieve higher\nratesthantheoriginalliftedrank-metriccodesof[5].How ever,\nwe believe that these rate improvements are minute from a\npractical perspective.\nREFERENCES\n[1] T. Etzion and A. Vardy, “Error-correcting codes in proje ctive space,”\nIEEE Intern. Symp. on Inform. Theory , pp. 871–875, July 2008.\n[2] E. Gabidulin and M. Bossert, “Codes for network coding,” IEEE Intern.\nSymp. on Inform. Theory , pp. 867–870, July 2008.\n[3] F. Manganiello, E. Gorla, and J. Rosenthal, “Spread code s and spread\ndecoding in network coding,” IEEE Intern. Symp. on Inform. Theory ,\npp. 881–885, July 2008.\n[4] A.Kohnertand S.Kurz,“Construction oflargeconstant d imension codes\nwith a prescribed minimum distance,” in MMICS, 2008, pp. 31–42.\n[5] R. K¨ otter and F. R. Kschischang, “Coding for errors and e rasures in\nrandom network coding,” IEEE Trans. on Inform. Theory , vol. 54, no. 8,\npp. 3579–3591, Aug. 2008.\n[6] D. Silva, F. R. Kschischang, and R. K¨ otter, “A rank-metr ic approach\nto error correction in random network coding,” IEEE Trans. on Inform.\nTheory, vol. 54, pp. 3951–3967, Sep. 2008.\n[7] D. Silva and F. R. Kschischang, “On metrics for error corr ection in\nnetwork coding,” 2008, submitted for publication. [Online ]. Available:\nhttp://arxiv.org/abs/0805.3824\n[8] Tuvi Etzion and N. Silberstein, “Error-correcting code s in projective\nspaces via rank-metric codes and Ferrers diagrams,” 2009. [ Online].\nAvailable: http://arxiv.org/abs/0807.4846v4\n[9] E.M.Gabidulin, “Theory of codes with maximum rank dista nce,”Probl.\nInform. Transm , vol. 21, no. 1, pp. 1–12, 1985.\n[10] R. Roth, “Maximum-rank array codes and their applicati on to crisscross\nerror correction,” IEEE Trans. on Inform. Theory , vol. 37, no. 2, pp.\n328–336, Mar 1991.\n[11] G. Richter and S. Plass, “Fast decoding of rank-codes wi th rank errors\nand column erasures,” IEEE Intern. Symp. on Inform. Theory , 2004.\n[12] P.Loidreau, “A Welch-Berlekamp like algorithm for dec oding Gabidulin\ncodes,” in WCC, 2005, pp. 36–45.\n[13] L. Tolhuizen, “The generalized Gilbert-Varshamov bou nd is implied by\nTuran’s theorem [code construction],” IEEE Trans. on Inform. Theory ,\nvol. 43, no. 5, pp. 1605–1606, Sep 1997.\n[14] R. Varshamov, “A class of codes for asymmetric channels and a problem\nfrom the additive theory of numbers,” IEEE Trans. on Inform. Theory ,\nvol. 19, no. 1, pp. 92–95, Jan 1973.\n[15] T. Kløve, “Error correction codes for the asymmetric ch annel,” Math.\nInst. Univ. Bergen, Tech. Rep., 1981.\n[16] V. P. Shilo, “New lower bounds of the size of error-corre cting codes for\nthe Z-channel,” Cybernetics and Sys. Anal. , vol. 38, pp. 13–16, 2002."    },    {        "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": "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": "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": "0905.3242v1.Eigenvalue_asymptotics__inverse_problems_and_a_trace_formula_for_the_linear_damped_wave_equation.pdf",        "content": "arXiv:0905.3242v1  [math.SP]  20 May 2009EIGENVALUE ASYMPTOTICS, INVERSE PROBLEMS\nAND A TRACE FORMULA FOR THE LINEAR\nDAMPED WAVE EQUATION\nDENIS BORISOV AND PEDRO FREITAS\nAbstract. We determine the general form of the asymptotics for\nDirichlet eigenvalues of the one–dimensional linear damped wave\noperator. Asaconsequence,weobtainthatgivenaspectrumcor re-\nsponding to a constant damping term this determines the damping\nterm in a unique fashion. We also derive a trace formula for this\nproblem.\n1.Introduction\nConsider the one–dimensional linear damped wave equation on the\ninterval (0 ,1), that is,\n(1.1)\n\nwtt+2a(x)wt=wxx+b(x)w, x ∈(0,1), t >0\nw(0,t) =w(1,t) = 0, t > 0\nw(x,0) =w0(x), wt(x,0) =w1(x), x∈(0,1)\nThe eigenvalue problem associated with (1.1) is given by\nuxx−(λ2+2λa−b)u= 0, x∈(0,1), (1.2)\nu(0) =u(1) = 0, (1.3)\nand has received quite a lot of attention in the literature since the pa -\npers of Chen et al. [CFNS] and Cox and Zuazua [CZ]. In the first of\nthese papers the authors derived formally an expression for the a symp-\ntotic behaviour of the eigenvalues of (1.2), (1.3) in the case of a zer o\npotential b, whichwaslaterprovedrigorouslyinthesecondoftheabove\nDate: November 17, 2018.\n2000Mathematics Subject Classification. Primary 35P15; Secondary 35J05.\nD.B. was partially supported by RFBR (07-01-00037) and gratefully acknowl-\nedges the support from Deligne 2004 Balzan prize in mathematics. D.B . is\nalso supported by the grant of the President of Russia for young s cientist and\ntheir supervisors (MK-964.2008.1) and by the grant of the Preside nt of Russia\nfor leading scientific schools (NSh-2215.2008.1) P.F. was partially sup ported by\nFCT/POCTI/FEDER. .\n12 DENIS BORISOV AND PEDRO FREITAS\npapers. Following this, there were several papers on the subject which,\namong other things, extended the results to non–vanishing b[BR], and\nshowed that it is possible to design damping terms which make the\nspectral abscissa as large as desired [CC]. In [F2] the second autho r of\nthe present paper addressed the inverse problem in arbitrary dime n-\nsion giving necessary conditions for a sequence to be the spectrum of\nan operator of this type in the weakly damped case. As far as we\nare aware, these are the only results for the inverse problem asso ciated\nwith (1.2), (1.3). Other results for the n−dimensional problem include,\nfor instance, the fact that in that case the decay rate is no longer de-\ntermined solely by the spectrum [L], a study of some particular case s\nwhere the role of geometric optics is considered [AL], the asymptotic\nbehaviour of the spectrum [S] and the study of sign–changing damp ing\nterms [F1].\nThe purpose of the present paper is twofold. On the one hand,\nwe show that problem (1.2), (1.3) may be addressed in the same way\nas the classical Sturm–Liouville problem in the sense that, although\nthis is not a self–adjoinf problem, the methods used for the former\nproblemmaybeappliedherewithsimilarresults. Thisideawasalready\npresent in both [CFNS] and [CZ]. Here we take further advantage of\nthis fact to obtain the full asymptotic expansion for the eigenvalue s\nof (1.2), (1.3) (Theorem 1). Based on these similarities, we were also\nled to a (regularized) trace formula in the spirit of that for the Stur m–\nLiouville problem (Theorem 4).\nOn the other hand, the idea behind obtaining further terms in the\nasyptotics was to use this information to address the associated in verse\nspectral problem of finding all damping terms that give a certain spe c-\ntrum. Our main result along these lines is to show that in the case of\nconstant damping there is no other smooth damping term yielding the\nsame spectrum (Corollary 2). Namely, we obtain the criterion for th e\ndamping term to be constant. Note that this is in contrast with the\ninverse (Dirichlet) Sturm–Liouville problem, where for each admissible\nspectrum there will exist a continuum of potentials giving the same\nspectrum [PT]. In particular this result shows that we should expect\nthe inverse problem to be much more rigid in the case of the wave\nequation than it is for the Sturm–Liouville problem. This should be\nunderstood in the sense that, at least in the case of constant dam ping,\nit will not be possible to perturb the damping term without disturbing\nthe spectrum, as is the case for the potential in the Sturm–Liouville\nproblem.\nThe plan of the paper is as follows. In the next section we set the\nnotation and state the main results of the paper. The proof of theEIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 3\nasymptotics of the eigenvalues is done in Sections 3 and 4, where in\nthe first of these we derive the form of the fundamental solutions of\nequation (1.2), while in the second we apply a shooting method to\nthese solutions to obtain the formula for the eigenvalues as zeros o f an\nentire function – the idea is the same as that used in [CZ]. Finally, in\nSection 5 we prove the trace formula.\n2.Notation and results\nIt is easy to check that if λis an eigenvalue of the problem (1.2),\n(1.3), then λis also an eigenvalue of the same problem. In view of this\nproperty, we denote the eigenvalues of this problem by λn,n/ne}ationslash= 0, and\norder them as follows\n.../lessorequalslantImλ−2/lessorequalslantImλ−1/lessorequalslantImλ1/lessorequalslantImλ2/lessorequalslant...\nwhile assuming that λ−n=λn. We also suppose that possible zero\neigenvalues are λ±1=λ±2=...=λ±p= 0. Ifp= 0, the problem\n(1.2), (1.3) has no zero eigenvalues. For any function f=f(x) we\ndenote/an}bracketle{tf/an}bracketri}ht:=/integraltext1\n0f(x)dx.\nTheorem 1. Suppose a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1. The\neigenvalues of (1.2), (1.3) have the following asymptotic b ehaviour as\nn→ ±∞:\nλn=πni+m−1/summationdisplay\nj=0cjn−j+O(n−m), (2.1)\nwere the cj’s are numbers which can be determined explicitly. In par-\nticular,\nc0=−/an}bracketle{ta/an}bracketri}ht, c1=/an}bracketle{ta2+b/an}bracketri}ht\n2πi, (2.2)\nc2=1\n2π2/bracketleftbigg\n/an}bracketle{ta(a2+b)/an}bracketri}ht−/an}bracketle{ta/an}bracketri}ht/an}bracketle{ta2+b/an}bracketri}ht+a′(1)−a′(0)\n2/bracketrightbigg\n. (2.3)\nA straightforward consequence of the fact that the spectrum d eter-\nmines the average as well as the L2norm of the damping term (as-\nsumingbfixed) is that the spectrum corresponding to the constant\ndamping determines this damping uniquely.\nCorollary 2. Assume that a∈C3[0,1],λnare the eigenvalues of the\nproblem (1.2), (1.3), the function b∈C2[0,1]is fixed, and the formula\n(2.1) gives the asymptotics for these eigenvalues. Then the function\na(x)is constant, if and only if\nc2\n0= 2πic1−/an}bracketle{tb/an}bracketri}ht,4 DENIS BORISOV AND PEDRO FREITAS\nin which case a(x)≡ −c0.\nIn the same way, the asymptotic expansion allows us to derive other\nspectral invariants in terms of the damping term a. However, these\ndo not have such a simple interpretation as in the case of the above\nconstant damping result.\nCorollary 3. Suppose b≡0,ai(x) =a0(x) +/tildewideai(x),i= 1,2, where\na0(1−x) =a0(x),/tildewideai(1−x) =−/tildewideai(x),/tildewideai,a0∈C4[0,1], and for a=ai\nthe problems (1.2), (1.3) have the same spectra. Then\n/an}bracketle{t/tildewidea2\n1/an}bracketri}ht=/an}bracketle{t/tildewidea2\n2/an}bracketri}ht,/an}bracketle{ta0/tildewidea2\n1/an}bracketri}ht=/an}bracketle{ta0/tildewidea2\n2/an}bracketri}ht\nis valid.\nFrom Theorem 1 we have that the quantity Re( λn−c0) behaves as\nO(n−2) asn→ ∞. This means that the series\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) = 2∞/summationdisplay\nn=1Re(λn−c0)\nconverges. In the following theorem we express the sum of this ser ies in\nterms of the function a. This is in fact the formula for the regularized\ntrace.\nTheorem 4. Leta∈C3[0,1],b∈C2[0,1]. Then the identity\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) =a(0)+a(1)\n2−/an}bracketle{ta/an}bracketri}ht\nholds.\n3.Asymptotics for the fundamental system\nInthissectionweobtaintheasymptoticexpansionforthefundame n-\ntal system of the solutions of the equation (1.2) as λ→ ∞,λ∈C. This\nis done by means of the standard technique described in, for instan ce,\n[E, Ch. IV, Sec. 4.2, 4.3], [Fe, Ch. II, Sec. 3].\nWe begin with the formal construction assuming the asymptotics to\nbe of the form\n(3.1) u±(x,λ) = e±λx±xR\n0φ±(t,λ)dt\n,\nwhere\n(3.2) φ±(x,λ) =m/summationdisplay\ni=0φ(±)\ni(x)λ−i+O(λ−m−1), m/greaterorequalslant1.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 5\nIn what follows we assume that a∈Cm+1[0,1],b∈Cm[0,1].\nWe substitute the series (3.1), (3.2) into (1.2) and equate the coef -\nficients of the same powers of λ. It leads us to a recurrent system of\nequations determining φ(±)\niwhich read as follows:\nφ(±)\n0=a, (3.3)\nφ(±)\n1=−1\n2(±a′+a2+b), (3.4)\nφ(±)\ni=−1\n2/parenleftBigg\n±φ(±)\ni−1′+i−1/summationdisplay\nj=0φ(±)\njφ(±)\ni−j−1/parenrightBigg\n, i/greaterorequalslant2. (3.5)\nThe main aim of this section is to prove that there exist solutions to\n(1.2) having the asymptotics (3.1), (3.2). In other words, we are g oing\nto justify these asymptotics rigorously. We will do this for u+, the case\nofu−following along similar lines.\nLet us write\nUm(x,λ) = eλx+mP\ni=0λ−ixR\n0φ(+)\ni(t)dt\n.\nIn view of the assumed smoothness for aandbwe conclude that Um∈\nC2[0,1]. It is also easy to check that\n(3.6)U′′\nm−λ2Um−2λaUm+bUm=λ−meλxfm(x,λ), x∈[0,1],\nUm(0) = 1, U′\nm(0) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i,\nwhere the function fmsatisfies the estimate\n|fm(x,λ)|/lessorequalslantCm\nuniformly for large λandx∈[0,1]\nWe consider first the case Re λ/greaterorequalslant0. Differentiating the function u+\nformally we see that\nu′\n+(0,λ) =λ+φ+(0,λ) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i+O(λ−m−1).\nLet\nA0(λ) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i,\nandu+(x,λ) be the solution to the Cauchy problem for the equation /diamondsolid\n(1.2) subject to the initial conditions\nu+(0,λ) = 1, u′\n+(0,λ) =A0(λ).6 DENIS BORISOV AND PEDRO FREITAS\nWe introduce one more function wm(x,λ) =u+(x,λ)/Um(x,λ). This\nfunction solves the Cauchy problem\n(U2\nmw′\nm)′+λ−mUmeλxfmwm= 0, x∈[0,1],\nwm(0,λ) = 1, w′\nm(0,λ) = 0.\nThe last problem is equivalent to the integral equation\nwm(x,λ)+λ−m(Km(λ)wm)(x,λ) = 1,\n(Km(λ)wm)(x,λ) :=x/integraldisplay\n0U−2\nm(t1)t1/integraldisplay\n0Um(t2)eλt2fm(t2,λ)wm(t2,λ)dt2dt1.\nSince Re λ/greaterorequalslant0 for 0/greaterorequalslantt2/greaterorequalslantt1/greaterorequalslant1, the estimate\n|U−2\nm(t1,λ)Um(t2,λ)eλt1|/lessorequalslantCm\nholds true, where the constant Cmis independent of λ,t1,t2. Hence,\nthe integral operator Km:C[0,1]→C[0,1] is bounded uniformly in λ\nlarge enough, Re λ/greaterorequalslant0. Employing this fact, we conclude that\nwm(x) = 1+O(λ−m), λ→ ∞,Reλ/greaterorequalslant0,\nin theC2[0,1]-norm. Hence, the formula (3.1), where\n(3.7) φ+(x,λ) =m−1/summationdisplay\ni=0φ(+)\ni(x)λ−i+O(λ−m),\ngives the asymptotic expansion for the solution of the Cauchy prob lem\n(1.2), (3.6) as λ→ ∞, Reλ/greaterorequalslant0.\nSuppose now that Re λ/lessorequalslant0. LetA1(λ),A2(λ) be functions having\nthe asymptotic expansions\nA1(λ) =λ+m/summationdisplay\ni=0λ−i1/integraldisplay\n0φ(+)\ni(x)dx, A 2(λ) =λ+m/summationdisplay\ni=0φ(+)\ni(1)λ−i.\nWe define the function /tildewideu+(x,λ) as the solution to the Cauchy problem\nfor equation (1.2) subject to the initial conditions\n/tildewideu+(1,λ) = eA1(λ),/tildewideu′\n+(1,λ) =A2(λ)eA1(λ).\nIn a way analogous to the arguments given above, it is possible to\ncheck that the function /tildewideu+has the asymptotic expansion (3.1) in the\nC2[0,1]-norm as λ→+∞, Reλ/lessorequalslant0. Hence,/tildewideu+(0,λ) = 1 +O(λ−m)\nfor eachm/greaterorequalslant1. In view of this identity we conclude that the function\nu+(x,λ) :=/tildewideu+(x,λ)//tildewideu+(0,λ) is a solution to (1.2), satisfies the condi-\ntionu+(0,λ) = 1, and has the asymptotic expansion (3.1), where the\nasymptotics for φ+is given in (3.7).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 7\nFor convenience we summarize the obtained results in\nLemma 3.1. Leta∈Cm+1[0,1],b∈Cm[0,1]. There exist two linear\nindependent solutions to the equation (1.2) satisfying the initial con-\nditionu±(0,λ) = 1and having the asymptotic expansions (3.1) in the\nC2[0,1]-norm as λ→ ∞,λ∈C, where\nφ±(x,λ) =m−1/summationdisplay\ni=0φ(±)\ni(x)λ−i+O(λ−m).\n4.Asymptotics of the eigenvalues\nThis section is devoted to the proof of Theorem 1 and Corollaries 2\nand 3. We assume that a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1.\nLetu=u(x,λ) be the solution to (1.2) subject to the initial condi-\ntionsu(0,λ) = 0,u′(0,λ) = 1. Denote γ0(λ) :=u(1,λ). The function\nγ0is entire, and its zeros coincide with the eigenvalues of the problem\n(1.2), (1.3). It follows from Lemma 3.1 that, for λlarge enough the\nfunction u(x,λ) can be expressed in terms of u±by\nu(x,λ) =u+(x,λ)−u−(x,λ)\nu′\n+(0,λ)−u′\n−(0,λ).\nThe denominator is non-zero, since due to (3.1)\nu′\n+(0,λ)−u′\n−(0,λ) = 2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1), λ→ ∞.\nThus, for λlarge enough\n(4.1) γ0(λ) =u+(1,λ)−u−(1,λ)\nu′\n+(0,λ)−u′\n−(0,λ).\nLemma 4.1. Fornlarge enough, the set\nQ:={λ:|Reλ|< πn+π/2,|Imλ|< πn+π/2}\ncontains exactly 2neigenvalues of the problem (3.1), (3.3).\nProof.Letγ1(λ) :=γ0(λ)eλ+/angbracketlefta/angbracketright. The zeros of γ1are those of γ0(λ). For\nλlarge enough we represent the function γ1(λ) as /diamondsolid\nγ1(λ) =γ2(λ)+γ3(λ), γ2:=e2(λ+a(0))−1\n2(λ+a(0)),\nγ3(λ) =−γ2(λ)/tildewideφ+(0,λ)+/tildewideφ−(0,λ)+2(1+ λ−1a(0))(1−eλ−1/angbracketlefteφ+(·,λ)/angbracketright)\n2λ(λ+a(0))+/tildewideφ+(0,λ)+/tildewideφ−(0,λ)\n+eλ−1/angbracketlefteφ+(·,λ)/angbracketright−e−λ−1/angbracketlefteφ−(·,λ)/angbracketright\n2(λ+a(0))+λ−1(/tildewideφ+(0,λ)+/tildewideφ−(0,λ)),8 DENIS BORISOV AND PEDRO FREITAS\n/tildewideφ±(x,λ) :=λ−1(φ±(x,λ)−a(x)).\nItisclear thatfor λlargeenoughthefunction γ3(λ) satisfies anuniform\ninλestimate\n|γ3(λ)|/lessorequalslantC|λ|−2/parenleftbig\n|γ2(λ)|+1/parenrightbig\n.\nOne can also check easily that /diamondsolid\n|γ2(λ)|/greaterorequalslantC|λ|, λ∈∂K,\nifnis large enough. These two last estimates imply that |γ3(λ)|/lessorequalslant\n|γ2(λ)|asλ∈∂K, ifnislargeenough. ByRouch´ etheoremweconclude\nthat for such nthe function γ1has the same amount of zeros inside\nQas the function γ2does. Since the zeros of the latter are given by\nπni−/an}bracketle{ta/an}bracketri}ht,n/ne}ationslash= 0, this completes the proof. /square\nProof of Theorem 1. Assume first that a∈C2[0,1],b∈C1[0,1]. As\nwas mentioned above, the eigenvalues of problem (1.2), (1.3) are th e\nzeros of the function γ0(λ) = 0. It follows from Lemma 4.1 that these\neigenvalues tend to infinity as n→ ∞. By Lemma 3.1, for λlarge\nenough the equation γ0(λ) = 0 becomes\ne2λ+/angbracketleftφ+(·,λ)+φ−(·,λ)/angbracketright= 0\nwhich may be rewritten as\n(4.2) 2 λ+/an}bracketle{tφ+(·,λ)+φ−(·,λ)/an}bracketri}ht= 2πni, n∈Z.\nIf we now replace φ±by the leading terms of their asymptotic expan-\nsions we obtain\n2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1) = 2πni, (4.3)\nλ=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞.\nHence, the eigenvalues behave as λ∼πni−/an}bracketle{ta/an}bracketri}htfor large n. Moreover,\nit follows from Lemma 4.1 that it is exactly the eigenvalue λnwhich\nbehaves as\nλn=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞.\nIt follows from this identity and (4.3) that\nλn=πni−/an}bracketle{ta/an}bracketri}ht+O(n−1), n→ ∞,\nand we complete the proof in the case m= 1. Ifm= 2, we substitute\nthe above identity and (3.1) into (4.2) and get\nλn+/an}bracketle{ta/an}bracketri}ht+1\nλn/an}bracketle{tφ(+)\n1+φ(−)\n1/an}bracketri}ht+O(λ−2\nn) =πni,\nλn=πni−/an}bracketle{ta/an}bracketri}ht−/an}bracketle{tφ(+)\n1+φ(−)\n1/an}bracketri}ht\nπni+O(n−2).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 9\nThe last formula and the identities (3.4) yield formulas (2.2) for c0and\nc1. Repeating the described procedure one can easily check that the\nasymptotics (2.1), (2.2) hold true. /square\nProof of Corollary 2. The coefficients c0,c1in the asymptotics (2.1)\nare determined by the formulas (2.2) and, by the Cauchy-Schwarz in-\nequality, we thus obtain\nc2\n0=/an}bracketle{ta/an}bracketri}ht2/lessorequalslant/an}bracketle{ta2/an}bracketri}ht= 2πic1−/an}bracketle{tb/an}bracketri}ht,\nwith equality if and only if a(x) is a constant function. This fact\ncompletes the proof. /square\nProof of Corollary 3. Itfollowsfrom(2.2),(2.3)that /an}bracketle{ta2\n1/an}bracketri}ht=/an}bracketle{ta2\n2/an}bracketri}ht,/an}bracketle{ta3\n1/an}bracketri}ht=\n/an}bracketle{ta3\n2/an}bracketri}ht. Now we check that\n/an}bracketle{ta2\ni/an}bracketri}ht=/an}bracketle{ta2\n0/an}bracketri}ht+/an}bracketle{t/tildewidea2\ni/an}bracketri}ht,/an}bracketle{ta3\ni/an}bracketri}ht=/an}bracketle{ta3\n0/an}bracketri}ht+3/an}bracketle{ta0/tildewidea2\ni/an}bracketri}ht, i= 1,2,\nand arrive at the statement of the theorem. /square\n5.Regularized trace formulas\nIn this section we prove Theorem 4. We follow the idea employed in\nthe proof of the similar trace formula for the Sturm-Liouville operat ors\nin [LS, Ch. I, Sec. 14].\nWe begin by defining the function\nΦ(λ) :=λ2p∞/productdisplay\nn=p+1/parenleftbigg\n1−λ\nλn/parenrightbigg/parenleftbigg\n1−λ\nλn/parenrightbigg\n.\nThe above product converges, since\n/parenleftbigg\n1−λ\nλn/parenrightbigg/parenleftbigg\n1−λ\nλn/parenrightbigg\n= 1+λ2−2λReλn\n|λn|2,\nand by Theorem 1 we have\n(5.1)|λn|2=π2n2−2πic1+c2\n0+O(n−2),\nReλn=c0+O(n−2)\nasn→+∞. Proceeding in the same way as in the formulas (14.8),\n(14.9) in [LS, Ch. I, Sec. 14], we obtain\nΦ(λ) =C0Ψ(λ)sinhλ\nλ,\nΨ(λ) :=∞/productdisplay\nn=1/parenleftbigg\n1−π2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg\n,10 DENIS BORISOV AND PEDRO FREITAS\nC0:= (πn)2p∞/productdisplay\nn=p+1π2n2\n|λn|2.\nIn what follows we assume that λis real, positive and large. In the\nsame way as in [LS, Ch. I, Sec. 14] it is possible to derive the formula\n(5.2) lnΨ( λ) =−∞/summationdisplay\nk=11\nk∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n.\nOur aim is to study the asymptotic behaviour of lnΨ( λ) asλ→+∞.\nEmploying the same arguments as in the proof of Lemma 14.1 and in\nthe equation (14.11) in [LS, Ch. I, Sec. 14], we arrive at the estimate\n∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n/lessorequalslantckλk∞/summationdisplay\nn=11\n(π2n2+λ2)k\n/lessorequalslantckλk+∞/integraldisplay\n0dt\n(π2t2+λ2)k=ck\nλk+∞/integraldisplay\n0dz\n(π2z2+1)k/lessorequalslantck+1\nλk,\n∞/summationdisplay\nk=31\nk∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n=O(λ−3), λ→+∞, (5.3)\nwherecis a constant independent of kandn. Let us analyze the\nasymptotic behaviour of the first two terms in the series (5.2). As\nk= 1, we have\n(5.4)∞/summationdisplay\nn=1π2n2−|λn|2+2λReλn\nπ2n2+λ2=∞/summationdisplay\nn=1π2n2−|λn|2−2πic1+c2\n0\nπ2n2+λ2\n+/parenleftbig\n2πic1−c2\n0+2λc0/parenrightbig∞/summationdisplay\nn=11\nπ2n2+λ2\n+2λ−1S−2λ−1∞/summationdisplay\nn=1π2n2(Reλn−c0)\nπ2n2+λ2,\nwhereS:=∞/summationdisplay\nn=1(Reλn−c0).\nTaking into account (5.1), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1π2n2−|λn|2−2πic1+c2\n0\nπ2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\nn2(π2n2+λ2)\n=π2\n63+λ2−3cothλ\nλ4/lessorequalslantCλ−2,EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 11\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(Reλn−c0)π2n2\nπ2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\nπ2n2+λ2/lessorequalslantCλ−1,\nwhere the constant Cis independent of λ. Here we have also used the\nformula\n(5.5)∞/summationdisplay\nn=11\nπ2n2+λ2=λcothλ−1\n2λ2=λ−1−λ−2\n2+O(λ−1e−2λ)\nasλ→+∞. We employ this formula to calculate the remaining terms\nin (5.4) and arrive at the identity\n(5.6)\n∞/summationdisplay\nn=1π2n2−|λn|2+2λReλn\nπ2n2+λ2=c0+/parenleftbigg\n2S−c0−c2\n0\n2+iπc1/parenrightbigg\nλ−1+O(λ−2),\nasλ→+∞. Fork= 2 we proceed in the similar way,\n∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg2\n=∞/summationdisplay\nn=1(π2n2−|λn|2)2\n(π2n2+λ2)2\n−2λ∞/summationdisplay\nn=1(π2n2−|λn|2)Reλn\n(π2n2+λ2)2+4λ2c2\n0∞/summationdisplay\nn=11\n(π2n2+λ2)2\n+4λ2∞/summationdisplay\nn=1(Reλn)2−c2\n0\n(π2n2+λ2)2.\nBy differentiating (5.5) we obtain\n∞/summationdisplay\nn=11\n(π2n2+λ2)2=λcothλ−2−λ2(1−coth2λ)\n4λ4.\nThis identity and (5.1) yield that as λ→+∞\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(π2n2−|λn|2)2\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(π2n2−|λn|2)Reλn\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(Reλn)2−c2\n0\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3.\nHence,\n(5.7)∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg2\n=c2\n0λ−1+O(λ−2)12 DENIS BORISOV AND PEDRO FREITAS\nasλ→+∞. It follows from (5.2), (5.3), (5.6), (5.7) that\nlnΨ(λ) =−c0−(2S−c0+iπc1)λ−1+O(λ−2),\nΦ(λ) =C0e−c0sinhλ\nλ/bracketleftbig\n1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig\n(5.8)\nasλ→+∞. It follows from (4.1) and Lemma 3.1 that for λlarge\nenough the estimate /diamondsolid\n|γ0(λ)|/lessorequalslantC|λ|−1e|λ|\nholds true. Hence, the order of the entire function γ0(λ) is one. In view\nof Theorem 1 we also conclude that the series∞/summationtext\nn=p+1|λn|−2converges\nand therefore the genus of the canonical product associated wit hγ0is\none. We apply Hadamard’s theorem (see, for instance, [Le, Ch. I, Sec.\n10, Th. 13]) and obtain that\nγ0(λ) = eP(λ)Φ(λ), P(λ) =α1λ+α0+2∞/summationdisplay\nn=p+1|λn|−2Reλn,\nwhereα1,α0are some numbers. Hence, due to (5.8), it follows that γ0\nbehaves as\nγ0(λ) =C0eP(λ)sinhλ\nλ/bracketleftbig\n1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig\n,\nasλ→+∞. On the other hand, Lemma 3.1 and (4.1) imply that\nγ0(λ) =eλ+/angbracketlefta/angbracketright\n2λ/bracketleftbig\n1+(/an}bracketle{tφ(+)\n1/an}bracketri}ht−a(0))λ−1+O(λ−2)/bracketrightbig\n+O(λ−1e−λ),\nasλ→+∞. Comparing the last two identities yields α1= 0,\nC0eα0−c0+2∞P\nn=1|λn|−2Reλn= e/angbracketlefta/angbracketright\nand\n−(2S−c0+iπc1) =/an}bracketle{tφ(+)\n1/an}bracketri}ht−a(0).\nIt now follows from (2.2), (3.4) that\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) = 2S=c0+a(0)−/an}bracketle{tφ(+)\n1/an}bracketri}ht−iπc1=a(0)+a(1)\n2−/an}bracketle{ta/an}bracketri}ht,\ncompleting the proof of Theorem 4.\nAcknowledgments\nThis work was done during the visit of D.B. to the Universidade de\nLisboa; he is grateful for the hospitality extended to him. P.F. would\nlike to thank A. Laptev for several conversations of this topic.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 13\nReferences\n[AL] M. Asch and G. Lebeau, The spectrum of the damped wave oper ator for\na bounded domain in R2,Exp. Math. 12(2003), 227-241.\n[BR] A. Benaddi and B. Rao, Energy decay rate of damped wave equ ations\nwith indefinite damping, J. Differential Eq. 161(2000), 337–357.\n[CC] C. Castro and S. Cox, Achieving arbitrarily large decay in the dam ped\nwave equation, SIAM J. Control Optim. 39(2001), 1748–1755.\n[CFNS] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun. Exponential d ecay of\nenergy of evolution equations with locally distributed damping. SIAM J.\nAppl. Math. 51(1991), 266–301.\n[CZ] S. Cox and E. Zuazua, The rate at which energy decays in a damp ed\nstring.Comm. Part. Diff. Eq. ,19(1994), 213–243.\n[E] A. Erd´ elyi. Asymptotic expansions. Dover Publications Inc., N.Y. 1956.\n[Fe] M.V. Fedoryuk, Asymptotic analysis: linear ordinary differential equa-\ntions. Berlin: Springer-Verlag. 1993.\n[F1] P. Freitas, On some eigenvalueproblems relatedto the waveequ ation with\nindefinite damping, J. Differential Equations ,127(1996), 320–335.\n[F2] P.Freitas,Spectralsequencesforquadraticpencilsandthe inversespectral\nproblem for the damped wave equation, J. Math. Pures Appl. 78(1999),\n965–980.\n[L] G. Lebeau, ´Equations des ondes amorties, S´ eminaire sur les ´Equations\naux D´ eriv´ ees Partielles, 1993–1994,Exp. No. XV, 16 pp., ´Ecole Polytech.,\nPalaiseau, 1994.\n[Le] B.Ya. Levin. Distribution of zeros of entire functions. Providen ce, R.I.:\nAmerican Mathematical Society. 1964.\n[LS] B.M. Levitan, I.S. Sargsjan. Introduction to spectral theor y: Selfadjoint\nordinarydifferentialoperators.TranslationsofMathematicalMo nographs.\nVol. 39. Providence, R.I.: American Mathematical Society. 1975.\n[PT] J. P¨ oschel and E. Trubowitz, Inverse spectran theory, Pu re and Applied\nMathematics, Vol. 130, Academic Press, London, 1987.\n[S] J.Sj¨ ostrand, Asymptoticdistributionofeigenfrequenciesfo rdampedwave\nequations, Publ. Res. Inst. Math. Sci. 36(2000), 573–611.\nDepartment of Physics and Mathematics, Bashkir State Pedag ogi-\ncal University, October rev. st., 3a, 450000, Ufa, Russia\nE-mail address :borisovdi@yandex.ru\nDepartment ofMathematics, Faculdade de Motricidade Human a (TU\nLisbon) andGroup of Mathematical Physics of the University of Lis-\nbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1 649-003\nLisboa, Portugal\nE-mail address :freitas@cii.fc.ul.pt"    },    {        "title": "0905.4544v2.Hydrodynamic_theory_of_coupled_current_and_magnetization_dynamics_in_spin_textured_ferromagnets.pdf",        "content": "Hydrodynamic theory of coupled current and magnetization dynamics in\nspin-textured ferromagnets\nClement H. Wong and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics\nin metallic ferromagnets. The collective spin density couples to the spin current through a U(1)\nBerry-phase gauge \feld determined by the local texture and dynamics of the magnetization. We\ndetermine phenomenologically the dissipative corrections to the equation of motion for the electronic\ncurrent, which consist of a dissipative spin-motive force generated by magnetization dynamics and\na magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque\non the magnetic texture follows from the Onsager principle. We investigate the e\u000bects of thermal\n\ructuations and \fnd that electronic dynamics contribute to a nonlocal Gilbert damping tensor in\nthe Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including\nmagnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.\nPACS numbers: 72.15.Gd,72.25.-b,75.75.+a\nI. INTRODUCTION\nThe interaction of electrical currents with magnetic\nspin texture in conducting ferromagnets is presently a\nsubject of active research. Topics of interest include\ncurrent-driven magnetic dynamics of solitons such as do-\nmain walls and magnetic vortices,1,2,3,4as well as the\nreciprocal process of voltage generation by magnetic\ndynamics.5,6,7,8,9,10,11,12This line of research has been\nfueled in part by its potential for practical applications\nto magnetic memory and data storage devices.13Funda-\nmental theoretical interest in the subject dates back at\nleast two decades.5,6,14It was recognized early on6that\nin the adiabatic limit for spin dynamics, the conduction\nelectrons interact with the magnetic spin texture via an\ne\u000bective spin-dependent U(1) gauge \feld that is a local\nfunction of the magnetic con\fguration. This gauge \feld,\non the one hand, gives rise to a Lorentz force due to\n\\\fctitious\" electric and magnetic \felds and, on the other\nhand, mediates the so-called spin-transfer torque exerted\nby the conduction electrons on the collective magnetiza-\ntion. An alternative and equivalent view is to consider\nthis force as the result of the Berry phase15accumulated\nby an electron as it propagates through the ferromagnet\nwith its spin aligned with the ferromagnetic exchange\n\feld.8,10,16In the standard phenomenological formalism\nbased on the Landau-Lifshitz-Gilbert (LLG) equation,\nthe low-energy, long-wavelength magnetization dynamics\nare described by collective spin precession in the e\u000bective\nmagnetic \feld, which is coupled to electrical currents via\nthe spin-transfer torques. In the following, we develop\na closed set of nonlinear classical equations governing\ncurrent-magnetization dynamics, much like classical elec-\ntrodynamics, with the LLG equation for the spin-texture\n\\\feld\" in lieu of the Maxwell equations for the electro-\nmagnetic \feld.\nThis electrodynamic analogy readily explains various\ninteresting magnetoelectric phenomena observed recently\nin ferromagnetic metals. Adiabatic charge pumping bymagnetic dynamics17can be understood as the gener-\nation of electrical currents due to the \fctitious electric\n\feld.18In addition, magnetic textures with nontrivial\ntopology exhibit the so-called topological Hall e\u000bect,19,20\nin which the \fctitious magnetic \feld causes a classical\nHall e\u000bect. In contrast to the classical magnetoresis-\ntance, the \rux of the \fctitious magnetic \feld is a topo-\nlogical invariant of the magnetic texture.6\nDissipative processes in current-magnetization dynam-\nics are relatively poorly understood and are of central\ninterest in our theory. Electrical resistivity due to quasi-\none-dimensional (1D) domain walls and spin spirals have\nbeen calculated microscopically.21,22,23More recently, a\nviscous coupling between current and magnetic dynam-\nics which determines the strength of a dissipative spin\ntorque in the LLG equation as well the reciprocal dis-\nsipative spin electromotive force generated by magnetic\ndynamics, called the \\ \fcoe\u000ecient,\"2was also calcu-\nlated in microscopic approaches.3,24,25Generally, such\n\frst-principles calculations are technically di\u000ecult and\nrestricted to simple models. On the other hand, the num-\nber of di\u000berent forms of the dissipative interactions in the\nhydrodynamic limit are in general constrained by sym-\nmetries and the fundamental principles of thermodynam-\nics, and may readily be determined phenomenologically\nin a gradient expansion. Furthermore, classical thermal\n\ructuations may be easily incorporated in the theoretical\nframework of quasistationary nonequilibrium thermody-\nnamics.\nThe principal goal of this paper is to develop a (semi-\nphenomenological) hydrodynamic description of the dis-\nsipative processes in electric \rows coupled to magnetic\nspin texture and dynamics. In Ref. 11, we drew the anal-\nogy between the interaction of electric \rows with quasis-\ntationary magnetization dynamics with the classical the-\nory of magnetohydrodynamics. In our \\spin magnetohy-\ndrodynamics,\" the spin of the itinerant electrons, whose\n\rows are described hydrodynamically, couples to the lo-\ncal magnetization direction, which constitutes the col-\nlective spin-coherent degree of freedom of the electronicarXiv:0905.4544v2  [cond-mat.mes-hall]  16 Nov 20092\n\ruid. In particular, the dissipative \fcoupling between\nthe collective spin dynamics and the itinerant electrons\nis loosely akin to the Landau damping, capturing cer-\ntain kinematic equilibration of the relative motion be-\ntween spin-texture dynamics and electronic \rows. In our\nprevious paper,11we considered a special case of incom-\npressible \rows in a 1D ring to demonstrate the essential\nphysics. In this paper, we establish a general coarse-\ngrained hydrodynamic description of the interaction be-\ntween the electric \rows and textured magnetization in\nthree dimensions, treating the itinerant electron's degrees\nof freedom in a two-component \ruid model (correspond-\ning to the two spin projections of spin-1 =2 electrons along\nthe local collective magnetic order). Our phenomenology\nencompasses all the aforementioned magnetoelectric phe-\nnomena.\nThe paper is organized as follows. In Sec. II, we use a\nLagrangian approach to derive the semiclassical equation\nof motion for itinerant electrons in the adiabatic approx-\nimation for spin dynamics. In Sec. III, we derive the\nbasic conservation laws, including the Landau-Lifshitz\nequation for the magnetization, by coarse-graining the\nsingle-particle equation of motion and the Hamiltonian.\nIn Sec. IV, we phenomenologically construct dissipative\ncouplings, making use of the Onsager reciprocity princi-\nple, and calculate the net dissipation power. In particu-\nlar, we develop an analog of the Navier-Stokes equation\nfor the electronic \ruid, focusing on texture-dependent\ne\u000bects, by making a systematic expansion in nonequi-\nlibrium current and magnetization consistent with sym-\nmetry requirements. In Sec. V, we include the e\u000bects of\nclassical thermal \ructuations by adding Langevin sources\nto the hydrodynamic equations, and arrive at the central\nresult of this paper: A set of coupled stochastic di\u000ber-\nential equations for the electronic density, current, and\nmagnetization, and the associated white-noise correlators\nof thermal noise. In Sec. VI, we apply our results to\nspecial examples of rotating and spinning magnetic tex-\ntures, calculating magnetic texture resistivity and mag-\nnetic dynamics-generated currents for a magnetic spiral\nand a vortex. The paper is summarized in Sec. VII and\nsome additional technical details, including a microscopic\nfoundation for our semiclassical theory, are presented in\nthe appendices.\nII. QUASIPARTICLE ACTION\nIn a ferromagnet, the magnetization is a symmetry-\nbreaking collective dynamical variable that couples to the\nitinerant electrons through the exchange interaction. Be-\nfore developing a general phenomenological framework,\nwe start with a simple microscopic model with Stoner in-\nstability, which will guide us to explicitly construct some\nof the key magnetohydrodynamic ingredients. Within a\nlow-temperature mean-\feld description of short-ranged\nelectron-electron interactions, the electronic action isgiven by (see appendix A for details):\nS=Z\ndtd3r^ y\u0014\ni~@t+~2\n2mer2\u0000\u001e\n2+\u0001\n2m\u0001^\u001b\u0015\n^ :(1)\nHere, \u0001( r;t) is the ferromagnetic exchange splitting,\nm(r;t) is the direction of the dynamical order param-\neter de\fned by ~h^ y^\u001b^ i=2 =\u001asm,\u001asis the local spin\ndensity, and ^ (r;t) is the spinor electron \feld operator.\nFor the short-range repulsion U > 0 discussed in ap-\npendix A, \u0001( r;t) = 2U\u001as(r;t)=~and\u001e(r;t) =U\u001a(r;t),\nwhere\u001a=h^ y^ iis the local particle number density.\nFor electrons, the magnetization Mis in the opposite di-\nrection of the spin density: M=\r\u001asm, where\r <0 is\nthe gyromagnetic ratio. Close to a local equilibrium, the\nmagnetic order parameter describes a ground state con-\nsisting of two spin bands \flled up to the spin-dependent\nFermi surfaces, with the spin orientation de\fned by m.\nWe will focus on soft magnetic modes well below the\nCurie temperature, where only the direction of the mag-\nnetization and spin density are varied, while the \ructu-\nations of the magnitudes are not signi\fcant. The spin\ndensity is given by \u001as=~(\u001a+\u0000\u001a\u0000)=2 and particle den-\nsity by\u001a=\u001a++\u001a\u0000, where\u001a\u0006are the local spin-up/down\nparticle densities along m.\u001ascan be essentially constant\nin the limit of low spin susceptibility.\nStarting with a nonrelativistic many-body Hamilto-\nnian, the action (1) is obtained in a spin-rotationally\ninvariant form. However, this symmetry is broken by\nspin-orbit interactions, whose role we will take into ac-\ncount phenomenologically in the following. When the\nlength scale on which m(r;t) varies is much greater than\nthe ferromagnetic coherence length lc\u0018~vF=\u0001, where\nvFis the Fermi velocity, the relevant physics is captured\nby the adiabatic approximation. In this limit, we start\nby neglecting transitions between the spin bands, treat-\ning the electron's spin projection on the magnetization\nas a good quantum number. (This approximation will\nbe relaxed later, in the presence of microscopic spin-\norbit or magnetic disorder.) We then have two e\u000bec-\ntively distinct species of particles described by a spinor\nwave function ^ 0, which is de\fned by ^ =^U(R)^ 0. Here,\n^U(R) is an SU(2) matrix corresponding to the local spa-\ntial rotationR(r;t) that brings the z-axis to point along\nthe magnetization direction: R(r;t)z=m(r;t), so that\n^Uy(^\u001b\u0001m)^U= ^\u001bz. The projected action then becomes:\nS=Z\ndtZ\nd3r^ 0y\"\n(i~@t+ ^a)\u0000(\u0000i~r\u0000^a)2\n2me\n\u0000\u001e\n2+\u0001\n2^\u001bz\u0015\n^ 0\u0000Z\ndtF[m];(2)\nwhere\nF[m] =A\n2Z\nd3r(@im)2(3)\nis the spin-texture exchange energy (implicitly summing\nover the repeated spatial index i), which comes from the3\nterms quadratic in the gauge \felds that survive the pro-\njection. In the mean-\feld Stoner model, the ferromag-\nnetic exchange sti\u000bness is A=~2\u001a=4me. To broaden our\nscope, we will treat it as a phenomenological constant,\nwhich, for simplicity, is determined by the mean electron\ndensity.26The spin-projected \\\fctitious\" gauge \felds are\ngiven by\na\u001b(r;t) =i~h\u001bj^Uy@t^Uj\u001bi;\na\u001b(r;t) =i~h\u001bj^Uyr^Uj\u001bi: (4)\nChoosing the rotation matrices ^U(m) to depend only on\nthe local magnetic con\fguration, it follows from their\nde\fnition that spin- \u001bgauge potentials have the form:\na\u001b=\u0000@tm\u0001amon\n\u001b(m); a\u001bi=\u0000@im\u0001amon\n\u001b(m);(5)\nwhere amon\n\u001b(m)\u0011 \u0000i~h\u001bj^Uy@m^Uj\u001bi. We show in Ap-\npendix B the well known result (see, e.g., Ref. 27) that\namon\n\u001bis the vector potential (in an arbitrary gauge) of\na magnetic monopole in the parameter space de\fned by\nm:\n@m\u0002amon\n\u001b(m) =q\u001bm; (6)\nwhereq\u001b=\u001b~=2 is the monopole charge (which is ap-\npropriately quantized).\nBy noting that the action (2) is formally identical to\ncharged particles in electromagnetic \feld, we can imme-\ndiately write down the following classical single-particle\nLagrangian for the interaction between the spin- \u001belec-\ntrons and the collective spin texture:\nL\u001b(r;_r;t) =me_r2\n2+_r\u0001a\u001b(r;t) +a\u001b(r;t); (7)\nwhere _ris the spin-\u001belectron (wave-packet) velocity. To\nsimplify our discussion, we are omitting here the spin-\ndependent forces due to the self-consistent \felds \u001e(r;t)\nand \u0001( r;t), which will be easily reinserted at a later\nstage. See Eq. (29).\nThe Euler-Lagrange equation of motion for v=\n_rderived from the single-particle Lagrangian (7),\n(d=dt)(@L\u001b=@_r) =@L\u001b=@r, gives\nme_v=q\u001b(e+v\u0002b): (8)\nThe \fctitious electromagnetic \felds that determine the\nLorentz force are\nq\u001bei=@ia\u001b\u0000@ta\u001bi=q\u001bm\u0001(@tm\u0002@im);\nq\u001bbi=\u000fijk@ja\u001bk=q\u001b\u000fijk\n2m\u0001(@km\u0002@jm):(9)\nThey are conveniently expressed in terms of the tensor\n\feld strength\nq\u001bf\u0016\u0017\u0011@\u0016a\u001b\u0017\u0000@\u0017a\u001b\u0016=q\u001bm\u0001(@\u0017m\u0002@\u0016m) (10)\nbyei=fi0andbi=\u000fijkfjk=2.\u000fijkis the antisymmet-\nric Levi-Civita tensor and we used four-vector notation,de\fning@\u0016= (@t;r) anda\u001b\u0016= (a\u001b;a\u001b). Here and\nhenceforth the convention is to use Latin indices to de-\nnote spatial coordinates and Greek for space-time coor-\ndinates. Repeated Latin indices i;j;k are, furthermore,\nalways implicitly summed over.\nIII. SYMMETRIES AND CONSERVATION\nLAWS\nA. Gauge invariance\nThe Lagrangian describing coupled electron transport\nand collective spin-texture dynamics (disregarding for\nsimplicity the ordinary electromagnetic \felds) is\nL(rp;vp;m;@\u0016m)\n=X\np \nmev2\np\n2+vp\u0001a\u001b+a\u001b!\n\u0000A\n2Z\nd3r(@im)2\n=X\np \nmev2\np\n2+v\u0016\npa\u001b\u0016!\n\u0000A\n2Z\nd3r(@im)2:(11)\nv\u0016\np\u0011(1;vp),vp=_r, and\u001bhere is the spin of indi-\nvidual particles labelled by p. The resulting equations\nof motion satisfy certain basic conservation laws, due to\nspin-dependent gauge freedom, space-time homogeneity,\nand spin isotropicity.\nFirst, let us establish gauge invariance due to an ambi-\nguity in the choice of the spinor rotations ^U(r;t)!^U^U0.\nOur formulation should be invariant under arbitrary di-\nagonal transformations ^U0=e\u0000ifand ^U0=e\u0000ig^\u001bz=2on\nthe rotated fermionic \feld ^ 0, corresponding to gauge\ntransformations of the spin-projected theory:\n\u000ea\u001b\u0016=~@\u0016fand\u000ea\u001b\u0016=\u001b~@\u0016g=2; (12)\nrespectively. The change in the Lagrangian density is\ngiven by\n\u000eL=j\u0016@\u0016fand\u000eL=j\u0016\ns@\u0016g; (13)\nrespectively, where j=j++j\u0000andjs=~(j+\u0000j\u0000)=2\nare the corresponding charge and spin gauge currents.\nThe action S=R\ndtd3rLis gauge invariant, up to sur-\nface terms that do not a\u000bect the equations of motion,\nprovided that the four-divergence of the currents vanish,\nwhich is the conservation of particle number and spin\ndensity:\n_\u001a+r\u0001j= 0;_\u001as+r\u0001js= 0: (14)\n(The second of these conservation laws will be relaxed\nlater.) Here, the number and spin densities along with\nthe associated \rux densities are\n\u001a=X\npnp\u0011\u001a++\u001a\u0000;\nj=X\npnpvp\u0011\u001av; (15)4\nand\n\u001as=X\npq\u001bnp\u0011~\n2(\u001a+\u0000\u001a\u0000);\njs=X\npq\u001bnpvp\u0011\u001asvs; (16)\nwherenp=\u000e(r\u0000rp) and\u001bp=\u0006for spins up and down.\nIn the hydrodynamic limit, the above equations deter-\nmine the average particle velocity vand spin velocity\nvs, which allows us to de\fne four-vectors j\u0016= (\u001a;\u001av)\nandj\u0016\ns= (\u001as;\u001asvs). Microscopically, the local spin-\ndependent currents are de\fned, in the presence of electro-\nmagnetic vector potential aand \fctitious vector potential\na\u001b, by\nme\u001a\u001bv\u001b= Reh y\n\u001b(\u0000i~r\u0000a\u001b\u0000ea) \u001bi; (17)\nwheree<0 is the electron charge.\nB. Angular and linear momenta\nOur Lagrangian (11) contains the dynamics of m(r)\nthat is coupled to the current. In this regard, we note\nthat the time component of the \fctitious gauge poten-\ntial (B4),a\u001b=\u0000~@t'(1\u0000\u001bcos\u0012)=2, is a Wess-Zumino\naction that governs the spin-texture dynamics.4,6,28The\nvariational equation m\u0002\u000emL= 0 gives:\n\u001as(@t+vs\u0001r)m+m\u0002\u000emF= 0: (18)\nTo derive this equation, we used the spin-density con-\ntinuity equation (14) and a gauge-independent identity\nsatis\fed by the \fctitious potentials: their variations with\nrespect to mare given by\n\u000ema\u001b\u0016(m;@\u0016m) =q\u001bm\u0002@\u0016m; (19)\nwhere\n\u000em\u0011@\n@m\u0000X\n\u0016@\u0016@\n@(@\u0016m): (20)\nOne recognizes that Eq. (18) is the Landau-Lifshitz (LL)\nequation, in which the spin density precesses about the\ne\u000bective \feld given explicitly by\nh\u0011\u000emF=\u0000A@2\nim: (21)\nEquation (18) also includes the well-known reactive spin\ntorque:\u001c= (js\u0001r)m,3which is evidently the change\nin the local spin-density vector due to the spin angular\nmomentum carried by the itinerant electrons. One can\nformally absorb this spin torque by de\fning an advective\ntime derivative Dt\u0011@t+vs\u0001r, with respect to the\naverage spin drift velocity vs.\nEquation (18) may be written in a form that explicitly\nexpresses the conservation of angular momentum:27,29\n@t(\u001asmi) +@j\u0005ij= 0; (22)where the angular-momentum stress tensor is de\fned by\n\u0005ij=\u001asvsjmi\u0000A(m\u0002@jm)i: (23)\nNotice that this includes both quasiparticle and collective\ncontributions, which stem respectively from the trans-\nport and equilibrium spin currents.\nThe Lorentz force equation for the electrons, Eq. (8),\nin turn, leads to a continuity equation for the kinetic\nmomentum density.6To see this, let us start with the\nmicroscopic perspective:\n@t(\u001avi) =@tX\npnpvp=X\np( _npvp+np_vp): (24)\nUsing the Lorentz force equation for the second term, we\nhave:\nmeX\npnp_vp=X\npq\u001bnp(ei+\u000fijkbkvpj) =X\npq\u001bnpfi\u0016v\u0016\np\n=\u001asm\u0001(@tm\u0002@im) +\u001asvsjm\u0001(@jm\u0002@im)\n= (@im)\u0001(\u000emF) =\u0000A(@im)\u0001(@2\njm); (25)\nutilizing Eq. (18) to obtain the last line. Coarse-graining\nthe \frst term of Eq. (24), in turn, we \fnd:\nX\np_npvp=\u0000@jX\np\u000e(r\u0000rp)vpivpj!\u0000@jX\n\u001b\u001a\u001bv\u001biv\u001bj:\n(26)\nPutting Eqs. (25) and (26) together, we can \fnally write\nEq. (24) in the form:\nme@t(\u001avi) +@j \nTij+meX\n\u001b\u001a\u001bv\u001biv\u001bj!\n= 0;(27)\nwhere\nTij=A\u0014\n(@im)\u0001(@jm)\u0000\u000eij\n2(@km)2\u0015\n(28)\nis the magnetization stress tensor.6\nA spin-dependent chemical potential ^ \u0016=^K\u00001^\u001agov-\nerned by local density and short-ranged interactions can\nbe trivially incorporated by rede\fning the stress tensor\nas\nTij!Tij+\u000eij\n2^\u001aT^K\u00001^\u001a: (29)\nIn our notation, ^ \u0016= (\u0016+;\u0016\u0000)T, ^\u001a= (\u001a+;\u001a\u0000)Tand ^Kis\na symmetric 2\u00022 compressibility matrix in spin space,\nwhich includes the degeneracy pressure as well as self-\nconsistent exchange and Hartree interactions. In general,\nEq. (29) is valid only for su\u000eciently small deviations from\nthe equilibrium density.\nUsing the continuity equations (14), we can combine\nthe last term of Eq. (27) with the momentum density\nrate of change:\n@t(\u001a\u001bv\u001bi) +@j(\u001a\u001bv\u001biv\u001bj) =\u001a\u001b(@t+v\u001b\u0001r)v\u001bi;(30)5\nwhich casts the momentum density continuity equation\nin the Euler equation form:\nmeX\n\u001b\u001a\u001b(@t+v\u001b\u0001r)v\u001bi+@jTij= 0: (31)\nWe do not expect such advective corrections to @tto\nplay an important role in electronic systems, however.\nThis is in contrast to the advective-like time derivative\nin Eq. (18), which is \frst order in velocity \feld and is\ncrucial for capturing spin-torque physics.\nC. Hydrodynamic free energy\nWe will now turn to the Hamiltonian formulation and\nconstruct the free energy for our magnetohydrodynamic\nvariables. This will subsequently allow us to develop a\nnonequilibrium thermodynamic description. The canon-\nical momenta following from the Lagrangian (11) are\npp\u0011@L\n@vp=mevp+ap;\n\u0019\u0011@L\n@_m=X\npnp@a\u001b\n@_m=X\npnpamon\n\u001b(m): (32)\nNotice that for our translationally-invariant system, the\ntotal linear momentum\nP\u0011X\nppp+Z\nd3r(\u0019\u0001r)m=meX\npvp; (33)\nwhere we have used Eq. (5) to obtain the second equality,\ncoincides with the kinetic momentum (mass current) of\nthe electrons. The latter, in turn, is equivalent to the lin-\near momentum of the original problem of interacting non-\nrelativistic electrons, in the absence of any real or \fcti-\ntious gauge \felds. See appendix A. While Pis conserved\n(as discussed in the previous section and also follows now\nfrom the general principles), the canonical momenta of\nthe electrons and the spin-texture \feld, Eqs. (32), are\nnot conserved separately. As was pointed out by Volovik\nin Ref. 6, this explains anomalous properties of the lin-\near momentum associated with the Wess-Zumino action\nof the spin-texture \feld: This momentum has neither\nspin-rotational nor gauge invariance. The reason is that\nthe spin-texture dynamics de\fne only one piece of the\ntotal momentum, which is associated with the coherent\ndegrees of freedom. Including also the contribution as-\nsociated with the incoherent (quasiparticle) background\nrestores the proper gauge-invariant momentum, P, which\ncorresponds to the generator of the global translation in\nthe microscopic many-body description.\nPerforming a Legendre transformation to Hamiltonianas a function of momenta, we \fnd\nH[rp;pp;m;\u0019] =X\npvp\u0001pp+Z\nd3r_m\u0001\u0019\u0000L\n=X\np(pp\u0000a\u001b)2\n2me+A\n2Z\nd3r(@im)2\n\u0011E+F; (34)\nwhereEis the kinetic energy of electrons and Fis the\nexchange energy of the magnetic order. As could be\nexpected,Eis the familiar single-particle Hamiltonian\ncoupled to an external vector potential. According to\na Hamilton's equation, the velocity is conjugate to the\ncanonical momentum: vp=@H=@ pp. We note that ex-\nplicit dependence on the spin-texture dynamics dropped\nout because of the special property of the gauge \felds:\n_m\u0001@_ma\u001b=a\u001b. Furthermore, according to Eq. (19), we\nhavem\u0002\u000emE= (js\u0001r)m, so the LL Eq. (18) can be\nwritten in terms of the Hamiltonian (34) as11\n\u001as_m+m\u0002\u000emH= 0: (35)\nSo far, we have included in the spin-texture equa-\ntion only the piece coupled to the itinerant electron de-\ngrees of freedom. The purely magnetic part is tedious\nto derive directly and we will include it in the usual LL\nphenomenology.29To this end, we rede\fne\nF[m(r)]!F+F0; (36)\nby adding an additional magnetic free energy F0[m(r)],\nwhich accounts for magnetostatic interactions, crystalline\nanisotropies, coupling to external \felds, as well as energy\nassociated with localized dorforbitals.30Then the to-\ntal free energy (Hamiltonian) is H=E+F, and we in\ngeneral de\fne the e\u000bective magnetic \feld as the thermo-\ndynamic conjugate of m:h\u0011\u000emH. The LL equation\nthen becomes\n%s_m+m\u0002h= 0; (37)\nwhere%sis the total e\u000bective spin density. To enlarge\nthe scope of our phenomenology, we allow the possibility\nthat%s6=\u001as. For example, in the s\u0000dmodel, an extra\nspin density comes from the localized d-orbital electrons.\nMicroscopically, %s@tmterm in the equation of motion\nstems from the Wess-Zumino action generically associ-\nated with the total spin density.\nIn the following, it may sometimes be useful to separate\nout the current-dependent part of the e\u000bective \feld, and\nwrite the purely magnetic part as hm\u0011\u000emF, so that\nh=hm\u0000m\u0002(js\u0001r)m (38)\nand Eq. (37) becomes:\n%s_m+ (js\u0001r)m+m\u0002hm= 0: (39)6\nFor completeness, let is also write the equation of motion\nfor the spin- \u001bacceleration:\nme(@t+v\u001b\u0001r)v\u001bi=q\u001b[m\u0001(@tm\u0002@im)\n+v\u001bjm\u0001(@jm\u0002@im)]\u0000r\u0016\u001b;(40)\nretaining for the moment the advective correction to\nthe time derivative on the left-hand side and reinserting\nthe force due to the spin-dependent chemical potential,\n^\u0016=^K\u00001^\u001a. These equations constitute the coupled re-\nactive equations for our magneto-electric system. The\nHamiltonian (free energy) in terms of the collective vari-\nables is (including the elastic compression piece)\nH[\u001a\u001b;p\u001b;m] =X\n\u001bZ\nd3r\u001a\u001b(p\u001b\u0000a\u001b)2\n2me\n+1\n2Z\nd3r^\u001aT^K\u00001^\u001a+F[m]; (41)\nwhere p\u001b=mev\u001b+a\u001bis the spin-dependent momentum\nthat is locally averaged over individual particles.\nD. Conservation of energy\nSo far, our hydrodynamic equations are reactive, so\nthat the energy (41) must be conserved: P\u0011_H=_E+\n_F= 0. The time derivative of the electronic energy Eis\n_E=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001b_v\u001b+ _\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001bj_v\u001bj\u0000@j(\u001a\u001bv\u001bj)\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u001a\u001bv\u001bj[me(@t+v\u001b\u0001r)v\u001bj+@j\u0016\u001b]\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001(e+v\u001b\u0002b)\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001e=Z\nd3rjs\u0001e: (42)\nThe change in the spin-texture energy is given, according\nto Eq. (39), by\n_F=Z\nd3r_m\u0001\u000emF=Z\nd3r_m\u0001hm\n=Z\nd3r_m\u0001[%sm\u0002_m+m\u0002(js\u0001r)m)]\n=\u0000Z\nd3rjs\u0001e: (43)\nThe total energy is thus evidently conserved, P= 0.\nWhen we calculate dissipation in the rest of the paper,\nwe will omit these terms which cancel each other. The\ntotal energy \rux density is evidently given by\nQ=X\n\u001b\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\nv\u001b: (44)IV. DISSIPATION\nHaving derived from \frst principles the reactive cou-\nplings in our magneto-electric system, summed up in\nEqs. (39)-(41), we will proceed to include the dissipa-\ntive e\u000bects phenomenologically. Let us focus on the lin-\nearized limit of small deviations from equilibrium (which\nmay be spin textured), so that the advective correction\nto the time derivative in the Euler Eq. (40), which is\nquadratic in the velocity \feld, can be omitted. To elimi-\nnate the quasiparticle spin degree of freedom, let us, fur-\nthermore, treat halfmetallic ferromagnets, so that \u001a=\u001a+\nand\u001as=q\u001a, whereq=~=2 is the electron's spin.31From\nEq. (40), the equation of motion for the local (averaged)\ncanonical momentum is:32\n_p=q\n\u001aj\u0002b\u0000r\u0016; (45)\nin a gauge where a\u001b= 0, so that _p=me_v\u0000qe.33\n\u0016=\u001a=K. The Lorentz force due to the applied (real)\nelectromagnetic \felds can be added in the obvious way\nto the right-hand side of Eq. (45). Note that since we\nare now interested in linearized equations close to equi-\nlibrium,\u001ain Eq. (45) can be approximated by its (ho-\nmogeneous) equilibrium value.\nIntroducing relaxation through a phenomenological\ndamping constant (Drude resistivity)\n\r=me\n\u001a\u001c; (46)\nwhere\u001cis the collision time, expressing the \fctitious\nmagnetic \feld in terms of the spin texture, Eq. (45) be-\ncomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji: (47)\nAdding the phenomenological Gilbert damping34to\nthe magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert\nequation:\n%s(_m+\u000bm\u0002_m) =h\u0002m; (48)\nwhere\u000bis the damping constant. Eqs. (47) and\n(48), along with the continuity equation, _ \u001a=\u0000r\u0001j,\nare the near-equilibrium thermodynamic equations for\n(\u001a;p;m) and their respective thermodynamic conjugates\n(\u0016;j;h) = (\u000e\u001aH;\u000epH;\u000emH). This system of equations of\nmotion may be written formally as\n@t0\n@\u001a\np\nm1\nA=b\u0000[m(r)]0\n@\u0016\nj\nh1\nA: (49)\nThe matrix ^\u0000 depends on the equilibrium spin texture\nm(r). By the Onsager reciprocity principle, \u0000 ij[m] =\nsisj\u0000ji[\u0000m], wheresi=\u0006if theith variable is even\n(odd) under time reversal.7\nIn the quasistationary description of a nonequilibrium\nthermodynamic system, the entropy S[\u001a;p;m] is for-\nmally regarded as a functional of the instantaneous ther-\nmodynamic variables, and the probability of a given con-\n\fguration is proportional to eS=kB. If the heat conduc-\ntance is high and the temperature Tis uniform and con-\nstant, the instantaneous rate of dissipation P=T_Sis\ngiven by the rate of change in the free energy, P=_H=R\nd3rP:\nP=\u0000\u0016_\u001a\u0000h\u0001_m\u0000j\u0001_p=\u000b%s_m2+\rj2; (50)\nwhere we used Eq. (47) and expressed the e\u000bective \feld\nhas a function of _mby taking m\u0002of Eq. (48):\nh=%sm\u0002_m\u0000\u000b%s_m: (51)\nNotice that the \fctitious magnetic \feld bdoes not con-\ntribute to dissipation because it does not do work.\nSo far, there is no dissipative coupling between the\ncurrent and the spin-texture dynamics, and the macro-\nscopic equations obey the global time-reversal symme-\ntry. However, we know that dissipative couplings ex-\nists due to the misalignment of the electron's spin with\nthe collective spin texture and spin-texture resistivity.3,22\nFollowing Ref. 11, we add these well-known e\u000bects phe-\nnomenologically by making an expansion in the equations\nof motion to linear order in the nonequilibrium quanti-\nties _mandj. To limit the number of terms one can write\ndown, we will only add terms that are spin-rotationally\ninvariant and isotropic in real space (which disregards,\nin particular, such e\u000bects as the angular magnetoresis-\ntance and the anomalous Hall e\u000bect). To second order in\nthe spatial gradients of m, there are only three dissipa-\ntive phenomenological terms with couplings \u0011,\u00110, and\f\nconsistent with the above requirements, which could be\nadded to the right-hand side of Eq. (47).35The momen-\ntum equation becomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji\n\u0000\u0011(@km)2ji\u0000\u00110@im\u0001(j\u0001r)m\u0000q\f_m\u0001@im:(52)\nIt is known that the \\ \fterm\" comes from a misalignment\nof the electron spin with the collective spin texture, and\nthe associated dephasing. It is natural to expect thus\nthat the dimensionless parameter \f\u0018~=\u001cs\u0001, where\u001cs\nis a characteristic spin-dephasing time.3The \\\u0011terms\"\nevidently describe texture-dependent resistivity, which\nis anisotropic with respect to the gradients in the spin\ntexture along the local current density. Such term are\nalso naturally expected, in view of the well-known giant-\nmagnetoresistance e\u000bect,36in which noncollinear magne-\ntization results in electrical resistance. The microscopic\norigin of this term is due to spin-texture misalignment,\nwhich modi\fes electron scattering.\nThe total spin-texture-dependent resistivity can be putinto a tensor form:\n\rij[m] =\u000eij\u0002\n\r+\u0011(@km)2\u0003\n+\u00110@im\u0001@jm\n+q\n\u001am\u0001(@im\u0002@jm): (53)\nThe last term due to \fctitious magnetic \feld gives a Hall\nresistivity. Note that ^ \r[m] = ^\rT[\u0000m], consistent with\nthe Onsager theorem. We can \fnally write Eq. (47) as:\n_pi=\u0000\rij[m]jj\u0000@i\u0016\u0000q\f_m\u0001@im: (54)\nAs was shown in Ref. 11, since the Onsager relations\nrequire thatb\u0000[m] =b\u0000[\u0000m]Twithin the current/spin-\ntexture \felds sector, there must be a counterpart to the\n\fterm above in the magnetic equation, which is the well-\nknown dissipative \\ \fspin torque:\"\n%s(_m+\u000bm\u0002_m) =h\u0002m\u0000q\fm\u0002(j\u0001r)m:(55)\nThe total dissipation Pis now given by\nP=\u000b%s_m2+ 2q\f_m\u0001(j\u0001r)m+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u00110[(j\u0001r)m]2\n=\u000b%s\u0014\n_m+q\f\n\u000b%s(j\u0001r)m\u00152\n+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u0012\n\u00110\u0000(q\f)2\n\u000b%s\u0013\n[(j\u0001r)m]2: (56)\nThe second law of thermodynamics requires the total dis-\nsipation to be positive, which puts some constraints on\nthe allowed values of the phenomenological parameters.\nWe can easily notice, however, that the dissipation (56)\nis guaranteed to be positive-de\fnite if\n\u0011+\u00110\u0015(q\f)2\n\u000b%s; (57)\nwhich may serve as an estimate for the spin-texture re-\nsistivity due to spin dephasing. This is consistent with\nthe microscopic \fndings of Ref. 23.\nV. THERMAL NOISE\nAt \fnite temperature, thermal agitation causes \ruc-\ntuations of the current and spin texture, which are cor-\nrelated due to their coupling. A complete description\nrequires that we supplement the stochastic equations of\nmotion with the correlators for these \ructuations. It\nis convenient to regard these \ructuations as being due\nto the stochastic Langevin \\forces\" ( \u000e\u0016;\u000ej;\u000eh) on the\nright-hand side of Eq. (49). The complete set of \fnite-\ntemperature hydrodynamic equations thus becomes:\n_\u001a=\u0000r\u0001~j;\n_p+q\f_mirmi=\u0000^\r[m]~j\u0000r~\u0016;\n%s(1 +\u000bm\u0002)_m=~h\u0002m\u0000q\fm\u0002(~j\u0001r)m:(58)8\nwhere (~\u0016;~j;~h) = (\u0016+\u000e\u0016;j+\u000ej;h+\u000eh). The simplest\n(while possibly not most realistic) case corresponds to\na highly compressible \ruid, such that K!1 . In this\nlimit,\u0016=\u001a=K!0 and the last two equations com-\npletely decouple from the \frst, continuity equation. In\nthe remainder of this section, we will focus on this special\ncase. The correlations of the stochastic \felds are given\nby the symmetric part of the inverse matrix b\u0007 =\u0000b\u0000\u00001,37\nwhich is found by inverting Eq. (58) (reduced now to a\nsystem of two equations):\n~j=\u0000^\r\u00001(_p+q\f_mirmi);\n~h=%sm\u0002_m\u0000\u000b%s_m\u0000q\f(~j\u0001r)m: (59)\nWriting formally these equations as (after substituting ~j\nfrom the \frst into the second equation)\n\u0012~j\n~h\u0013\n=\u0000b\u0007[m(r)]\u0012\n_p\n_m\u0013\n; (60)\nwe immediately read out for the matrix elements\nb\u0007(r;r0) =b\u0007(r)\u000e(r\u0000r0):\n\u0007ji;ji0(r) =(^\r\u00001)ii0;\n\u0007ji;hi0(r) =q\f(^\r\u00001)ik@kmi0;\n\u0007hi0;ji(r) =\u0000q\f(^\r\u00001)ki@kmi0;\n\u0007hi;hi0(r) =\u000b%s\u000eii0+%s\u000fii0kmk\n\u0000(q\f)2(@kmi)(^\r\u00001)kk0(@k0mi0):(61)\nAccording to the \ructuation-dissipation theorem, we\nsymmetrize b\u0007 to obtain the classical Langevin\ncorrelators:37\nh\u000eji(r;t)\u000eji0(r0;t0)i=T=gii0;\nh\u000eji(r;t)\u000ehi0(r0;t0)i=T=q\fg0\nik@kmi0;\nh\u000ehi(r;t)\u000ehi0(r0;t0)i=T=\u000b%s\u000eii0\n\u0000(q\f)2gkk0(@kmi)(@k0mi0); (62)\nwhereT= 2kBT\u000e(r\u0000r0)\u000e(t\u0000t0) and\n^g= [^\r\u00001+ (^\r\u00001)T]=2;^g0= [^\r\u00001\u0000(^\r\u00001)T]=2 (63)\nare, respectively, the symmetric and antisymmetric parts\nof the conductivity matrix ^ \r\u00001. The short-ranged, \u000e-\nfunction character of the noise correlations in space stems\nfrom the assumption of high electronic compressibility.\nContrast this to the results of Ref. 11 for incompressible\nhydrodynamics. A presence of long-ranged Coulombic\ninteractions and plasma modes would also give rise to\nnonlocal correlations. These are absent in our treatment,\nwhich disregards ordinary electromagnetic phenomena.\nFocusing on the microwave frequencies !characteris-\ntic of ferromagnetic dynamics, it is most interesting to\nconsider the regime where !\u001c\u001c\u00001. This means that\nwe can employ the drift approximation for the \frst of\nEqs. (59):\n_pi=me_vi\u0000qei\u0019\u0000qei=q_m\u0001(m\u0002@im): (64)Substituting this _pin Eq. (59), we can easily \fnd a closed\nstochastic equation for the spin-texture \feld:\n%s(1 +\u000bm\u0002)_m+m\u0002\u001c$_m= (hm+\u000eh)\u0002m;(65)\nwhere we have de\fned the \\spin-torque tensor\"\n\u001c$=q2(^\r\u00001)kk0(m\u0002@km\u0000\f@km)\n\n(m\u0002@k0m+\f@k0m): (66)\nThe antisymmetric piece of this tensor modi\fes the e\u000bec-\ntive gyromagnetic ratio, while the more interesting sym-\nmetric piece determines the additional nonlocal Gilbert\ndamping:\n\u000b$=\u001c$+\u001c$T\n2%s=q2\n%sG$; (67)\nwhere\nG$=gkk0\u0002\n(m\u0002@km)\n(m\u0002@k0m)\u0000\f2@km\n@k0m\u0003\n+\fg0\nkk0[(m\u0002@km)\n@k0m\u0000@km\n(m\u0002@k0m)]:\n(68)\nIn obtaining Eq. (65) from Eqs. (59), we have separated\nthe reactive spin torque out of the e\u000bective \feld: h=\nhm\u0000qm\u0002(j\u0001r)m. (The remaining piece hmthus re\rects\nthe purely magnetic contribution to the e\u000bective \feld.)\nThe total stochastic magnetic \feld entering Eq. (65),\n\u000eh=\u000eh+qm\u0002(\u000ej\u0001r)m; (69)\ncaptures both the usual magnetic Brown noise38\u000eh\nand the Johnson noise spin-torque contribution39\u000ehJ=\nqm\u0002(\u000ej\u0001r)mthat arises due to the substitution j=~j\u0000\u000ej\nin the reactive spin torque q(j\u0001r)m. Using correla-\ntors (62), it is easy to show that the total e\u000bective \feld\n\ructuations \u000ehare consistent with the nonlocal e\u000bec-\ntive Gilbert damping tensor (68), in accordance with the\n\ructuation-dissipation theorem applied directly to the\npurely magnetic Eq. (65).\nTo the leading, quadratic order in spin texture, we can\nreplacegkk0!\u000ekk0=\randg0\nkk0!0 in Eq. (68). This ad-\nditional texture-dependent nonlocal damping (along with\nthe associated magnetic noise) is a second-order e\u000bect,\nphysically corresponding to the backaction of the magne-\ntization dynamics-driven current on the spin texture.11\nIt should be noted that in writing the modi\fed LLG\nequation (55), we did not systematically expand it to\ninclude the most general phenomenological terms up to\nthe second order in spin texture. We have only included\nextra spin-torque terms, which are required by the On-\nsager symmetry with Eq. (52). The second-order Gilbert\ndamping (68) was then obtained by solving Eqs. (52) and\n(55) simultaneously. (Cf. Refs. 11,40.) This means in\nparticular, that this procedure does not capture second-\norder Gilbert damping e\u000bects whose physical origin is\nunrelated to the longitudinal spin-transfer torque physics\nstudied here. One example of that is the transverse spin-\npumping induced damping discussed in Refs. 41.9\nVI. EXAMPLES\nA. Rigidly spinning texture\nTo illustrate the \u0011resistivity terms in the electron's\nequation of motion (52), we \frst consider 1D textures.\nTake, for example, the case of a 1D spin helix m(z)\nalong thezaxis, whose spatial gradient pro\fle is given by\n@zm=\u0014^ z\u0002m, where\u0014is the wave vector of the spatial\nrotation and m?^ z. See Fig. 1. It gives anisotropic re-\nsistivity in the xyplane,r(\u0011)\n?, and along the zdirection,\nr(\u0011)\nk:\nr(\u0011)\n?=\u0011(@zm)2=\u0011\u00142; r(\u0011)\nk= (\u0011+\u00110)\u00142: (70)\nFIG. 1: (Color online) The transverse magnetic helix, @zm=\n\u0014^ z\u0002m, with texture-dependent anisotropic resistivity (70).\nWe assume here translational invariance in the transverse ( xy)\ndirections. Spinning this helix about the vertical zaxis gen-\nerates the dissipative electromotive forces f(\f)\nz, which is spa-\ntially uniform and points everywhere along the zaxis. A\nmagnetic spiral, @zm=\u0014^'\u0002m=\u0014^\u0012, spinning around the z\naxis, on the other hand, produces a purely reactive electromo-\ntive forceez, as discussed in the text, which is oscillatatory\nin space along the zaxis.\nThe \fctitious electric \feld and dissipative \fforce re-\nquire magnetic dynamics. A general texture globally ro-\ntating clockwise in spin space in the xyplane according\nto_m=\u0000!^ z\u0002m(which may be induced by applying a\nmagnetic \feld along the zdirection) generates an electric\n\feld\nei= (m\u0002_m)\u0001@im=\u0000!(m\u0002^ z\u0002m)\u0001@im\n=\u0000!@imz=\u0000!@icos\u0012 (71)and a\fforce\nf(\f)\ni=\u0000\f_m\u0001@im=\f!^ z\u0001(m\u0002@im)\n=\f!sin2\u0012@i'; (72)\nwhere (\u0012;') denote the position-dependent spherical an-\ngles parametrizing the spin texture. The reactive force\n(71) has a simple interpretation of the gradient of the\nBerry-phase15accumulation rate [which is locally deter-\nmined by the solid angle subtended by m(t)]. In the\ncase of the transverse helix discussed above, \u0012=\u0019=2,\n'=\u0014z\u0000!t, so thatez= 0 whilef(\f)\nz=\u0000\f!\u0014 is \fnite.\nAs an example of a dynamical texture that does not\ngenerate f(\f)while producing a \fnite e, consider a spin\nspiral along the zaxis, described by @zm=\u0014^'\u0002m=\u0014^\u0012,\nand rotating in time in the manner described above. It is\nclear geometrically that the change in the spin texture in\ntime is in a direction orthogonal to its gradients in space.\nSpeci\fcally, \u0012=\u0014z,'=\u0000!t, so thatf(\f)\nz= 0 while the\nelectric \feld is oscillatory, ez=!\u0014sin\u0012.\nB. Rotating spin textures\nWe show here that a vortex rotating about its core in\norbital space generates a current circulating around its\ncore, as well as a current going radially with respect to\nthe core. Consider a spin texture with a time depen-\ndence corresponding to the real-space rotation clockwise\nin thexyplane around the origin, such that m(r;t) =\nm(r(t);0) with _r=!^ z\u0002r=!r^\u001e, where we use polar co-\nordinates (r;\u001e) on the plane normal to the zaxis in real\nspace [to be distinguished from the spherical coordinates\n(\u0012;') that parametrize min spin space], we have\n_m= (_r\u0001r)m=!@\u001em: (73)\nForm(r;\u001e) in polar coordinates, the components of the\nelectric \feld are,\ner=!m\u0001(@\u001em\u0002@rm); e\u001e= 0; (74)\nwhile the components of the \fforce are\nf(\f)\nr=\u0000\f!(@rm)\u0001(@\u001em); f(\f)\n\u001e=\u0000\f!(@\u001em)2\nr:(75)\nIn order to \fnd the \fctitious electromagnetic \felds, we\nneed to calculate the following tensors (which depend on\nthe instantaneous spin texture):\nbij\u0011m\u0001(@im\u0002@jm) = sin\u0012(@i\u0012@j'\u0000@j\u0012@i');\ndij\u0011@im\u0001@jm=@i\u0012@j\u0012+ sin2\u0012@i'@j': (76)\nAs an example, consider a vortex centered at the ori-\ngin in thexyplane with winding number 1 and positive\npolarity, as shown in Fig. 2. Its angular coordinates are\ngiven by\n'= (\u001e+!t) +\u0019\n2; \u0012=\u0012(r); (77)10\nwhere\u001e= arg( r) and\u0012is a rotationally invariant func-\ntion such that \u0012!0 asr!0 and\u0012!\u0019=2 asr!1 .\nEvaluating the tensors in equation (76) for this vortex in\npolar coordinates gives drr= (@r\u0012)2,d\u001e\u001e= (sin\u0012=r)2,\ndr\u001e= 0, andbr\u001e=\u0000(@rcos\u0012)=r. The radial electric\n\feld is then given by\ner=\u0000!rbr\u001e=!@rcos\u0012: (78)\nThe\fforce is in the azimuthal direction:\nf(\f)\nr= 0; f(\f)\n\u001e=\u0000\f!rd\u001e\u001e=\u0000\f!sin2\u0012\nr: (79)\nWe can interpret this force as the spin texture \\dragging\"\nthe current along its direction of motion. Notice that the\nforces in Eqs. (78) and (79) are the negative of those in\nEqs. (71) and (72), as they should be for the present case,\nsince the combination of orbital and spin rotations of our\nvortex around its core leaves it invariant, producing no\nforces.\nFIG. 2: Positive-polarity magnetic vortex con\fguration pro-\njected on the xyplane. mhas a positive (out-of-plane) z\ncomponent near the vortex core. Rotating this vortex about\nthe origin in real space generates the current in the xyplane\nshown in Fig. 3.\nThe total resistivity tensor (53) is (in the cylindrical\ncoordinates)\n^\r=\r+\u0011(drr+d\u001e\u001e) +\u00110^d+q\n\u001a^b=\u0012\n\rr\r?\n\u0000\r?\r\u001e\u0013\n;(80)\nwhere\n\rr=\r+ (\u0011+\u00110)(@r\u0012)2+\u0011\u0012sin\u0012\nr\u00132\n;\n\r\u001e=\r+\u0011(@r\u0012)2+ (\u0011+\u00110)\u0012sin\u0012\nr\u00132\n;\n\r?=\u0000q\n\u001a@rcos\u0012\nr: (81)Here, the two diagonal components, \rrand\r\u001e, describe\nthe (dissipative) anisotropic resistivity, while the o\u000b-\ndiagonal component, \r?, captures what is called the\ntopological Hall e\u000bect.19\nIn the drift approximation, Eq. (64), the current-\ndensity \feld j=jr^ r+j\u0012^\u0012is given by\nj= ^\r\u00001q(e+f(\f));\u0012\njr\nj\u001e\u0013\n=q!^\r\u00001\u0012@rcos\u0012\n\u0000\fsin2\u0012=r\u0013\n=\u0000q!sin\u0012\n\rr\r\u001e+\r2\n?\u0012\n\r\u001e\u0000\r?\n\r?\rr\u0013\u0012\n@r\u0012\n\fsin\u0012=r\u0013\n:(82)\nMore explicitly, we may consider a pro\fle \u0012=\u0019(1\u0000\ne\u0000r=a)=2, whereais the radius of the vortex core. The\ncorresponding current (82) is sketched in Fig. 3.\nFIG. 3: We plot here the current in Eq. (82) (all parameters\nset to 1). Near the core, the current spirals inward and charges\nbuild up at the center (which is allowed for our compressible\n\ruid).\nWe note that the \fctitious magnetic \feld \u000fijkbjk=2\npoints everywhere in the zdirection, its total \rux\nthrough the xyplane being given by\nF=Z\nd\u001edr (rbr\u001e) =\u0000Z\nd\u001edr (@\u001e'@rcos\u0012) = 2\u0019:\n(83)\nNote that the integrand is just the Jacobian of the map\nfrom the plane to the sphere de\fned by the spin-texture\n\feld:\n(\u0012(r);'(r)) :R2!S2: (84)\nThis re\rects the fact that the \fctitious magnetic \rux is\ngenerally a topological invariant, corresponding to the \u00192\nhomotopy group of the mapping (84).6,42\nC. Anisotropic resistivity of a 3D spiral\nConsider the texture described by @im=\u0014i^ z\u0002m,\nwhere the spatial rotation stays in the xyplane, but the11\nwave vector\u0014can be in any direction. The spin texture\nforms a transverse helix in the zdirection and a planar\nspiral in the xandydirections. Fig. 4 shows such a\ncon\fguration for \u0014pointing along ( x+y+z)=p\n3. The\n\fctitious magnetic \feld bvanishes, but the anisotropic\nresistivity still depends nontrivially on the spin texture:\n\rij=\u0002\n\r+\u0011(@km)2\u0003\n\u000eij+\u00110@im\u0001@jm\n= (\r+\u0011\u00142)\u000eij+\u00110\u0014i\u0014j; (85)\nwhich, according to j= ^\r\u00001E, would give a transverse\ncurrent signal for an electric \feld applied along the Carte-\nsian axesx,y, orz.\nFIG. 4: (Color online) A set of spin spirals which is topo-\nlogically trivial because r\u0012= 0 (and equivalent to the spin\nhelix, Fig. 1, up to a global real-space rotation), hence the\n\fctitious magnetic \feld b, Eq. (76), is zero. There is, how-\never, an anisotropic texture-dependent resistivity with \fnite\no\u000b-diagonal components, Eq. (85).\nVII. SUMMARY\nWe have developed semi-phenomenologically the hy-\ndrodynamics of spin and charge currents interacting with\ncollective magnetization in metallic ferromagnets, gener-\nalizing the results of Ref. 11 to three dimensions and\ncompressible \rows. Our theory reproduces known re-\nsults such as the spin-motive force generated by mag-\nnetization dynamics and the dissipative spin torque, al-\nbeit from a di\u000berent viewpoint than previous microscopic\napproaches. Among the several new e\u000bects predicted,\nwe \fnd both an isotropic and an anisotropic texture-\ndependent resistivity, Eq. (53), whose contribution to theclassical (topological) Hall e\u000bect should be described on\npar with that of the \fctitious magnetic \feld. By calculat-\ning the dissipation power, we give a lower bound on the\nspin-texture resistivity as required by the second law of\nthermodynamics. We \fnd a more general form, includ-\ning a term of order \f, of the texture-dependent correction\nto nonlocal Gilbert damping, predicted in Ref. 11. See\nEq. (68).\nOur general theory is contained in the stochastic hy-\ndrodynamic equations, Eqs. (58), which we treated in\nthe highly compressible limit. The most general situ-\nation is no doubt at least as rich and complicated as\nthe classical magnetohydrodynamics. A natural exten-\nsion of this work is the inclusion of heat \rows and re-\nlated thermoelectric e\u000bects, which we plan to investigate\nin a future work. Although we mainly focused on the\nhalfmetallic limit in this paper, our theory is in principle\na two-component \ruid model and allows for the inclu-\nsion of a fully dynamical treatment of spin densities and\nassociated \rows.31Finally, our hydrodynamic equations\nbecome amenable to analytic treatments when applied to\nthe important problem of spin-current driven dynamics\nof magnetic solitons, topologically stable objects that can\nbe described by a small number of collective coordinates,\nwhich we will also investigate in future work.\nAcknowledgments\nWe are grateful to Gerrit E. W. Bauer, Arne Brataas,\nAlexey A. Kovalev, and Mathieu Taillefumier for stimu-\nlating discussions. This work was supported in part by\nthe Alfred P. Sloan Foundation and the NSF under Grant\nNo. DMR-0840965.\nAPPENDIX A: MANY-BODY ACTION\nWe can formally start with a many-body action, with\nStoner instability built in due to short-range repulsion\nbetween electrons:25\nS[\u0016 \u001b(r;t); \u001b(r;t)] =Z\nCdtZ\nd3r\n\u0014\n^ +\u0012\ni~@t+~2\n2mer2\u0013\n^ \u0000U\u0016 \"\u0016 # # \"\u0015\n;(A1)\nwhere time truns along the Keldysh contour from \u00001\nto1and back. \u0016 \u001band \u001bare mutually independent\nGrassmann variables parametrizing fermionic coherent\nstates and ^ += (\u0016 \";\u0016 #) and ^ = ( \"; #)T. The four-\nfermion interaction contribution to the action can be de-\ncoupled via Hubbard-Stratonovich transformation, after12\nintroducing auxiliary bosonic \felds \u001eand\u0001:\neiSU=~= exp\u0012\n\u0000i\n~Z\nCdtZ\nd3rU\u0016 \"\u0016 # # \"\u0013\n=Z\nD[\u001e(r;t);\u0001(r;t)] exp\u0012i\n~Z\nCdtZ\nd3r\n\u0014\u001e2\n4U\u0000\u00012\n4U\u0000\u001e\n2^ +^ +\u0001\n2^ +^\u001b^ \u0015\u0013\n:(A2)\nIn obtaining this result, we decomposed the interaction\ninto charge- and spin-density pieces:\n\u0016 \"\u0016 # # \"=1\n4(^ +^ )2\u00001\n4(^ +m\u0001^\u001b^ )2; (A3)\nwhere mis an arbitrary unit vector. It is easy to\nshow thath\u001e(r;t)i=Uh^ +(r;t)^ (r;t)iandh\u0001(r;t)i=\nUh^ +(r;t)^\u001b^ (r;t)i, when properly averaging over the\ncoupled quasiparticle and bosonic \felds.\nThe next step in developing mean-\feld theory is to\ntreat the Hartree potential \u001e(r;t) and Stoner exchange\n\u0001(r;t)\u0011\u0001(r;t)m(r;t) \felds in the saddle-point approx-\nimation. Namely, the e\u000bective bosonic action\nSe\u000b[\u001e(r;t);\u0001(r;t)] =\u0000i~lnZ\nD[^ +;^ ]ei\n~S(^ +;^ ;\u001e;\u0001)\n(A4)\nis minimized, \u000eSe\u000b= 0, in order to \fnd the equations\nof motion for the \felds \u001eand\u0001. In the limit of suf-\n\fciently low electron compressibility and spin suscepti-\nbility, the charge- and spin-density \ructuations are sup-\npressed, de\fning mean-\feld parameters \u0016\u001eand \u0016\u0001. Since\na constant \u0016\u001eonly shifts the overall electrochemical po-\ntential, it is physically inconsequential. Our theory is de-\nsigned to focus on the remaining soft (Goldstone) modes\nassociated with the spin-density director m(r;t), while\n\u001e(r;t) and \u0001( r;t) are in general allowed to \ructuate\nclose to their mean-\feld values \u0016\u001eand \u0016\u0001, respectively.\nThe saddle-point equation of motion for the collective\nspin direction m(r;t) follows from \u000emSe\u000b[m] = 0, after\nintegrating out electronic degrees of freedom. Because\nof the noncommutative matrix structure of the action\n(A2), it is still a nontrivial problem. The problem sim-\npli\fes considerably in the limit of large exchange split-\nting \u0001, where we can project spins on the local magnetic\ndirection m. This lays the ground to the formulation dis-\ncussed in Sec. II, where the collective spin-density \feld\nparametrized by the director m(r;t) interacts with the\nspin-up/down free-electron \feld. The resulting equations\nof motion constitute the self-consistent dynamic Stoner\ntheory of itinerant ferromagnetism.\nIn the remainder of this appendix, we explicitly show\nthat the semiclassical formalism developed in Secs. II-\nIII B is equivalent to a proper \feld-theoretical treatment.\nThe equation of motion for the spin texture follows from\nextremizing the e\u000bective action with respect to variations\ninm. Because of the constraint on the magnitude of m,\nits variation can be expressed as \u000em=\u000e\u0012\u0002m, with\u000e\u0012being an arbitrary in\fnitesimal vector, so that the\nequation of motion is given by m\u0002\u000emSe\u000b= 0:\n0 =m\u0002\u000emSe\u000b\n=1\nZZ\nD[^ +;^ ] (m\u0002\u000emS)ei\n~S[^ +;^ ;\u001e;\u0001]\n=X\n\u001b\u0016(m\u0002\u000ema\u001b\u0016)\u001c@S\n@a\u001b\u0016\u001d\n\u0000m\u0002\u000emF; (A5)\nwhereZ=R\nD[^ +;^ ]ei\n~S[^ +;^ ;\u001e;\u0001]and we have used\nthe path-integral representation of the vacuum expecta-\ntion value. a\u001b\u0016are the spin-dependent gauge potentials\n(4) andFthe spin exchange energy, appearing after we\nproject spin dynamics on the collective \feld \u0001. Equa-\ntion (A5) may be expressed in terms of the hydrody-\nnamic variables of the electrons. De\fning spin-dependent\ncharge and current densities, j\u0016\n\u001b= (\u001a\u001b;j\u001b), by\n\u001a\u001b=\u001c@S\n@a\u001b\u001d\n=h\u0016 \u001b \u001bi;\nj\u001b=\u001c@S\n@a\u001b\u001d\n=1\nmeRe\n\u0016 \u001b(\u0000i~r\u0000a\u001b) \u001b\u000b\n=\u001a\u001bv\u001b;\n(A6)\nEq. (A5) reduces to the Landau-Lifshitz Eq. (18). Min-\nimizing action (A4) with respect to the \u001eand \u0001 \felds\ngives the anticipated self-consistency relations:\n\u001e(r;t) =Uh^ +(r;t)^ (r;t)i=U(\u001a++\u001a\u0000);\n\u0001(r;t) =Uh^ +(r;t)^\u001bz^ (r;t)i=U(\u001a+\u0000\u001a\u0000):(A7)\nAPPENDIX B: THE MONOPOLE GAUGE FIELD\nLet (\u0012;') be the spherical angles of m, the direction\nof the local spin density, and ^ \u001f\u001bbe the spin up/down\n(\u001b=\u0006) spinors given by, up to a phase,\n^\u001f+(\u0012;') =\u0012\ncos\u0012\n2\nei'sin\u0012\n2\u0013\n;\n^\u001f\u0000(\u0012;') = ^\u001f+(\u0019\u0000\u0012;'+\u0019) =\u0012\nsin\u0012\n2\n\u0000ei'cos\u0012\n2\u0013\n:(B1)\nThe spinors are related to the spin-rotation matrix ^U(m)\nby ^\u001f\u001b=^Uj\u001bi. The gauge \feld in mspace, which enters\nEq. (5), is thus given by\namon\n\u001b(\u0012;') =\u0000i~^\u001fy\n\u001b@m^\u001f\u001b=~\n2\u00121\u0000\u001bcos\u0012\nsin\u0012\u0013\n^';(B2)\nwhere we used the gradient on a unit sphere: @m=\n^\u0012@\u0012+^'@'=sin\u0012. The magnetic \feld corresponding to\nthis vector potential [extended to three dimensions by\na(m)!a(\u0012;')=m] is given on the unit sphere by\n@m\u0002amon\n\u001b=@m\u0002(a'^') =m\nsin\u0012@\u0012(sin\u0012a') =\u001b~\n2m:\n(B3)13\nIt follows from Eqs. (5) and (B2) that the spin-dependent\nreal-space gauge \felds are given by\na\u001b\u0016=\u0000~\n2@\u0016'(1\u0000\u001bcos\u0012): (B4)\nNotice that the \u001b=\u0006monopole \feld (B2), as well as the\nabove gauge \felds, are singular on the south/north pole(corresponding to the Dirac string). This is what allows a\nmagnetic \feld with \fnite divergence. 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Polyakov, JETP Lett. 22, 503\n(1975)."    },    {        "title": "0905.4650v2.Hamilton_cycles_in_random_geometric_graphs.pdf",        "content": "arXiv:0905.4650v2  [math.PR]  8 Nov 2012The Annals of Applied Probability\n2011, Vol. 21, No. 3, 1053–1072\nDOI:10.1214/10-AAP718\nc/circlecopyrtInstitute of Mathematical Statistics , 2011\nHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS\nBy J´ozsef Balogh1, B´ela Bollob ´as2, Michael Krivelevich3,\nTobias M ¨uller4and Mark Walters\nUniversity of California and University of Illinois,\nUniversity of Cambridge and University of Memphis, Tel Aviv U niversity,\nCentrum voor Wiskunde en Informatica, and Queen Mary,\nUniversity of London\nWe prove that, in the Gilbert model for a random geometric\ngraph, almost every graph becomes Hamiltonian exactly when it first\nbecomes 2-connected. This answers a question of Penrose.\nWe also show that in the k-nearest neighbor model, there is a\nconstant κsuch that almost every κ-connected graph has a Hamilton\ncycle.\n1. Introduction. In this paper we mainly consider one of the frequently\nstudied models for random geometric graphs, namely the Gilb ert model.\nSuppose that Snis a√n×√nbox and that Pis a Poisson process in it\nwith density 1. The points of the process form the vertex set o f our graph.\nThere is a parameter rgoverning the edges: two points are joined if their\n(Euclidean) distance is at most r.\nHaving formed this graph we can ask whether it has any of the st andard\ngraph properties, such as connectedness. As usual, we shall only consider\nthese for large values of n. More formally, we say that G=Gn,rhas a prop-\nReceived April 2009; revised January 2010.\n1This material is based upon work supported by NSF CAREER Gran t DMS-07-45185\nand DMS-06-00303, UIUC Campus Research Board Grants 09072 a nd 08086 and OTKA\nGrant K76099.\n2Supported in part by NSF Grants DMS-05-05550, CNS-0721983 a nd CCF-0728928\nand ARO Grant W911NF-06-1-0076.\n3Supported in part by USA-Israel BSF Grant 2006322, by Grant 1 063/08 from the\nIsrael Science Foundation and by a Pazy memorial award.\n4Supported in part by a VENI grant from Netherlands Organisat ion for Research\n(NWO). The results in this paper are based on work done while a t Tel Aviv University,\npartially supported through an ERC advanced grant.\nAMS 2000 subject classifications. 05C80, 60D05, 05C45.\nKey words and phrases. Hamilton cycles, random geometric graphs.\nThis is an electronic reprint of the original article published by the\nInstitute of Mathematical Statistics inThe Annals of Applied Probability ,\n2011, Vol. 21, No. 3, 1053–1072 . This reprint differs from the original in\npagination and typographic detail.\n12 J. BALOGH ET AL.\nertywith high probability (abbreviated to whp) if the probability that Ghas\nthis property tends to one as ntends to infinity.\nPenrose [ 10] proved that the threshold for connectivity is πr2=logn. In\nfact he proved the following very sharp result: suppose πr2=logn+αfor\nsomeconstant α.Thentheprobabilitythat Gn,risconnectedtendsto e−e−α.\nHe also generalized this result to find the threshold for κ-connectivity\nforκ≥2: namely πr2=logn+(2κ−3)loglog n. [Since the reader may be\nsurprised that this formula does not work for κ=1 we remark that this is\ndue to boundary effects: the threshold for κ-connectivity is the maximum of\ntwo quantities: log n+(κ−1)loglog ntoκ-connect the central points and\nlogn+(2κ−3)loglog ntoκ-connect the points near the boundary. If one\nworked on thetorus instead of thesquare, then these boundar yeffects would\ndisappear.]\nMoreover, he found the “obstruction” to κ-connectivity. Suppose we fix\nthe vertex set (i.e., the point set in Sn) and “grow” r. This gradually adds\nedges to the graph. For a monotone graph property PletH(P) denote the\nsmallest rfor which the graph on this point set has the property P. Penrose\nshowed that\nH(δ(G)≥κ)=H(connectivity( G)≥κ)\nwhp: that is, as soon as the graph has minimum degree κit isκ-connected\nwhp.\nHealsoconsideredthethresholdfor GtohaveaHamiltoncycle.Obviously\nanecessary condition isthat thegraphis2-connected. Inth enormal(Erd˝ os–\nR´ enyi) randomgraphthisisalsoasufficientcondition inthe following strong\nsense. If we add edges to the graph one at a time, then the graph becomes\nHamiltonian exactly when it becomes 2-connected (see [ 5,8,9] and [14]).\nPenrose asked whether the same is true for a random geometric graph. In\nthis paper we prove the following theorem answering this que stion.\nTheorem 1. Suppose that G=Gn,ris the two-dimensional Gilbert model.\nThen\nH(Gis 2-connected )=H(Ghas a Hamilton cycle )\nwhp.\nCombining this with Penrose’s results mentioned above we se e that, if\nπr2=logn+loglog n+α, then the probability that Ghas a Hamilton cycle\ntends to e−e−α−√πe−α/2(the second term in the exponent is the contribution\nfrom points near the boundary of the square).\nSomepartialprogresshasbeenmadeonthisquestionpreviou sly.Petit [ 13]\nshowed that if πr2/logntends to infinity, then Gis, whp, Hamiltonian, andHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 3\nD´ ıaz, Mitsche and P´ erez [ 7] proved that if πr2>(1+ε)lognfor some ε>0\nthenGis Hamiltonian whp. (Obviously, Gis not Hamiltonian if πr2<logn\nsince whp Gis not connected!) Finally usinga similar method to [ 7] together\nwith significant case analysis, Balogh, Kaul and Martin [ 4] proved for the\nspecial case of the ℓ∞norm in two dimensions that the graph does become\nHamiltonian exactly when it becomes 2-connected.\nOur proof generalizes to higher dimensions and to other norm s. The\nGilbert model makes sense with any norm and in any number of di men-\nsions: we let Sd\nnbe thed-dimensional hypercube with volume n. We prove\nthe analog of Theorem 1in this setting.\nTheorem 2. Suppose that the dimension d≥2and/bar⌈bl·/bar⌈bl, ap-norm for\nsome1≤p≤∞, are fixed. Let G=Gn,rbe the resulting Gilbert model. Then\nH(Gis 2-connected )=H(Ghas a Hamilton cycle )\nwhp.\nThe proof is very similar to that of Theorem 1. However, there are some\nsignificant extra technicalities.\nTogiveanideawhytheseoccurconsiderconnectivity intheG ilbertmodel\nin the cube S3\nn(with the Euclidean norm). Let Abe the volume of a sphere\nof radius r. We count the expected number of isolated points in the proce ss\nwhich are away from the boundary of the cube. The probability a point\nis isolated is e−A, so the expected number of such points is ne−A, so the\nthreshold for the existence of a central isolated point is ab outA=logn.\nHowever, consider the probability that a point near a face of the cube is\nisolated: there are approximately n2/3such points, and the probability that\nthey are isolated is about e−A/2(since about half of the sphere about the\npoint is outside the cube S3\nn). Hence, the expected number of such points is\nn2/3e−A/2, so the threshold for the existence of an isolated point near a face\nis about A=4\n3logn. In other words isolated points are much more likely\nnear the boundary. These boundary effects are the reason for ma ny of the\nextra technicalities.\nWe remark that Theorem 2is trivially true for d=1: indeed, if Gis 2-\nconnected then there are two vertex disjoint paths from the l eft-most vertex\nto the right-most vertex. By adding any remaining vertices t o one of these\npaths these two paths form a Hamilton cycle.\nThek-nearest neighbor model. We also consider a second model for ran-\ndom geometric graphs: namely the k-nearest neighbor graph. In this model\nthe initial setup is the same as in the Gilbert model: the vert ices are given\nby a Poisson process of density one in the square Sn, but this time each\nvertex is joined to its knearest neighbors (in the Euclidean metric) in the\nbox. This naturally gives rise to a k-regular directed graph, but we form a4 J. BALOGH ET AL.\nsimple graph G=Gn,kby ignoring the direction of all the edges. It is easily\nchecked that this gives us a graph with degrees between kand 6k.\nXue and Kumar [ 15] showed that there are constants c1,c2such that if\nk < c1logn, then the graph Gn,kis, whp, not connected, and that if k >\nc2lognthenGn,kis, whp, connected. Balister et al. [ 1] proved reasonably\ngood bounds on the constants: namely c1= 0.3043 and c2= 0.5139, and\nlater [3] proved that there is some critical constant csuch that if k=c′logn\nforc′<c, then the graph is disconnected whp, and if k=c′lognforc′>c,\nthen it is connected whp. Moreover, in [ 2], they showed that in the latter\ncase the graph is s-connected whp for any fixed s∈N.\nWe would like to prove a sharp result like the above; that is, t hat as soon\nas the graph is 2-connected it has a Hamilton cycle. However, we prove only\nthe weaker statement that some (finite) amount of connectivi ty is sufficient.\nExplicitly, we show the following.\nTheorem 3. Suppose that k=k(n), thatG=Gn,kis the two-dimensional\nk-nearest neighbor graph (with the Euclidean norm) and that Gisκ-connected\nforκ=5·107whp. Then Ghas a Hamilton cycle whp.\nAnalogous results could be proved in higher dimensions and f or other\nnorms but we do not do so here.\nBinomial point process. To conclude this section we briefly mention a\nclosely related model: instead of choosing the points in Snaccording to a\nPoisson process of density one we choose npoints uniformly at random, and\nthen form the corresponding graph. This new model is very clo sely related\nto our first model (the Gilbert model). Indeed, Penrose origi nally proved\nhis results for the Binomial Point Process but it is easy to ch eck that this\nimplies them for the Poisson Process.\nIt is very easy to modify our proof to this new model. Indeed, i n very\nbroad terms each of our arguments consists of two steps: first we have an\nessentially trivial lemma that says the random points are “r easonably” dis-\ntributed, and then we have an argument saying that if the poin ts are reason-\nably distributed and the resulting graph is two-connected t hen the resulting\ngraph necessarily has a Hamilton cycle. The second of these s teps is entirely\ndeterministic, so only the essentially trivial lemma needs modifying.\n2. Proof of Theorem 1.We divide the proof into five parts: first we tile\nthesquare Snwithsmallsquaresinastandardtessellation argument.Sec ond\nwe identify “difficult” subsquares. Roughly, these will be sq uares containing\nonly a few points, or squares surrounded by squares containi ng only a few\npoints. Third we prove some lemmas about the structure of the difficult\nsubsquares. In stage 4 we deal with the difficult subsquares. F inally we use\nthe remaining easy subsquares to join everything together.HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 5\nStage 1: Tessellation. Letr0=/radicalbig\n(logn)/π(soπr2\n0=logn), and let rbe\nthe random variable H(Gis 2-connected). Let s=r0/c=c′√lognwherec\nis a large constant to be chosen later (1000 will do). We tesse llate the box\nSnwith small squares of side length s. Whenever we talk about distances\nbetween squares we will always be referring to the distance b etween their\ncenters. Moreover, we will divide all distances between squ ares bys, so, for\nexample, a square’s four nearest neighbors all have distanc e one.\nBy Penrose’s result [ 11] mentioned in the Introduction we may assume\nthat (1−1/2c)r0< r <(1+ 1/2c)r0: formally the collection of point sets\nwhich do not satisfy this has measure tending to zero as ntends to infinity,\nand we ignore this set.\nHence points in squares at distancer−√\n2s\ns≥r0−2s\ns=c−2 are always\njoined, and points in squares at distancer+√\n2s\ns≤r0+2s\ns=c+2 are never\njoined.\nStage 2: The “difficult” subsquares. We call a square fullif it contains\nat least Mpoints for some Mto be determined later (107will do), and\nnonfullotherwise. Let N0be the set of nonfull squares. We say two nonfull\nsquares are joined if their ℓ∞distance is at most 4 c−1 and define Nto be\nthe collection of nonfull components.\nFirst we bound the size of the largest component of nonfull sq uares (here,\nand throughout this paper, we use size to refer to the number o f vertices in\nthe component).\nLemma 4. For any M, the largest component of nonfull squares in the\nabove tesselation has size at most\nU=⌈π(c+2)2⌉\nwhp.\nAlso, the largest component of nonfull squares including a s quare within c\nof the boundary of Snhas size at most U/2whp. Finally, there is no nonfull\nsquare within distance Ucof a corner whp.\nProof. We shall make use of the following simple result: suppose tha t\nGis any graph with maximal degree ∆, and vis a vertex in G. Then the\nnumber of connected subsets of size nofGcontaining vis at most ( e∆)n\n(see, e.g., Problem 45 of [ 6]).\nHence, the number of potential components of size Ucontaining a par-\nticular square is at most ( e(8c)2)Uso, since there are less than nsquares,\nthe total number of such potential components is at most n(e(8c)2)U. The\nprobability that a square is nonfull is at most 2 s2Me−s2/M!. Hence, the\nexpected number of components of size at least Uis at most\nn(2s2Me−s2(e(8c)2)/M!)U≤n/parenleftbigg\n2(logn)Me(8c)2\nM!/parenrightbiggU\nexp/parenleftbigg\n−(c+2)2logn\nc2/parenrightbigg\n,6 J. BALOGH ET AL.\nwhich tends to zero as ntends to infinity; that is, whp, no such component\nexists.\nFor the second part there are at most 4 c√nsquares within distance cof\nthe boundary of Sn, and the result follows as above.\nFinally, there are only 4 U2c2squares within distance Ucof a corner. Since\nthe probability that a square is nonfull tends to zero we see t hat there is no\nsuch square whp. /square\nNote that this is true independently of Mwhich is important since we\nwill want to choose Mdepending on U.\nIn the rest of the argument we shall assume that there is no non full\ncomponent of size greater than U, no nonfull component of size U/2 within\ncof an edge and no nonfull square within Ucof a corner.\nBetween these components of nonfull squares there are numer ous full\nsquares. To define this more precisely let /hatwideGbe the graph with vertex set the\nsmall squares, and where each square is joined to all others w ithin (c−2)\nof this square (in the Euclidean norm). Since the probabilit y a square is in\nN0(i.e., is nonfull) is 1 −o(1), the graph /hatwideG\\N0has one giant component\nconsisting of almost all the squares. We call this component sea. (We give\nan equivalent formal definition just before Corollary 8.)\nThe idea is that it is trivial to find a cycle visiting every poi nt of the pro-\ncess in a square in the sea, and that we can extend this cycle to a Hamilton\ncycle by adding each nonfull component (and any full squares cut off by it)\none at a time. However, it is easier to phrase the argument by s tarting with\nthe difficult parts and then using the sea of full squares.\nStage 3: The structure of the difficult subsquares. Consider one com-\nponentN∈Nof the nonfull squares, and suppose that it has size u. By\nLemma4we know u<U. We will also consider N2c: the 2c-blow-up of N:\nthat is the set of all squares with ℓ∞distance at most 2 cfrom a square in N.\nNow some full squares may be cut off from the rest of the full squ ares by\nnonfull squares in N. More precisely the graph /hatwideG\\Nhas one component\nA=A(N) consisting of all but at most a bounded number of squares (si nce\nwe have removed at most Usquares from /hatwideG). We call Acthecutoffsquares.\nWe split the cutoff squares into two classes: those with a neig hbor inA\n(in/hatwideG) which we think of as being “close” to A, and the rest, which we\nshall call farsquares. All the close squares must be in N(since otherwise\nthey would be part of A). However, we do not know anything about the far\nsquares: they may be full or nonfull. See Figure 1for a picture.\nLemma 5. No two far squares are more than ℓ∞distance c/10apart.\nRemark. This does not say whp since we are assuming this nonfull\ncomponent has size at most U.HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 7\nFig. 1. A small part of Sncontaining the nonfull component Nand the corresponding\nsetA, far squares and close squares. It also shows the two vertex d isjoint paths from the\nfar squares to Aand the path joining Q2toQ1(see stage 4).\nProof. Suppose not.\nSuppose, first, that no point of Nis within cof the edge of Sn, and that\nthe two far squares are at horizontal distance at least c/10. Then consider\nthe left-most far square. All squares which are to the left of this and with\ndistancetothissquarelessthan( c−2)mustbecloseandthusin N.Similarly\nwith the right-most far square. Also at least ( c−2) squares [in fact nearly\n2(c−2)] in each of at least c/10 columnsbetween theoriginal two farsquares\nmust be in N. This is a total of about π(c−2)2+(c−2)c/10>Uwhich is\na contradiction (provided we chose creasonably large).\nIf there is a point of Nwithincof the boundary, then the above argument\ngives more than U/2 nonfull squares. Indeed, either it gives half of each part\nof the above construction, or it gives all of one end and all th e side parts.\nThis contradicts the second part of our assumption about the size of nonfull\ncomponents.\nWe do not need to consider a component near two sides: it canno t be\nlarge enough to be near two sides. It also cannot go across a co rner, since\nno square within distance Ucof a corner is nonfull. /square\nThis result can also be deduced from a result of Penrose, as we do in the\nnext section. We have the following instant corollary.\nCorollary 6. The graph /hatwideGrestricted to the far squares is complete.8 J. BALOGH ET AL.\nCorollary 7. The set of cutoff squares Acis contained in Nc(the\nc-blow-up of N). In particular, the set Γ(Ac)of neighbors in /hatwideGofAcis\ncontained in N2c.\nProof. Suppose Ac/\\⌉}atio\\slash⊆Nc. Letxbe a square in Ac\\Nc. First,xcannot\nbe a neighbor of any square in Aorxwould also be in A; that is, xis a far\nsquare.\nNow, let ybe any square with ℓ∞distance c/5 fromx. The square y\ncannot be in Nsince then xwould be in Nc. Therefore, ycannot be a\nneighbor of any square in Asince then it would be in Aand, since xandy\nare joined in /hatwideG,xwould be in A; that is, yis also a far square. Hence, xand\nyare both far squares with ℓ∞distance c/5 which contradicts Lemma 5./square\nInparticular,Corollary 7tellsusthatthesetsofsquarescutoffbydifferent\nnonfull components and all their neighbors are disjoint (ob viously the 2 c-\nblow-ups are disjoint).\nWe now formally define the sea/tildewideA=/intersectiontext\nN∈NA(N). We show later (Corol-\nlary11) that/tildewideAis connected and, thus, that this is the same as our earlier\ninformal definition. The following corollary is immediate f rom Corollary 7.\nCorollary 8. For any N∈Nwe have /tildewideA∩N2c=A(N)∩N2c.\nThe final preparation we need is the following lemma.\nLemma 9. The setN2c∩Ais connected in /hatwideG.\nSince the proof will be using a standard graph theoretic resu lt, it is con-\nvenient to define one more graph /hatwideG1: again the vertex set is the set of small\nsquares, but this time each square is joined only to its four n earest neigh-\nbors;that is, /hatwideG1istheordinarysquarelattice. Weneedtwoquickdefinitions .\nFirst, for a set E∈/hatwideG1we define the boundary ∂1EofEto be set of vertices\ninEcthat are neighbors (in /hatwideG1) of a vertex in E. Second, we say a set\nEin/hatwideG1isdiagonally connected if it is connected when we add the edges\nbetween squares which are diagonally adjacent (i.e., at dis tance√\n2) to/hatwideG.\nThe lemma we need is the following; since its proof is short we include it\nhere for completeness. (It is also an easy consequence of the unicoherence of\nthe square (see, e.g., page 177 of [ 12]).)\nLemma 10. Suppose that Eis any subset of /hatwideG1withEandEccon-\nnected. Then ∂1Eis diagonally connected: in particular, it is connected in /hatwideG.\nProof. LetFbe the set of edges of /hatwideG1fromEtoEc, and let F′be\nthe corresponding set of edges in the dual lattice. Consider the setF′as a\nsubgraph of the dual lattice. It is easy to check that every ve rtex has even\ndegree except vertices on the boundary of /hatwideG1. Thus we can decompose F′\ninto pieces, each of which is either a cycle or a path starting and finishing atHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 9\nthe edge of /hatwideG1. Any such cycle splits /hatwideG1into two components, and we see\nthat one of these must be exactly Eand the other Ec. ThusF′is a single\ncomponent in the dual lattice, and it is easy to check that imp lies that ∂1E\nis diagonally connected. /square\nProof of Lemma 9.Consider /hatwideG1\\N2c. This splits into components\nB1,B2,...,B m. By definition each Biis connected. Moreover, each Bc\niis\nalso connected. Indeed, suppose x,y∈Bc\ni. Then there is an xypath in/hatwideG1.\nIf this is contained in Bc\niwe are done. If not then it must meet N2c, but\nN2cis connected. Hence we can take this path until it first meets N2c, go\nthrough N2cto the point where the path last leaves N2cand follow the path\non toy. This gives a path in Bc\ni.\nHence, by Lemma 10, we see that each ∂1Biis connected in /hatwideGfor each i\n(where∂1denotes the boundary in /hatwideG1). Obviously ∂1Bi⊂N2c.\nAs usual, for a set of vertices Vlet/hatwideG[V] denote the graph /hatwideGrestricted\nto the vertices in V.\nClaim. Any two vertices in/uniontextm\ni=1∂1Biare connected in /hatwideG[A∩N2c].\nProof. Suppose not. Without loss of generality assume that, for som e\nk<m,/hatwideG[/uniontextk\ni=1∂1Bi] is connected and that no other ∂1Biis connected via a\npath to it. Pick x∈B1andy∈Bm. Bothxandyare inA(since they are\nnot inN2candAc⊂N2cby Corollary 7).\nHence there is a path from xtoyinA. Consider the last time it leaves/uniontextk\ni=1Bi. The path then moves around in N2cbefore entering some Bjwith\nj >k. This gives rise to a path in A∩N2cfrom a point in/uniontextk\ni=1∂1Bito a\npoint in ∂1Bj, contradicting the choice of k./square\nWe now complete the proof of Lemma 9. To avoid clutter we shall say\nthat two points are joinedif they are connected by a path. Suppose that\nx,y∈A∩N2c. SinceAis connected there is a path in Afromxtoy. If the\npath is contained in N2cwe are done. If not, consider the first time the path\nleavesN2c. It must enter one of the Bi, crossing the boundary ∂1Bi. Hence\nxis joined to some w∈∂1BiinA∩N2c. Similarly, by considering the last\ntime the path is not in N2cwe see that yis joined to some z∈∂1Bjfor\nsomej. However, since the claim showed that wandzare joined in A∩N2c,\nwe see that xandyare joined in A∩N2c./square\nCorollary 11. The set of sea squares /tildewideAis connected in /hatwideG.\nProof. Given two squares x,yin/tildewideA, pick a path in /hatwideGfromxtoy. Now\nfor each nonfull component Nin turn do the following. If the path misses\nN2cdo nothing. Otherwise let wbe the first point on the path in N2candz\nbe the last point in N2c. Replace the xypath by the path xw, any path wz\ninA(N)∩N2cand then the path zy.10 J. BALOGH ET AL.\nAt each stage the modification ensured that the path now lies i nA(N).\nAlso, the only vertices added to the path are in N2cwhich is disjoint from\nall the previous N′\n2c, and thus from all previous sets A(N′). Hence, when we\nhave done this for all nonfull components the path lies in eve ryA(N′), that\nis, in/tildewideA. Hence, /tildewideAis connected. /square\nStage 4: Dealing with the difficult subsquares. We deal with each nonfull\ncomponent N∈Nin turn. Fix one such component N.\nLet us deal with the far squares first. There are three possibi lities: the\nfar squares contain no points at all, they contain one point i n total or they\ncontain more than one point. In the first case, do nothing and p roceed to\nthe next part of the argument.\nIn the second case, by the 2-connectivity of G, we can find two vertex\ndisjointpathsfromthissinglevertex v1topointsinsquaresin A.Inthethird\ncase pick two points v1andv2in the far squares. Again by 2-connectivity\nwe can find vertex disjoint paths from these two vertices to po ints in squares\ninA.\nSuppose that the path from v1meetsAin square Q1at point q1and\nthe other path (either from v2or the other path from v1again) meets A\nin square Q2at point q2. LetP1,P2be the squares containing the previous\npoints on these paths. Since no two points in squares at (Eucl idean) distance\n(c+2) are joined we see that P1is within ( c+2) ofQ1. SinceP1/∈Awe\nhave that some square on a shortest P1Q1path in/hatwideG1is inNand thus that\nQ1∈N2c. Similarly Q2∈N2c. Combining we see that both Q1andQ2are\ninN2c∩A. By Lemma 9, we know that N2c∩Ais connected in /hatwideGso we\ncan find a path from Q1toQ2inN2c∩Ain/hatwideG. This “lifts” to a path in G\ngoing from q2to a point other than q1inQ1using at most one vertex in\neach subsquare on the way and never leaving N2c.\nConstruct a path starting and finishing in Q1by joining together the\nfollowing paths:\n1. the path from q1tov1;\n2. a path starting at v1going round all points in the far region (except any\nsuch points on the q1v1orq2v2paths) finishing back at v2. (Corollary 6\nguarantees the existence of such a path.) We omit this piece i f there is\njust one far vertex;\n3. the path v2toq2;\n4. the path from q2through the sea back to Q1constructed above.\nSinceQ1∈A∩N2c, by Corollary 8we have that Q1∈/tildewideA. Combining, we\nhave a path starting and finishing in the same subsquare of the sea/tildewideA(i.e.,\nQ1) containing all the vertices in the far region.\nNext we deal with the close squares: we deal with each close sq uarePin\nturn. Since Pis a close square we can pick Q∈AwithPQjoined in /hatwideG. InHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 11\nthe following we ignore all points that we have used in the pat h constructed\nabove and any points already used when dealing with other clo se squares.\nIf the square Phas no point in it we ignore it. If it has one point in it,\nthen join that point to two points in Q.\nIf it has two or more points in it then pick two of them x,y: and pick two\npointsuvinQ(we choose Mlarge enough to ensure that we can find these\ntwo unused points in Q, see below). Place the path formed by the edge ux\nround all the remaining unused vertices in the cutoff square fi nishing at y\nand back to the square Qwith the edge yvin the cycle we are constructing.\nThe square Qis a neighbor of P∈Acso, by Corollary 7is inN2c. Since\nQis also in Awe see, by Corollary 8as above, that Q∈/tildewideA.\nWhen we have completed this construction we have placed ever y vertex\nin a cutoff square on one of a collection of paths, each of which starts and\nfinishes at the same square in the sea (although different paths may start\nand finish in different squares in the sea).\nWe use at most 2 U+2 vertices from any square in A=A(N) when doing\nthis, so, provided that M >2U+2+(2c+1)2, there are at least (2 c+1)2\nunused vertices in each square of Awhen we finish this. Moreover, obviously\nthe only squares touched by this construction are in N2c, and for distinct\nnonfull components these are all disjoint. Hence, when we ha ve done this for\nevery nonfull component N∈Nthere are at least (2 c+1)2unused vertices\nin each square of the sea /tildewideA.\nStage 5: Using the subsquares in the sea to join everything to gether. It\njust remains to string everything together. This is easy. Si nce, by Corol-\nlary11, the sea of squares /tildewideAis connected, there is a spanning tree for /tildewideA. By\ndoubling each edge we can think of this as a cycle, as in Figure 2. This cycle\nvisits each square at most (2 c+1)2times. (In fact, by choosing a spanning\ntree such that the sum of the edge lengths is minimal we could a ssume that\nFig. 2. A tree of subsquares and its corresponding tree cycle.12 J. BALOGH ET AL.\nit visits each vertex at most six times but we do not need this. ) Convert this\ninto a Hamilton cycle as follows. Start at an unused vertex in a square of the\nsea. Move to any (unused) vertex in the next square in the tree cycle. Then,\nif this is the last time the tree cycle visits this square, vis it all remaining\nvertices and join in all the paths constructed in the first par t of the argu-\nment, then leave to the next square in the tree cycle. If it is n ot the last time\nthe tree cycle visits this square, then move to any unused ver tex in the next\nsquare in the tree cycle. Repeat until we complete the tree cy cle. Then join\nin any unused vertices and paths to this square constructed e arlier before\nclosing the cycle.\n3. Higher dimensions. We generalise the proof in the previous section\nto higher dimensions and any p-norm. Much of the argument is the same,\nin particular, essentially all of stages four and five. We inc lude details of all\ndifferences but refer the reader to the previous section where the proof is\nidentical.\nStage 1: Tessellation. We work in the d-dimensional hypercube Sd\nnof\nvolumen(for simplicity we will abbreviate hypercube to cube in the f ollow-\ning). As mentioned in the Introduction , we no longer have a nice formula\nfor the critical radius: the boundary effects dominate.\nInstead, we consider the expected number of isolated vertic esE=E(r).\nWe need a little notation: let Ardenote the set {x∈Sd\nn:d(x,A)≤r}and\n|·|denote Lebesgue measure.\nWe have E=/integraltext\nSdnexp(−|{x}r|)dx. Letr0=r0(n) be such that E(r0)=1.\nAs before fix ca large constant to be determined later, and let s=r0/c. It\nis easy to see that rd\n0=Θ(logn) andsd=Θ(logn). We tile the cube Sd\nnwith\nsmall cubes of side length s.\nAs before, let r=H(Gis 2-connected). By Penrose (Theorems 1.1 and 1.2\nof [11] or Theorems 8.4 and 13.17 of [ 12]) the probability that r /∈[r0(1−\n1/2c),r0(1+1/2c)] tends to zero and we ignore all these point sets. (Note\nthat these two of Penrose’s results are not claimed for p=1. However, since\nfor anyε >0 we can pick p >1 such that B1(r)⊂Bp(r)⊂B1((1 +ε)r)\n[whereB1(r) andBpdenote the l1andlpballs of radius r, resp.], the above\nbound on rforp=1 follows from Penrose’s results for p>1.)\nThis time any two points in cubes at distancer−s√\nd\ns≥r0−ds\ns=c−dare\njoined, and no points in cubes at distancer+s√\nd\ns≤r0+ds\ns=c+dare joined.\nStage 2: The “difficult” subcubes. Exactly as before we define nonfull\ncubes to be those containing at most Mpoints, and we say two are joined\nif they have ℓ∞distance at most 4 c−1.\nWe wishto prove aversion of Lemma 4. However, we have several possible\nboundaries: for example, in three dimensions we have the cen ter, the faces,\nthe edges and the corners. We call a nonfull component contai ning a cubeHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 13\nQbadif it consists of at least (1+1 /c)|Qr0|/sdcubes. (Note a component\ncan be bad for some cubes and not others.)\nLemma 12. The expected number of bad components tends to zero as n\ntends to infinity. In particular there are no bad components wh p.\nProof. The number of connected sets of size Ucontaining a particular\ncube is at most ( e(8c)d)U. The probability that a cube is nonfull is at most\n2sdMe−sd/M!. Since min {|Qr0|:cubesQ}=Θ(logn) andsd=Θ(logn), the\nexpected number of bad components is at most\n/summationdisplay\ncubesQ(2sdMe−sd(e(8c)d)/M!)(1+1/c)|Qr0|/sd\n=/summationdisplay\ncubesQ(2sdM(e(8c)d)/M!)(1+1/c)|Qr0|/sdexp(−(1+1/c)|Qr0|)\n=o(1)/summationdisplay\ncubesQexp(−|Qr0|)\n≤o(1)/integraldisplay\nSdnexp(−|{x}r0|)dx\n=o(1)E(r0)\n=o(1). /square\n(Again, note that this is true independently of M.)\nFrom now on we assume that there is no bad component.\nStage 3: The structure of the difficult subcubes. In this stage we will need\none extra geometric result of Penrose, a case of Proposition 5.15 of [12] (see\nalso Proposition 2.1 of [ 11]).\nProposition 13. Suppose dis fixed and that /bar⌈bl·/bar⌈blis ap-norm for some\n1≤p≤∞. Then there exists η>0such that if F⊂Od(the positive orthant\ninRd) is compact with ℓ∞diameter at least r/10, andxis a point of Fwith\nminimal l1norm; then |Fr|≥|F|+|{x}r|+ηrd.\nWe begin this stage by proving Lemma 5for this model.\nLemma 14. No two far cubes are more than ℓ∞distance c/10apart.\nProof. Supposenot. Then let Fbethe set of far cubes, let xbea point\nofFclosest to a corner in the l1norm and let Qbe the cube containing x\n(or any of the possibilities if it is on the boundary between c ubes). We know\nthat all the cubes within ( c−d) of a far cube are not in A. Hence all such\ncubes which are not far must be close, and thus nonfull.14 J. BALOGH ET AL.\nThe number of close cubes is at least\n|F(c−2d)s\\F|\nsd≥|{x}(c−2d)s|+η((c−2d)s)d\nsdby Proposition 13\n≥|Q(c−3d)s|+ηrd\n0/2\nsdprovided cis large enough\n=|Q(1−3d/c)r0|+ηrd\n0/2\nsd\n≥(1−3d/c)d|Qr0|+ηrd\n0/2\nsd\n>(1+1/c)|Qr0|\nsdprovided cis large enough.\nThis shows that the component is bad which is a contradiction ./square\nCorollaries 6,7and8hold exactly as before. Lemma 9also holds, we just\nneed to replace Lemma 10by the following higher-dimensional analogue.\nNote that, even in higher dimensions we say two squares are di agonally\nconnected if their centers have distance√\n2.\nLemma 15. Suppose that Eis any subset of /hatwideG1withEandEccon-\nnected. Then ∂1Eis diagonally connected: in particular, it is connected in /hatwideG.\nRemark. Again the final conclusion of connectivity in /hatwideGis an easy\nconsequence of unicoherence, this time of the hypercube.\nProof. LetIbe a (diagonally connected) component of ∂1E. We aim\nto show the I=∂1Eand, thus, that ∂1Eis diagonally connected.\nClaim. Suppose that Cis any circuit in /hatwideG1. Then the number of edges\nofCwith one end in Eand the other end in Iis even.\nProof. We say that a circuit is contractible to a single point using t he\nfollowing operations. First, we can remove an out and back ed ge. Second, we\ncan do the following two-dimensional move. Suppose that two consecutive\nedges of the circuit form two sides of a square; then we can rep lace them by\nthe other two sides of the square keeping the rest of the circu it the same.\nFor example, we can replace ( x,y+1,/vector z)→(x+1,y+1,/vector z)→(x+1,y,/vector z) in\nthe circuit by ( x,y+1,/vector z)→(x,y,/vector z)→(x+1,y,/vector z).\nNext we show that Cis contractible. Let w(C) denote the weight of the\ncircuit: that is, the sum of all the coordinates of all the ver tices inC. We\nshow that, if Cis nontrivial, we can apply one of the above operations and\nreducew. Indeed, let vbe a vertex on Cwith maximal coordinate sum, andHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 15\nsuppose that v−andv+are the vertices before and after von the circuit. If\nv−=v+then we can apply the first operation removing vandv+from the\ncircuit which obviously reduces w. If not, then both v−andv+have strictly\nsmaller coordinate sums than v, and we can apply the second operation\nreducing wby two. We repeat the above until we reach the trivial circuit .\nNow, let Jbe the number of edges of Cwith an end in each of EandI.\nThe first operation obviously does not change the parity of J. A simple\nfinite check yields the same for the second operation. Indeed , assume that\nwe are changing the path from ( x,y+1),(x+1,y+1),(x+1,y) to (x,y+\n1),(x,y),(x+1,y). LetFbe the set of these four vertices. If no vertex of\nIis inF, then obviously Jdoes not change. If there is a vertex of IinF,\nthen, by the definition of diagonally connected, F∩I=F∩∂1E. Hence the\nparity of Jdoes not change. [It is even if ( x,y+1) and ( x+1,y) are both\ninEor both in Ecand odd otherwise.] /square\nSupposethat thereis some vertex v∈∂1E\\Iand that u∈Eis aneighbor\nofv. Lety∈Iandx∈Ebeneighbors. Since EandEcare connected we can\nfindpaths PxuandPvyinEandEc, respectively. Thecircuit Pxu,uv,Pvy,yx\ncontains a single edge from EtoIwhich contradicts the claim. /square\nTo complete this stage observe that Corollary 11holds as before.\nStage 4: Dealing with the difficult subcubes, and Stage 5: Usin g the sub-\ncubes in the sea to join everything together. These two stages go through\nexactly as before [with one trivial change: replace (2 c+1)2by (2c+1)d].\nThis completes the proof of Theorem 2.\n4. Proof of Theorem 3.In this section we prove Theorem 3. Once again,\nthe proof is very similar to that in Section 2. We shall outline the key\ndifferences, and emphasise why we are only able to prove the wea ker version\nof the result.\nStage 1: Tessellation. The tessellation is similar to before, but this time\nsome edges may be much longer than some nonedges.\nLetk=H(Gisκ-connected) be the smallest kthatGn,kisκ-connected.\nSinceGis connected we may assume that 0 .3logn<k <0.52logn(see [1]\nand [2]). Letr−be such that any two points at distance r−are joined\nwhp; for example, Lemma 8 of [ 1] implies that this is true provided πr2\n−≤\n0.3e−1−1/0.3logn, so we can take r−=0.035√logn.\nLetr+be such that no edge in the graph has length more than r+. Then,\nagain by Lemma 8 of [ 1], we have\nπr2\n+≤4e(1+0.52)logn\nwhp, so we can take r+=2.3√logn≤66r−.\nFrom here on, we ignore all point sets with an edge longer than r+or a\nnonedge shorter than r−.16 J. BALOGH ET AL.\nLets=r−/√\n8. We tessellate the box Snwith small squares of side\nlengths. (Since we are proving only this weaker result our tesselati on does\nnot need to be very fine.) By the choice of sand the bound on r−any two\npoints in neighboring or diagonally neighbouring squares a re joined in G.\nAlso, by the bound on r+no two points in squares with centers at distance\nmore than (66√\n5+2)s<150sare joined. Let D=104; we have that no two\npoints in squares with centers distance Dsapart are joined.\nStage 2: The “difficult” subsquares. We call a square fullif it contains\nat leastM=109points and nonfullotherwise. We say two nonfull squares\nare joined if they are at ℓ∞distance at most 2 D−1.\nFirst we bound the size of the largest component of nonfull sq uares.\nLemma 16. The largest component of nonfull squares has size less than\n7000 whp.\nProof. The number of connected subgraphs of /hatwideGof size 7000 con-\ntaining a particular square is at most ( e(4D)2)7000, so, since there are less\nthannsquares, the total number of such connected subgraphs is at m ost\nn(e(4D)2)7000.Theprobabilitythatasquareisnonfullisatmost2 s2Me−s2/M!.\nHence, the expected numberof components of nonfull squares of size at least\n7000 is at most\nn(2s2Me−s2(e(4D)2)/M!)7000\n≤n/parenleftbigg\n2/parenleftbigg(0.035)2logn\n8/parenrightbiggMe(4D)2\nM!/parenrightbigg7000\nexp/parenleftbigg−7000(0.035)2logn\n8/parenrightbigg\n,\nwhich tends to zero as ntends to infinity [since 7000( −0.035)2/8>1.07>1];\nthat is, whp, no such component exists. /square\nIn the rest of the argument we shall assume that there is no non full\ncomponent of size greater than 7000.\nStage 3: The structure of the difficult subsquares. As usual we fix one\ncomponent Nof the nonfull squares, and suppose that it has size u(so we\nknowu<7000). This time we define /hatwideGto be the graph on the small squares\nwhere each square is joined to its eight nearest neighbors (i .e., adjacent and\ndiagonal). Let A=A(N) be the giant component of G\\N, and again split\nthe cutoff squares into close and far dependingwhether they h ave a neighbor\n(in/hatwideG) inA.\nBy the vertex isoperimetric inequality in the square there a re at most\nu2/2 squares in Ac\\Nso|Ac|≤u2/2+u<2.5·107.\nNext we prove a result similar to Corollary 7.\nLemma 17. The set of cutoff squares Acis inND(whereD=104as\nabove).HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 17\nFig. 3. Two paths from one cutoff square to the sea together with the pa th from the\nmeeting point in Q2to the square Q1.\nProof. Suppose not, and that Qis a square in Acnot inND. Then all\nsquares within ℓ∞distance of Qat mostDare not in N. Hence they must\nbe inAc(since otherwise there would be a path from Qto a square in Anot\ngoing through any square in N). Hence |Ac|>D2=108which contradicts\nLemma16./square\nFinally, we need the analogue of Lemma 9whose proof is exactly the\nsame.\nLemma 18. The setND∩Ais connected in /hatwideG.\nStage 4: Dealing with the difficult subsquares. Let us deal with these cut-\noff squares now. From each cutoff square that contains at least two vertices,\npick any 2 vertices, and from each cutoff square that contains a single vertex\npick that vertex with multiplicity two. We have picked at mos t 5·107ver-\ntices, so since Gisκ=5·107connected we can simultaneously find vertex\ndisjoint paths from each of our picked vertices to vertices i n squares in A\n(two paths from those vertices that are repeated).\nWe remark that these are not just single edges; these paths ma y go\nthrough other cutoff squares.\nCall the first point of such a path which is in Aameeting point , and the\nsquare containing this point a meeting square.\nFix a cutoff square and let v1,v2be the two vertices picked above from\nthis square (let v1=v2if the square only contains one vertex). This cutoff\nsquare has two meeting points, say q1andq2in subsquares Q1andQ2,\nrespectively. Sincethelongest edgeis at most r+, bothQ1andQ2areinND.\nSinceA∩NDis connected in /hatwideGwe construct a path in the squares in A∩ND\nfrom the meeting point in Q2to a vertex in Q1using at most one vertex in\neach subsquare on the way, and missing all the other meeting p oints. This\nis possible since each full square contains at least M=109vertices.18 J. BALOGH ET AL.\nConstruct a path starting and finishing in Q1containing all the (unused)\nvertices in this cutoff square by joining together the follow ing paths:\n1. the path from q1tov1;\n2. a path starting at v1going round all points in the cutoff square finishing\nback atv2(omit this piece if there is just one far vertex);\n3. the path v2toq2;\n4. the path from q2through A∩NDback toQ1constructed above.\nDo this for every cutoff square. For each cutoff square this con struction\nuses at most two vertices from any square in A. Moreover, it obviously only\ntouches squares in ND. Since nonfull squares in distinct components are at\ndistance at least 2 Dthe squares touched by different nonfull components\nare distinct. Thus in total we have used at most 4 ·107vertices in any square\nin the sea, and since M=109there are many (we shall only need 8) unused\nvertices left in each full square in the sea.\nStage 5: Using the subsquares in the sea to join everything to gether. This\nis exactly the same as before.\n5. Comments on the k-nearest neighbor proof. We start by giving some\nreasonswhytheproofinthe k-nearestneighbormodelonlyyields theweaker\nTheorem 3. The first superficial problem is that we use squares in the tes se-\nlation which are of “large” size rather than relatively smal l as in the proof\nof Theorem 1, (in other words we did not introduce the constant cwhen\nsettingsdepending on r).\nObviously we could have introduced this constant. The difficu lty when\ntrying to mimic the proof of Theorem 1is the large difference between r−\nandr+, which corresponds to having a very large number of squares ( many\ntimesπc2) in our nonfull component N. This means that we cannot easily\nprove anything similar to Lemma 5. Indeed, a priori, we could have two far\nsquares with πc2nonfull squares around each of them.\nA different way of viewing this difficulty is that, in the k-nearest neighbor\nmodel, the graph /hatwideGon the small squares does not approximate the real\ngraphGvery well, whereas in the Gilbert model it is a good approxima tion.\nThus, it is not surprising that we only prove a weaker result.\nThis is typical of results about the k-nearest neighbor model; the results\ntend to be weaker than for the Gilbert model. This is primaril y because\nthe obstructions tend to be more complex; for example, the ob struction\nfor connectivity in the Gilbert model is the existence of an i solated vertex.\nObviously in the k-nearest neighbor model we never have an isolated vertex;\nthe obstruction must have at least k+1 vertices.\nExtensions of Theorem 3.When proving Theorem 3we only used two\nfacts about the random geometric graph. First, that any two p oints at dis-\ntancer−=0.035√lognare joined whp. Secondly, that the ratio of r+(theHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 19\nlongest edge) to r−(the shortest nonedge) was at most 60 whp. Obviously,\nwe could prove the theorem (with different constants) in any gr aph with\nr−=Θ(√logn) andr+/r−bounded. This includes higher dimensions and\ndifferent norms and to different shaped regions instead of Sn(e.g., to disks\nor toruses). Indeed, the only place we used the norm was in obt aining the\nbounds on r+andr−in stage 1 of the proof.\nIndeed, it also generalizes to irregular distributions of v ertices provided\nthat theabove boundson r−andr+hold.For example, it holdsinthesquare\nSnwhere the density of points in the Poisson Process decrease l inearly from\n10 to 1 across the square.\n6. Closing remarks and open questions. A related model where the re-\nsult does not seem to follow easily from our methods is the dir ected version\nof thek-nearest neighbor graph. As mentioned above, the k-nearest neigh-\nbor model naturally gives rise to a directed graph, and we can ask whether\nthis has a directed Hamilton cycle. Note that this directed m odel is signif-\nicantly different from the undirected. For example, it is like ly (see [1]) that\nthe obstruction todirected connectivity (i.e., theexiste nce of adirected path\nbetween any two vertices) is a single vertex with in-degree z ero; obviously\nthis cannot occur in the undirected case where every vertex h as degree at\nleastk. In some other random graph models a sufficient condition for t he\nexistence of a Hamilton cycle (whp) is that there are no verti ces of in-degree\nor out-degree zero. Of course, in thedirected k-nearest neighbor model every\nvertex has out-degree kso we ask the following question.\nQuestion. Let/vectorG=/vectorGn,kbe the directed k-nearest neighbor model. Is\nH(/vectorGhas a Hamilton cycle )=H(/vectorGhas no vertex of in-degree zero )\nwhp?\nIt is obvious that the bound on connectivity in the k-nearest neighbor\nmodel can be improved, but the key question is “should it be tw o?” We\nmake the following natural conjecture:\nConjecture. Suppose that k=k(n)such that the k-nearest neighbor\ngraphG=G(k,n)is a2-connected whp. Then, whp, Ghas a Hamilton cycle.\nAcknowledgments. Some of the results published in this paper were ob-\ntained in June 2006 at the Institute of Mathematics of the Nat ional Univer-\nsity of Singapore during the program “11 Random Graphs and Re al-world\nNetworks.” J. Balogh, B. Bollob´ as and M. Walters are gratef ul to the Insti-\ntute for its hospitality.\nREFERENCES\n[1]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2005). Connectivity of\nrandomk-nearest-neighbour graphs. Adv. in Appl. Probab. 371–24.MR213515120 J. BALOGH ET AL.\n[2]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2009). Highly con-\nnectedrandom geometric graphs. Discrete Appl. Math. 157309–320. MR2479805\n[3]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2009). A critical\nconstant for the k-nearest neighbor model. Adv. in Appl. Probab. 411–12.\nMR2514943\n[4]Balogh, J. ,Kaul, H. andMartin, R. A threshold for random geometric graphs\nwith a Hamiltonian cycle. Unpublished Manuscript.\n[5]Bollob´as, B.(1984). The evolution of sparse graphs. In Graph Theory and Combi-\nnatorics (Cambridge, 1983) 35–57. Academic Press, London. MR0777163\n[6]Bollob´as, B.(2006).The Art of Mathematics: Coffee Time in Memphis . Cambridge\nUniv. Press, New York. MR2285090\n[7]D´ıaz, J.,Mitsche, D. andP´erez, X. (2007). Sharp threshold for Hamiltonicity\nof random geometric graphs. SIAM J. Discrete Math. 2157–65 (electronic).\nMR2299694\n[8]Koml´os, J.andSzemer´edi, E.(1983). Limit distribution for the existence of Hamil-\ntonian cycles in a random graph. Discrete Math. 4355–63.MR0680304\n[9]Korshunov, A. D. (1977). A solution of a problem of P. Erd˝ os and A. R´ enyi abou t\nHamilton cycles in non-oriented graphs. Metody Diskr. Anal. Teoriy Upr. Syst.,\nSb. Trudov Novosibirsk 3117–56 (in Russian).\n[10]Penrose, M. D. (1997). The longest edge of the random minimal spanning tree .\nAnn. Appl. Probab. 7340–361. MR1442317\n[11]Penrose, M. D. (1999). On k-connectivity for a geometric random graph. Random\nStructures Algorithms 15145–164. MR1704341\n[12]Penrose, M. (2003).Random Geometric Graphs .Oxford Studies in Probability 5.\nOxford Univ. Press, Oxford. MR1986198\n[13]Petit, J. (2001). Layout Problems. Ph.D. thesis, Univ. Polit` ecnica de Catalunya.\n[14]P´osa, L.(1976). Hamiltonian circuits in random graphs. Discrete Math. 14359–364.\nMR0389666\n[15]Xue, F. andKumar, P. R. (2004). The number of neighbors needed for connectivity\nof wireless networks. Wireless Networks 10169–181.\nJ. Balogh\nDepartment of Mathematics\nUniversity of Illinois\nUrbana, Illinois 61801\nUSA\nE-mail: jobal@math.uiuc.eduB. Bollob ´as\nDepartment of Pure Mathematics\nand Mathematical Statistics\nUniversity of Cambridge\nCambridge, CB3 0WB\nUnited Kingdom\nE-mail: b.bollobas@dpmms.cam.ac.uk\nM. Krivelevich\nSchool of Mathematical Sciences\nTel Aviv University\nRamat Aviv 69978\nIsrael\nE-mail: krivelev@post.tau.ac.ilT. M¨uller\nCentrum voor Wiskunde\nen Informatica\nP.O. Box 94079\n1090 GB Amsterdam\nThe Netherlands\nE-mail: tobias@cwi.nl\nM. Walters\nSchool of Mathematical Sciences\nQueen Mary, University of London\nLondon, E1 4NS\nUnited Kingdom\nE-mail: M.Walters@qmul.ac.uk"    },    {        "title": "0905.4779v2.Ferromagnetic_resonance_linewidth_in_ultrathin_films_with_perpendicular_magnetic_anisotropy.pdf",        "content": "arXiv:0905.4779v2  [cond-mat.mtrl-sci]  15 Jun 2009Beaujour et al.\nFerromagnetic resonance linewidth in ultrathin films with p erpendicular magnetic\nanisotropy\nJ-M. Beaujour,1D. Ravelosona,2I. Tudosa,3E. 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. 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Magn.\n40, 2 (2004)."    },    {        "title": "0908.0481v1.Time_domain_detection_of_pulsed_spin_torque_damping_reduction.pdf",        "content": "arXiv:0908.0481v1  [cond-mat.mes-hall]  4 Aug 2009\n/CC/CX/D1/CT /CS/D3/D1/CP/CX/D2 /CS/CT/D8/CT\r/D8/CX/D3/D2 /D3/CU /D4/D9/D0/D7/CT/CS /D7/D4/CX/D2 /D8/D3/D6/D5/D9/CT /CS/CP/D1/D4/CX/D2/CV /D6/CT/CS/D9\r/D8/CX/D3/D2/C4/D3/D2/CV/CU/CT/CX /CH /CT1/B8 /CB/CP/D1/CX/D6 /BZ/CP/D6/DE/D3/D21/B8 /CA/CX\r /CW/CP/D6/CS /BT/BA /CF /CT/CQ/CQ1/B8 /C5/CP/D6/CZ /BV/D3 /DA/CX/D2/CV/D8/D3/D22/B8 /CB/CW/CT/CW/DE/CP/CP/CS /C3/CP/CZ /CP2/B8 /CP/D2/CS /CC/CW/D3/D1/CP/D7 /C5/BA /BV/D6/CP /DB/CU/D3/D6/CS1\n1/BW/CT/D4 /CP/D6/D8/D1/CT/D2/D8 /D3/CU /C8/CW/DD/D7/CX\r/D7 /CP/D2/CS /BT/D7/D8/D6 /D3/D2/D3/D1/DD /CP/D2/CS /CD/CB/BV /C6/CP/D2/D3 \r /CT/D2/D8/CT/D6/B8/CD/D2/CX/DA/CT/D6/D7/CX/D8/DD /D3/CU /CB/D3/D9/D8/CW /BV/CP/D6 /D3/D0/CX/D2/CP/B8 /BV/D3/D0/D9/D1/CQ/CX/CP/B8 /CB/BV /BE/BL/BE/BC/BK/B8 /CD/CB/BT/BA\n2/CB/CT /CP/CV/CP/D8/CT /CA /CT/D7/CT /CP/D6 \r/CW/B8 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(ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  1 Nonlinear viscoelastic wave propagation:   \nan extension of Nearly Constant Attenuation (NCQ)  models.  \nNicolas Delé pine1, Luca Lenti2, Guy Bonnet3, Jean -François Semblat, A.M.ASCE4 \n \nSubject headings : Inelasticity;  Viscoelasticity;  Damping ; Wave propagation ; Earthquake \nengineering ; Ground motion ; Nonlinear response ; Finite element method . \n \nAbstract  \nHysteretic damping is often model ed by means of  linear viscoelastic approaches such as  \n“nearly constant Attenuation (NCQ) ” models. These models do not take into acco unt nonlinear \neffects either on the stiffness or on the damping, which are well known features of soil \ndynamic behavior . The aim of this paper is to propose a mechanical model involving nonlinear \nviscoelastic behavior  for isotropic materials. This model simultaneously takes into account \nnonlinear elasticity and nonlinear damping. On the one hand, t he shear modulus is a f unction of \nthe excitation level ; on the other , the description of viscosity is based on a general ized \nMaxwell body involving non -linearity . This formulation is implemented in to a 1D finite \nelement approach  for a dry soil. The validation of the model shows its ability to retrieve low \namplitude ground motion response. For larger excitation levels, the analysis of seismic wave \npropagation in a n onlinear soil layer over an elastic bedrock  leads to results which are \nphysically satisfactory  (lower amplitudes, larger time delays, higher frequency content) . \n1 Introduction  \nThe analysis of seismic wave propagation in alluvial basins is complex  since vario us \nphenomena are involved at different scales  (Semblat and Pecker, 2009) : resonance at the scale \nof the wh ole basin (Bard and Bouchon, 1985;  Paoluci, 1999; Semblat et al., 2003), surface \nwaves generation at the basin e dges (Bard and Riepl -Thomas, 2000;  Bozzano et al., 2008; \nKawase, 2003; Moeen -Vaziri and Trifunac, 1988 ; Semblat  et al. , 2000, 2005; Sánchez -Sesma \nand Luzón, 1995 ), soil nonlinear behavior  at the geote chnical scale ( Bonilla et al., 2006 ; Iai et \nal., 1995;  Kramer, 1996 ). Handling these different  features of seismic wave propagation at the \nsame time may be important because the interaction between, for instance, surface wave \ngener ation and shear modulus degradation  may be significant . The impact on the amplification \nprocess could th us be very larg e and complex . \nNonlinear constitutive equations are  very important in the case of strong ground moti on since \nthe mechanical behavior  of many soils depend s on the excitation level and on the loading \nhistory. In this work, the attention is focused on the asp ects of nonlinear behavior  of dry \nisotropic soils submitted to dynamic loadings. Various approaches are available to model the \ndependence of the mechanical features of soils on the excitation level: equivalent linear model \nand non linear cyclic constitutive  equations (including plasticity).  \nThe equivalent linear model approximat es the problem in the linear range using an iterative \nproce dure (Schnabel et al., 1972) . Since this model leads to over -damped higher frequency \ncomponents, r ecent researches  improved it by introducing both frequency or mean stress \ndependencies of the soil properties (Sugito, 1995; Kausel and Assimaki, 2002). Several \n                                                 \n1 Université Paris -Est, LCPC, presently at: IFP , 1 & 4 av. de Bois -Préau , 92852 Rueil -Malmaison Cedex,  France, \nnicolas.delepine@ifp.fr  \n2 Université Paris -Est, LCPC, 58 bd Lefebvre, 75732 Paris Cedex 15, France, lenti@lcpc.fr  \n3 Université Paris -Est, Champs sur Marne, France, bonnet@univ -paris -est.fr \n4 Université Paris -Est, LCPC, 58 bd Lefebvre, 75732 Paris Cedex 15, France, semblat@lcpc.fr (correspond . \nauthor)  Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  2 comparisons involving such models were proposed by Bonilla et al. (2006)  and Kwok et al. \n(2008) . \nConcerning nonlinear ity, some  models  are based on both the “hyperbolic law” , for describing \nshear modulus reduction cu rves, and on the Masing criterion  (Masing, 1926) for the description \nof unloading and reloading phases . Such models  have been widely developed (Matasovic,  \n1993; M atasovic and Vucetic, 1995). However, these models generally need large \ncomputational efforts and often lack of a strong mechanical basis , e.g. thermodynamics  \n(Lemaître and Chaboche, 1992 ). Some other models are fully el astoplastic (Aubry et al., 1982 ; \nPrevost, 1985; Gyebi and Dasgupta , 1992 ) or include dependence on confining  pressure \n(Hashash and Park, 2001;  Park and Hashash, 2004) and pore pressure (Bonilla et al., 2005) . \nHowever , their use  for large scale wave propagation analyses is limited as a conseq uence of the \nlarge number of parameters needed  and the frequency/wavelength range to investigate . \nIn this paper, a 3D nonlinear viscoelastic model is proposed. This model simultaneously \nfollows a  nonlinear elastic law and a nonlinear viscous law to investi gate the ground response \nto strong seismic excitation . \n2 Mechanical formulation  of the model  \n2.1 3D linear viscoelasticity  \n2.1.1 General formulation  \nThe 3D formulation of the viscoelastic model sta rts from the following relation : \nij=sij+pij (1) \nwhere ij, sij, ij and p are the Cauchy stress tensor, the deviatoric stress tensor, the Kronecker \nunit tensor and the volumetric tension  respectively . For an isotropic material , we can write : \np=K ekk (2) \nwhere K and ekk are the bulk modulus and  the volumetric strain respectively . The relation \nbetween the components of the deviatoric stress tensor s and the shear deviatoric strain tensor e \nin the case of linear viscoelasticity is formulated in the frequency domain as simply as:   \nsij ()=2M()eij() (3) \nsij(), eij() are the Fourier transforms of the components of the deviatoric stress and strain \ntensors. M() is the complex -valued , frequency -dependent, viscoelastic modulus from which \nwe can define the specific attenuation Q-1 in the following way  (Bourbié  et al. , 1987 ; Semblat \nand Pecker, 2009 ):  \n2=Q-1()\nIm(M())/Re(M()) (4) \nwhere  is the damping ratio and Re and Im are the real and imaginary parts of a complex \nvariable (resp.).  \n2.1.2 NCQ models  \nThis family of models is defined in term of the quality factor  Q. A nearly constant Q in a broad \nfrequency range and for a given  strain  level  is introduced . Biot ( 1958) first demonstrated t hat a \ncausal form of hysteretic damping can be simulated by viscoelastic cell s in parallel. Liu et al. \n(1976) construct ed such  model s by direct superposition of Zener cells (standard  solid) . \nEmmerich and Korn (1987 ) improved and extended the Padé approxima tion (Day and Minster, \n1984)  by considering  a generalized n-cells Maxwell body  (Fig. 1, left) . Mozco and Kristek \n(2005) pro ved the equivalence of  the models  of Liu and Emmerich and Korn. The \nimplementation proposed by Emmerich and Korn is used in the follo wing . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  3 \nRt()\nMU\nMRM\nt=0 timeaMl a M/llM \nFig. 1. Generalized Maxwell body with viscosities \nl lMa/.  and elastic moduli \nMal.  for \neach rheological cell (left). Typical relaxation function R(t) (right) with MU the unrelaxed \nmodulus and MR=M U-M the relaxed modulus.  \n \nThe generalized Maxwell model leads to the frequency dependent complex modulus  (variables \nwith bracket are not tensorial):  \n\n\n\n\n\n\n\n\nn\nlln\nll l l\nU\nyi y\nM M\n1)0,(1)( )()0,(\n1) /(\n1 )(\n\n (5) \nMU is the unrelaxed (instantaneous) modulus and MR is the relaxed (long term) modulus \n(Fig. 1, right ). The y(l,0) variabl es characteri ze the rheological model  and are calculated by \nmeans of an optimization method in order to obtain a nearly constant attenuation in a given  \nfrequency range (see  Appendix ). \nUsing Eqs. (4) and (5), the quality factor has the following expression:  \n\n \n \nn\nl ll\nln\nl ll\nl\nyy\nMMQ\n12\n)(2\n)(\n)0,(12\n)()(\n)0,(\n1\n)/(1)/(1)/(1/\n)( Re)( Im)(\n\n\n (6) \nThe \n)(l  frequencies characterize  each individual rheological cell (see Appendix) . \nThe constitutive equations  for the linear viscoelastic model are  thus: \n\n \nn\nll ij U ij t te M ts\n1)()( )( 2)( \n (7) \nand \n)(\n1)( )(\n1)0,()0,(\n)( )()( )( te\nyyt tij n\nlll\nl l l l\n\n  \n (8) \n \n \nwhere (l)(t) are rela xation parameters physically related to the anelastic deformation of the lth-\ncell (Fig.  1, left) . \nFig. 2 displays the attenuation  curve (a),  2=Q-1, and the phase velocity  (b), Vph, as functions \nof frequency.  These graphs  are obtained considering  3 Zener’s  cells which are equivalent to \ngeneralized Maxwell cells  (Fig. 1, left). The attenuation  is nearly constant,  2=Q-1=0.05 , in the \nfrequency range 0.1 -10Hz . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  4 \n00.010.020.030.040.050.06\n180190200210220phase velocity (m/s ) atten uation Q-1\nf req ue ncy (Hz )10-210-1100101102 \nFig. 2. Generalized Maxwell body appro ximating a Nearly Constant Quality factor Q 20 (top), \nin the frequency range 0.1 -10Hz, and corresponding phase velocity V ph (bottom ). The target \nphase velocity, |V ph|=200m/s, is chosen at a frequency of 1Hz.  \n2.2 3D nonlinear viscoelastic  model  \n2.2.1 Principles of th e nonlinear model  \nIn order to describe the soil’s shear modulus and damping variations with the excitation level, \nan elastic potential function and a dissipation function depending on the magnitude of the \nsecond invariant of the strain tensor are introduce d. The description of viscosity is based on a \nNearly Constant Attenuation model able to fulfil the causality principle for seismic wave \npropagation (dispersive materials). Owing to the frequent use of this model within the \ngeophysical community, it is usua lly called Nearly Constant Quality Factor (or “NCQ”) model. \nAt the same time, it leads to a constant value of the damping factor at low strains over a broad \nfrequency range of engineering interest (Kjartansson, 1979). The model is well -adapted to time \ndoma in formulations (some alternative numerical strategies are available ( Carcione et al., \n2002;  Munjiza et al., 1998 ; Semblat, 1997 )). \nIn the NCQ model, we introduce a dependence on the excitation level in order to consider an \nincreasing damping ratio suggest ed from earthquakes records and geotechnical data (Iai et al ., \n1995; Vucetic, 1990). This dependence is controlled during the 3D stress -strain path by the \nvariation of the second order invariant of the strain tensor.  \n2.2.2 Formulation of the e xtended NCQ model  (“X-NCQ”) \nTo account for non linear behavior  of soils in the case of any 3D stress -strain path, Eq. (7) is \nextended as fo llows : \n\n  \nn\nll l ij U ij Jyt teJ M ts\n12 )( )( 2 ))(,( )()( 2)( \n (9) \nwhere J2 is the second invariant o f the deviatoric strain tensor, defined from the following  \nrelations:  \n32'\n1 '\n2 2II J\n (10) \nwith the 2 first invariants of the strain tensor:  \n)('\n1 traceI\n (11) \nand Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  5 \n)(212 '\n2  trace I (12) \n \nIn addition, the shear modulus is assumed to change during the global stress -strain path \naccording to the following relation : \n)( 1 )(2 0, 2 J M J MU U \n (13) \nwith \n21\n221\n2\n2\n1)(\nJJJ\n\n\n (14) \nand where MU,0 denotes the unrelaxed  modulus  characterizing the instantaneous response of the \nsoil at small  strains and  is a parameter quantifying  its nonlinear behavior  for larger strains.  \n \nThe octahedral strain\noct  is now introduced : \n21\n22Joct\n (15) \n \nIt leads to  \n) ( 1 ) (0, oct U oct U M M   \n (16) \nwhere:  \n2/ 12/) (\noctoct\noct\n (17) \n \nSuch a dependence of the nonlinear elastic modulus on the octahedral strain also implies a \nstrain dependence for the variables y(l) and (l). Determination of damping ratio has been \nperform ed by Strick (1967) using wave propagation measurements.  Formulations for the \ndependence of the damping ratio on the shear strain modulus have been  proposed by Hardin \nand Drnevich  (1972 ). In the case of 3D loadings, di fferent authors (El Hosri  et al. , 1984; Heitz, \n1992; Bonnet and Heitz , 1994) proposed an extension of , such as:  \n) () ( ) (oct 0 max 0 oct  \n (18) \nwhere 0andmaxcharacterize  the dissipated  energy in the small and larger strain  range s \nrespectively . Typical  MU()=G() and ()curves are proposed  in Fig. 3. \n \nThe damping ratio  and the  attenuation  Q-1 are now related by:  \n) (21\noct Q\n (19) Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  6 \n10-510-410-310-210-100.20.40.60.81\nshear strain 00 . 0 50 . 10 . 1 50 . 20 . 2 5\n G( )/G0 \nFig. 3. Typical nonlinear dynamic properties of soils: shear modulus reduction (solid) and \ndamping increase (dashed) with increasing shear strain.  \n2.2.3 Features of the extended NCQ model  \nIn paragraph 2.1.1 , the solution of Eq. (9) in the limit of low excitation levels has been found. \nFor low octahedral strain s, we can consider that:  \n01\n02Q\n (20) \nand \n0 0, )0 ( G M MU oct U \n (21) \nFor every other  value of the induced strain, the Q-1 facto r increases with  strain according to \nEq. (19). This change has no influence on the frequenc y range  in which Q-1 is constant.  In \nother words, in Eq. (6) only the variables y(l,0) change to account for the variation of the \ndamping with strain. We therefore introduce a strain variation of the variables y(l) with strain in \nthe following form : \n)0,( )( ) () (l oct oct l y c y \n (22) \nUsing Eqs. (4), (15) and (17), for every level of induc ed octahedral strain, Eqs. (5) and (34) can \nbe rewritten in the following form , respectively : \n\n\n\n\n\n \n\n\nn\nll octn\nll l l oct\noct U oct\ny ci y c\nM M\n1)0,(1)( )()0,(\n) (1) /( ) (\n1) ( ) ,(\n \n \n \n\n\nn\nl ll\nl oct oct y c Q\n12\n)()(\n)0,(1\n/ 1/) () ,(\n\n (23) \n \n \n \n(24) \nwhere, using Eq. (18), c(|oct|) is given by:  \n\n \n) ( 1) ( ) ,() (\n00 max\n01\n01\noctoct oct\noctQQc \n\n (25) \nFor every level of induced octahedral strain, Eq. (8) can be written in the more general  form : \n)(\n) (1) ()( )(\n1)0,()0,(\n)( )()( )( te\ny cy ct tij n\nll octl oct\nl l l l\n\n \n\n \n(26) \nThe latter expression and Eq. (16) are used to solve  Eq. (9) in the time domain.  Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  7 2.3 Synthesis : 1D case  \nFor a unidirectional propagating shear wave,  |oct| is equal to 2 ||, where  is the shear strain. \nEquation (16) can be written in the form:  \n1)( )(0GG MU\n (27) \n \nIn this case, Eq. (27) expresses a  hyperbolic law for the reduction of the shear modulus as the \none proposed by Hardin and Drnevich (1972). As a cons equence, the following equation for \nthe function c(|oct|) is obtained:  \n\n\n\n\n\n\n\n\n\n11)(\n00 maxc\n (28) \nwhere max and 0are two  constant rheological experimental values. At every time, the values \nassociated to the functions (l)(t) are obtained by solving the following equations:  \n)(\n)(1)()( )(\n1)0,()0,(\n)( )()( )( te\ny cy ct tn\nlll\nl l l l\n\n \n\n \n(29) \nwhere the variables y(l,0) are known, given by formula (6) for the lower strain Q-1 value.  \n \nFinal ly, the rheological Eq. (9) is used for the considered 1D case :  \n\n \nn\nll lyt teGts\n1)( )(0))(,( )(12)( \n (30) \n \n3 Validation of the model for cyclic loadings  \nThe nonlinear model will be validated for 1D cyclic loadings first (homogeneous stress -strain \nstate) directly solving Eq s (28), (29) and (30) . The analysis of seismic wave propagation will \nbe considered  afterwards.  \n \nThe c yclic loading s correspond to sinusoidal excitations at various strain levels. The nonlinear \nparameter is chosen as =1000 and the elastic shear modulus is G 0=80MPa. The relaxation \nparameters may then be computed considering Eqs (28) and (29 ) with the following asymptotic \ndamping values : 0=0.025 and max=0.25. In Fig . 4, some of t he results (obtained at 10Hz) are \ndisplayed  as stress -strain loops for max=10-5, 10-4, 5.10-4 and 10-3. For each case, the secant \nshear modulus G is calculated and  normalized by G0 (the ratio r=G/G0 is given in each curve).  \n \nThe first case  (Fig.  4, top left),  correspond ing to max=10-5 and r=0.99 , leads to a nearly linear \nresponse with an elliptical stress -strain loop. In the 2nd case, max=10-4 and r=0.91 (Fig.  4, top \nright), the area of the loop  is larger and there is a slight decrease of the shear modulus. For the \nlargest excitations (max=5.10-4; r=0.77 ) and (max=10-3; r=0.50 ) (Fig.  4, bottom), the nonlinear \neffects are obvious since the stress -strain loops are strongly modified  (secant  modulus , area, \netc). Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  8 \n0\n00\n0-1000-50005001000 (Pa)\n \n  \n-2.10402.104- 10-5\n-5.10-4-10-4\n-10-310-5\n5.10-410-4\n10-3\n (Pa)\n- 4.104- 2.10402.1044.104\n-500005000\nr=0.99 r=0. 91\nr=0.77 r=0.5NM\n1st loading \nFig. 4. Stress -strain curves from cyclic loadings  of variable maximum amplitudes (r=G/G 0) at \n10Hz: nonlinear extended NCQ model (solid) and 1st loading curve (dashed) . \n \nFrom these loops, it is straightforward to derive the secant shear modulus as a function of \nmaximum shear strain. For each loading level, the dissipation may also be quantified by \ncalculating the ratio between the area of the stress -strain loop and the strain energy estimated \nfrom the first loading curve  (up to the maximum shear strain max). The damping ratio  may be \neasily derived from this energy ratio as a function of maximum shear strain  (Kramer, 1996) . \nThe actual G(max) and (max) curves are then compared to the theoretical curves in Fig.  5. The \neffective shear modulus (solid) is very close from the theoretical one  (dotted). For the damping \nratio, the difference is larger for large shear strains, but the effective dissipation increases as \nexpected.  \n \n00.10.20.30.4\n02468x 107\n10-510-410-310-2theoretical G( ) & ( )\nNM:  actual G( ) & ( )  \n  G( ) (Pa) ()\n \nFig. 5. Comparison of the shear modulus and damping values (%) of the extended NCQ model  \nunder cyclic loadings (solid)  with the theoretical variations predicted by Eq s. (18) and (27) \n(dashed) . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  9 Numerical implementation  (FEM)  \nThe mechanical model described above is introduced into the framework of the Finite el ement \nmethod, for the case of a  unidirectional s hear loading. Let us consider a  homogeneous layer \nover an elastic bedrock as depicted in Fig. 6. \n0\nhalluvial layer\nV =200m/s\n=2000kg/mL\nL3N\nN-1\nN-2\n3\n2\n1e( N- 1)/ 2\ne1=0\n V (2v -v )1 =S S input\nZ(m)bedrock\nV =400m/s\n=2000kg/mS\nS3\n \nFig. 6. 1D soil layer over an elastic bedrock: finite element discretization an d absorbing \nboundary condition at the interface.  \n \nThe domain is divided into (N-1)/2 linear quadratic finite elements, each of  the N nodes having \n1 degree of freedom  (horizontal motion) . Using square brackets […]  and braces {…} to denote \nmatrices and vecto rs, the discretiz ed equation of motion can be written in the following form at \neach time step (n+1)t: \n \n\n    \n   \nmax 1n )( )()( )(1 1 1 1 1\n,1    ;   )u( )( )()]([ ][ ][\nl l Ht tF u uK vC aM\nl l l ln n n n n\n\n (31) \nwhere [ M], [C] and  [K(un+1)] represent the mass, the radiation condition at the bedrock/layer  \ninterface (elastic substratum), and the stiffness matrix  respectively . {an+1}, {vn+1} and  {un+1} \nare the acceleration, velocity and displacement vector  respectively , while { Fn+1} is the vector \nof external  force s at the inter face. (l) and (l) are the relaxatio n parameters and central \nfrequencies of the rheological cells (resp.), H(l)(un+1) corresponds to the right hand -side term in \nEq. (29) and lmax is the total number of cells included in the model  (lmax=3 herein) . \nFor the time integration, an extension of the Newmark formulation is used,  namely an  \nunconditionally stable implicit -HHT scheme  (Hughes, 1987) . This scheme  allows a control \nof the highe r frequencies generated during the propagation  (Semblat and Pecker, 2009) . At \neach time step, the Newton -Raphson iterative algorithm is adopted to deal with the nonlinear \nnature of the f irst equation in system (31). The Crank -Nicolson procedure (Zienkewicz, 2005) \nis simultaneously used in order to estimate the (l)(t) variables in the first order differential \nequation s (system  (31), bottom ). \n4 Modeling wave propagation in the nonlinear range  \n4.1 Nonlinear layered model  \nWe performed two different types of simulations: linear attenuating model ( denoted “ LM”) and \nnonlinear extended NCQ model ( denoted “ NM”). For the  first one (0=max=2.5% and ), \nthe mechanical and dissipative properties of the material do not depend on the excitation level \nwhile, in the second case ( 0=2.5%, max=25% and  ), both elastic and dissipative \nproperties are function of the induced str ain as  shown in Figs 3 and 5. \nFor both models, we  performed simulations for a 20 m deep soil layer over a n elastic bedrock, \nwith a velocity contrast of 2 and an absorbing condition at the bottom of the layer (Fig. 6). The Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  10 excitations considered hereafter thus cor respond to the incident wavefield at the top of the \nbedrock.  \n4.2 Sinusoidal incident wavefield  \nIn this section, the incident wavefield  is a double  sine-shaped  acceleration wavelet similar to \nthat proposed by Mavroeidis and Papageorgiou (Mavroeidis and Papageor giou, 2003; Semblat \nand Pecker, 2009). It is defined by the following equation:  \n\n\n\n\n00sin) sin()(tt ta\n  with  \n0 02f  and \nHz f30  (32) \nThe total duration of the resulting signal is about 2 seconds.  \nIn Fig. 7, taking into account the velo city contrast, a comparison is shown in terms of \nacceleration time histories and corresponding Fourier spectra at the top of the soil layer for two \nexcitati on levels  (0.5 and 0.75 m/s2). The nonlinear time histories involve  propagation  time \ndelay s when  compared to the linear ones, as it can be easily observed  by comparing the peaks \narrival ti mes for both models in  Fig. 7. In the latter case, the Fourier spectra of t he nonlinear \nsignals indicate:  \n1) a significant decrease o f the spectral amplitude , with increasing excitation level, for the \nmain  frequency components of the input signal;   \n2) the generatio n of higher frequenc y peaks which are not  contained in the input signal \n(around 3 and 5 times the predominant excitation freque ncy). Such higher frequency \ncomponents are larger for stronger excitations  (bottom)  ; \n3) a frequency shift of the largest peaks to lower frequencies for increasing excitations . \nThe shear strain at the center of the layer is also plotted in Fig.  8 (left) for b oth excitation \nlevels. Similar time delays are observed in the time -histories. From the stress -strain paths \n(Fig.  8, right), the reduction of the shear modulus and the energy dissipation are found to be \nlarger for peaks of increasing amplitudes. The larges t effect is obtained for the strongest \nexcitation (Fig.  8, bottom right).  \n \n-1.5-1-0.500.511.5\n0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.500.10.20.30.4\n0 5 10 1500.10.20.30.4NM: LM: \n=1000 \n0=2.5%0=2.5%\nm a x=25%\ntime (s )accele ration (m/s )2\nFFT (m/s) FFT (m/s)acce leration (m/s)2\nfrequency (Hz)\n \nFig. 7. Accelerations (left) and corresponding Fourier spectra (right) at the top of the soil layer, \nfor 2 values of t he maximum input acceleration on bedrock 0.5 (top) and 0.75 m/s2 (bottom) : \nlinear (LM , dotted ) and nonlinear (NM , solid ) simulations.  \n Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  11 \n-1-0.500.51x 1 0-3(t)\n0 1 2 3-1-0.500.51x 1 0-3\ntime (s)(t)-505x 104() (Pa)\n-1 -0.5 0 0.5 1\nx 10-3-505x 104\n() (Pa)LM\nNM\n1st loading \nFig. 8. Shear strains (left) and stress -strain loops (righ t) at the middle of the soil layer, for 2 \nvalues of the maximum input acceleration o n bedrock 0.5 (top)  and 0.75 m/s2 (bottom) : linear \n(LM, dotted ) and nonlinear (NM , solid ) simulations.  \n4.3 Real seismic input  \n4.3.1 Linear and nonlinear simulations  \nWe use the same m odel as in the previous case ( Fig. 6) but the incident wavefield now \ncorresponds to  the horizontal  acceler ation  recorded at Topanga station during the 1994 M6.7 \nNorthridge ea rthquake ( Fig. 9, top). In the linear case , the results are displayed in terms of  time \nhistory and Fourier spectrum  in Fig. 9 (2nd line ). For the nonlinear case, two different values \nof the nonlinear parameter are chosen: =300 (Fig.  9, 3rd line ) and =600 (Fig.  9, bottom). \nFrom the results of the linear case (2nd line), the incident wa vefield is found to be significantly \namplified at the free surface  in terms of Peak Ground Acceleration  (30%) . Comparing the \nlinear and the nonlinear responses, p eak amplitude s in the time histories and the spectra  appear \nto be  modified. The results of the  nonlinear cases lead to lower amplitudes at intermediate \nfrequencies, whereas nonlinear responses at higher frequencies are generally larger  (Fig. 9, \nright ). It is nevertheless difficult to assess the influence of the nonlinearities for each individual \npeak. A time -frequency analysis is thus proposed in the next section.  \n \nIn the case of the seismic excitation, the stress -strain loops are plotted in Fig.  10 for the linear \nand nonlinear models. When compared to the linear case (Fig.  10 left), the nonlinear c ases \n(Fig.  10 center and right) lead to a strong modulus decrease and a large dissipation increase. \nThe difference between both  values is also significant (e.g. larger loops) showing stronger \nnonlinear effects for the largest  value ( r=0.27 for =600 and r=0.47 for =300).  \n Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  12 \n4\n2\n0\n-2\n-4acc. (m /s²) acc. (m/s²) acc. (m/s²)\n16 18 20 22 24 26 28 30\ntime (s)input input\nlinear linear\n=300 =300\n=600 =600\n \nFFT (m/s) FF T (m /s) FFT (m/s) FFT (m/s)\n0 2 4 6 8 10\nfrequency (Hz)4\n2\n0\n-2\n-4\n4\n2\n0\n-2\n-4acc. (m/s²)4\n2\n0\n-2\n-43\n3\n3\n32\n2\n2\n21\n1\n1\n10\n0\n0\n0 \nFig. 9. Accelerations at the free surface  for the M6.7  Northridge earthquake: time -histories \n(left) and related spectra (right) ; measured signal at Topanga station (top),  linear  simulation  \n(2nd line ) and nonlinear simulations with =300 ( 3rd line ) and =600 (bottom).  \n \n-3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3x 10-3x 10-3x 10-3\n-1.5-1-0.500.511.5x 105\n   (Pa)lin ear =300 =600\n \nFig. 10. Stres s-strain curves at the middle of the soil layer for the M6.7  Northridge earthquake: \nlinear case (left) and nonlinear cases with =300 (center) and =600 (right).  \n \nTime -frequency analysis  \nThe analysis will now be performed in different f requency bands a s defined in Fig.  11. In this \nfigure, the spectral amplitudes are found to be similar for the linear and nonlinear cases in \nfrequency bands (a) and (c), whereas bands (b) and (d) evidence significant differences. These \nfrequency bands are the following: (a) [0 -2.5Hz], (b) [2.5 -4.3Hz], (c) [4.3 -6.3Hz] and (d) [6.3 -\n20Hz]. The time -histories have been (Butterworth -) filtered in each frequency band to make \nthe comparison between the linear and nonlinear cases easier.  Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  13 \nlinear\nnonlinear\n( =600)(a) (b) (c) (d)\n2.0\n1.0\n0.00.0 2 . 0 4.0 6.0 8.0 10.0 12.0 1 4.03.0FFT (m/s)\nfrequency (Hz) \nFig. 11. Fourier spectra of the accelerations at the top of the soil layer ( Fig. 9) for the M6.7  \nNorthridge earthquake in the case of linear (LM, dotted) and nonlinear simulations ( =600, \nsolid) . \n \nThe f iltered accelerograms related to each frequency band are displayed in Fig.  12. The filtered \nlinear time -histories are plotted on the left whereas the nonlinear ones ( =600) are located on \nthe right part.  The comparison of the filtered accelerograms  lead to the following conclusions:  \n1) Frequency band s (a) and (c) : the peak amplitudes of the filtered time -histories in the \nlinear and nonlinear cases are similar. It may also be noticed in the spectra plotted in \nFig. 11. \n2) Frequency band (b) : the discrepancy  between both time -histories is large since the \nlinear response may be 30% larger than the nonlinear one. Such a difference may be \ndirectly seen in the spectra (Fig.  11). \n3) Frequency band (d) : the nonlinear response is now larger than the linear one (up to \n40%) due to the influence of higher order harmonics generated by nonlinear models \n(Van Den Abeele , 2000).  \nFor strong seismic motion, the nonlinear ground response may then be smaller or larger than \nthe linear one depending on the excitation level as well as  the frequency content of the input \nmotion. The nonlinear properties of the soil are also an important governing parameter of its \nseismic response.  \n5 Conclusions  \nA 3D nonlinear viscoelastic model (“extended NCQ”) is proposed to approximate the \nhysteretic behavior  of alluvial deposits undergoing seismic excitations. Such nonlinear features \nas the reduction of shear modulus and the increase of damping are controlled by the variations \nof the 2nd invariant of the strain tensor during multidimensional loading. In the case of a \nunidirectional shear loading, nonlinearity is controlled by only one shear strain component: \nnonlinear elasticity by a hyperbolic law and viscosity by a NCQ model with nonlinear features \n(nearly frequency constant but strain amplitude depende nt). \nThis model allows to account for the generation of higher order harmonics shown in the \nnonlinear case for 1D simulations. At the same time, a reduction of the spectral amplitudes and \na shift to lower frequencies were found for increasing motion amplit udes. The interest of the \nsimplified nonlinear “X-NCQ” model proposed herein is to reduce the computational cost for \nthe analysis of strong seismic motion  in 2D/3D alluvial basins  (small number of constitutive \nparameters) . \n Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  14 \n16 162.02.0\n1.01.01.0\n-0 .5\n1.0\n0.00.00 .5\n-1 .00.0\n0.0\n-1 .0-1 .0\n-2 .0\n-1 .0\n18 18 20 20 22 22 24 24 26 26 28 28 30 30(a) linear\n(b) linear\n(c) linear\n(d) linear(a) = 600\n(b) =600\n(c) = 600\n(d) =600acceleration (m/s )2acceleration (m/s )2acceleration (m/s )2acceleration (m/s )2\ntime (s) time (s) \nFig. 12. Accelerations at the top of the soil layer for the M6.7  Northridge earthquake in the \ncase of linear (LM, dotted) and nonlinear simulations ( =600, solid) filtered in different  \nfrequency bands  defined in Fig.  11. \n \nFor example, in the 1D case, the reduction of she ar modulus is controlled by a  hyperbolic law \nwith only one parameter estimated from the experimental knowledge of the G(γ) curve. As a \nconsequence,  the dissipation properties are directly derived from the hy perbolic law and from \ntwo other characteristic parameters responsible for the min imum and maximum loss of energy  \nat lower and larger strain levels , 0and max. These are sufficient to give an overall description \nof the unloading and reloading phases durin g the seismic sequence . Combined with the \nnonlinear properties of the soil  in the simplified model , the frequency content of the seismic \ninput has an important influence on strong ground motions . Finally, the proposed model will \nallow  future computations i n the case of 2D or either 3D alluvial basins for which the \namplification is generally found to be much larger than predicted through 1D analyses (Chaillat \net al., 2009; Chávez -García et al., 1999 ; Fäh et al., 1994;  Gélis et al., 2008 ; Lenti et al., 200 9; Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  15 Moeen -Vaziri and Trifunac, 1988 ; Sánchez -Sesma and Luzón, 1995 ; Semblat  et al. , 2000 , \n2005 ). Several authors proposed some 2D/1D aggravation factors (Makra et al., 2005; Semblat \nand Pecker, 2009 ), but it is probably not sufficient for strong seismic motion s involving \nsignificant nonlinerities in the soil response . \n \nAPPENDIX : \nEmmerich and Korn’s method to find the optim al parameters of the  linear viscoelastic “NCQ” \nmodel  is presented in this appendix . \nWe consider the viscoelastic model depic ted in Fig. 1 (left). To estimate the a (l) coefficients, a \nnormal ization condition is introduced : \n)( )0,( l\nRl aMMy\n (33) \nThe (l)/(l-1) ratio being chosen constant , Eq. (6) is simplified  as: \n\n\nn\nll ll\nly Q2\n)()(\n)0,(1\n/ 1/)(\n\n (34) \nThe y (l,0) quantities are estimated by using Eq. (34): writing it for different  and for several \nfixed values of (l) and taking the first term equal to a given const ant value, the obtained \nalgebraic linear system can be solved by a least -squares algorithm. An example of the result of \nthis pr ocedure is displayed in Fig. 2: in the case of =2.5% (Q=20) and a velocity of 200m/s. A \nnormalization condition allows to choose  a target phase velocity (200m/s) at a given reference \nfrequency (1Hz in the example).  \nFor more details, the readers may refer to Emmerich and Korn (1987).  \n \nAcknowledgements : \nThe authors would like to thank  Luis F. Bonilla (IRSN) for fruitful discussions. This work was \npartly funded by the French National Research Agency in the framework of the “ QSHA ” \nresearch project (“Quantitative Seismic Hazard Assessment”).  \n \nNotations:  \nThe following symbols are used in this paper : \n \n{a} = acceleration vector (FEM)  \n[C] = damping matrix (FEM)  \nc(|oct|) = weighting function for non linear damping  \ne = shear  deviatoric strain tensor  \neij() = Fourier transforms of the components of the deviatoric strain  \nekk = volumetric strain  \n{F} = external force vector (FEM)  \nf = frequency  \nG0 = (unrelaxed ) shear modulus at low strains  \nI’1 = first invariant  of the strain tensor  \nI’2 = second invariant  of the strain tensor  \nJ2 = second invariant of the deviatoric strain tensor  \nK = bulk modulus  \n[K] = tangent stiffness matrix (FEM)  \nM() = complex vis coelastic modulus  \n[M] = mass matrix (FEM)  \nMR = relaxed modulus  \nMU = unrelaxed modulus  Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009  \nDelépine, Lenti, Bonnet, Semblat  16 MU,0 = unrelaxed modulus  at low strains  \np = volumetric tension  \nQ = quality factor  \nQ-1\n = specific attenuation  \ns = shear  deviatoric stress tensor  \nsij = components of the de viatoric stress tensor  \nsij() = Fourier transforms of the components of the deviatoric stress  \n{u} = displacement vector (FEM)  \n{v} = velocity vector (FEM)  \ny(l,0) = relaxation parameters  of the viscoelastic cells for low excitation levels  \n = scalar paramete r characterizing the modulus reduction  \noct = octahedral strain  \nij = Kronecker unit tensor components  \nM = difference between the relaxed and unrelaxed modul i \nl(t)  = relaxation functions  \n0 = minimum damping at low strains  \nmax = maximum damping at large strains  \nij = components of the Cauchy stress tensor  \n = function characterizing the modulus reduction  \n = circular frequency  \n \nREFERENCES  \nAubry  D., Hujeux J.C., Lassoudire F., Meimon  Y. 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However, we show that the optical  vortex trap can exhibit \ncomplex force constants, implying that th e trapping must be stabilized by ambient \ndamping. At different damping levels, particle shows remarkably different dynamics, such as stable trapping, periodic  and aperiodic orbital motions. \n Optical tweezers is a powerful tool for trapping mesoscopic objects. Applications \nrange from the trapping and cooling of atoms, to large molecules such as DNA, and to microscopic particles and biol ogical objects. As new methods  to create beam profiles \nare being introduced, more exotic beams are used to trap particles and many of them, \nknown as optical vortex (OV), carry angular  momentum (AM). We will show that \nalthough the conventional stiffness constant approach works for a Gaussian beam, the \ntheory of trapping by an OV is more co mplex and interesting. In the conventional \napproach, optical traps are us ually characterized by three stiffness constants along the \nthree principal axes, which are usually taken to be the Cartesian axes (e.g., x-polarized, z-propagating beam ). Although such approach is pre tty accurate in describing the \nordinary optical tweezers, a mo re rigorous treatment reveals that the principal axes are \nnot necessarily given by the Cartesian axes and, for an OV , the pr incipal axes are not \neven “real”. The rigorous theoretical proced ure to obtain the principal axes is to \ndiagonalize the force constant matrix ,/ij light i jKf x=∂∂  at equilibrium, where , light if  \nand xi are, respectively, the i-th Cartesian component of th e optical force and particle \ndisplacement away from the equilibrium positio n. It is the eigenvalues of the force \nconstant matrix that give the eigen force constants (EFC, or trap stiffness), while the eigenmodes determine the principal axes. We shall apply the force constant matrix formalism to study OV trapping [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, 19,20,21,22,23,24], and we see that the \ndifference between the conventional appr oach and the rigorous treatment is a \nqualitative one, to the extent that the EF Cs can be complex numbers and we have to \nabandon concepts such as pa rabolic potential for the tr ansverse directions. By \nanalyzing the stability  and simulating the dynamics of a particle trapped by OVs, it is \nfound that the trapping stabilit y of OVs generally depends on the ambient damping. In \nparticular, in the presence of AM, the op tical trapping may exhibit a fascinating \nvariety of phenomena ranging from “opt o-hydrodynamic” trapping (where the \ntrapping is stabilized by the ambient damp ing) to supercritical Hopf bifurcation \n(where a periodic orbit is created as ambient damping decreases). \nTo illustrate the basic idea, let us starts from the linear stability analysis. \nConsider a trapped particle, near an equilibrium trapping (zero-force) position, the \noptical and damping forces [25] are \n22/ / F m d x dt K x d x dt γ =∆ ≈ ∆ − ∆I K K, where m is \nthe mass of the particle, x∆K is the particle’s displacement from the equilibrium, \nKx∆IK is the optical force, and γ is the ambient damping constant. The eigenvaluesiK’s of the force matrix KI\n are precisely the EFCs and the eigenvectors of \nKI\nare the eigenmodes. For a trapping beam propagating along ˆz, the force constant \nmatrix has the general form  \n 0\n0ad\nKg b\nefc⎡ ⎤\n⎢ ⎥=⎢ ⎥\n⎢ ⎥⎣ ⎦I\n, (1) \nwhere all elements are real numbers. The elements ,/light xf z∂∂and,/light yf z∂∂  are \nzeros by symmetry, because th ere is no induced force along the transverse plane as the \nparticle is displaced along ˆz. The diagonal elements a, b, and c characterize three \nrestoring forces, which are usually take n as the three stiffness constant in \nconventional approach. Off diagonal elements d and g characterize the rotational \ntorques: i.e. as the particle is displaced along x (y), it experiences a torque that \nmanifests as a force along the y (x) direction. As an OV carries orbital AM, the energy \nof the beam propagates along a helical path. There is a rotating energy flux in the \ntransverse plane that would exert a torque on the particle, implying non-zero d and g. \nOn the other hand, 0 dg==  for beams that carry no AM. \n By diagonalizing Eq. (1), we obtain three EFCs: axialKc=  and \n2() 4/ 2transKa b a b d g±⎡⎤=+ ± −+⎣⎦. Conservative mechanical systems can be \ndescribed by a potential energy U, and their force constant ma trix is a real symmetric \nmatrix (i.e.2/ij i j jiKU x x K=−∂ ∂ ∂ = ), which then follows that  all eigenvalues are real. \nHowever, optical force is non-conservative as the particle can exchange energy with \nthe beam, and its KI\n is in general a real but non-sy mmetric matrix. Consequently, its \neigenvalues can carry conjugate  pair of complex numbers. It can be seen that when \n 24( )dg a b−> − , (2) \ntransK± are a conjugate pair of complex numb ers. Thus, in general, an optically trapped particle may not be characterized by three real force constants if the beam \ncarries AM. For beams that have no AM, 0 dg==, all the EFCs are real numbers. On \nthe other hand, an OV beam, well known for its AM carrying char acteristic, leads to \nnon zero d and g, and thus (2) can be fulfilled under certain conditions. The existence \nof complex EFCs implies that the common notion of a parabolic  potential in an \noptical trap is no longer meaningful. Wh ether complex EFC occurs depend on the \ncompetition between the beam asymmetry and the AM. Equation (2) cannot be \nfulfilled when | | ab− is large. Since a (b) is the restoring force constant for the x- \n(y-) direction, a large | | ab− implies a large asymmetry between the two coordinate \naxes. The underlining physics is  that for large asymmetry, the beam can pin a particle \nto one of its axis, preventing it  from “falling” into the OV . On the contrary, if there is \nweak or no asymmetry (| | 0 ab−\u0011), the trapped particle is not tied to the coordinate \naxis, and will thus be swiped by the OV . As a result, its AM and energy will \naccumulate. If there is no dissipation ( 0 γ=) the particle will orbit around the beam \ncenter with increasing speed and eventually escape from the trap [26]. It is therefore \nconcluded that in general the OV trapping cannot be achieved solely by light. It is the \ndissipation in the suspending medium that k eeps the particle’s kinetic energy and AM \nbounded, rendering the particle tr apped [26]. Such a state of  OV trapping is believed \nto be what the experiments observed, instead  of the pure gradient  force trapping. For \nthe case of circularly polarized beams (L G or Gaussian), cylindrical symmetry \nmandates that| | 0 ab−= , implying that the transverse EF Cs are always complex. This \nmeans that circularly pol arized beam cannot trap without dissipation. More \nmathematical details can be found in on-line materials [34]. \nWe now proceed to show concrete examples in which the EFCs are indeed \ncomplex. We model the incident trappi ng beam by using the highly accurate \ngeneralized vector Debye integral [27,28], where the focusing of the incident laser beam by the high numerical aperture (N.A.) object lens is treate d using geometrical \noptics, and then the focused field near the focal region is obtained using the angular \nspectrum representations. The use of geom etrical optics in beam focusing is fully \njustified as the lens is macroscopic in si ze, and all remaining parts of our theory \nemploys classical electromagnetic optics. W ith the strongly focused beam given by \nthe vector Debye integral, the Mie theory is then applied to calculate the scattered field, and then the Maxwell stress tensor formalism is applied to compute the optical force. [29,30] We note that the formalisms  we use have been proven to agree well \nwith experiments [28,31,32,33]. Fig. 1\n(a) shows transK± for an LG beam focused by \na high N.A. water immersion objective. Th e beam is linearly polarized with \nwavelength 1064 nmλ= , topological charge l=1, N.A.=1.2, and filling factor f=1. \nThe trapping is in water (21.33waterε= ) and the sphere is made of polystyrene \n(21.57sphereε=  and mass density31050 kg mρ−= ). The axial EFC is not plotted, as it \nis always real and negative, indicating that  the particle can be trapped along the axial \ndirection solely by gradient for ces. It can be clearly seen from  Fig. 1(a) that at certain \nranges of particle sizes, Im{ } 0transK±≠, and this numerical results manifest the \nassertion that the OV trap cannot be characterized by th ree real force constants in \ngeneral. At these particle sizes, *\ntrans transKK+ −= , and the two curves corresponding to \nRe{ }transK± merge together. Note that only the absolute value of Im{ }transK± is \nplotted in Fig. 1(a). When the EFCs are all real numbers, the behavior of the trapped \nparticle is qualitatively sim ilar to that of the ordinary optical trapping by conventional \noptical tweezers. This corresponds to the scen ario that either the beam’s AM is weak \n(small d and g), or the asymmetry of the beam (| | ab−) is large. We note that the \nexistence of region where Im{ } 0transK±≠ implies that in low viscosity media, only \nparticles of certain sizes can be tra pped. For complex EFCs, the eigenmodes corresponding to the co mplex EFCs are [34]: \n Im( )Re( )sin[Re( ) ]()\nIm( ) cos[Re( ) ]tVtxt A e\nVtφ\nφ±±± −Ω\n±±\n±±⎧ ⎫ Ω+ ⎪ ⎪∆= ⎨ ⎬+Ω +⎪ ⎪ ⎩⎭K\nKK , (3) \nwhere VK is the eigenvector co rresponding to eigenvalue Ki of the force constant \nmatrix KI\n. {},Aφ±± are to be determined from initial conditions, and  \n 22 1 / 4\n22 1 / 4Re( ) ( ) sin( / 2) / 2 ,\nIm( ) ( ) cos( / 2) / 2 ,RI\nRIm\nmδ\nγδ±\n±Ω=∆+ ∆\nΩ=± ∆+ ∆∓ (4) \n ()\n()1\n1tan / , if 0,\ntan / , if 0,IR R\nIR Rδπ−\n−⎧∆∆∆ > ⎪=⎨−∆ ∆ ∆ <⎪⎩ (5) \nwhere 4 Im{ }Ii mK∆= and 24R e {} .Ri mKγ∆= +  The modes are stable if and only \nif both Im( ) 0±Ω> . If{} Re 0iK>, one of the two modes is always unstable such that \nupon small perturbation, the particle will  spiral outward and leave the trap. \nIf{} Re 0iK<, the mode is unstable for {} {} Im / Recritical i i mK K γγ<= . \nHowever this equilibrium can be stabilized by increasing γ to beyond criticalγ . We \nlabel this kind of mode as quasistable modes, in which the stability of the modes \ndepends on the ambient damping. The complex modes described by Eq. (3) correspond to spiral motions, which means that  the particle is absorbing AM from the \nbeam. The converse is also true: these spir al modes can exist only when the particle \ncan absorb AM.  \nWe note in Fig. 1 that in our specific example,\n{} Re 0transK±> for sphere \nradius 0.36 R mµ<  (radius of the intensity ring ~ 0.33 mµ, see Fig. 2(a)), which \nmeans that small dielectric particles are uns table, as reported in experiments [14,35]. \nThe small dielectric particles are attrac ted by the high intensity ring and under \nsufficient damping, these small particles will orbit along the ring [5].  On the other hand, {} Re 0iK< for 0.36R mµ> . The sphere is bigger than the intensity ring, so \nthat the gradient force drives the sphere to the beam center. A phase diagram for the \noptically trapped particle is given in Fig. 1(b). At 0.36amµ= , criticalγ→∞  \nas{} Re 0transK±→ . The equilibrium point at (, ) ( 0 , 0 )xy=  is unstable for \n0.36amµ<  at any values of damping. For 0.36µm a> , the white (shaded) region \nwhere criticalγγ>  (criticalγγ< ) is the regime where the damping is sufficient \n(insufficient) to stabilize th e particle. We note that when  the EFC is real, no damping \nis required for stability since 0criticalγ=. According to Stoke’s law, the damping \nconstant of water and air are, respectively, 421.9 10  (pN s/ m ) Rµµ ×  \nand223.3 10  (pN s/ m ) Rµµ × . The damping of water is much larger than criticalγ  \nplotted in Fig. 1(b), and thus unless one  uses high laser power, one shall observe \nstable trapping in water, in agreement with  existing experiments. The damping of air \nis of the same order of magnitude of criticalγ  plotted in Fig. 1(b) , consequently for an \nexperiment conducted in air, one shall be able to see the transition between the stable \nand unstable state, depending on the laser power employed.  \nIt is now clear that a particle trapped by an OV is stable if criticalγγ>  and \nunstable ifcriticalγγ< . Nevertheless, the case of criticalγγ≈  is non-hyperbolic (the \nlinear term vanishes), and thus the higher order terms ar e important. In that case, we \nnumerically integrate th e full equation of motion 22/ /light md x d t f d x d t γ ∆= − ∆KK K, \nusing an adaptive time-step Runge-Kutta-Ver ner algorithm [29]. Fi g. 2(a) shows the \nfield intensity on the focal plane for a right circularly polarized LG  beam, with a dark \ncentral spot and a high intensity ring of radius ~0.33 µm. Fig. 2(b)-(f) show the \ntrajectories of a 1- µm -diameter particle illuminated by the LG beam (power=550 \nmW), in the order of decreas ing damping. When there is strong damping, as shown in Fig. 2(b) where 550 pN s/ m γ µµ= , the trapped sphere exhi bits damped oscillation \nupon small perturbation and settles into a stable equilibrium position. For weaker \ndamping, the sphere initially spirals outward , and then settles into  a periodic circular \norbit (see Fig. 2(c) where 110 pN s/ m γ µµ= ). Such bifurcation of a stable \nequilibrium into an unstable equilibrium and a stable periodic orbit is known as a \nsupercritical Hopf bifurcation [29]. If we further reduce the damping, the radius of the \ncircular orbit increases, as show n in Fig. 2(d) where 55 pN s/ m γ µµ= . If the damping \ndecreases further, the particle goes into an  exotic orbit around the intensity ring, as \nshown in Fig. 2(e) where 5.5 pN s/ m γ µµ= . When there is no damping (see Fig. \n2(f), 0 pN s/ mγµµ= ), the particle initia lly fluctuates around the equilibrium with \nincreasing amplitude, and eventually escapes from the trap due to the accumulation of \nAM. If a small imaginary part is introduced into the dielectric constant of the particle, the introduced absorption will compete with  the light scattering, reducing the amount \nof AM that is transferred to the orbital motion of the particle, and the particle will now spin along its own axis. In other words, small absorption may in fact favor the \ntransverse trapping, though it degrades the axial trapping. \nOur analysis reveals that for an AM carrying beam, its EFCs can be complex \nnumbers. In the case of complex EFCs, when there is sufficient (insufficient) damping a particle can (cannot) be stably trapped. Th ere is an intermediate range of damping in \nwhich the particle will be driven into e xotic periodic or aperiodic orbital motions. \nFinally, we note that as the ambient damping force plays an important role in the OV \ntrapping, it should be more accurately termed “opto-hydrodynamic trapping”.  \nWe have also applied the stability analys is to other types of  focused beams [34] \nwith different N.A., and we find that we  can observe complex EFCs whenever the \nbeam carries AM. \nThis work is supported by Hong Kong RGC grant 600308. ZFL was supported by NSFC (Grant number 10774028), PCSIRT, an d MOE of China (B06011). Jack Ng \nwas partly supported by N SFC (Grant number 10774028).  \n012-4-20(a)\n   \n Radius (µm)Ki ( pN µm-1 mW-1)\n Re( Ktransverse+)\n Re( Ktransverse-)\n |Im( Ktransverse)|\n0120300600\n   \nγ ( pN µm-1 µs )\nRadius (µm)(b)\n \nFig. 1 The incident beam is a linearly polarized LG beam with 1064 nmλ= , l=1, f=1, \nand N.A.= 1.2.  (a) The transverse EFCs. (b) Phase diagram for a particle trapped at a \npower of 1W. The white (red) regions are unstable (stable). The black line \nmarkscriticalγ .  \nFig. 2 (a) The focal plane intensity (arbitrary units) of a right polarized LG beam \nwith 1064 nmλ= , l=1, f=1, and N.A.=1.2. (b)-(f)  The trajectory (blue) of a \n1-micron-diameter particle trapped by a 550 mW beam. Th e red dotted lines are the \napproximate radius of the intensity ring of  the trapping beam. The damping constants \nγ for each panel, in unit of pN s/ m µµ, are given by (b)550, (c)110, (d)55, (e)5.5, \nand (f)0. The arrows in (b) and (f) indicate the direction of motion. \n \n                                                 \n1 A. T. O’Neil and M. J. Padgett, Opt. Commun. 185 139 (2000). \n2 H. Rubinsztein-Dunlop et al. , Adv. Quantum Chem.  30, 469 (1998). \n3 K. T. Gahagan and G. A. Swartzlander, JOSA B  15, 524 (1998) ; ibid, ibid 16, 533 \n(1999).  \n4 L. Allen et al. , Phys. Rev. A  45, 8185 (1992). ibid, Optical Angular Momentum  (IOP \nPublishing, London, 2003). \n5 V . Garces-Chavez et al. , Phys. Rev. A  66, 063402 (2002). \n6 A. T. O’Neil et al. , Phys. Rev. Lett. 88, 053604 (2002).  \n7 M. Funk et al. , Opt. Lett. 34, 139 (2009). \n8 J. Courtial et al. , Opt. Comm.  144, 210 (1997).  \n9 S. M. Barnett and L. Allen,  Opt. Commun.  110, 670 (1994).                                                                                                                                             \n10 A. T. O’Neil and M. J. Padgett, Opt. Commun. 185 139 (2000). \n11 H. He et al. , Phys. Rev. Lett. 75, 826 (1995). \n12 M. E. J. Friese et al. , Phys. Rev. A  54, 1593 (1996). \n13 M. E. J. Friese et al. , Appl. Opt. 35, 7112 (1996).  \n14 N. B. Simpson et al. , Opt. Lett.  22, 52 (1997). \n15 N. B. Simpson et al. , J. of Modern Opt. 45, 1943 (1998). \n16 A. T. O’Neil and M. J. Padgett, Opt. Comm. 185, 139 (2000). \n17 A. Jesacher et al. , Opt. Exp.  12, 4129 (2004). \n18 K. Ladavac and D. G. Grier, Opt. Exp.  12, 1144 (2004). \n19 D. G. Grier, Nature  424, 810 (2003). \n20 K. C. Neuman and S. M. Block, Rev. of Sci. Instrum. 75, 2787 (2004). \n21 K. Dholakia et al. , nanotoday  1, 18 (2006). ibid, Chapter 6 of Advances in Atomic, \nMolecular, and Optical Physics, V olume 56 (2008). \n22 D. Cojoc et al., Micro. Eng.  78-79 , 125-131 (2005).  \n23 Adrian Alexandrescu, Dan Cojoc, and Enzo Di Fabrizio, Phys. Rev. Lett. 96, \n243001 (2006). \n24 D. S. Bradshaw and D. L. Andrew, Opt. Lett.  30, 3039 (2005). \n25 Here we neglect the thermal fluctuation, as it is small compare to the optical force \nfor an intense laser. \n26 N. R. Heckenberg et al. , “Mechanical effects of optical vortices,” M Vasnetsov \n(ed.) Optical Vortices (Horizons in World Physics) 228 (Nova Science Publishers, \n1999) pp 75-105. \n27 L. Novotny and B. Hecht, Principles of Nano Optics (Cambridge University Press, \nNew York, 2006). \n28 Y.  Z h a o  et al. , Phys. Rev. Lett.  99, 073901 (2007). \n29 J. Ng et al. , Phys. Rev. B  72, 085130 (2005). \n30 M. I. Antonoyiannakis and J. B. Pendry, Phys. Rev. B  60, 01631829 (1999). \n31 A. Rohrbach, Phys. Rev. Lett.  95, 168102 (2005). \n32 N. B. Viana et al. , Appl. Phys. Lett.  88, 131110 (2006); ibid, Phys. Rev.  E 75, \n021914 (2007). \n33 A. A. Neves, et al. , Opt. Exp. 14, 13101 (2006). \n34 See EPAPS Document No. []  for a discussion on (I) linear stability analysis, (II) \nthe eigen force constant for various type of trapping beams, and (III) the optical trapping by cylindrically symmetric beams. \n35 For the trapping of a strongly absorptive pa rticle, the particle is unstable along the \naxial direction. Accordingly, other forces, su ch as a repulsive force from a substrate, \nare needed to stabilize the particle. However, in this letter, we consider the transverse \noptical trapping, thus our conclusion is va lid irrespective to the nature of the axial \ntrapping. Appendix I: Linear Stability Analysis  \nIn this appendix, we give more details a bout the formalism on the linear stability \nanalysis for a particle trapped by an arbitrary incident light beam. \nA. Linearized equation of motion \nWe denote the displacement of the particle  away from the equilibrium position by the \nposition vector (,,)x xyz∆=∆∆∆K. The equation of motion of the particles are given \nby \n 2\n2()lightdx d xmf xdt dtγ∆ ∆=∆ −K KKK, (I.1) \nwhere m is the mass of the particle, ()lightf x∆KK is the optical force, γ is the damping \nconstant for the particle in the suspending medium. The frictional term in (I.1) is \nadded to account for the Stoke’s drag be tween the particle and the suspending \nmedium, and we have deliberately neglected th e Brownian term in (I.1), as it is of \nsignificance only at low laser power. If the displacement x∆K is small compare to the \nwavelength of inci dent light (| | xλ∆< <K), it is possible to simplify (I.1) with a linear \napproximation with respect to the displacemen t. The linearized equation of motion is  \n 2\n2dx d xmK xdt dtγ∆ ∆≈∆ −K KIK, (I.2) \nwhere  \n \n0()()light j\njk\nkxfKx\n∆=∂=∂∆ KKKI\n (I.3) \nis the force constant matrix. We note that the zero-th order term in (I.2) vanishes, \nbecause we are expanding the force near an equilibrium where (0 ) 0 .lightfx∆= =K KKK \nIntroducing the transformation \n xVη∆=IKK, (I.4) \nwhere 'isη  are the normal coordinate s, and the columns of VI\n are the eigenvectors of KI\n so that KI\n is diagonalized with eigenvalues Ki’s: \n 1ˆˆT\nii i\niVK V x x K−=∑III\n. (I.5) \nHere ˆix is the unit vector along the Cartesian axes. In a conservative mechanical \nsystem that can be desc ribed by a potential energy U, the force constant matrix is \nsymmetric (i.e.2/ij i j jiKU x x K=−∂ ∂ ∂ = ) and can only give real negative or real \npositive eigen force constants. However, optical force is non-conservative, and KI\n is \nin general a real valued non-symmetric matrix (i.e.ij jiKK≠ ). As such, its eigenvalues \nand their corresponding eigenvectors, can be  real numbers or a conjugate pair of \ncomplex numbers. After substituting (I.4) into (I.2), the equation of motion is now \ndecoupled into three independent equations: \n 2\n2ii\niiddmKdt dtη ηηγ=− . (I.6) \nEquation (I.6) is a second order linear ordi nary differential equation, which may be \nsolved by the standard technique of substituting \n 0iit\nii eηηΩ= . (I.7) \nwhere 0iη and iΩ are independent of time. It turns out that the solution to (I.6) can \nbe categorized according to the eigenvalues iK or equivalently the natural frequency \nof the eigenmode defined as  \n 0 /ii KmΩ=− . (I.8) B. Types of Eigenmodes \n1. Unstable mode characterized by an imaginary natural frequency  \nIf Ki is real and positive, the corresponding na tural frequency is purely imaginary and \nthe corresponding solution is \n 22 22(/ 2 ) (/ 2 ) 00/2()mt mt iitm\nii i ixt e V A e B eγγγ−+ Ω + Ω−⎡⎤∆= +⎢⎥⎣⎦K L,      ( I . 9 )  \nwhere Ai and Bi are unknown constants to be determ ined from the initial conditions. \nThe mode is unstable because (I.9) diverges with time. \n2. Stable mode characterized by real natural frequency  \nIf Ki is real and negative, the natural freque ncy is purely real and the motions of the \nparticles are that of a damp ed harmonic oscillator. For 22\n0 (/ 2)i mγ>Ω , \n 22 22(/ 2 ) (/ 2 ) 00/2()mt mt iitm\nii i ixt e V A e B eγγγ−− Ω − Ω−⎡ ⎤∆= +⎢ ⎥⎣ ⎦K K, (I.10) \nwhere Ai and Bi are unknown constants to be determined from initial conditions. The \noscillation is over damped. For 22\n0 (/ 2)i mγ=Ω, \n []/2()tm\nii i ixte V A B tγ−∆= +K K, (I.11) \nwhere Ai and Bi are unknown constants to be determined from initial conditions. The \noscillation is critically damped. For22\n0 (/ 2)i mγ<Ω, \n /2 2 2\n0 () s i n ( / 2 )tm\nii i i ixt A e V m tγγ φ− ⎡ ⎤ ∆= Ω − +⎣ ⎦K K, (I.12) \nwhere Ai and φi are unknown constants to be determined from initial conditions. The \noscillation is under damped. \nThe trajectories of the solutions (I.10), (I.11) and (I.12) ar e all bounded as time \nincreases, accordingly they are all stable. \n3. Complex mode characterized by  a complex natural frequency \nAs the force constant matrix is non-sy mmetric, a complex conjugate pair of eigenvalues can occur. To obtain the trajector ies associated with the conjugate pair of \neigenvalues iK and *\niK, it suffices to consider only Ki where Im{  Ki }>0. The \nsolutions associated with *\niKare the same as that of Ki. The solutions are \n [ ] [ ] { }Im( )( ) Re( )sin Re( ) Im( )cos Re( )it\nii i i i a i i i axt a e V t V t φ φ+−Ω\n++ +∆= Ω + + Ω +KK K(I.13) \n [ ] [ ] { }Im( )Re( )sin Re( ) Im( )cos Re( )it\nii i i i b i i i bxb e V t V t φ φ−−Ω\n−− −∆= Ω+ + Ω+KK K (I.14) \nwhere { } ,, ,i i ia ibabφφ  are unknown constants to be determined from initial \nconditions, \n {} {} ( )\n{} {} ( )1/42 22 2\n1/ 42 22 2(4 R e) 1 6 I m s i n / 2\nRe( )2\n(4 R e) 1 6 I m c o s / 2\nIm( )2ii i\ni\nii i\nimK m K\nm\nmK m K\nmγδ\nγγ δ±\n±⎡⎤++⎣⎦Ω=\n⎡⎤±+ +⎣⎦Ω=∓\n (I.15) \nand \n {}\n{}{}\n{}\n{}{}12\n2\n12\n24I mtan                 if    4 Re4R e\n4I mtan         if    4 Re\n4R ei\ni\ni\ni\ni\ni\nimKmKmK\nmKmK\nmKγγ\nδ\nπγ\nγ−\n−⎧\n> ⎪+⎪=⎨\n⎪−<⎪+⎩ (I.16) \na) Complex unstable mode \n If Re{ Ki}>0, ( )ixt+∆K is spiraling inward to the equilibrium, whereas ( )ixt−∆K is \nspiraling outward and its displacement diverg es with time. Consequently, an optically \ntrapped particle having a complex Ki with positive real part is unstable and we denote \nthis kind of solution as  complex unstable mode. \nb) Quasi-stable mode \n If Re{ Ki}<0, ( )ixt+∆K is spiraling inward to the equilibrium. Here ( )ixt−∆K \nrequires some attention. The mode is spiraling outward if  Im( )\nRe( )i\ncritical\nimK\nKγγ<= , (I.17) \nbut spiraling inward ifcriticalγγ> . \nWe denote this kind of solu tion as quasi-stable, where the stability depends on the \ndamping provided by the environm ent. We note that the point criticalγγ= is \nnon-hyperbolic, which simply means the lin ear term of the equation of motion \nvanishes, and the higher order terms are need ed. As discussed in the main text, linear \nstability analysis is not sufficie nt to determine the stability atcriticalγγ≈ . Consequently, \nreal time dynamics simulations are performe d and the results are presented in Fig. 2 \nof the main text. Appendix II: The eigen force constant  \nfor various types of trapping beams \nA. The general form of force consta nt matrix and eigen force constants \nFor an incident trapping beam, the general form of the force constant matrix for the \ntrapped particle is \n 0\n0ad\nKg b\nefc⎡ ⎤\n⎢ ⎥=⎢ ⎥\n⎢ ⎥⎣ ⎦I\n, (II.1) \nwhere a, b, c, d, e, f, and g are real numbers, and () /ij light i jKf x=∂∂K\n. Two of the \ncomponents in Eq. (II.1), Kxz and Kyz, are zero, because there is no induced force \nalong the transverse plane as the particle is displaced along the z axis. Here, we \nassume that the optical system including th e focusing lens does not change the axial \nsymmetry of the beam. By diagonalizing KI\n, we obtained the eigen force constants: \n 2,\n() 4/ 2 .axial\ntransKc\nKa b a b d g±=\n⎡ ⎤ =+ ± −−⎣ ⎦ (II.2) \nWhen \n 24( )dg a b−< − , (II.3) \nthe eigen force constants are all real numbers, and thus the nature of the optical \ntrapping by such beam will be qualitatively si milar to that of the conventional optical \ntweezers, i.e. all the eigen vibrational modes are stable  modes (see Appendix I). \nHowever, when \n 24( )dg a b−> − , (II.4) \ntransK± are conjugate pair of complex numbers, and thus they correspond to the \ncomplex unstable mode or quasi-stable mode (see Appendix I).  B. Numerical computation of eigen force constants for a variety of trapping \nbeams. \nIn this section, we present the numerica lly computed eigen force constants for a \nparticle in water trapped by a variety of different incident trapping beams. The \nincident trapping beams include (1) a linear polarized Gaussian beam (Fig. II.1), (2) a \ncircularly polarized Gaussian beam  (Fig. II.2), (3) a linear polarized \nLaguerre-Gaussian beam (Fig. II.3), (4) a ri ght circularly polarized Laguerre-Gaussian \nbeam (Fig. II.4), and (5) a left circularly  polarized Laguerre-Gaussian beam (Fig. \nII.5). \n012-4-20\nRadius (µm)Ki ( pN µm-1 mW-1)  Re( Ktransverse1)\n Im(Ktransverse1)\n Re( Ktransverse2)\n Im(Ktransverse2)\n Re( Kaxial)\n Im(Kaxial)\n \nFig. II.1. The eigen force constants for a particle (21.57sphereε= ) trapped by a linear \npolarized Gaussian beam with f=1, and N.A.= 1.2 in water (21.33waterε= ). 012-4-20\nRadius (µm)Ki ( pN µm-1 mW-1)  Re( Ktransverse1)\n Im(Ktransverse1)\n Re( Ktransverse2)\n Im(Ktransverse2)\n Re( Kaxial)\n Im(Kaxial)\n \nFig. II.2. The eigen force constants for a particle (21.57sphereε= ) trapped by a \ncircularly polarized Gaussian beam with f=1, and N.A.= 1.2 in water (21.33waterε= ). \n012-4-20\nRadius (µm)Ki ( pN µm-1 mW-1)  Re( Ktransverse1)\n Im(Ktransverse1)\n Re( Ktransverse2)\n Im(Ktransverse2)\n Re( Kaxial)\n Im(Kaxial)\n \nFig. II.3. The eigen force constants for a particle (21.57sphereε= ) trapped by a linear \npolarized Laguerre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water \n(21.33waterε= ). \n 012-4-202\nRadius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1)\n Im(Ktransverse1)\n Re( Ktransverse2)\n Im(Ktransverse2)\n Re( Kaxial)\n Im(Kaxial)\n \nFig. II.4. The eigen force constant for a particle (21.57sphereε= ) trapped by a right \ncircularly polarized Lague rre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water \n(21.33waterε= ). \n012-4-20\nRadius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1)\n Im(Ktransverse1)\n Re( Ktransverse2)\n Im(Ktransverse2)\n Re( Kaxial)\n Im(Kaxial)\n \nFig. II.5. The eigen force constants for a particle (21.57sphereε= ) trapped by a left \ncircularly polarized Lague rre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water \n(21.33waterε= ).C. Trapping beams that carry no angular momentum \nFor an incident trapping beam that carries no angular moment, 0 dg==as there is \nno rotating energy flux on the transverse plane (see main text). Accordingly, Eq. (II.4) \ncan never be fulfilled, and thus the eigen force constants are always real numbers. \nFrom another perspective, the complex eige n force constants can only occur when the \nparticle is allowed to excha nge its angular momentum with  the beam (see main text). \nSince the beam carries no angular moment um, complex eigen force constant should \nnot occur. \nThe force constant matrix reduces to  \n 00\n00a\nKb\nefc⎡⎤\n⎢⎥=⎢⎥\n⎢⎥⎣⎦I\n, (II.5) \nand the corresponding eige n force constants are \n 1\n2,\n,\n,axial\ntransverse\ntransverseKc\nKaKb=\n=\n= (II.6) \nwhich are indeed real. The eigenvalues of a linearly polarized Gaussian beam, which \ncarries no angular momentum, are plotted in Fi g. II.1. Clearly, its eigenvalues are real \nnumbers. The nature of optical trapping by a beam that carries no angular momentum \nwill be qualitatively similar to that  of the conventional optical tweezers. \n D. Cylindrically symmetric trapping beams that carry angular momentum \nA cylindrically symmetric optical vortex beam propagates along a helical path, which \ncan drive the trapped particle to rotate (so 0 dg=−≠ ). Moreover, owing to the \ncylindrical symmetry, ab=. As such, the condition Eq. (II.4) is always fulfilled. \nConsequently, the corresponding transverse  eigen force constants are always a \nconjugate pair of complex numbers. \nTo show this explicitly, consider a cylindr ically symmetric optical vortex, such as \na circularly polarized beam (Gaussian or Laguerre-Gaussian ). It can be shown that a = \nb, as the restoring force acting on the particle when it is displaced along the x axis is \nequal to that of the y axis. Moreover, dg=− because the induced torque when the \nparticle is displaced along the x axis is equal to that of the y axis. Finally, ef=\nbecause the induced force along the z axis when the particle is displaced along the x \naxis is equal to that of the y axis. Substituting these expressions into (II.1), we obtain \n 0\n0ad\nKd a\nee c⎡ ⎤\n⎢ ⎥=−⎢ ⎥\n⎢ ⎥⎣ ⎦I\n, (II.7) \nand the corresponding eige n force constants are \n ,\n.axial\ntransKc\nKa i d±=\n=± (II.8) \nFrom Eq. (II.8), we see that complex eigen force constants occur whenever 0d≠, as \nfor any angular momentum carrying beam. In fact 0d≠ indicates that there are \nangular momentum exchange between the beam and the particle, because complex \neigenvalues can exist only when the tr apped particle can exchange angular \nmomentum with the beam (see main text). It  is clear from (II.8) that the equilibrium \ncannot be solely characterized by real optic al force constants. Loosely speaking, a \nparticle in a cylindrically symmetric optical vortex can be considered as \nsimultaneously experiencing a radial restoring force characterized by Re( )transKa= and a torque about the beam’s  axis characterized by Im( )transKd=. \n Fig. II.2, Fig. II.4, and Fig. II.5 show Ki versus the radius of the trapped sphere, \nfor a circularly polarized Gaussian beam, a right circularly polarized \nLaguerre-Gaussian beam, and a left circul arly polarized Laguerre-Gaussian beam, \nrespectively. Before entering the objective lens, the non-fo cused circularly polarized \nGaussian beam carries spin angular moment um due to its polarization state, but not \norbital angular momentum. After the beam is being strongly focused by the objective \nlens, part of its spin angular momentum is converted to orbital angular momentum \n(see main text). The left and right circul arly polarized Laguerre-Gaussian beams have \nboth spin and orbital angular momentum, in the former (later) case, the two forms of \nangular momentum are in opposite (same) dire ction. After focusing,  the spin angular \nmomentum is partially convert ed to orbital angular momentum. In the left (right) \ncircular polarization case, th e resultant angular momentum is small (large), owing to \nthe cancellation (reinforcemen t) between the spin and orbital angular momentum. \nConsequently, Im{ Ktrans} is the largest (smallest) for the right (left) circularly \npolarized Laguerre-Gaussian be am in general, because the spin and orbital angular \nmomentum are reinforcing (cancelling) each others. \nFor all three beams, 'axialKs  are always real and nega tive, indicating that the \nparticle can be trapped along the axial di rection due to gradient forces. On the \ncontrary, transK± are a conjugate pairs of complex numbers. For the Gaussian beam, \n{} Re 0transK±<, indicating that the particle can always be stabilized by introducing \nsufficient damping. On the other hand, for the Laguerre-Gaussian beams, \n{} Re 0transK±>for particles that are smaller than th e intensity ring of the beam, which \nmeans that small dielectric particles are uns table, as reported in experiments. Small \ndielectric particles are attracted toward in tensity maxima. Under sufficient damping, these small particles will be orbiting along the high intensity ring of the beam. On the \nother hand, {} Re 0transK±<for large particle. The spheres are bigger than the intensity \nring, so that the gradient force drives the spheres to the beam center. A phase diagram \nis given in Fig. II.6, for a right circularly polarized  LG beam with wavelength\n1064 nmλ= , topological charge l=1, numerical aperture N.A. =1.2, and filling factor \nf=1. The trapped sphere is in water (21.33waterε= ) and it has dielectric constant\n21.57sphereε=  and mass density31050 kg mρ−= . At radius 0.39R mµ= , criticalγ→∞  \nas {} Re 0iK→. The equilibrium point at (, ) ( 0 , 0 )xy=  is unstable for 0.39R mµ<  \nfor any values of damping. The particle will be trapped in the ring of the beam instead. \nFor 0.39R mµ> , the white region (criticalγγ> ) is the regime where sufficient damping \ncan stabilize the particle, and the shaded region (criticalγγ< ) is where the damping is \ninsufficient to stabilize the particle. \nCompare Fig. II.6 with Fig. 2(b) of the main text, there two major differences. \nFirstly, criticalγ  is always greater than zero for the right circular polarization, whereas \nfor the linear polarization,  criticalγ  can be zero for some particle sizes. This is \nbecause the linear polarization does not posse ss cylindrical symmetric, therefore the \ncondition Eq. (II.4) cannot always be fulfilled.  Secondly, in general, the magnitude of \ncriticalγ  for the right polarization is greater than that of the linear polarization. This is \nbecause the angular momentum of the right  circularly polarized beam comes from \nboth the spin and orbital angular momentum that are reinforcing each other, whereas that of the linearly polarized beam comes fr om the orbital angular momentum only. In \nboth polarization, the envelope for \ncriticalγ  increases linearly for large particle in \ngeneral (see the blue dotted line in Fig. II.6) . This is because the envelope of the force, \nand thus that of the eigen for ce constants, are proportional to R2 (i.e. proportional to the geometrical cross section) for large particle. Then, according to (I.17), the \nenvelope of criticalγ  increases linearly. \n \n01205001000\n  \nγ ( pN µm-1 µs )\nRadius (µm)(b)\n \nFig. II.6 Phase diagram for a particle tra pped at a power of 1W. The white (shaded) \nregions are unstable (stable). The black line iscriticalγ . The incident beam is a right \ncircularly polarized LG beam with 1064 nmλ= , l=1, f=1, and N.A.= 1.2. \n "    },    {        "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": "0910.1477v2.Fast_domain_wall_propagation_under_an_optimal_field_pulse_in_magnetic_nanowires.pdf",        "content": "arXiv:0910.1477v2  [cond-mat.mtrl-sci]  2 Feb 2010Fast domain wall propagation under an optimal field pulse in m agnetic nanowires\nZ. Z. Sun and J. Schliemann\nInstitute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany\n(Dated: December 2, 2018)\nWeinvestigatefield-drivendomainwall (DW)propagation in magneticnanowiresintheframework\nof the Landau-Lifshitz-Gilbert equation. We propose a new s trategy to speed up the DW motion\nin a uniaxial magnetic nanowire by using an optimal space-de pendent field pulse synchronized with\nthe DW propagation. Depending on the damping parameter, the DW velocity can be increased by\nabout two orders of magnitude compared to the standard case o f a static uniform field. Moreover,\nunder the optimal field pulse, the change in total magnetic en ergy in the nanowire is proportional\nto the DW velocity, implying that rapid energy release is ess ential for fast DW propagation.\nPACS numbers: 75.60.Jk, 75.75.-c, 85.70.Ay\nRecently the study of domain wall (DW) motion in\nmagnetic nanowires has attracted a great deal of atten-\ntion, inspired both by fundamental interest in nanomag-\nnetism as well as potential industrial applications. Many\ninteresting applications like memory bits[1, 2] or mag-\nnetic logic devices[3] involve fast manipulation of DW\nstructures, i.e. a large magnetization reversal speed.\nIngeneral,themotionofaDWcanbedrivenbyamag-\nnetic field [4–6] and/or a spin-polarized current[7–11].\nAlthough the DW dynamics in systems of higher spatial\ndimension can be very complicated, some simple but im-\nportant results were obtained by Schryer and Walker for\neffectively one-dimensional (1D) situations[12]: At low\nfield (or current density), the DW velocity vis linear in\nthe field strength HuntilHreaches a so-called Walker\nbreakdown field Hw[12]. Within this linear regime, DW\npropagates as a rigid object. For H > H w, the DW\nloses its rigidity and develops a complex time-dependent\ninternal structure. The velocity can even oscillate with\ntime due to the “breathing” of the DW width. The time-\naveraged velocity ¯ vdecreases with the increase of H, re-\nsulting in a negative differential mobility. ¯ vcan be again\nlinear with Happroximately when H≫Hw. The pre-\ndictedv-Hcharacteristic is in a good agreement with\nexperimental results on permalloy nanowires[4–6]. Re-\ncently a general definition of the DW velocity proper for\nany types of DW dynamics has been also introduced[13].\nFor a single-domain magnetic nanoparticle (called\nStoner particle), an appropriate time-dependent but spa-\ntially homogeneous field pulse can substantially lower\nthe switching field and increase the reversal speed\nsince it acts as an energy source enabling to overcome\nthe energy barrier for switching the spatially constant\nmagnetization[14, 15]. In the present letter, we inves-\ntigate the dynamics of a DW in a magnetic nanowire\nunder a field pulse depending both on time and space.\nAs a result, such a pulse, synchronized with the DW\npropagation, can dramatically increase the DW velocity\nby typically two orders compared with the situation of a\nconstant field. Moreover, the total magnetic energy typi-\ncally decreases with a rate being proportional to the DWvelocity, i.e. the external field source can even absorb\nenergy from the nanowire.\nA\nBzx\ny\nz\nx\nDWy\nFIG. 1: A schematic diagram of two dynamically equivalent\n1D magnetic nanowire structures. (A) Easy axis is along the\nwire axis (z-axis); (B) Easy axis (z-axis) ⊥the wire axis (x-\naxis). The region between two dashed lines denotes the DW\nregion.\nA magnetic nanowirecan be described as an effectively\n1D continuum of magnetic moments along the wire axis\ndirection. Magnetic domains are formed due to the com-\npetitionbetweentheanisotropicmagneticenergyandthe\nexchangeinteractionamongadjacentmagneticmoments.\nLet us first concentrate on the case of a uniaxial mag-\nnetic anisotropy: Two dynamically equivalent configura-\ntions of 1D uniaxial magnetic nanowires are schemati-\ncally shown in Fig. 1. Type A shows the wire axis to\nbe also the easy-axis (z-axis). Type B shows the easy\naxis (z-axis) is perpendicular to the wire axis (x-axis).\nAlthough our results described below apply to both con-\nfigurations, we will focus in the following on type B.\nThe spatio-temporal dynamics of the magnetization den-\nsity/vectorM(x,t) is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation[16]\n∂/vectorM\n∂t=−|γ|/vectorM×/vectorHt+α\nMs/parenleftBigg\n/vectorM×∂/vectorM\n∂t/parenrightBigg\n,(1)\nwhere|γ|= 2.21×105(rad/s)/(A/m) the gyromag-\nnetic ratio, αthe Gilbert damping coefficient, and Ms\nis the saturation magnetization density. The total ef-\nfective field /vectorHtis given by the variational derivative of\nthe total energy with respect to magnetization, /vectorHt=2\n−(δE/δ/vectorM)/µ0, whereµ0the vacuum permeability. The\ntotal energy E=/integraltext∞\n−∞dxε(x) can be written as an inte-\ngral over an energy density (per unit section-area),\nε(x) =−KM2\nz+J/bracketleftBigg/parenleftbigg∂θ\n∂x/parenrightbigg2\n+sin2θ/parenleftbigg∂φ\n∂x/parenrightbigg2/bracketrightBigg\n−µ0/vectorM·/vectorH,\n(2)\nwherexis the spatial variable in the wire direc-\ntion. Here K,Jare the coefficients of energetic\nanisotropy and exchange interaction, respectively, and\n/vectorHis the external magnetic field. Moreover, we have\nadopted the usual spherical coordinates, /vectorM(x,t) =\nMs(sinθcosφ,sinθsinφ,cosθ) where the polar angle\nθ(x,t) andthe azimuthalangle φ(x,t) depend onposition\nand time.\nHence, the total field /vectorHtconsists of three parts: the\nexternal field /vectorH, the intrinsic uniaxial field along the\neasy axis /vectorHK= (2KMz/µ0)ˆz, and the exchange field\n/vectorHJwhich reads in spherical coordinates as[12, 17],\nHJ\nθ=2J\nµ0Ms∂2θ\n∂x2−Jsin2θ\nµ0Ms/parenleftbigg∂φ\n∂x/parenrightbigg2\n,\nHJ\nφ=2J\nµ0Mssinθ∂\n∂x/parenleftbigg\nsin2θ∂φ\n∂x/parenrightbigg\n. (3)\nFollowing Ref. [12], let us focus on DW structures ful-\nfilling∂φ/∂x= 0, i.e. all the magnetic moments rotate\naround the easy axis synchronously. Then the dynamical\nequations take the form\nΓ˙θ=α/parenleftbigg\nHθ−KMs\nµ0sin2θ+2J\nµ0Ms∂2θ\n∂x2/parenrightbigg\n+Hφ,\nΓsinθ˙φ=αHφ−Hθ+KMs\nµ0sin2θ−2J\nµ0Ms∂2θ\n∂x2,(4)\nwhere we have defined Γ ≡(1 +α2)|γ|−1, andHi(i=\nr,θ,φ) are the three components of the external field\nin spherical coordinates. In the absence of an exter-\nnal field, an exact solution for a static DW is given by\ntanθ(x)\n2= exp(x/∆)where∆ =/radicalbig\nJ/(KM2s)isthewidth\nof the DW. We note that a static DW can exist in a con-\nstant field only if the field component along the easy axis\nis zero,Hz= 0. In fact, according to Eqs. (4) static solu-\ntions need to fulfill Hφ= 0 [implying φ= tan−1(Hy/Hx)\nis spatially constant] and\n2J\nµ0Ms∂2θ\n∂x2−KMs\nµ0sin2θ+Hθ= 0 (5)\nor, upon integration,\nJ\nµ0Ms/parenleftbigg∂θ\n∂x/parenrightbigg2\n+KMs\n2µ0cos2θ+Hr(θ) = constant .(6)\nConsidering the two boundaries at θ= 0(x→ −∞) and\nθ=π(x→+∞) for the DW, we conclude Hr(0) =Hr(π), which requires Hz= 0. In this case, the station-\nary DW solutions under a transverse field are described\nasx=/integraltext\n[/radicalBig\n(KM2ssin2θ−µ0MsHsinθ)/J]−1dθ.\nThus, when an external field with a component along\nthe easy axis is applied to the nanowire, the DW is ex-\npected to move. We use a travelling-wave ansatzto de-\nscribe rigid DW motion[12],\ntanθ(x,t)\n2= exp/parenleftbiggx−vt\n∆/parenrightbigg\n, (7)\nwhere the DW velocity vis assumed to be constant. Sub-\nstituting this trial function into Eq. (4), the dynamic\nequations become\nΓsinθv=−∆(αHθ+Hφ),Γsinθ˙φ=αHφ−Hθ.(8)\nEq. (8) describes the dependence of the linear velocity v\nand the angular velocity ˙φon the external field /vectorH. Our\nfollowing results discussion will be based on Eqs. (8).\nLet us firstturn to the caseofastatic field caseapplied\nalong the easy axis (z-axis in type B of Fig. 1), Hθ=\n−Hsinθ,Hφ= 0. Here we recover the well-known static\nsolution for a uniaxial anisotropy[18],\nv=|γ|∆H\nα+α−1, (9)\nwhere the azimuthal angle φ(t) =φ(0)+|γ|Ht/(1+α2) is\nspatially constant (i.e. ∂φ/∂x= 0) and increases linearly\nwith time.\nLet us now allow the applied external field to depend\nboth on space and time. Our task is to design, under a\nfixed field magnitude H, an optimal field configuration\n/vectorH(x,t) to increase the DW velocity as much as possible.\nFrom Eqs. (8), we find a manifold of solutions of specific\nspace-time field configurations described by a parameter\nu,\nHr(x,t) =Hcosθ, H θ(x,t) =−Hsinθ//radicalbig\n1+u2,\nHφ(x,t) =−uHsinθ//radicalbig\n1+u2.(10)\nThe velocities vand˙φreads\nv=|γ|∆H\n1+α2α+u√\n1+u2,˙φ=|γ|H\n1+α21−αu√\n1+u2.(11)\nThe previous static field case is recovered for u= 0. The\nmaximum of the velocity vmwith regard to uus reached\nforu= 1/α,\nvm=|γ|∆H√\n1+α2, (12)\nwhere the angular velocity is zero, ˙φ= 0. On the other\nhand,˙φattains a maximum for u=−α, where, in turn,\nthe linear velocity vanishes. In Fig. 2 we have plotted3\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s61/s50/s48/s110/s109\n/s72/s61/s49/s48/s48/s79/s101\n/s32/s32/s118/s32/s40/s109/s47/s115/s41\n/s117/s32 /s61/s48/s46/s49\n/s32 /s61/s48/s46/s50\n/s32 /s61/s48/s46/s53\n/s32 /s61/s48/s46/s56\nFIG. 2: (Color online) The DW propagation velocity vver-\nsus the parameter uat the different damping values α=\n0.1,0.2,0.5,0.8. The other parameters are chosen as ∆ =\n20nmandH= 100Oe.\nthe dependence of the velocity on the parameter ufor\ndifferent damping strengths and typical values for the\nDW width ∆ and the magnitude Hof the external field.\nTo understand the physical meaning of the maximum\nvelocityvm, we note that, according to Eqs. (8), the field\ncomponents HθandHφare required to be proportional\nto sinθto ensure the constant velocity under the rigid\nDW approximation. Moreover,at u= 1/αwe haveHθ=\nαHφ, and from the identity\n(αHθ+Hφ)2+(αHφ−Hθ)2= (1+α2)(H2−H2\nr),(13)\nwe conclude that the term ( αHθ+Hφ) is maximal re-\nsulting in a maximal velocity according to Eqs. (8). As\na result, the new velocity under the optimal field pulse is\nlargerby a factor of vm/v=√\n1+α2/α≈1/αcompared\nto a constant field with the same field magnitude. To\ngive a practical example, the typical value for the damp-\ning parameter in permalloy is α= 0.01 which results in\nan increase of the DW velocity by a factor of 100.\nIt is instructive to also analyze the optimal field pulse\naccording to Eq. (10) with u= 1/αin its cartesian com-\nponents,\nHx(x,t) =Hsin2θ(1−α//radicalbig\n1+α2)/2,\nHy(x,t) =−Hsinθ//radicalbig\n1+α2, (14)\nHz(x,t) =H(cos2θ+αsin2θ//radicalbig\n1+α2),\nwhereθfollowsthewave-likemotiontanθ(x,t)\n2= exp(x\n∆−\n|γ|H√\n1+α2t). In Fig. 3 we plotted these quantities at t= 0\naround the DW center where the main spatial variation\nof the pulse occurs. Note that the space-dependent field\ndistribution should move with the same speed vmsyn-\nchronizedwiththeDWpropagation. NeartheDWcenter\nthe components HxandHzare (almost) zero whereas a\nlargetransversecomponent Hyis required to achievefast\nDW propagation. Qualitatively speaking, the transverse\nfield causes a precessionof the magnetization resulting inits reversal. This finding is consistent with recent micro-\nmagnetic simulations showing that the DW velocity can\nbe largely increased by applying an additional transverse\nfield[19].\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s61/s48/s46/s48/s49\n/s61/s50/s48/s110/s109\n/s72/s61/s49/s48/s48/s79/s101\n/s32/s32/s72\n/s120/s44/s121/s44/s122/s32/s40/s79/s101/s41\n/s120/s32/s40/s110/s109/s41/s32/s72\n/s120\n/s32/s72\n/s121 \n/s32/s72\n/s122\nFIG. 3: (Color online) The x,y,zcomponents of the optimal\nfield pulse. The parameters are chosen as α= 0.01, ∆ =\n20nmfor permalloy[6]. The field magnitude is H= 100Oe.\nIt is also interesting to study the energy variation un-\nder the optimal field pulse,\ndE\ndt=−µ0/integraldisplay+∞\n−∞dx/parenleftBigg\n∂/vectorM\n∂t·/vectorHt+/vectorM·∂/vectorH\n∂t/parenrightBigg\n≡Pα+Ph.\n(15)\nThe first term Pαis the intrinsic damping power due to\nall kinds of damping mechanisms described by the phe-\nnomenologicalparameter α. According to the LLG equa-\ntionPα=−µ0α\n|γ|Ms/integraltext+∞\n−∞dx(∂M\n∂t)2is always negative[14],\nimplying an energy loss. Phis the external power due to\nthe time-dependent external field. From Eq. (11), both\npowers are obtained as\nPα=−2α\n1+α2µ0|γ|Ms∆H2, (16)\nPh=2α(√\n1+u2−1)−2u\n(1+α2)√\n1+u2µ0|γ|Ms∆H2,(17)\nsuch that the total energy change rate is\ndE\ndt=−2µ0MsHv=−2(α+u)µ0|γ|Ms∆H2\n(1+α2)√\n1+u2.(18)\nNote that the intrinsic damping power is independent of\nthe parameter uand always negative, whereas the total\nenergy change rate is proportional to the negative DW\nvelocity. Thus, for positive velocities ( u >−α) the total\nmagnetic energy decreases while it grows for negative ve-\nlocities (u <−α). In the former case energy is absorbed\nby the external field source while in the latter case the\nfield source provides energy to the system. The optimal\nfield source helps to rapidly release or gain magnetic en-\nergywhichisessentialforfastDWmotion. Thisaspectis\nverydifferent from the reversalof a Stoner particle where\nthe time-dependent field is always needed to provide en-\nergy to the system to overcome the energy barrier[14].4\nMoreover, our new strategy of employing space-\ndependent field pulses can also be applied to uniaxial\nanisotropies of arbitrary type: Let w(θ) be the uniaxial\nmagnetic energy density. The static DW solution in the\nabsence of an external field reads x=/integraltext\nχ−1(θ)dθ, where\nχ(θ) =/radicalbig\n[w(θ)−w0]/J. (19)\nHerew0is the minimum energy density for magneti-\nzation along the easy axis. By performing analogous\nsteps as before, we obtain the the optimal velocity as\nvm=|γ|H√\n1+α2χmax, whereχmaxdenotes the maximum of\nχ(θ) throughout all θ.\nOn the other hand, our approach is not straightfor-\nwardlyextended tothecaseofamagneticwirewith biax-\nial anisotropy. To see this, consider, a biaxial anisotropy\nεi=−KM2\nz+K′M2\nxwhere the coefficients K,K′cor-\nrespond to the easy and hard axis, respectively[12]. The\nLLG equations read\nΓ˙θ=α/parenleftbigg\nHθ−KMs\nµ0sin2θ−K′Ms\nµ0sin2θcos2φ\n+2J\nµ0Ms∂2θ\n∂x2/parenrightbigg\n+Hφ+K′Ms\nµ0sinθsin2φ,\nΓsinθ˙φ=αHφ−Hθ+KMs\nµ0sin2θ−2J\nµ0Ms∂2θ\n∂x2\n+K′Ms\nµ0sin2θcos2φ+αK′Ms\nµ0sinθsin2φ.(20)\nLet us assume φ(x,t) =φ0is a constant deter-\nmined by the applied field. Substituting the travelling-\nwaveansatztanθ(x,t)\n2= exp/parenleftbigx−vt\n∆/parenrightbig\n, where now ∆ =/radicalbig\nJ/(K+K′cos2φ0)/Ms, into Eqs. (20) we obtain\nΓsinθv=−∆(αHθ+Hφ+K′Mssinθsin2φ0/µ0),\n(21)\nαK′Mssinθsin2φ0/µ0+(αHφ−Hθ) = 0.(22)\nFor a static field along z-axis Hθ=−Hsinθ,Hφ= 0, the\nsolution is just the Walker’s result v=|γ|∆H/α(Note\nhere ∆ also depends on H)[12]. To implement our new\nstrategy, we need to find the maximum of the right-hand\nside of Eq. (21) under two constraints of Eq. (22) and\nEq. (13) with HθandHφbeing proportional to sin θ.\nThe unique solution to this problem is indeed a constant\nfield along the z-axis which is thus the optimal field con-\nfiguration.\nIn summary, our theory is general and can be applied\nto a magnetic nanowire with a uniaxial anisotropy which\ncan be from shape, magneto-crystalline or the dipolar\ninteraction. The experimental challenge of our proposal\nis obviously the generation of a field pulse focused on\nthe DW region and synchronized with its motion. How-\never, the field sourcesynchronizationvelocity can be pre-\ncalculated from the material parameters. As for the re-\nquired localized field (See Fig. 3), we propose to employa ferromagnetic scanning tunneling microscope (STM)\ntip to produce a localized field perpendicular to the wire\naxis[20] and use a localized current to produce an Oer-\nsted field along the wire axis[21]. Moreover, such re-\nquired localized fields may also be produced by nano-\nferromagnetswithstrongferromagnetic(orantiferromag-\nnetic) coupling to the nanowire. We also point out that,\nalthough the field source typically does not consume en-\nergy but gain energy from the magnetic nanowire, the\npulse source may still require excess energy to overcome\neffects such as defects pinning, which is not included in\nour model. At last, the generalization of the strategy be-\nyond the rigid DW approximation, and to DW motion\ninduced by spin-polarized current will also be attractive\ndirection of future research.\nZ.Z.S. thanks the Alexander von Humboldt Founda-\ntion (Germany) for a grant. This work has been sup-\nported by Deutsche Forschugsgemeinschaft via SFB 689.\n[1] R. P. Cowburn, Nature (London) 448, 544 (2007).\n[2] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[3] D. A. Allwood, et al., Science 309, 1688 (2005).\n[4] T. Ono, et al., Science, 284, 468 (1999).\n[5] D. Atkinson, et al., Nature Mater. 2, 85 (2003).\n[6] G. S. D. Beach, et al., Nature Mater. 4, 741 (2005); G.\nS. D. Beach, et al., Phys. Rev. Lett. 97, 057203 (2006);\nJ. Yang, et al., Phys. Rev.B 77014413 (2008).\n[7] M. Klaui, et al., Phys. Rev. Lett. 94, 106601 (2005).\n[8] M. Hayashi, et al., Phys. Rev. Lett. 96, 197207 (2006);\nL. Thomas, et al., Nature (London) 443, 197 (2006); M.\nHayashi, et al., Phys. Rev. Lett. 98, 037204 (2007).\n[9] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[11] A. Thiaville, et al., Europhys. Lett. 69, 990 (2005).\n[12] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[13] X.R.Wang, P.Yan, andJ.Lu, Europhys.Lett. 86, 67001\n(2009); X. R. Wang, et al., Ann. Phys. (N.Y.) 324, 1815\n(2009).\n[14] Z. Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205\n(2006); Phys. Rev. B 73, 092416 (2006); ibid.74, 132401\n(2006).; X. R. Wang and Z. Z. Sun, Phys. Rev. Lett. 98,\n077201 (2007).\n[15] X. R. Wang, et al., Europhys. Lett. 84, 27008 (2008).\n[16] Z. Z. Sun and X. R. Wang, Phys. Rev. B 71, 174430\n(2005), and references therein.\n[17] M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008).\n[18] A. P. Malozemoff and J. C. Slonczewski, Domain Walls\nin Bubble Materials , (Academic, New York, 1979).\n[19] M. T. Bryan, et al., J. Appl. Phys. 103, 073906 (2008).\n[20] T. Michlmayr, et al., J. Appl. Phys. 99, 08N502 (2006).\n[21] T. Michlmayr, et al., J. Phys. D: Appl. Phys. 41, 055005\n(2008)."    },    {        "title": "0910.3776v1.Bifurcation_and_chaos_in_spin_valve_pillars_in_a_periodic_applied_magnetic_field.pdf",        "content": "arXiv:0910.3776v1  [cond-mat.other]  20 Oct 2009APS/123-QED\nBifurcation and chaos in spin-valve pillars in a periodic ap plied magnetic field\nS. Murugesh1∗and M. Lakshmanan2†\n1Department of Physics & Meteorology, IIT-Kharagpur, Khara gpur 721 302, India and\n2Centre for Nonlinear Dynamics, School of Physics, Bharathi dasan University, Tiruchirapalli 620 024, India\n(Dated: November 10, 2018)\nWe study the bifurcation and chaos scenario of the macro-mag netization vector in a homogeneous\nnanoscale-ferromagnetic thin film of the type used in spin-v alve pillars. The underlying dynamics\nis described by a generalized Landau-Lifshitz-Gilbert (LL G) equation. The LLG equation has\nan especially appealing form under a complex stereographic projection, wherein the qualitative\nequivalence of an applied field and a spin-current induced to rque is transparent. Recently chaotic\nbehavior of such a spin vector has been identified by Zhang and Li using a spin polarized current\npassing through the pillar of constant polarization direct ion and periodically varying magnitude,\nowing to the spin-transfer torque effect. In this paper we sho w that the same dynamical behavior\ncan be achieved using a periodically varying applied magnet ic field, in the presence of a constant\nDC magnetic field and constant spin current, which is technic ally much more feasible, and demon-\nstrate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is\nnoted that in the presence of a nonzero crystal anisotropy fie ld chaotic dynamics occurs at much\nlower magnitudes of the spin-current and DC applied field.\nBolstered by the importance of Giant Magneto\nResistance (GMR), a sequence of experimental\nand theoretical developments in the last few years\non current induced switching of magnetization in\nnanoscale ferromagnets has thrown open several\nprospects in next generation magnetic memory\ndevices. The direct role of spin polarized current,\nas against the traditional applied field, in control-\nling spin dynamics has brought in the possibility\nof new types of current-controlled memory de-\nvices and microwave resonators. The system un-\nder consideration is primarily a nanoscale spin-\nvalve pillar structure, with one freeferromagnetic\nlayer and another pinnedlayer separated by a non-\nferromagnetic conducting layer. The behavior of\nthe dynamical quantity of interest, the magneti-\nzation field in the free layer, is well modeled by an\nextended Landau-Lifshitz equation with Gilbert\ndamping, which is a fascinating nonlinear dynam-\nical system. The free layer is usually assumed\nto be of single magnetic domain. Owing to the\nhighly nonlinear nature of the LLG equation it is\nimperative to study the chaotic dynamical regime\nof the magnetization field. Indeed, several recent\nexperiments have exclusively focused on chaos as-\npect of the system. In this article we have shown\nthat a small applied periodically varying (AC)\nmagnetic field, in the presence of a constant spin-\ncurrent and a steady applied magnetic field, can\ninduce parametric regimes displaying a broad va-\nriety of dynamics and period doubling route to\nchaos. A numerical study of the effects of a non-\n∗Electronic address: murugesh@phy.iitkgp.ernet.in\n†Electronic address: lakshman@cnld.bdu.ac.inzero anisotropy field reveals chaotic dynamics at\nmuch lower magnitudes of the spin-current and\napplied DC field. This could be an important\nfactor to consider in microwave resonator appli-\ncations of spin-valve pillars.\nI. INTRODUCTION\nFollowing the success of GMR, the next major devel-\nopment in classical computer technology is expected to\nbe through MRAMs (1; 2; 3). Apart from a manifold\nreduction in power consumption, being inherently non-\nvolatile in nature, they also bring in the prospect of\ncomputers that need not be rebooted. Understandably,\nfabrication of MRAMs has been a major thrust area of\nresearch in the last two decades. Typically, the mem-\nory unit consists of two nanoscale magnetic films sepa-\nrated by a spacer conductor/semiconductor medium and\nworks on the principle of GMR. The imminent prospects\nin the recording media industry has prompted a breadth\nof development in the field. Besides the eminent role\nas a memory unit, the possibility of a single MRAM as\na logical unit has also been proposed (4; 5). One no-\ntable technological hiccup in fabricating large MRAM\ngrids is the extent to which the applied magnetic field\ncan be localized. This imposes limitations on the ef-\nficiency with which an individual unit can be manipu-\nlated. The applied field required for the purpose is in\nmost cases the Oersted field generated through electrical\ncurrents. Asignificantstepforward,inbypassingthelim-\nitation on field localization, occurred when Slonczewski\nandBergerindependentlyenvisagedamoredirectrolefor\n(spin polarized) current on magnetization(6; 7). They\npredicted that the angular momentum acquired by the\nspin polarized current can interact with the magnetiza-\ntion vector, and thus a suitable spin polarized current2\ncan possibly flip its direction. This phenomenon has\nbeen well established in a series of experiments in the\nlast decade and referred to as the spin-torque effect in\nthe literature(8; 9; 10; 11; 12; 13; 14; 15). Interest-\ningly, the effect has a simple semiclassical description in\nthe form of an extended Landau-Lifshitz equation with\nGilbert damping (6; 7; 16). Several proposals have ap-\npeared in the last few years suggesting an increased role\nof the spin current and the associated torque in the even-\ntually expected version of the MRAM.\nOneimportantassumptionoftenmadein mostofthese\nstudies is that the magnetization in the film is homoge-\nneous, or at least enough well defined that one can con-\nsider the film to be a mono-domain layer. This homo-\ngeneity assumption effectively nulls the Heisenberg in-\nteraction between neighboring spins, and allows one to\ntreat the system as a single spin unit. As the size of\nthe magnetic film is increased, this approximation is ex-\npected to fail. Indeed, chaotic behavior has been ob-\nserved due to spin field inhomogeneity at lateral sizes of\norder 60 −130nm(17; 18). Besides, it is well realized\nexperimentally that spin-transfer can induce microwave\noscillations(17; 19; 20; 21). The possible role of such\ncurrent controlled microwave oscillators in the nanoscale\nhas been well realized (17). For higher power levels that\nare practically desired it is natural to look at a series of\nsuch coupled spin-valve oscillators. Studies on synchro-\nnization of networks of such coupled nano-spin trans-\nfer oscillators, each one modeled on the extended LLG\nequation have been carried out both experimentally and\nnumerically in an effort to enhance the emitted power\n(22; 23; 24; 25).\nSince theextended LLGequationis highlynonlinearin\nnature, a detailed study of the underlying nonlinear dy-\nnamics in spin-valve structures becomes inevitable. The\nstability diagram based on Melnikov theory in the space\nof the external field along the direction of anisotropy and\nthe strength of spin torque due to a DC current was ob-\ntained in detail by Bertotti et. al.,in (26). Following\nthis, it was shown by Z. Li, Y. C. Li and S. Zhang that\nan applied alternating current can induce three broad\ntypes of dynamics, vis−a−visChaos, Multiply periodic\nand Periodic (27). Qualitatively similar features are also\nnoted when the effects of nearest neighbor interactions\nare included in a long one dimensional ferromagnet with\na spin torque term. Indeed the ferromagnetic chain ex-\nhibits both periodic and chaotic behavior in the presence\nof an applied AC spin current. (28). The multiply pe-\nriodic behavior refers to the case where, with change in\nparameter, the system moves from periodic to multiply\nperiodic behavior but not leading to chaos. It must be\nmentioned at this point that the last two types ‘multiply\nperiodic’ and ‘periodic’ are also referred to in the litera-\nture as ‘modification’ and ‘synchronization’(27; 29). The\nusage here is deliberate as the periods do not have a di-\nrect relation to the period of the applied magnetic field.\nIt was also shown that in the space of the applied exter-\nnal field and the strength of the spin current the systemexhibits chaotic behavior for a range of values within the\nboundary predicted by Melnikov theory (29).\nThe possibility of chaotic oscillations in monodomain\nsingle nanoscale spin-valve requires further in depth\nstudy of underlying bifurcation and chaos scenarios, in-\ncluding the detailed phase diagrams, at least for two rea-\nsons: (i) Chaotic oscillations in spin transfer oscillators\nmay be unfavorable from a practical/technological point\nof view and such oscillations need to be suppressed by\nminimal intervention using one of the several control-\nling techniques developed in the field of chaotic dynam-\nics (30; 31), where already such a study has been made\n(32). (ii) The previously mentioned possibility of syn-\nchronization of coupled spin-valve oscillators raises com-\nplex problems which are new in spintronics and is related\nto the topic of chaos synchronization in dynamical sys-\ntems (30; 31; 33), similar to synchronization of chaoti-\ncally evolving networks of Josephson junctions, laser sys-\ntems and nonlinear oscillators. As synchronization of\nchaotic oscillators is considered as a potential technique\nfor secure communication including cryptography, there\nexistspossibleapplicationsofnetworksofnanoscalespin-\nvalve structures along these lines, although this may be\ncomplicated due to the presence of several parameters in\nthe system. Consequently the study of the full nonlinear\ndynamics of spin transfer oscillators will be of consider-\nable significance. It may be noted that chaosin magnetic\nsystems, such as yttrium garnet, driven by external fields\nhave been extensively studied in the past (34; 35; 36).\nNumerical experiments on a model of thin magnetic film\nwherein spin wave excitations induced by spin-current\nlead to chaos have also been studied in (37). However,\nchaosinnano-spinvalvegeometryisreasonablynew with\npotential new applications as discussed above.\nIn this paper, we study the chaotic dynamics of the\nmagnetization vector in a single domain current driven\nspin-valve pillar, induced by a periodically varying (AC)\napplied magnetic field in the presence of a constant spin\ncurrent and steady (DC) magnetic field, using the ex-\ntended LLG equation as the model for the system. Mak-\ning use of a complex stereographic variable, we observe\nthat the spin current induced torque is effectively equiv-\nalent to an applied magnetic field. Following this ob-\nservation, we show numerically that a periodically alter-\nnating field can lead to chaotic behavior of the magne-\ntization vector, which is similar to that of an alternat-\ning spin-current induced torque, studied in Li, Li, and\nZhang (27). It will be shown that the order of magni-\ntude of the applied alternating field required for chaotic\nmotion is substantially lower, within practically achiev-\nable limits, compared to the alternating current magni-\ntudes shown in (27; 29). This is expected to be helpful\nin applications such as resonators, as an AC magnetic\nfield is much more feasible practically than a AC spin-\npolarized current (although, when it comes to DC fields,\nthe current induced DC Oersted fields are more cumber-\nsome to produce in spin valve geometries compared to\nDC spin currents). We study the dynamics in an infi-3\nnite thin film, both with a vanishing and non-vanishing\nanisotropy field. In the setup we shall consider the mon-\nodomain spin layer influenced by a DC spin-current, and\nboth a DC and AC applied magnetic field. Although\nthe applied magnetic field is qualitatively equivalent to\na suitable spin-current(38), we employ both a DC spin-\ncurrent and applied magnetic field as the magnitudes at\nwhich chaos is observed is quite high, and hence difficult\nto achieve exclusively using either of the two. It may be\nnoted that the chaotic dynamics studied here is induced\nby an AC applied magnetic field, and is phenomenologi-\ncally different from the spin field inhomogeneity induced\nchaotic behavior that is observed in (17). Periodic dy-\nnamics is possible even in the absence of an alternating\ncurrent, or field, as has been noticed in ferromagnetic\nfilms induced by a spin-current (19). Futher it has been\nreported therein that the current magnitudes at which\nperiodic behavior is seen share a linear relation of nega-\ntive slope with the oscillationfrequencies. Our numerical\nresults based on the extended LLG model are shown to\nbe in agrement with these observations. Interestingly,\nin the presence of a non-zero in plane easy-axis crystal\nanisotropy field (taken along the zdirection), the chaotic\ndynamical regime is observed at much lower magnitudes\nof the DC applied field and spin-current. This could be\nan aspect to factor in while designing spin-valve based\nmicrowave resonators.\nThe paper is organized as follows. In Section 2, we\ndetail the various interactions, including spin-transfer\ntorque, that make up the extended LLG equation.\nFurther, we introduce a complex stereographic vari-\nable equivalent to the macro-magnetization vector, and\nrewrite the LLG equation in this variable. As will be\nshown, this elucidates the role of the spin transfer torque\nas equivalent to an applied magnetic field. In Section\n3, we present our numerical results which demonstrate\nchaoticdynamics ofthe magnetizationvectorin the pres-\nence of a periodic applied field. Here we shall consider\ntwocases,namelyresponseinthepresenceandabsenceof\na crystal anisotropy field. As will be noticed, the chaotic\nregimes in the two cases are significantly different. In\nSection 4 we conclude with a discussion on our results.\nII. THE EXTENDED LANDAU-LIFSHITZ EQUATION\nAND COMPLEX REPRESENTATION\nA typical spin-valve pillar used in most experiments is\nschematicallyshowninFIG 1. Acurrentpassingthrough\nthis ferromagnet acquires a spin polarization in the ˆ zdi-\nrection. The thickness of the spacer conductor medium\nshould be less than the spin diffusion length of the polar-\nized current. The polarized current then passing through\nthe free layer causes a change in the magnetization vec-\ntorˆS, an effective torque referred to as the spin transfer\ntorque. Interestingly, it has been realized that semiclas-\nsically the phenomenon can be described by an extensionj\nPinned layer Conductor Thin film ConductorS S p\nxz\ny~100 nm 2−10 nm\nFIG. 1 A schematic figure of a spin-valve pillar. The cross\nsection of the free layer is roughly 5000 nm2.ˆSis the magne-\ntization vector in the free layer, and is the dynamical quant ity\nof interest. ˆSPis the direction of polarization of the spin cur-\nrent.\nto the LLG equation, (6; 7)\ndˆS\ndt=−γˆS×/vectorHeff+λˆS×dˆS\ndt−γ aˆS×(ˆS׈Sp),(1)\nHere,ˆS={S1,S2,S3}is the unit vector along the di-\nrection of the magnetization vector in the ferromagnetic\nfilm, which isthe dynamicalvariableofinterest, γthe gy-\nromagnetic ratio, and λthe dissipation coefficient. /vectorHeff\nin Eq. (1) is the effective field due to exchange inter-\naction, anisotropy, demagnetization and an applied field\n(see (38) for details):\n/vectorHeff=DS0∇2ˆS+κ(ˆS·ˆ e/bardbl)ˆ e/bardbl+/vectorHdemag+/vectorBapplied,(2)\nˆ e/bardblbeing the unit vector along the direction of the\nanisotropy axis. The demagnetization field is obtained\nas a solution of the differential equation\n∇·/vectorHdemag=−4πS0∇·ˆS, (3)\nand/vectorBappliedis the applied magnetic field on the sample.\nThe last term in Eq. (1) is the additional term describing\nthe spin-transfer torque, and the parameter ‘ a’ depends\nlinearly on the current density j.ˆSpis the direction of\nmagnetization of the polarizer, i.e., the polarization of\nthe spin current.\nIn this study we shall assume the magnetization to be\nhomogeneous. This effectively nulls the exchange inter-\naction, while Eq. (3) can be immediately solved to give\nthe demagnetization field as\n/vectorHdemagnetization =−4πS0(N1S1ˆ x+N2S2ˆ y+N3S3ˆ z),\n(4)\nwhereNi,i= 1,2,3 are conveniently chosen such that\nN1+N2+N3= 1 (after suitable rescaling of the magni-\ntude of the spin). For a spherical sample N1=N2=N3,\nand the demagnetization term is effectively null in Eq.\n(1). In the next sections we study chaotic dynamics\nshown by the system when an alternating magnetic field\nis applied.\nRewriting Eq. (1) using the complex stereographic\nvariable (39; 40)\nΩ≡S1+iS2\n1+S3, (5)4\nprovidesamorecomprehensiblepictureoftheroleofspin\ntransfer torque. For the spin valve system, the direction\nof polarization of the spin-polarized current ˆSpremains\na constant, and lies in the plane of the film. Without loss\nof generality, we call this the direction ˆ zin the internal\nspin space, i.e., ˆSp=ˆ z. Asmentioned in Sec. 2, wedisre-\ngard the exchange term. We choose the applied externalfield also in the ˆ zdirection, i.e., /vectorBapplied={0,0,ha3}.\nDefining\nˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl}(6)\nand upon using Eq. (5) in Eq. (1), we get\n(1−iλ)˙Ω =−γ(a−iha3)Ω+iS/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n−iγ4π S0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω\n−N1\n2(1−Ω2−|Ω|2)Ω−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n,(7)\nwhereS/bardbl=ˆS·ˆ e/bardbl. Using Eq. (5) and Eq. (6), S/bardbl, and\nthus Eq. (7), can be written entirely in terms of Ω. For\nfurther details of derivation of Eq. (7) see Ref. (38).\nIt is interesting to note that in this representation the\nspin-transfertorque,proportionaltotheparameter a, ap-\npears only in the first term in the right hand side of Eq.\n(7) as an addition to the applied magnetic field ha3but\nwith a prefactor i. Thus the spin torque term can be\nconsidered as an effective applied magnetic field (38). It\nwas further explicitly shown in (38), in the absence of the\ncrystal anisotropy field, that the switching time due to\nthe spin-torque will be shorter by a factor λ, compared\nto that of magnetic field induced switching. Further, the\nspin-torque produced the dual effect of precession and\ndissipation even in the presence of the external applied\nfield. The nature of switching of magnetization for other\ntypes of materials can be investigated by analyzing Eq.\n(7), and for typical materials this has been carried out in\n(38).\nIII. CHAOTIC DYNAMICS\nMagnetization reversal in a spin mono-domain layer in\nthepresenceofbothasteadyappliedmagneticfieldanda\nsteady polarized current corresponds to a rather compli-\ncated phasediagram,as revealedin (26). Using Melnikov\ntheory, it was also shown therein that the magnetization\nvector also has limit cycles for a range of values of the\nparameters, with frequency in the microwave range. The\ndynamical quantity in question, the magnetization vec-\ntorˆS, is two dimensional, owing to its constant (unit)\nmagnitude. Hence, chaotic behavior is ruled out. How-\never, making the applied field, or current, time depen-\ndent is one way of increasing the effective dimensionality\nof the system to three and hence introduce a possibility\nof chaotic dynamics. Following (26) it has been shown in\n(27) that a small alternating current can produce a vari-\nety of dynamics, namely, Multiply periodic, Periodic and\nChaos. It is also noticed that, along with a steady spincurrent of order 250 Oeand a steady applied magnetic\nfield of the same order, inclusion of a small alternating\nspin polarized current leads to chaotic dynamics (29).\nThe dynamical similarity of the applied field and the\nspin-torque was noted in Section II. In this section we\nshow numerically that an applied AC field can also pro-\nduce diverse dynamical scenarios and point out the ad-\nvantagesin using a periodically varyingapplied magnetic\nfield instead of an alternating current. i.e., in Eq. (1) (or\nequivalently Eq. (7)) we take\n/vectorBapplied={0,0,ha3+haccosωt}. (8)\nIt will be notedthat chaoticdynamicsis possibleatmuch\nlower DC applied field strengths and spin current in the\npresence of an anisotropy field.\nFor the film geometry, the film is taken to be in the\ny−zplane, and the demagnetization field perpendicular\nto the film, the ˆ xdirection, i.e.,\nHdemagnetization =−4πS0S1ˆ x. (9)\nThe saturation magnetization is taken to be that of\npermalloy, so that 4 πS0= 8400Oe. Two different scenar-\nios, one without anisotropy, and anotherwith an in-plane\neasy axis anisotropy of strength κ= 45Oealong the z\ndirection, are investigated. All the numerical results that\nfollow have been obtained by directly simulating Eq. (1)\nfor the vector ˆS. The same results were confirmed using\nEq. (7) as well.\nA. Regions of Chaos in the presence of an applied\nalternating field\nAs a first step we show below (FIG 2) the regions\nof chaos, or regions of positive Lyapunov exponent, in\nthe space of the DC magnetic field and the DC current\nfor an alternating applied magnetic field of magnitude\n10Oeand frequency 15 ns−1, first without an anisotropy\nfield,κ= 0, (FIG 2a), and then with anisotropy field,5\n(a)\nha3(Oe)a (Oe)\n800700600500400300200100280\n270\n260\n250\n240\n230\n220\n210\n(b)\nha3(Oe)a (Oe)\n1800 1400 1000 600 200400\n350\n300\n250\n200\n150\n100\n50\n0\nFIG. 2 Regions of chaos in the a−ha3space, for an applied\nalternating magnetic field of amplitude hac= 10Oeand fre-\nquencyω= 15ns−1, a) without anisotropy field, κ= 0,\nand b) with anisotropy field of strength κ= 45Oealong the\nzdirection. The dark regions indicate values for which the\ndynamics is chaotic, i.e, regions where the largest Lyapuno v\nexponentispositive. Ina)chaosis rarelynoticedforlower val-\nues ofha3. The other parameters are N1= 1,N2=N3= 0,\n4πS0= 8400Oe. The points are plotted at intervals of 5 Oe\nalong both axes, and hence the figure offers only limited res-\nolution in the dark(chaotic) regions.\nκ= 45Oe, (FIG 2b). The regions are obtained by di-\nrect numerical simulation using the model in Eq. (1), or\nequivalentlyEq. (7). ThedarkregionsinFIG2represent\nparameter values when the largest Lyapunov exponent is\npositive.\n1. Case a: No anisotropy ( κ= 0)\nThe similarity of FIG 2a with that of regions of chaos\nfor an alternating spin current must be noted (see FIG 1\nin (29)). The figure is a demonstration of the qualitative\nequivalence of the applied field and current induced spin-\ntorque in describing the gross dynamical scenario. From\nFIG 2a, chaotic behavior of spins is observed for spin-\ncurrent magnitudes in the range of a= 200−300Oe, and\nDC magneticfieldsabove100 Oe. These valuesof‘ a’ cor-\nrespond to spin-current magnitudes of over 1012A/cm2,\nwhich is at the higher end of the presently achievable\nlevels.\nIn FIG 3 we present the bifurcation diagram showingthe period doubling route to chaos as the DC current\nis varied. These diagrams are obtained by plotting the\nminimum values for one of the components of the spin,\nin this case S1, over severalperiods of time for each value\nofain the given range. From the data in FIG 3a, the\nfirst five period doubling bifurcations are seen to occur\natan= 268.4845,267.7723,267.6055,267.5685,267.5605.\nConsequently, the ratio δn= (an−an−1)/(an−1−an+1)\ntakesvalues4 .2698,4.5081,4.625,clearlyapproachingthe\nFeigenbaum constant. The Lyapunov spectrum for this\nrange of ‘ a′is shown as inset. On comparison, we notice\nthe largest Lyapunov exponent is positive for values of\nawhere the dynamics is chaotic in FIG 3a. A similar\ncheck has been made for FIG 3b, which again shows the\nperiod doubling route in the presence of an anisotropy\nfield. Period doubling route to chaos is also noticed as\nthe strength of the steady applied magnetic field is var-\nied(keeping the current and frequency fixed), and when\na (Oe)λ\n268.6268267.40\n-4\n-8\n-12(a)\na (Oe)S1minimum\n268.6268.4268.2268267.8267.6267.4-0.994\n-0.995\n-0.996\n-0.997\n-0.998\n(b)\na (Oe)S1minimum\n250.5250 249.5249 248.5-0.984\n-0.986\n-0.988\n-0.99\n-0.992\n-0.994\n-0.996\nFIG. 3 Period doubling route to chaos as ais varied. The\nfigure is a plot of the minimum values of S1over several pe-\nriods for the given parameter values a) without anisotropy,\nand b) with anisotropy of κ= 45Oealong the zdirection.\nThe applied DC field ha3= 200Oe. All the other parameters\nremain the same as in FIG 2. The corresponding Lyapunov\nspectrum is shown as an inset.6\nthe frequency of the applied field is varied (keeping cur-\nrent and magnetic field strength steady), respectively.\n(a)\na (Oe)ω(ns−1)\n60055050045040035030025045\n40\n35\n30\n25\n20\n15\n10\n5\n(b)\na (Oe)ω(ns−1)\n60055050045040035030025045\n40\n35\n30\n25\n20\n15\n10\n5\nFIG. 4 Regions of chaos(red dots) and periodicity (blue\nwings) in theparameter space of DCcurrent‘ a′andfrequency\n‘ω′, a) without anisotropy and b)with anisotropy ( κ= 45Oe)\nalong the zdirection. The left over regions show multiply pe-\nriodicbehavior. All other parameters remain the same as in\nFIG 3. The power spectrum at the two dark points in (a)\n(255,25) and (280 ,25) are shown in FIG 5 (a) and (b), re-\nspectively.\n2. Case b: Non-zero anisotropy( κ= 45Oe)\nContrary to the observation in FIG 2a, in the pres-\nence of a non-zero easy axis anisotropy, chaos is noted\nat much lower values of the spin current and DC applied\nfield (FIG 2b). FIGs 3b shows period doubling bifur-\ncation scenario in the presence of anisotropy. In FIG\n3b the current magnitude is varied, keeping the mag-\nnetic field strength and frequency of the AC component\nfixed. Similar period doubling route to chaos is also no-\nticed as a) the magnitude of the steady magnetic field is\nvaried (keeping current and frequency fixed), and when\nthe frequency of the AC component of the applied mag-netic field is changed (keeping the current and magnetic\nfield strengths constant). In either case, an easy axis\nanisotropy of magnitude κ= 45Oeis chosen along the z\ndirection, which is also the polarization direction of the\nspin current. The figures corresponding to these results\nare, however, not presented in here.\nB. Periodic, Multiply periodic and Chaotic dynamics\nIn the presence of an AC spin-current induced torque,\nit is known that as the frequency of the spin-current is\nvaried the system exhibits three distinct phases wherein\nthe dynamicsispredominantlyeither- Periodic, Multiply\nperiodicorChaotic(27). Herein we show that an applied\nAC magnetic field of magnitude 10 Oe, instead of a AC\ncurrent, results in the three dynamical phases as the fre-\nquency of the applied field is varied for a constant value\nof the applied DC field.\nFIG 4a shows regions of periodic (blue wings), and\nchaotic (red stem) dynamics in the parameter space of\nthe spin current magnitude and the frequency of the ap-\npliedmagneticfield. Multiplyperiodicbehaviorisseenin\ntheunshadedregions. Hereagainthesimilaritywith that\nofFIG1in(27)maybenoted. Thisfurtherillustratesthe\nqualitative similarity of the spin current induced torque\nwith that of the applied field. Our numerical results fur-\nther show a number of wing like bands in the ω−aspace\nwhere periodic behavior is noticed. This is clearly ab-\nsent with a periodic spin-current as seen in (27). The\npowerspectrum correspondingto two specific points, one\nchaotic and the other periodic (indicated with dark dots\nin FIG 4a), are shown in FIG 5. Periodic behavior is no-\nticed even in the limit ω→0 forcertain values of a, while\nmultiply periodic behavior is noticed for the other inter-\nmediate values. The power spectrum corresponding to\na = 280Oe\n20010008\n4\n0\n-4\n-8\na = 255Oe\nω(ns−1)log(Power)\n2502001501005008\n6\n4\n2\n0\n-2\n-4\n-6\n-8\nFIG. 5 The power spectrum distribution corresponding tope-\nriodic,a= 280Oe(inset), and chaotic, a= 255Oe, scenarios\nin FIG 4a. The first peak in the inset is seen at ω= 25ns−1.\nThe anisotropy is taken zero, and all other parameters are th e\nsame as in FIG 4a.7\nthese current values are shown in FIG. 6. Such a behav-\nior, induced by a spin-currentof constantmagnitude, has\nbeen noticed in (19). Indeed, the current magnitudes, a,\nwhere periodic behavior is seen to vary linearly (with a\nnegative slope) with the corresponding periods (see FIG.\n6 inset), in further agreement with (19).\nHowever,in the presenceof the anisotropyfield chaotic\nregime is much more pronounced and wider, as seen in\nFIG 4b. For a much lower value of the DC applied field\n(ha3= 20Oe) there appear wide bands in the ω−aspace\nwhere periodic behavior is exclusively noted at low fre-\nquencies (FIG 7). As the frequency is increased multiple\nperiodicity islands appear in between periodicbands, and\nfor even higher frequencies the dynamics is largely one of\nmultiply periodic type. No chaotic dynamics is noted\nin the parameter range chosen. These thick periodicity\nbands present better regions to choose in applications\nsuch as the microwave resonator discussed earlier, while\nchaos synchronization studies can be carried out in the\nchaos regimes.\nIV. DISCUSSION AND CONCLUSION\nUsing the complex stereographic variable to represent\nthe spin vector, and rewriting the modified Landau-\nLifshitz-Gilbert equation, we have shown that the spin-\ncurrent induced torque is qualitatively equivalent to an\napplied field. Using this equivalence we have shown that\nan applied AC magnetic field in the presence of con-\nstant spin current and DC applied magnetic field can\nlead to varied dynamical scenarios including chaos. We\nhave explicitly demonstrated numerically the chaotic be-\nhavior for a range of values of the parameters. The sys-\n182Oe190Oe198Oe199Oe\n207Oe\n215Oe 241OeAng.freq. (ns−1)a(Oe)\n2.31.81.3240\n220\n200\n180\nAng.freq. (ns−1)Power\n2.42.221.81.61.41.210.25\n0.2\n0.15\n0.1\n0.05\n0\nFIG. 6 The power spectrum distribution in the limit ω= 0,\nat certain values of a(indicated on each spectrum) where\nperiodic behavior is noted. Multiply periodic behavior is n o-\nticed for other values of ain the range shown. The current\nmagnitudes vary linearly and decrease with the frequency of\noscillation ( inset).hac= 0, while all other parameters are the\nsame as in FIG. 3.a (Oe)ω(GHz)\n4003503002502001501005045\n40\n35\n30\n25\n20\n15\n10\n5\nFIG. 7 Regions of multiply periodic dynamics for the system\nwith the DC applied field fixed at ha3= 20Oe, and non-zero\nanisotropy. All the other parameters remain the same as in\nFIG 4b. Synchronization is noted in the unshaded regions,\nwhile chaotic dynamics is not noticed in the parameter range\nshown in the figure. Islands of multiply periodic behavior ap -\npear between regions of periodic behavior for low frequenci es.\nFor higher frequencies, the dynamics is exclusively multip ly\nperiodic.\ntem also exhibits regular periodic behavior for a different\nrange of values. It is now realized that such nanoscale\nmonodomain layers can find application as resonators,\nthrough periodic oscillations induced by an alternating\nspin-current. The resultspresentedhereprovideanalter-\nnativemethod throughoscillationsinduced byanapplied\nmagnetic field. It is further noticed that the range of\nthe chaotic regime strongly depends on the presence of a\ncrystal anisotropy field. In the presence of an anisotropy\nfield chaotic behavior is noticed for much lower values\nof the DC field and spin-current, which are more suited\nfor chaos synchronization studies. However, there are re-\ngions in the ω−aspace where regular periodic motion is\nmore robust and presents a better alternative in applica-\ntions. In a future study we will present the possibility of\nchaos synchronization in spin-valve structures.\nACKNOWLEDGMENTS\nS.M. wishes to thank DST, India, for funding through\nthe FASTTRACK scheme. The work of M.L. forms part\nof a DST-IRHPA research project and is supported by a\nDST-Ramanna fellowship.\nReferences\n[1] S. A. Wolf, A. Y. Chtchelkanova and D. M. Treger, IBM\nJ. Res. Dev. 50, 101 (2006).\n[2] R. K. Nesbet, IBM J. Res. Dev. 42, 53 (1998).8\n[3] C. H. Tsang, R. E. Fontana, Jr., T. Lin, D. E. Heim,\nB. A. Gurnet and M. L. Williams, IBM J. Res. Dev. 42,\n103 (1998).\n[4] W. C. Black, Jr. and B. Das, J. Appl. Ph. 87, 6674\n(2000).\n[5] A. Ney, C. Pampuch, R. Koch and K. H. Ploog, Nature\n425, 485 (2003).\n[6] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[7] J. C. Slonczewski, J. Mag. Mag. Mat. 159, L1 (1996).\n[8] D. Berkov and J. Miltat, J. Mag. Mag. Mat. 320, 1238\n(2008).\n[9] M. D. Stiles and J. Miltat, Topics Appl. Physics 101, 225\n(2006).\n[10] E. B. Myers, D. C. 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Lett. 53,\n2497 (1984)."    },    {        "title": "0910.4684v2.Two_bodies_gravitational_system_with_variable_mass_and_damping_antidamping_effect_due_to_star_wind.pdf",        "content": "arXiv:0910.4684v2  [physics.class-ph]  20 May 2011Two bodies gravitational system with\nvariable mass and damping-anti damping\neffect due to star wind\nG.V. L´ opez∗and E. M. Ju´ arez\nDepartamento de F´ ısica de la Universidad de Guadalajara,\nBlvd. Marcelino Garc´ ıa Barrag´ an 1421, esq. Calzada Ol´ ım pica,\n44430 Guadalajara, Jalisco, M´ exico\nPACS: 45.20.D−,45.20.3j, 45.50.Pk, 95.10.Ce, 95.10.Eg, 96.30.Cw,\n03.67.Lx, 03.67.Hr, 03.67.-a, 03.65w\nJuly, 2010\nAbstract\nWe study two-bodies gravitational problem where the mass of one of the\nbodies varies and suffers a damping-anti damping effect due to st ar wind\nduring its motion. A constant of motion, a Lagrangian and a Ha miltonian\nare given for the radial motion of the system, and the period o f the body is\nstudied using the constant of motion of the system. An applic ation to the\ncomet motion is given, using the comet Halley as an example.\n∗gulopez@udgserv.cencar.udg.mx\n11 Introduction\nThere is not doubt that mass variable systems have been relevant s ince the founda-\ntion of the classical mechanics and modern physics too [0] which have been known as\nGylden-Meshcherskii problems [1]. Among these type of systems on e could mention:\nthe motion of rockets [2], the kinetic theory of dusty plasma [3], prop agation of elec-\ntromagnetic waves in a dispersive-nonlinear media [4], neutrinos mass oscillations\n[5], black holes formation [6], and comets interacting with solar wind [7]. T his last\nsystem belong to the so called ”gravitational two-bodies problem” w hich is one of\nthe most studied and well known system in classical mechanics [8]. In t his type of\nsystem, oneassumes normally that themasses of the bodiesarefix ed andunchanged\nduring the dynamical motion. However,when one is dealing with comets , beside to\nconsider its mass variation due to the interaction with the solar wind, one would\nlike to have an estimation of the the effect of the solar wind pressure on the comet\nmotion. This pressure may produces a dissipative-antidissipative eff ect on its mo-\ntion. The dissipation effect must be felt by the comet when this one is a pproaching\nto the sun (or star), and the antidissipation effect must be felt by t he comet when\nthis one is moving away from the sun.\nIn previous paper [14]. a study was made of the two-bodies gravitat ional prob-\nlem with mass variation in one of them, where we were interested in the difference\nof the trajectories in the spaces ( x,v) and (x,p). In this paper, we study the two-\nbodies gravitational problem taking into consideration the mass var iation of one\nof them and its damping-anti damping effect due to the solar wind. Th e mass of\nthe other body is assumed big and fixed , and the reference system of motion is\nchosen just in this body. In addition, we will assume that the mass los t is expelled\nfrom the body radially to its motion. Doing this, the three-dimensiona l two-bodies\nproblem is reduced to a one-dimensional problem. Then, a constant of motion, the\nLagrangian, and the Hamiltonian are deduced for this one-dimension al problem,\nwhere a radial dissipative-antidissipative force proportional to th e velocity square\nis chosen. A model for the mass variation is given, and the damping-a nti damping\neffect is studied on the period of the trajectories, the trajector ies themselves, and\nthe aphelion distance of a comet. We use the parameters associate d to comet Halley\nto illustrate the application of our results.\n22 Equations of Motion.\nNewton’s equations of motion for two bodies interacting gravitation ally, seen from\narbitrary inertial reference system, and with radial dissipative-a ntidissipative force\nacting in one of them are given by\nd\ndt/parenleftbigg\nm1dr1\ndt/parenrightbigg\n=−Gm1m2\n|r1−r2|3(r1−r2) (1a)\nand\nd\ndt/parenleftbigg\nm2dr2\ndt/parenrightbigg\n=−Gm1m2\n|r2−r1|3(r2−r1)−γ\n|r1−r2|/bracketleftbiggd|r1−r2|\ndt/bracketrightbigg2\n(r2−r1),(1b)\nwherem1andm2are the masses of the two bodies, r1= (x1,y1,z1) and\nr2= (x2,y2,z2) are their vectors positions from the reference system, Gis the\ngravitational constant ( G= 6.67×10−11m3/Kg s2),γis the nonnegative constant\nparameter of the dissipative-antidissipative force, and\n|r1−r2|=|r2−r1|=/radicalbig\n(x2−x1)2+(y2−y1)2+(z2−z1)2\nis the Euclidean distance between the two bodies. Note that if γ >0 and\nd|r1−r2|/dt>0one has dissipation since the force acts against the motion of the\nbody, and for d|r1−r2|/dt<0one has anti-dissipation since the force pushes the\nbody. Ifγ <0 this scheme is reversed and corresponds to our actual situation with\nthe comet mass lost.\nIt will be assumed the mass m1of the first body is constant and that the mass\nm2of the second body varies. Now, It is clear that the usual relative, r, and center\nof mass,R, coordinates defined as r=r2−r1andR= (m1r1+m2r2)/(m1+m2) are\nnot so good to describe the dynamics of this system. However, one can consider the\ncase form1≫m2(which is the case star-comet), and consider to put our referenc e\nsystem just on the first body ( r1=˜0). In this case, Eq. (1a) and Eq. (1b) are\nreduced to the equation\nm2d2r\ndt2=−Gm1m2\nr3r−˙m2˙r−γ/bracketleftbiggdr\ndt/bracketrightbigg2\nˆr, (2)\nwhere one has made the definition r=r2= (x,y,z),ris its magnitude, r=/radicalbig\nx2+y2+z2, andˆr=r/ris the unitary radial vector. Using spherical coordinates\n(r,θ,ϕ),\nx=rsinθcosϕ , y=rsinθsinϕ , z=rcosθ , (3)\n3one obtains the following coupled equations\nm2(¨r−r˙θ2−r˙ϕ2sin2θ) =−Gm1m2\nr2−˙m2˙r−γ˙r2, (4)\nm2(2˙r˙θ+r¨θ−r˙ϕ2sinθcosθ) =−˙m2r˙θ , (5)\nand\nm2(2˙r˙ϕsinθ+r¨ϕsinθ+2r˙ϕ˙θcosθ) =−˙m2r˙ϕsinθ . (6)\nTaking ˙ϕ= 0 as solution of this last equation, the resulting equations are\nm2(¨r−r˙θ2) =−Gm1m2\nr2−˙m2˙r−γ˙r2, (7)\nand\nm2(2˙r˙θ+r¨θ)+ ˙m2r˙θ= 0. (8)\nFrom this last expression, one gets the following constant of motion (usual angular\nmomentum of the system)\nlθ=m2r2˙θ , (9)\nand with this constant of motion substituted in Eq. 7, one obtains th e following\none-dimensional equation of motion for the radial part\nd2r\ndt2=−Gm1\nr2−˙m2\nm2/parenleftbiggdr\ndt/parenrightbigg\n−γ\nm2˙r2+l2\nθ\nm2\n2r3. (10)\nNow, let us assume that m2is a function of the distance between the first and the\nsecond body, m2=m2(r). Therefore, it follows that\n˙m2=m′\n2˙r , (11)\nwherem′\n2is defined as m′\n2=dm2/dr. Thus, Eq. (10) is written as\nd2r\ndt2=−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2/parenleftbiggdr\ndt/parenrightbigg2\n, (12)\nwhich, in turns, can be written as the following autonomous dynamica l system\ndr\ndt=v;dv\ndt=−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2v2. (13)\nNote from this equation that m′\n2is always a non-positive function of rsince it\nrepresents the mass lost rate. On the other hand, γis a negative parameter in our\ncase.\n43 Constant of Motion, Lagrangian and Hamilto-\nnian\nA constant of motion for the dynamical system (13) is a function K=K(r,v) which\nsatisfies the partial differential equation [9]\nv∂K\n∂r+/bracketleftbigg−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2v2/bracketrightbigg∂K\n∂v= 0. (14)\nThe general solution of this equation is given by [10]\nK(x,v) =F(c(r,v)), (15a)\nwhereFis an arbitrary function of the characteristic curve c(r,v) which has the\nfollowing expression\nc(r,v) =m2\n2(r)e2γλ(r)v2+/integraldisplay/parenleftbigg2Gm1\nr2−2l2\nθ\nm2\n2r3/parenrightbigg\nm2\n2(r)e2γλ(r)dr , (15b)\nand the function λ(r) has been defined as\nλ(r) =/integraldisplaydr\nm2(r). (15c)\nDuring a cycle of oscillation, the function m2(r) can be different for the comet\napproaching the sun and for the comet moving away from the sun. L et us denote\nm2+(r)forthefirstcaseand m2−(r)forthesecondcase. Therefore, onehastwocases\nto consider in Eqs. (15a), (15b) and (15c) which will denoted by ( ±). Now, if mo\n2±\ndenotes the mass at aphelium (+) or perielium (-) of the comet, F(c) =c±/2mo\n2±\nrepresents the functionality in Eq. (15a) such that for m2constant and γequal\nzero, this constant of motion is the usual gravitational energy. T hus, the constant\nof motion can be chosen as K±=c(r,v)/2m0\n2±, that is,\nK±=m2\n2±(r)\n2mo\n2±e2γλ±(r)v2+V±\neff(r), (16a)\nwhere the effective potential Veffhas been defined as\nV±\neff(r) =Gm1\nmo\n2±/integraldisplaym2\n2±(r)e2γλ±(r)dr\nr2−l2\nθ\nmo\n2±/integraldisplaye2γλ±(r)dr\nr3(16b)\n5This effective potential has an extreme at the point r∗defined by the relation\nr∗m2\n2(r∗) =l2\nθ\nGm1(17)\nwhich is independent on the parameter γand depends on the behavior of m2(r).\nThis extreme point is a minimum of the effective potential since one has\n/parenleftBigg\nd2V±\neff\ndr2/parenrightBigg\nr=r∗>0. (18)\nUsing the known expression [11-13] for the Lagrangian in terms of t he constant of\nmotion,\nL(r,v) =v/integraldisplayK(r,v)dv\nv2, (19)\nthe Lagrangian, generalized linear momentum and the Hamiltonian are given by\nL±=m2\n2±(r)\n2mo\n2±e2γλ±(r)v2−V±\neff(r), (20)\np=m2\n2±(r)v\nmo\n2±e2γλ±(r), (21)\nand\nH±=mo\n2±p2\n2m2\n2±(r)e−2γλ±(r)+V±\neff(r). (22)\nThe trajectories in the space ( x,v) are determined by the constant of motion (16a).\nGiven the initial condition ( ro,vo), the constant of motion has the specific value\nK±\no=m2\n2±(ro)\n2mo\n2±e2γλ±(ro)v2\no+V±\neff(ro), (23)\nand the trajectory in the space ( r,v) is given by\nv=±/radicalBigg\n2mo\n2±\nm2\n2±(r)e−γλ±(r)/bracketleftbigg\nK±\no−V±\neff(r)/bracketrightbigg1/2\n. (24)\nNote that one needs to specify ˙θoalso to determine Eq. (9). In addition, one\nnormallywants toknowthetrajectoryintherealspace, thatis, t heacknowledgment\n6ofr=r(θ). Since one has that v=dr/dt= (dr/dθ)˙θand Eqs. (9) and (24), it\nfollows that\nθ(r) =θo+l2\nθ/radicalbig2mo\n2±/integraldisplayr\nrom2±(r)eγλ±(r)dr\nr2/radicalBig\nK±o−V±\neff(r). (25)\nThe half-time period (going from aphelion to perihelion (+), or backwa rd (-)) can\nbe deduced from Eq. (24) as\nT±\n1/2=1/radicalbig2mo\n2±/integraldisplayr2\nr1m2±(r)eγλ±(r)dr/radicalBig\nK±o−V±\neff(r), (26)\nwherer1andr2are the two return points resulting from the solution of the following\nequation\nV±\neff(ri) =K±\noi= 1,2. (27)\nOntheother hand, thetrajectory inthespace ( r,p) isdetermine by theHamiltonian\n(21), and given the same initial conditions, the initial poandH±\noare obtained from\nEqs. (21) and (22). Thus, this trajectory is given by\np=±/radicalBigg\n2m2\n2±(r)\nmo\n2±eγλ±(r)/bracketleftbigg\nH±\no−V±\neff(r)/bracketrightbigg1/2\n. (28)\nIt is clear just by looking the expressions (24) and (28) that the tr ajectories in the\nspaces (r,v) and (r,p) must be different due to complicated relation (21) between v\nandp(see reference [14]).\n4 Mass-Variable Model and Results\nAs a possible application, consider that a comet looses material as a r esult of the\ninteraction with star wind in the following way (for one cycle of oscillatio n)\nm2±(r) =/braceleftBiggm2−(r2(i−1))/parenleftbigg\n1−e−αr/parenrightbigg\nincoming (+)v <0\nm2+(r2i−1)−b/parenleftbigg\n1−e−α(r−r2i−1)/parenrightbigg\noutgoing (−)v >0\n7(29)\nwhere the parameters b >0 andα >0 can be chosen to math the mass loss rate in\nthe incoming and outgoing cases. The index ”i” represent the ith-se mi-cycle, being\nr2(i−1)andr2i−1the aphelion( ra) and perihelion( rp) points ( rois given by the initial\nconditions, and one has that m2−(ro) =mo). For this case, the functions λ+(r) and\nλ−(r) are given by\nλ+(r) =1\nαmaln/parenleftbigg\neαr−1/parenrightbigg\n, (30a)\nand\nλ−(r) =−1\nα(b−mp)/bracketleftBigg\nαr+ln/parenleftbig\nmp−b(1−e−α(r−rp))/parenrightbig/bracketrightBigg\n. (30b)\nwhere we have defined ma=m2(ra) andmp=m2(rp). Using the Taylor expansion,\none gets\ne2γλ+(r)=e2γr/ma/bracketleftbigg\n1−2γ\nαmae−αr+1\n22γ\nαma/parenleftbigg2γ\nαma−1/parenrightbigg\ne−2αr+.../bracketrightbigg\n,(31a)\nand\ne2γλ−(r)=e−2γr\n(b−mp)\n(mp−b)2γ\nα(mp−b)/bracketleftbigg\n1+2γ\nα(mp−b)e−α(r−rp)\nmp−b\n+1\n22γ\nα(mp−b)/parenleftbigg2γ\nα(mp−b)−1/parenrightbigge−2α(r−rp)\n(mp−b)2+.../bracketrightbigg\n(31b)\nThe effective potential for the incoming comet can be written as\nV+\neff(r) =/bracketleftbigg\n−Gm1ma\nr+l2\nθ\n2ma1\nr2/bracketrightbigg\ne2γr/ma+W1(γ,α,r), (32)\nand for the outgoing comet as\nV−\neff(r) =/bracketleftbigg\n−Gm1ma\nr+l2\nθ\n2ma1\nr2/bracketrightbigge2γr\n(mp−b)\n(mp−b)2γ\nα(mp−b)+W2(γ,α,r),(33)\nwhereW1andW2are given in the appendix.\n8We will use the data corresponding to the sun mass (1 .9891×1030Kg) and the\nHalley comet [15-17],\nmc≈2.3×1014Kg, r p≈0.6au, r a≈35au, l θ≈10.83×1029Kg·m2/s,(34)\nwith a mass lost of about δm≈2.8×1011Kgper cycle of oscillation. Although,\nthe behavior of Halley comet seem to be chaotic [18], but we will neglect this fine\ndetail here. Now, the parameters αand ”b” appearing on the mass lost model, Eq.\n(29), are determined by the chosen mass lost of the comet during t he approaching to\nthe sun and during the moving away from the sun (we have assumed t he same mass\nlost in each half of the cycle of oscillation of the comet around the sun ). Using Eqs.\n(32) and (33), Eq. (24), the trajectories can be calculated in the spaces (r,v) . Fig.\n1 shows these trajectories using δm= 2×1010Kg(orδm/m= 0.0087%) for γ= 0\nand (continuos line), and for γ=−3Kg/m(dashed line), starting both cases from\nthe same aphelion distance. As one can see on the minimum, dissipation causes to\nreduce a little bit the velocity of the comet , and the antidissipation inc reases the\ncomet velocity, reaching a further away aphelion point. Also, when o nly mass lost is\nconsidered ( γ= 0) the comet returns to aphelion point a little further away from th e\ninitial one during the cycle of oscillation. Something related with this eff ect is the\nchange of period as a function of mass lost ( γ= 0). This can be see on Fig. 2, where\nthe period is calculated starting always from the same aphelion point ( ra). Note\nthat with a mass lost of the order 2 .8×1011Kg(Halley comet), which correspond to\nδm/m=.12%, the comet is well within 75 years period. The variationofthe rat io of\nthe change of aphelion distance as a function of mass lost ( γ= 0) is shown on Fig.3.\nOn Fig. 4, the mass lost rate is kept fixed to δm/m= 0.0087%, and the variation of\nthe period of the comet is calculated as a function of the dissipative- antidissipative\nparameter γ <0 (using|γ|for convenience). As one can see, antidissipation always\nwins to dissipation, bringing about the increasing of the period as a fu nction of\nthis parameter. The reason seems to be that the antidissipation ac ts on the comet\nwhen this ones is lighter than when dissipation was acting (dissipation a cts when\nthe comet approaches to the sun, meanwhile antidissipation acts wh en the comet\ngoes away from the sun). Since the period of Halley comets has not c hanged much\nduring many turns, we can assume that the parameter γmust vary in the interval\n(−0.01,0]Kg/m. Finally, Fig. 5 shows the variation, during a cycle of oscillation, of\nthe ratio of the new aphelion ( r′\na) to old aphelion ( ra) as a function of the parameter\nγ.\n95 Conclusions and comments\nThe Lagrangian, Hamiltonian and a constant of motion of the gravita tional attrac-\ntion of two bodies were given when one of the bodies has variable mass and the\ndissipative-antidissipative effect of the solar wind is considered. By c hoosing the\nreference system in the massive body, the system of equations is r educe to 1-D\nproblem. Then, the constant of motion, Lagrangian and Hamiltonian were obtained\nconsistently. A model for comet-mass-variation was given, and wit h this model, a\nstudy was made of the variation of the period of one cycle of oscillatio n of the comet\nwhen there are mass variation and dissipation-antidissipation. When mass variation\nis only considered, the comet trajectory is moving away from the su n, the mass lost\nis reduced as the comet is farther away (according to our model), a nd the period of\noscillations becomes bigger. When dissipation-antidissipation is added , this former\neffect becomes higher as the parameter γbecomes higher.\nIt is important to mention that if instead of loosing mass the body wou ld had\nwinning mass, the period of oscillation of the system would decrease. One can\nimagine, for example, a binary stars system where one of the star is winning mass\nfromtheinterstellar space or fromtheother star companion. So, due to thiswinning\nmass, the period of the star would decrease depending on how much mass the star\nis absorbing.\n106 Appendix\nExpression for W1andW2:\nW1=Gm2\n2−\nmo\n2+/braceleftBigg\n−p(p−1)e(−4+p)αr\n2r+αpEi(αpr)−2αp(p−1)Ei/parenleftbig\n(−4+p)αr/parenrightbig\n+αp2(p−1)\n2Ei/parenleftbig\n(−4+p)αr/parenrightbig\n+p(p−1)\nr/bracketleftbig\ne(p−3)αr+3α(1−p)rEi/parenleftbig\n(p−3)αr/parenrightbig/bracketrightbig\n+p(p+3)\n2/bracketleftbigg\n−e(p−2)αr\nr+α(p−2)Ei/parenleftbig\n(p−2)αr/parenrightbig/bracketrightbigg\n+p+2\nr/bracketleftbig\ne(p−2)αr+α(p−1)rEi/parenleftbig\n(p−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n+l2\nθ\n2m2\n2+r2/braceleftBigg\np(p−1)\n2e(p−2)αr−pe(p−1)αr−αp(p−1)e(p−2)αr+αp(p−1)\n2epαr\n+α2p(p−1)r\n2e(p−2)αr+pαre(p−1)αr−p2αre(p−1)αr−p2α2r2Ei/parenleftbig\npαr/parenrightbig\n−α2(p−2)2p(p−1)r2\n2Ei/parenleftbig\n(p−2)αr/parenrightbig\n+pα2r2Ei/parenleftbig\n(p−1)αr/parenrightbig\n−2α2p2r2Ei/parenleftbig\n(p−1)αr/parenrightbig\n+p3α2r2Ei/parenleftbig\n(p−1)αr/parenrightbig/bracerightBigg\n(A1)\nwheremais the mass of the body at the aphelion, and we have made the definitio ns\np=2γ\nαma(A2)\nand the function Eiis the exponential integral,\nEi(z) =/integraldisplay∞\n−ze−t\ntdt (A3)\n11W2=Gm2\n2−\nmo\n2+/braceleftBigg\ne(q−2)αr\nr/bracketleftbigg\n1+q(q−1)\n2(mp+αq)e2qαr+2q\nmp+αqeqαr/bracketrightbigg\n+qαEi/parenleftbig\nqαr/parenrightbig\n−q(q−1)e2qαr\n(mp+αq)2r/bracketleftbig\ne(q−3)αr−α(q−3)rEi/parenleftbig\n(q−3)αr/parenrightbig/bracketrightbig\n+qeqαr\n(mp+αq)r/bracketleftbig\ne(q−3)αr−α(q−3)rEi/parenleftbig\n(q−3)αr/parenrightbig/bracketrightbig\n−2αEi/parenleftbig\n(q−2)αr/parenrightbig\n+αqEi/parenleftbig\n(q−2)αr/parenrightbig\n−q(q−1)αe2qαr\n(mp+αq)2Ei/parenleftbig\n(q−2)αr/parenrightbig\n+q2(q−1)αe2qαr\n2(mp+αq)2Ei/parenleftbig\n(q−2)αr/parenrightbig\n−4αeqαr\nmp+αqEi/parenleftbig\n(q−2)αr/parenrightbig\n+2q2αeqαr\n(mp+αq)rEi/parenleftbig\n(q−2)αr/parenrightbig\n+2\nr/bracketleftbig\ne(q−1)αr−(q−1)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig\n+qeqαr\n(mp+αq)r/bracketleftbig\ne(q−1)αr−(q−1)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n+l2\nθ\n2m2\n2+(mp+αq)q/braceleftBigg\n−qαeqαr\nr+q2α2Ei/parenleftbig\nqαr/parenrightbig\n+q(q−1)e(3q−2)αr\n2(mp+αq)2r2/bracketleftbig\n−1+2αr−qαr+(2−q)2α2r2e(2−q)αrEi/parenleftbig\n(q−2)αr/parenrightbig/bracketrightbig\n−qe(2q−1)αr\n(mp+αq)r2/bracketleftbig\n−1+αr+qαr+(q−1)2α2r2e(1−q)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n(A4)\nwherempis the mass of the body at the perihelion, and we have made the definit ion\nq=2γ\nα(mp−b)(A5)\n127 Bibliography\n0.G. L´ opez, L.A. Barrera, Y. Garibo, H. Hern´ andez, J.C. Salazar,\nand C.A. Vargas, One-Dimensional Systems and Problems Associated with Get-\nting Their Hamiltonians , Int. Jour. Theo. Phys., 43,10 (2004),1.\n1.H. Gylden, Die Bahnbewegungen in einem Systeme von zwei K¨ orpern in dem\nFalle, dass die Massen Ver¨ anderungen unterworfen sind ,\nAstron. Nachr., 109, no. 2593 (1884),1.\nI.V. Meshcherskii, Ein Specialfall des Gyldn’schen Problems (A. N. 2593) ,\nAstron. Nachr., 132, no. 3153 (1893),93.\nI.V. Meshcherskii, Ueber die Integration der Bewegungsgleichungen im Problem e\nzweier Krper von vernderlicher Masse ,\nAstron. Nachr., 159, no. 3807 (1902),229.\nE.O. Lovett, Note on Gyldn’s equations of the problem of two bodies with ma sses\nvarying with the time ,\nAstron. Nachr., 158, no. 3790 (1902), 337.\nJ.H. Jeans, Cosmogonic problems associated with a secular decrease of m ass,\nMNRAS, 85, no. 1 (1924),2.\nL.M. Berkovich, Gylden-Mescerskii problem ,\nCelestial Mechanics, 24(1981),407.\nA.A. Bekov, Integrable Cases and Trajectories in the Gylden-Meshchers kii Problem ,\nAstron. Zh., 66(1989),135.\nC. Prieto and J.A. Docobo, Analytic solution of the two-body problem with slowly\ndecreasing mass , Astron. Astrophys., 318(1997),657.\n2.A. Sommerfeld, Lectures on Theoretical Physics , Vol. I,\nAcademic Press (1964).\n3.A.G. Zagorodny, P.P.J.M. Schram, and S.A. Trigger, Stationary Velocity and\nCharge Distributions of Grains in Dusty Plasmas ,\nPhys. Rev. Lett., 84(2000),3594.\n4.O.T. Serimaa, J. Javanainen, and S. Varr´ o, Gauge-independent Wigner func-\ntions: General formulation ,\nPhys. Rev. A, 33, (1986), 2913.\n5.H.A. Bethe, Possible Explanation of the Solar-Neutrino Puzzle ,\n13Phys. Rev. Lett., 56, (1986),1305.\nE.D. Commins and P.H. Bucksbaum, Weak Interactions of\nLeptons and Quarks , Cambridge University Press (1983).\n6.F.W. Helhl, C. Kiefer and R.J.K. Metzler, Black Holes: Theory\nand Observation , Springer-Verlag (1998).\n7.P.W. Daly, The use of Kepler trajectories to calculate ion fluxes at mult i-\ngigameter distances from Comet ,\nAstron. Astrophys., 226(1989) 318.\n8.H. Goldstein, Classical Mechanics , Addison-Wesley, M.A., (1950).\n9.G. L´ opez, Partial Differential Equations of First Order and\nTheir Applications to Physics , World Scientific, 1999.\n10.F. John,Partial Differential Equations , Springer-Verlag\nNew York (1974).\n11.J.A. Kobussen, Some comments on the Lagrangian formalism for systems with\ngeneral velocity-dependent forces ,\nActa Phys. Austr. 51,(1979),193.\n12.C. Leubner, Inequivalent lagrangians from constants of the motion ,\nPhys. Lett. A 86,(1981), 2.\n13.G. L´ opez, One-Dimensional Autonomous Systems and Dissipative Syste ms,\nAnn. of Phys., 251,2 (1996), 372.\n14.G. L´ opez, Constant of Motion, Lagrangian and Hamiltonian of the Gravi ta-\ntional Attraction of Two Bodies with Variable Mass ,\nInt. Jour. Theo. Phys., 46, no. 4, (2007), 806.\n15.G. Cevolani, G. Bortolotti and A. Hajduk, Debris from comet Halley, comet’s\nmass loss and age ,\nIL Nuo. Cim. C, 10, no.5 (1987) 587.\n16.J.L. Brandy, Halley’s Comet: AD 1986 to 2647 BC ,\n14J. Brit. astron. Assoc., 92, no. 5 (1982) 209.\n17.D.C. Jewitt, From Kuiper Belt Object to Cometary Nucleus: The Missing\nUltrared Matter ,\nAstron. Jour., 123, (2002) 1039.\n18.B.V. Chirikov and V.V. Vecheslavov, Chaotic dynamics of Comet Halley\nAstron. Astrophys., 221, (1989) 146.\n151/Multiply10122/Multiply10123/Multiply10124/Multiply10125/Multiply10126/Multiply1012r/LParen1m/RParen1\n/Minus20000/Minus100001000020000v/LParen3m\ns/RParen3\nFigure 1: Trajectories in the ( r,v) space with δm/m= 0.009.0 0.1 0.2 0.3 0.4 0.5 0.6\nδm/m (%)100150200250300350400450500T (years)\nFigure 2: Period of the comet as a function of the mass lost ratio.\n160 0.1 0.2 0.3 0.4 0.5 0.6\nδm/m (%)012345δ ra/ra\nFigure 3: Ratio of aphelion distance change as a function of the mass lost rate.0 1 2 3 4 5 6 7\n| γ |708090100110120130140T (years)δm=2x1010\nKg(δm/m=0.0087%)\nFigure 4: Period of the comet as a function of the parameter γ.\n170 1 2 3 4 5 6 7\n| γ |11.21.41.61.82ra'/raδm/m=0.0087%\nFigure 5: Ratio of the aphelion increasing as a function of the parame terγ.\n18"    },    {        "title": "0911.1108v3.Bloch_oscillations_in_lattice_potentials_with_controlled_aperiodicity.pdf",        "content": "Bloch oscillations in lattice potentials with controlled aperiodicity\nStefan Walter,1,\u0003Dominik Schneble,1and Adam C. Durst1,y\n1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA\n(Dated: 25 March 2010)\nWe numerically investigate the damping of Bloch oscillations in a one-dimensional lattice poten-\ntial whose translational symmetry is broken in a systematic manner, either by making the potential\nbichromatic or by introducing scatterers at distinct lattice sites. We \fnd that the damping strongly\ndepends on the ratio of lattice constants in the bichromatic potential and that even a small concen-\ntration of scatterers can lead to strong damping. Moreover, collisional interparticle interactions are\nable to counteract aperiodicity-induced damping of Bloch oscillations. The discussed e\u000bects should\nreadily be observable for ultracold atoms in optical lattices.\nPACS numbers: 03.75.-b, 67.85.Hj, 72.10.Fk\nI. INTRODUCTION\nThe oscillatory motion of particles in a periodic po-\ntential, when subject to an external force, was predicted\nby Felix Bloch in 1928 [1]. Bloch oscillations were \frst\nobserved in the 1990s both in semiconductor superlat-\ntices [2] and in systems of laser-cooled atoms in optical\nlattices [3, 4]. Since then, they have also been studied\nin atomic quantum gases [5{8] and have found appli-\ncations in ultracold atomic-physics-based precision mea-\nsurements [9{12].\nBloch oscillations are absent in solids due to fast damp-\ning from scattering by defects and phonons. Their obser-\nvation in long-period semiconductor superlattices relies\non oscillation periods that are shorter than the charac-\nteristic scattering lifetime, and even in those systems,\ndamping of Bloch oscillations due to disorder is ubiq-\nuitous [13, 14]. In contrast, such damping is absent in\noptical lattice systems which are inherently defect-free,\nallowing for the observation of a large number of oscil-\nlations [6{8]. Damping can however be induced by the\nintroduction of disorder into the lattice potential, which\ncan in principle be done with various techniques [15{19].\nAlso, in the case of quantum gases, a damping of Bloch\noscillations arises from mean-\feld interactions between\nweakly interacting, Bose-condensed atoms [7, 8, 20{24].\nIn the presence of both interactions and disorder, a re-\nduction of disorder-induced damping due to screening of\ndisorder by the mean \feld has been predicted [25]. Ex-\nperiments with ultracold atoms thus not only constitute\na versatile testbed for behavior expected in solid-state\nsystems but may also display novel e\u000bects.\nRecently, the damping of Bloch oscillations in a Bose-\nEinstein condensate [26] has been observed for an optical\nlattice with a superimposed randomly corrugated opti-\ncal \feld. A related theoretical investigation of disorder-\ninduced damping of Bloch oscillations has considered\nthe case of Gaussian spatial noise [25]. In this paper,\n\u0003present address: Dept. of Physics, Univ. W urzburg, Germany\nypresent address: Photon Research Associates, Port Je\u000berson, NYwe numerically investigate the dynamics of an atomic\nwave packet in a potential with a systematically degraded\ntranslational symmetry, considering two scenarios: The\n\frst is based on the use of a weak bichromatic poten-\ntial [16] of a variable wavelength ratio, and the second\nconsiders scatterers (impurities) pinned at single sites of\nthe potential [17]. We \fnd that the damping strongly\ndepends on the ratio of lattice constants in the bichro-\nmatic case and that even a small concentration of scat-\nterers can lead to strong damping. We also include ef-\nfects of the mean-\feld interaction and \fnd that the rate\nat which damping of the Bloch oscillations occurs is re-\nduced, similar to the case of Gaussian disorder [25]. Both\ne\u000bects should be observable experimentally with existing\nultracold-atom technology.\nThis paper is organized as follows: After a brief discus-\nsion of fundamental aspects of tilted lattices in Sec. II,\nSection III investigates the in\ruence of aperiodicity on\nthe damping dynamics of Bloch oscillations. Sec. IV ad-\ndresses the interplay of aperiodicity and the interaction\nbetween atoms. Conclusions are given in Sec. V.\nII. TILTED PERIODIC POTENTIALS\nThe Hamiltonian for the motion of a particle in a one-\ndimensional periodic potential V(x) =V(x+a) with lat-\ntice constant apossesses a complete set of eigenfunctions\nthat obey Bloch's theorem '(x+a) =eika'(x). The\ncorresponding eigenvalues En(k) are periodic in momen-\ntum space and form energy bands, En(k+K) =En(k)\n(wherenis the band index, ~kthe quasimomentum,\nandK= 2\u0019=a the width of the \frst Brillouin zone).\nUnder the in\ruence of an externally applied constant\nforce (\\tilt\") F, the quasimomentum evolves as ~k(t) =\n~k0+Ft. Due to the periodicity of the energy bands,\nthis results in oscillations of the particle's group veloc-\nityvg;n(k) = (1=~)dEn(k)=dk. These oscillations occur\nwith a period TB= 2\u0019=!B=h=(aF) and a maximum\ndisplacement 2 AB;n= \u0001 n=Fin coordinate space, where\n\u0001n=jEn(K=2)\u0000En(0)jis the width of the n-th band.\nIn the following, we consider particles that are con-\n\fned to the lowest Bloch band, n=1, corresponding toarXiv:0911.1108v3  [cond-mat.quant-gas]  11 Apr 20102\n1\n0\nFIG. 1. Evolution of the momentum-space density j (k;t)j2\n(a) in a tilted periodic potential (b) in the presence of an\nadditional periodic potential, and (c) in the presence of two\nlocalized scatterers as discussed in the text. Additional mo-\nmentum components emerge in (b) and (c), broadening the\nmomentum distribution. The parameters in (b) are \r= 0:01\nand\u000b= 1=p\n5; the scatterers in (c) are spaced 11 sites apart.\nA detailed explanation is given in the text. The density of\nj (k;t)j2is normalized to 1; a corresponding color scale is\nshown on the right.\nsu\u000eciently deep potentials and small enough tilts such\nthat Zener tunneling [27] at the band edges is negligible.\nUsing the split-operator method [28], we perform numer-\nical simulations of the dynamics of Bloch oscillations of\na Gaussian wave packet\n (x;t= 0) =1\n(2\u0019\u001b2)1=4exp\u0014\n\u0000x2\n(2\u001b)2\u0015\n(1)\nthat evolves according to the Hamiltonian\nH=\u0000~2@2\nx\n2m+V0cos(Kx) +~V(x) +Fx (2)\nfor motion in a tilted periodic potential V(x) =\nV0cos(Kx) +Fxthat is modi\fed by a weak additional\npotential ~V(x).\nIII. EFFECTS OF MODIFIED PERIODICITY\nThis section investigates the in\ruence of the additional\npotential ~V(x), which either is a weak additional peri-\nodic potential with variable lattice constant, or arises\nfrom the local interaction with scatterers pinned at sin-\ngle sites of the tilted periodic potential. In the absence\nof~V(x), the energies of neighboring sites di\u000ber by a \fxed\namountFa=~!B, corresponding to a spatially homoge-\nneous phase di\u000berence \u0001 \u001e(t) =!Bt. In the presence of\n~V(x), \u0001\u001egenerally becomes position-dependent, leading\nto global dephasing and thus to a broadening of the wave\npacket in momentum space, as illustrated in Fig. 1.\n02468101214161820/Minus0.7/Minus0.6/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0\nt/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1\n0246810121416182001234\nt/LParen1units of TB/RParen1/LAngleBracket1Σ/MinusΣ 0/RAngleBracket1/LParen1units of 10/Minus3a/RParen1/LParen1b/RParen1FIG. 2. Damped Bloch oscillations in a bichromatic lattice\n(\r= 0:005,\u000b= 0:4). The collapse of the oscillations in\ncoordinate space (a) is accompanied by breathing-mode exci-\ntations of the wave packet (b), and vice versa (see also text).\nThe dashed line in (a) is the envelope Aexp(\u0000\u0011t2) +B.\nA. Bichromatic potentials\nA tunable bichromatic potential is generated by the\naddition of\n~V(x) =\rV0cos(\u000bKx ); (3)\nwith variable relative amplitude \r\u001c1 and lattice-\nconstant ratio \u000b. If\u000bis a rational number \u000b=p=q,\nthe total potential has a periodicity \u0003 = aq. If further-\nmore \u0003 exceeds the spatial range probed by the wave\npacket, the potential can be considered disordered.\nThe evolution of the wave packet (Eq. 1) typically ex-\nhibits collapses and revivals of center-of-mass oscillations\nthat are coupled to breathing-mode excitations at twice\nthe Bloch frequency as is shown in Fig. 2 (visible after\na transient phase immediately following the switch-on of\n~V). The presence of a collapse and a revival of the Bloch\noscillations is a consequence of the absence of dissipation\nin our model. To characterize the decay of Bloch oscil-\nlations, the numerical data for their initial decay, for a\ngiven\rand\u000b, are \ftted with the function\nf(t) =Aexp(\u0000\u0011t2) cos (!Bt) +B: (4)\nResults for the Gaussian-decay constant \u0011are shown in\nFig. 3.\nClearly an increase in the perturbation amplitude \r\nleads to an overall increase in the damping of the oscil-\nlations. However, the dependence of the damping on the\nratio\u000bis less trivial. For integer values of \u000b, the poten-\ntial retains its original periodicity and no dephasing of\nBloch oscillations occurs. It is also suppressed for half-\ninteger values of \u000b(i.e. forq= 2), which correspond to\na doubling of the lattice constant, and correspondingly a\nhalving of the Brillouin zone, and of the corresponding\nBloch period. For this case, the dynamics of the wave\npacket in the modi\fed band structure can easily be visu-\nalized. The shape of the band is essentially that of V(x),\nbut it is folded back into the new Brillouin zone, thus3\n/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder\n/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle0.0 0.5 1.0 1.5 2.00.000.020.040.060.080.100.12\nΑDamping constant Η/LParen1units of TB/Minus2/RParen1\n/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/Placeholder/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle1.49 1.5 1.5100.010.02\nFIG. 3. Damping constant \u0011as a function of the lattice-\nconstant ratio \u000b, for three di\u000berent depth ratios: \r= 0:005\n(squares),\r= 0:01 (diamonds) and \r= 0:013 (circles). The\ninset shows the behavior of the decay constant in the vicinity\nof\u000b= 1:5.\n0246810121416182000.0050.010.015\nt/LParen1units of TB/RParen1/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1 /VertBar1/LParen1a/RParen1\n1 43 2\n1 42 3\n/MinusΠ/Slash12 0 Π/Slash12\nk/LParen1units of 1 /Slash1a/RParen1E/LParen1k/RParen1/LParen1b/RParen1\n/LBrace1 /RBrace1\nFIG. 4. Wave packet in a bichromatic potential with \u000b=\n1=2;\r= 0:005. (a) Overlap jh (x;t= 0)j (x;t)ij, exhibiting\nthe original periodicity of TB, with small, growing, contri-\nbutions at odd multiples of TB=2. (b) Band structure of the\npotential. The inset shows the small gap at the Brillouin zone\nboundaries, with a calculated width ~\u0001 = 9:5\u000210\u00005Er. The\narrows with the corresponding numbers indicate the motion\nof the particle in kspace. The wave function completes one\nBloch cycle after TB, going from 1!2!3!4, with most\nof the wave function tunneling across the gap.\nforming a closed loop, with a small splitting ~\u0001/\rat\nthe new zone boundaries (cf. Fig. 4). In one Bloch cy-\ncle, most of the wave function tunnels through the tiny\ngap from the lower to the upper portion at one boundary\nand then back to the lower portion at the other bound-\nary (Bloch-Zener tunneling [29, 30]). The time needed\nfor one such cycle is TB, that is the Bloch period for the\nunperturbed potential V(x). This behavior can directly\nbe seen in the time dependence of the overlap of the wave\nfunction with the initial wave packet jh (x;0)j (x;t)ij,\nwhich has a periodicity of TB; contributions at odd mul-\ntiples ofTB=2 only grow very slowly over a large number\nof cycles.The dynamics of Bloch oscillations in Figs. 2 and 3 are\nfor a wave packet with width \u001b0= 20ain a lattice with\ndepthV0= 10Er[whereEr\u0011(~K=2)2=2mis the recoil\nenergy], and a small external force F= 0:011V0=a. In\ncoordinate space the amplitude of the Bloch oscillations\nisAB= 0:34a. The additional lattice ~V(x) is chosen\nto have a small relative amplitude \r= 0:005, which is\nsu\u000ecient to cause noticeable damping already after a few\nBloch cycles, depending on the lattice constant ratio \u000b.\nExperimentally, such a bichromatic potential is\nstraightforward to realize in the context of optical lat-\ntices, using two laser beams of di\u000berent wavelength. With\npresent-day tunable-laser technology, a large fraction of\nthe range of \u000bshown in Fig. 3 can be accessed.\nB. Scatterers on distinct sites\nFor scatterers pinned at a set of distinct sites fngof the\nlattice, we model the potential ~Vas a sum of Gaussians,\n~V(x) =X\nfng~Aexp\u0002\n\u0000(x\u0000xn)2=(2~\u001b2)\u0003\n: (5)\nFor the simulation, the parameters of the optical lattice\nand the tilt are chosen such that the size \u001bof the wave\npacket and its oscillation amplitude ABin the unper-\nturbed titled potential each cover a large number of lat-\ntice sites. A small number of scatterers are then ran-\ndomly placed within the range of the wave packet's mo-\ntion. The amplitude ~Aand width ~\u001bof the scatterers are\nchosen such that the valleys of the unperturbed poten-\ntialV0cos(Kx) are e\u000bectively \flled up where the scat-\nterers are located. Their e\u000bect on the Bloch oscillations\nis shown in Fig. 5. The general trend with an increas-\ning number of scatterers is a more rapid damping of the\nBloch oscillations, with the details of the dynamics de-\npending on their spatial arrangement. Clearly, already a\nsmall number of scatterers can lead to a rapid damping\nof Bloch oscillations.\nThe results shown in Fig. 5 are for a comparatively\nshallow lattice ( V0= 1:4Er) and a tilting force F=\n0:022Er=a, resulting in a large amplitude AB= 16a\nfor undamped Bloch oscillations. This condition for the\nscatterers is well ful\flled by setting ~ \u001b=\u0019a=20 and ~A=\nEr.\nIn the context of ultracold atoms, the placement of\nscatterers pinned at single lattice sites can be achieved,\nfor example by using atoms with two internal states in\nconjunction with a state-dependent lattice depth [31], or\nby using two atomic species in a species-dependent opti-\ncal lattice [32].\nIV. EFFECT OF INTERACTIONS\nThe discussion so far has been restricted to noninter-\nacting particles. We now consider the case of a Bose-4\n0 1 2 3 4/Minus30/Minus20/Minus100/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1\n10 1 2 3 4/Minus30/Minus20/Minus100\n2\n0 1 2 3 4/Minus30/Minus20/Minus100/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1\n3\n0 1 2 3 4/Minus30/Minus20/Minus100\n4\n0 1 2 3 4/Minus30/Minus20/Minus100\nt/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1\n5\n0 1 2 3 4/Minus30/Minus20/Minus100\nt/LParen1units of TB/RParen110\n0 1 2 3 400.250.50.751/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar1/LParen1b/RParen1\n1\n0 1 2 3 400.250.50.7512\n0 1 2 3 400.250.50.751/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar13\n0 1 2 3 400.250.50.7514\n0 1 2 3 400.250.50.751\nt/LParen1units of TB/RParen1/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar15\n0 1 2 3 400.250.50.751\nt/LParen1units of TB/RParen110\nFIG. 5. (a) Evolution of the packet position hx(t)iin the\npresence of randomly distributed scatterers. Increasing the\nnumber of scatterers (as indicated) increases the damping of\nBloch oscillations. Each curve in a plot represents a di\u000ber-\nent spatial con\fgurations. (b) Overlap with the initial wave\nfunction for the case of randomly distributed scatterers. The\nparameters for (a) and (b) are given in the text.\nEinstein condensate of Natoms with repulsive interpar-\nticle interaction U(xi\u0000xj) =g\u000e(xi\u0000xj). The evolution\nof the condensate wave function  (x;t) =p\nN \u001e(x;t)\nin the bichromatic potential is then determined by the\nmean-\feld Hamiltonian\nH=H0+\rV0cos(\u000bKx ) +gNj\u001e(x;t)j2: (6)\nThe simulation results (Fig. 6) show that an increase\nin the coupling strength gleads to a reduction of thedamping constant \u0011, up to a characteristic value gc, be-\nyond which an increase in gleads to an increase in \u0011.\nWe interpret the reduction of \u0011as a partial screening\nof the potential corrugations by the mean \feld [25] as g\nincreases, which is eventually overcompensated by mean-\n\feld-induced dephasing [21, 22].\n0 1 2 3 4 5 6 7 8/Minus0.7/Minus0.6/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0\nt/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1\n0.0 0.15 0.30 0.45 0.600.0000.0050.0100.0150.020\ng/LParen1units of Era/RParen1Damping constant Η/LParen1units of TB/Minus2/RParen1/LParen1b/RParen1\nFIG. 6. Interplay between aperiodicity and mean-\feld inter-\naction, showing (a) evolution for g= 0 (solid), g= 0:1Era\n(dotted) and g= 0:4Era(dashed), and (b) dependence\nof\u0011on the coupling constant g, reaching a minimum at\ngc= 0:33Era\n.\nThe parameters for the simulation are the same as\nthose in Sec. 3.1, leading to an associated coupling\nstrengthgc= 0:33Era. This value should be compared\nwith an e\u000bective one-dimensional interaction parameter\ng=g3D=(2\u0019a2\n?) for a trapped atomic Bose-Einstein con-\ndensate, where g3D= 4\u0019~2asN=m (with atomic s-wave\nscattering length asand atomic mass m), anda?\u0018R,\nwhereRis the Thomas-Fermi radius. Already for a small\ncondensate of87Rb atoms with N= 1\u0002104atoms and\nR= 5:3\u0016m in an isotropic 50 Hz trap [33], and an op-\ntical lattice with a= 532 nm, we obtain g\u00180:42Era,\nwhich is in the vicinity of gc. Hence, a signi\fcant mod-\ni\fcation of the damping rate in a bichromatic potential\ndue to mean-\feld e\u000bects can be expected; the coupling\ngdepends, for example on the atom number Nin the\ncondensate, which is variable. Alternatively, an inves-\ntigation of the interplay is possible for species in which\nthe mean-\feld interaction can be tuned via a Feshbach\nresonance. In this context we mention that Bloch oscil-\nlations with widely controllable mean-\feld interactions\nin a (monochromatic) optical lattice have recently been\ndemonstrated with cesium condensates [7, 8].\nV. CONCLUSIONS\nWe have numerically investigated the damping of\nBloch oscillations resulting from a controlled breakdown\nof the periodicity of the lattice potential. The e\u000bects dis-\ncussed here, including the e\u000bects of the mean-\feld inter-\naction on disorder-induced dephasing, should be readily\nobservable in experiments with ultracold atoms in optical\nlattices.5\nACKNOWLEDGMENTS\nWe thank D. Pertot and B. Gadway for a critical read-\ning of the manuscript. This work is supported by NSFGrant Nos DMR-0605919 (S.W. and A.C.D) and PHY-\n0855643 (D.S.) as well as a DAAD scholarship (S.W.).\n[1] F. Bloch, Z. Phys. A 52, 555 (1929).\n[2] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo,\nH. Kurz, and K. K ohler, Phys. Rev. Lett. 70, 3319\n(1993).\n[3] S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu,\nand M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).\n[4] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Sa-\nlomon, Phys. Rev. Lett. 76, 4508 (1996).\n[5] O. Morsch, J. H. M uller, M. Cristiani, D. Ciampini, and\nE. Arimondo, Phys. Rev. Lett. 87, 140402 (2001).\n[6] G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Mod-\nugno, and M. Inguscio, Phys. Rev. Lett. 92, 230402\n(2004).\n[7] M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl,\nG. Rojas-Kopeinig, and H.-C. N agerl, Phys. Rev. Lett.\n100, 080404 (2008).\n[8] M. Fattori, C. D'Errico, G. Roati, M. Zaccanti, M. Jona-\nLasinio, M. Modugno, M. Inguscio, and G. Modugno,\nPhys. Rev. Lett. 100, 080405 (2008).\n[9] R. Battesti, P. Clad\u0013 e, S. Guellati-Kh\u0013 elifa, C. Schwob,\nB. Gr\u0013 emaud, F. Nez, L. Julien, and F. Biraben, Phys.\nRev. Lett. 92, 253001 (2004).\n[10] I. Carusotto, L. Pitaevskii, S. Stringari, G. Modugno,\nand M. Inguscio, Phys. Rev. Lett. 95, 093202 (2005).\n[11] G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, Phys.\nRev. Lett. 97, 060402 (2006).\n[12] P. Clad\u0013 e, E. de Mirandes, M. Cadoret, S. Guellati-\nKh\u0013 elifa, C. Schwob, F. Nez, L. Julien, and F. Biraben,\nPhys. Rev. Lett. 96, 033001 (2006).\n[13] G. von Plessen, T. Meier, J. Feldmann, E. O. G obel,\nP. Thomas, K. W. Goossen, J. M. Kuo, and R. F. Kopf,\nPhys. Rev. B 49, 14058 (1994).\n[14] E. Diez, F. Dom\u0013 \u0010nguez-Adame, and A. S\u0013 anchez, Micro-\nelectron. Eng. 43-44 , 117 (1998).\n[15] P. Horak, J.-Y. Courtois, and G. Grynberg, Phys. Rev.\nA58, 3953 (1998).[16] R. B. Diener, G. A. Georgakis, J. Zhong, M. Raizen, and\nQ. Niu, Phys. Rev. A 64, 033416 (2001).\n[17] U. Gavish and Y. Castin, Phys. Rev. Lett. 95, 020401\n(2005).\n[18] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst,\nK. Sengstock, and K. Bongs, Phys. Rev. Lett. 96, 180403\n(2006).\n[19] D.-W. Wang, M. D. Lukin, and E. Demler, Phys. Rev.\nLett. 92, 076802 (2004).\n[20] B. Wu and Q. Niu, Phys. Rev. A 64, 061603(R) (2001).\n[21] M. Holthaus, J. Opt. B: Quantum Semiclassical Opt. 2,\n589 (2000).\n[22] D. Witthaut, M. Werder, S. Mossmann, and H. J. Ko-\nrsch, Phys. Rev. E 71, 036625 (2005).\n[23] C. Menotti, A. Smerzi, and A. Trombettoni, New J. Phys.\n5, 112 (2003).\n[24] M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev. A\n70, 043625 (2004).\n[25] T. Schulte, S. Drenkelforth, G. K. B uning, W. Ertmer,\nJ. Arlt, M. Lewenstein, and L. Santos, Phys. Rev. A 77,\n023610 (2008).\n[26] S. Drenkelforth, G. K. Buning, J. Will, T. Schulte,\nN. Murray, W. Ertmer, L. Santos, and J. J. 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Phys. 42, 215305\n(2009)."    },    {        "title": "0911.4628v1.Origin_of_adiabatic_and_non_adiabatic_spin_transfer_torques_in_current_driven_magnetic_domain_wall_motion.pdf",        "content": "arXiv:0911.4628v1  [cond-mat.mes-hall]  24 Nov 2009Origin of adiabatic and non-adiabatic spin transfer torque s in current-driven magnetic\ndomain wall motion\nJun-ichiro Kishine\nDepartment of Basic Sciences, Kyushu Institute of Technolo gy, Kitakyushu, 804-8550, Japan\nA. S. Ovchinnikov\nDepartment of Physics, Ural State University, Ekaterinbur g, 620083 Russia\n(Dated: 24 November 2009)\nAconsistent theorytodescribe thecorrelated dynamics ofq uantummechanical itinerantspins and\nsemiclassical local magnetization is given. We consider th e itinerant spins as quantum mechanical\noperators, whereas local moments are considered within cla ssical Lagrangian formalism. By appro-\npriately treating fluctuation space spanned by basis functi ons, including a zero-mode wave function,\nwe construct coupled equations of motion for the collective coordinate of the center-of-mass motion\nand the localized zero-mode coordinate perpendicular to th e domain wall plane. By solving them,\nwe demonstrate that the correlated dynamics is understood t hrough a hierarchy of two time scales:\nBoltzmann relaxation time τel, when a non-adiabatic part of the spin-transfer torque appe ars, and\nGilbert damping time τDW, when adiabatic part comes up.\nSpin torque transfer (STT) process is expected to rev-\nolutionize the performance of memory device due to non-\nvolatility and low-power consumption. To promote this\ntechnology, it is essential to make clear the nature of the\ncurrent-driven domain wall (DW) motion[1, 2]. Recent\ntheoretical [3, 4, 5, 6, 7, 8, 9, 10] and experimental[11]\nstudies have disclosed that the STT consists of two vec-\ntors perpendicular to the local magnetization m(x) and\ncan be written in general as N=c1∂xm+c2m×\n∂xm[3]. The c1andc2-terms respectively come from\nadiabatic[1, 4] and non-adiabatic [5] processes between\nconduction electrons and local magnetization, and the\nterminal velocity of a DW is controlled by not c1but\nsmallc2term. The origin of the c2term is ascribed to\nthe spatial mistracking of spins between conduction elec-\ntrons and local magnetization[5]. Behind appearance of\nthec2term is the so called transverse spin accumula-\ntion (TSA) of itinerant electrons generated by the elec-\ntric current[6, 7]. Now, any consistent theory should ex-\nplain how the adiabatic and non-adiabatic STT come up\nstarting with microscopic model. In particular, it should\nbe made clear how the TSA caused by the non-adiabatic\nSTTeventuallyleadstotranslationalmotionofthewhole\nDW. In this letter, to solve this highly debatable prob-\nlem, we propose a consistent theory to describe the cor-\nrelated dynamics of quantum mechanical itinerant spins\nand semiclassical local magnetization.\nWe consider a single head-to-head N´ eel DW through\na magnetic nanowire with an easy xaxis and a hard\nzaxis. Fee electrons travel along the DW axis ( x-\naxis). We describe a local spin by a semiclassical vec-\ntorS=Sn=S(sinθcosϕ,sinθsinϕ,cosθ) whereS=|S|\nand the polar coordinates θandϕare assumed to be\nslowly varying functions of one-dimensional coordinate x\n[Fig.1(a)]. The DW formation is described by the Hamil-\ntonian (energy per unit area) in the continuum limit,\nHDW=JS2\n2a/integraldisplay∞\n−∞dx/bracketleftBig\n(∂xn)2−λ−2ˆn2\nx+κ−2ˆn2\nz/bracketrightBig\n,(1)\nwhereais the cubic lattice constant, Jis the ferromag-\nnetic exchange strength, λ=/radicalbig\nJ/Kandκ=/radicalbig\nJ/K⊥respectively represent the single-ion easy and hard axis\nanisotropies measured in the length dimension. The\nstationary N´ eel wall ( θ0=π/2) is described by n0=\n(cosϕ0,sinϕ0,0) withϕ0(z) = 2arctan( ex/λ). In the in-\nfinite continuum system, the DW configuration has con-\ntinuous degeneracy labeled by the center of mass posi-\ntion,X, of the DW. This degeneracy apparently leads to\nrigidtranslationoftheDW,i.e., n0(x)→n0(x−X)[12].\nAs explicitly shown below, however, the translation in\noff-equilibrium accompanies internal deformation of the\nDW.\nThe creation operator of a conduction electron is writ-\nten in a spinor form as c†(x) = (c†\n↑(x),c†\n↓(x)). By per-\nforming the local gauge transformation c(x) =ˆU(x)¯c(x)\nwith the unitary operator ˆU(x) =eiˆσzϕ0(x)/2(ˆσzis a\n(a) \n(b)\nzx−y−\n0n\nxy\nTSAT1\nT2\nOPZAy\nxz0nx−y−\nX\n0\n00\nT2TT\nns'x\n'0\nȟx\nFIG. 1: (a) Stationary configuration of local spins ( n0) asso-\nciated with a single N´ eel wall. Labotatory frame x, y, zand\nlocal frame ¯ x,¯y, zare indicated. (b) Schematic view of the\ntransverse spin accumulation (TSA) of itinerant spin sand\nthe out-of-plane ( θ) zero-mode accumulation (OPZA) of local\nspinn. These magnetic accumulations respectively cause the\nnon-adiabatic torque T2and adiabatic torque T1.2\nPauli matrix) the quantization axis becomes parallel to\nthe local spin located at x. Assuming |a∂xϕ0(x)| ≃\na/λ≪1, i.e. wall thickness is much larger than atomic\nlattice constant, this procedure leads to the single-\nparticle Hamiltonian,\nHel=/planckover2pi12\n2m∗a/integraldisplay∞\n−∞dx/bracketleftbigg1\n2|∂x¯c|2+i(∂x¯c†)ˆAz¯c/bracketrightbigg\n+c.c,(2)\nwhere the effective mass of the conduction electron is\nm∗. The SU(2) gauge field[8, 13] is introduced as ˆAz≡\ni−1ˆU−1∂xˆU=−(∂xϕ0)ˆσz/2. The conduction electrons\nare assumed to interact with the local spins by a s-d\ncoupling represented in the form,\nHsd=−Jsd\na3/integraldisplay∞\n−∞dxˆs(x)·S(x−X),(3)\nwhereˆsandS=Snare respectively the spins of itin-\nerant and localized electrons .We treat ˆs(x) =1\n2c†ˆσcas\nfully quantum mechanical operator, while nis a semi-\nclassical vector.\nBoltzmann relaxation : let switch on the electric field\nEatt= 0. We introduce the Boltzmann relaxation\ntimeτeland the number density of the conduction elec-\ntronsfkσin the state k,σ. We assume that the devia-\ntionfromequilibriumFermi-Diracdistribution f0(εkσ) =\n[exp[(εkσ−µ)/kBT]+1]−1issmall, where εkσ(σ=↑,↓)\nis the single-particle energy, µis the chemical potential.\nUsing standard Boltzmann kinetic equation with relax-\nation time approximation[6], the distribution function is\nwritten as\nfkσ≃f0(εkσ)+eEτelvkσ∂f0(εkσ)\n∂εkσ,(4)\nwhere the electron charge is −eand the spin-dependent\nvelocity is vkσ≡/planckover2pi1−1∂εkσ/∂k. The spin-dependence of\nεkσoriginates from the SU(2) gauge fields ( ˆAz)↑↑and\n(ˆAz)↓↓. In the process of approaching to stationary cur-\nrentflowingstate aroundthe time t∼τel, aswewillshow\nexplicitly, the statistical average of the conduction elec-\ntron’s spin component perpendicular to the local quan-\ntization axis accumulates and acquires finite value. As\nschematically depicted in Fig.1(b), this process is ex-\nactly the TSA. The TSA causes an additional magnetic\nfield actingon the localspins and exertthe non-adiabatic\ntorque on the local spins.\nLocal spin dynamics : next we formulate dynamics of\nthe local spins coupled with the conduction electrons.\nWe introduce the δθ(x,t) (out-of-plane) and δϕ(x,t)\n(in-plane) fluctuations of the local spins around the sta-\ntionary DW configuration n0(x).We say “out-of-plane”\nand “in-plane” with respect to the DW plane. The\nfluctuations are spanned by the orthogonal basis func-\ntionsvqanduqasϕ(x) =ϕ0(x−X)+δϕ(x−X) and\nθ(x) =π/2+δθ(x−X),where\nδϕ(x) =/integraldisplay∞\n−∞dq ηq(t)vq(x), δθ(x)=/integraldisplay∞\n−∞dq ξq(t)uq(x).\n(5)\nAt this stage, Xis not a dynamical variable, but just\na parameter. The basis functions obey the Schr¨ odingerequations,/parenleftbig\nJS2/2/parenrightbig\n(−∂2\nx−2λ−2sin2ϕ0+λ−2)vq(x) =\nεϕ\nquq(x) and/parenleftbig\nJS2/2/parenrightbig\n(−∂2\nx−2λ−2sin2ϕ0+λ−2+κ−2)\nuq(x) =εθ\nquq(x).Bothθandϕmodes consist of a sin-\ngle bound state ( zero mode ) and continuum states ( spin-\nwave modes ). The dimensionless zero mode wave func-\ntions are given by u0(x) =v0(x) = Φ0(x), where\nΦ0(x)≡/radicalbigg\naλ\n2∂xϕ0(x) =/radicalbigga\n2λ1\ncosh(x/λ),(6)\nwith the corresponding energies respectively given by\nεθ\n0=JS2/(2κ2) andεϕ\n0= 0. The normalization is\ngiven by a−1/integraltext∞\n−∞dx[Φ0(x)]2= 1.Although to excite\nthe out-of-plane ( θ) zero mode costs finite energy gap\nεθ\n0coming from the hard-axis anisotropy, we still call\nthis “zero mode.” The spin-wave states have energy\ndispersions given by εθ\nq=1\n2JS2/parenleftbig\nq2+λ−2+κ−2/parenrightbig\nand\nεϕ\nq=1\n2JS2/parenleftbig\nq2+λ−2/parenrightbig\n. Because the zero mode and the\nspin-wave states are orthogonal to each other and sepa-\nrated by the anisotropy gaps, the spin-wave modes are\ntotally irrelevant to a low energy effective theory. There-\nfore, we ignore the spin-wave modes from now on.\nOut-of-plane zero-mode(OPZ) coordinate ξ0: in order\nto obtain the correct form of the dynamical Hamiltonian,\none has to regard the variable Xas a dynamical variable\nX(t) and replace the zero mode coordinate η0withX(t).\nFollowing this idea, the zero-mode fluctuations should be\ngiven by,\nϕ(x,t) =ϕ0[x−X(t)], (7)\nθ(x,t) =π/2+ξ0(t)Φ0[x−X(t)].(8)\nEq. (8) is a key ingredient of this letter, which has\nneverbeen explicitly treated sofar[14]. That is to say, we\nnaturally include the out-of-plane(OPZ) zero-mode, in\naddition to the in-plane ( ϕ) zero-mode replaced by X(t).\nThe zero-mode wave function Φ 0[x−X(t)] serves as the\nbasis function of the θ-fluctuations localized around the\ncenter of the DW and ξ0(t) is the OPZ coordinate . Now,\nour effective theory is fully described by two dynamical\nvariables X(t) andξ0(t) which naturally give physical\ncoordinates along the Hilbert space of orthogonal θand\nϕfluctuations. As we will see, we have ξ0(t)/ne}ationslash= 0 only\nfor inequilibrium current flowing state under E/ne}ationslash= 0 [Fig.\n2(a)].\nIt is here important to note an essential difference be-\ntween Tatara and Khono’s approach[8] and ours. Tatara\nand Khono used X(t) and the weighted average, θ0(t)\n=/integraltext∞\n−∞dx θ(x,t)sin2ϕ[x−X(t)], as dynamical vari-\nables. Later, they systematically used complex coordi-\nnateξ=eiϕtan(θ/2) and described the fluctuations in\nthe form ξ=e−u(x,t)+iϕ0+η[x−X(t)][9](their notation is\nreproduced by putting θ→π/2−θ,ϕ→ϕin our nota-\ntion). Inourunderstanding, thesedescriptionsinevitably\ncause redundant coupling between uandvmodes in Eq.\n(5). Actually, our natural choice of the dynamical vari-\nables is essential to appropriately derive relaxational dy-\nnamics described by the following equations of motion\ngiven by (12a) and (12b).\nEquations of motion of the DW : now, we construct\nan effective Lagrangian L=LDW+Lsdto describe the3\nxπ\n0\nXπ/2(a)\nj (b)\n(c)0\nȟ'0\nj OPZAȥX.\n0,\nˢk\nˢkJsd ȥ0ȍ0\nFIG. 2: (a) Spatial profile of the polar angles ϕ(x,t) =\nϕ0[x−X(t)] andθ(x,t) =π/2 +ξ0(t)Φ0[x−X(t)] in the\ncurrent flowing state. (b) Linear dependence of ξ0and˙X(t)\non thecurrentdensity j. (c) Single- particle propagation (rep-\nresented by solid line) with spin flip process by the s-d inter -\naction (represented by wavy line) which leads to the STT.\nDW motion and resultant equations of motion (EOM).\nUsing (7) and (8), the local spin counterpart is given by\nLDW=/planckover2pi1S\na3/integraltext∞\n−∞dx(cosθ−1) ˙ϕ−HDWexplicitly written\nas\nLDW=/planckover2pi1S\na3/parenleftBigg/radicalbigg\n2a\nλξ0+π/parenrightBigg\n˙X−JS2\n2κ2ξ2\n0.(9)\nTo understand the effect of the s-d coupling, it is use-\nful to note n[θ0+δθ,ϕ0+δϕ]≃n0−ezδθ−n0δθ2/2,\nwhere we dropped δϕbecause this degree of freedom is\neliminated by the global gauge fixing[14]. We have thus\ns-d Lagrangian,\nLsd=a−3JsdS/parenleftbig\nF0−S/bardblξ02/2/parenrightbig\n, (10)\nwhere,F0[X(t)]≡/integraltext∞\n−∞dxˆn0[x−X(t)]· /an}bracketle{ts(x,t)/an}bracketri}ht\nandS/bardbl[X(t)]≡/integraltext∞\n−∞dx{Φ0[x−X(t)]}2n0[x−X(t)]·\n/an}bracketle{ts(x,t)/an}bracketri}ht. Finally, to take account of dissipative\ndynamics, we use the Rayleigh dissipation function\nWRayleigh=α\n2/planckover2pi1S\na3/integraltext∞\n−∞dx˙n2explicitly written as\nWRayleigh=α\n2/planckover2pi1S\na3/parenleftbigg\na˙ξ2\n0+2\nλ˙X2/parenrightbigg\n,(11)\nwhereαis the Gilbert damping parameter. It is simple\nto write down the Euler-Lagrange-Rayleigh equations,\nd(∂L/∂˙qi)/dt−∂L/∂q i=−∂W/∂˙qi,for the dynami-\ncal variables q1=Xandq2=ξ0. We obtain the EOMs\nwhich contain the dynamical variables in linear order,\n/planckover2pi1/radicalbigg\n2a\nλ˙ξ0+JsdT⊥=−2α/planckover2pi1\nλ˙X, (12a)\n−/planckover2pi1/radicalbigg\n2a\nλ˙X+/parenleftbigga3JS\nκ2+JsdS/bardbl/parenrightbigg\nξ0=−α/planckover2pi1a˙ξ0,(12b)\nwhere the quantities\nT⊥≡ −∂F0\n∂X=/integraldisplay∞\n−∞dx ∂xϕ0[x−X(t)]/an}bracketle{t¯sy(x)/an}bracketri}ht,(13a)\nS/bardbl≡/integraldisplay∞\n−∞dxΦ2\n0[x−X(t)]/an}bracketle{t¯sx(x)/an}bracketri}ht, (13b)respectively give the non-adiabatic STT and longitudinal\nspin accumulation[7]. The statistical average of the con-\nduction electron’s spin component is denoted by /an}bracketle{t···/an}bracketri}ht.\nThe gauge-transformed spin variables are introduced by\n¯s(x) =ˆU−1[x−X(t)]ˆs(x)ˆU[x−X(t)] which has lo-\ncal quantization axis tied to the local spin at the po-\nsition of x−X(t). To obtain Eq.(13a), we used rela-\ntions∂xn0[x−X(t)] =−∂xϕ0(x)ez×n0[x−X(t)] and\n/an}bracketle{t¯sy/an}bracketri}ht=−/an}bracketle{tˆsx/an}bracketri}htsinϕ0+/an}bracketle{tˆsy/an}bracketri}htcosϕ0.The relation (13a) im-\nplies that the translation of the DW ( x→x−X) natu-\nrally gives rise to the TSA, /an}bracketle{t¯sy/an}bracketri}ht, along the local ¯ yaxis.\nThe appearance of /an}bracketle{t¯sy/an}bracketri}htcauses local magnetic moment\nwhich triggers the local spins to precess around the local\n¯yaxis and consequently produce finite deviation of the\npolar angle δθ=θ−θ0.It is seen that upon switching the\nexternal electric field, the deviation δθrelaxes to finite\nmagnitude in the stationary current-flowing state, i.e.,\nthe OPZ coordinate ξ0(t) accumulates and reaches finite\nterminal value ξ∗\n0. We call this process out-of-plane zero-\nmode accumulation(OPZA)as schematically depicted in\nFig.1(b). This effect is physically interpreted as appear-\nance of demagnetization field phenomenologically intro-\nduced by D¨ oring, Kittel, Becker[15], and Slonczewski[1].\nIt is also to be noted that we ignored the term ∂S/bardbl/∂X.\nThis simplification is legitimate for the case of of small\nsd-coupling.\nGilbert relaxation : coupled equations of motion (12a)\nand(12b)arereadilysolvedtogiverelaxationalsolutions,\nξ0=ξ∗\n0(1−e−t/τDW), V≡˙X=V∗(1−e−t/τDW),(14)\nwhere the OPZA reaches the terminal value,\nξ∗\n0=−1\nα/radicalbig\naλ/2JsdT⊥/parenleftbig\na3JSκ−2+JsdS/bardbl/parenrightbig≃ −α−1/radicalbigg\nλ\n2a/parenleftBigκ\na/parenrightBig2Jsd\nJST⊥,\n(15)\nand correspondingly the terminal velocity of the DW\nreachesV∗=−λ\n2α/planckover2pi1JsdT⊥.The relaxation time of the\nDW magnetization, τDW, is given by\nτDW=/planckover2pi1aα−1+α\nκ−2a3JS+JsdS/bardbl≃α−1/parenleftBigκ\na/parenrightBig2/planckover2pi1\nJS.(16)\nThis result clearly shows that the DW magnetization try\nto relax through the Gilbert damping toward the direc-\ntion of the newly established precession axis. We stress\nthat without the OPZ coordinate ξ0in Eqs. (12a) and\n(12b), onlytheterminalvelocityisavailableandthetran-\nsient relaxational dynamics is totally lost.\nAs depicted in Fig.2(a), the OPZA[Eq. (15)] gives rise\nto finite out-of-plane ( z) component of the local spin,\nnz(x,t) = cosθ≃1\n2α/parenleftBigκ\na/parenrightBig2Jsd\nJS1\ncosh[(x−X(t))/λ]T⊥.\n(17)\nThe resultant local spin S⊥=Seznz(x,t) gives the de-\nmagnetization field phenomenologically treated by Slon-\nczewski and gives rise to the adiabatic torque T1=\nc1∂xn(x) =c1(∂xϕ0)(−sinϕ0,cosϕ0,0). At the in-\nterface of the DW boundary, ϕ0=π/2 andT1=\nc1(∂xϕ0)(−1,0,0), i.e., the adiabatic torque rotate the\nlocal spin to counterclockwise direction when the electric4\ncurrent flows in the (1 ,0,0)-direction. As is clear from\nthe above discussion, this adiabatic torque is established\nafterthe stationary current-flowing [ j= (ne2τel/m∗)E]\nstate establishes the non-adiabatic torque, T⊥. Around\nthe time scale of t≃τel+τDW, the whole system (in-\ncluding conduction electrons and DW) reaches non-\nequilibrium but stationary state. In this state, the DW\nmagnetizations continuously feel the OPZA and macro-\nscopically rotate around it. This process exactly corre-\nsponds to stationary translation of the DW.\nComputation of T⊥: the final step is to compute an\nexplicit form of T⊥. By taking Fourier transform ¯ ckσ(t)\n=1√\nL/summationtext\nkeikx¯cσ(x,t), and retaining only the momentum\nconserving process, we have\nT⊥=1\n2/integraldisplayπ/a\n−π/adkReG<\nk↑,k↓(t,t), (18a)\nS/bardbl=a\n2π/integraldisplayπ/a\n−π/adkImG<\nk↑,k↓(t,t).(18b)\nHere, the expectation values are computed by using the\nlesser component of the path-oriented Green function\nG<\nkσ,k′σ′(t,t′) =i/an}bracketle{t¯c†\nk′σ′(t′)¯ckσ(t)/an}bracketri}ht, wheret(t′) is defined\non the upper (lower) branch of Keldysh contour. Since\nS/bardbldoes not play an essential role, we pay attention to an\nessential quantity T⊥. To evaluate the Green functions,\nwe perturbatively treat the s-d coupling and write down\nthe Dyson equation. Then, we truncate the Dyson equa-\ntion by using the Born approximation including the s-d\ncoupling in linear order which causes a single spin flip\nprocess [Fig.2(c)] and gives rise to off-diagonal compo-\nnent in spin space,\nG<\nk↑,k↓(t,t) =−iJsd\n2fk↑−fk↓\nεk↑−εk↓−i0.(19)\nTo obtain the explicit form of εkσ, we write the single-\nparticle Hamiltonian (2) in Fourier space and obtain\nHel=H0+Hgauge, whereH0represents free conduction\nandHgaugecomes form the second term in Eq. (2). By\nretaining only momentum conserving process, we have\nHel=/summationtext\nk,σεkσ¯c†\nkσ¯ckσ,whereεk↑,↓=/planckover2pi12(k∓δk)2/2m∗,\nwhere the shift of the Fermi wave numbers due to the\nbackground DW is given by δk=π/(2a).\nUsing Eqs. (4), (19), and (18a), we finally obtain the\nSTT which points in the z-direction, T1=T⊥ez, where\nits magnitude is given in a form,\nT⊥=1\n4Jsd\nkBT1\ncosh2[(ε0−µ)/2kBT]j\nj0,(20)\nwherej0= 4ne/planckover2pi1/(πam∗) andε0=/planckover2pi12π2/(8m∗a2) corre-\nspondstothechemicalpotentialathalf-filling. Wehavea\nmaster formula which gives relation between the current\ndensity and the terminal velocity of the DW,\nV∗=−1\n8αλJsd\n/planckover2pi1Jsd\nkBT1\ncosh2[(ε0−µ)/2kBT]j\nj0.(21)\nAs shown in Fig.2(b), we see there is no threshold for the\nvelocity, which is consistent with the result obtained byThiaville et al.[10]. Standard choice of parameters, j0≃\n1016[A·m−2],λ= 10−8[m],α= 10−2,j≃1011[A·m−2]\ngive a rough estimate V∗≃ −100(Jsd/kBT)2[m/s].Of\ncourse, to pursuit more quantitative result needs numer-\nical estimation of T⊥taking account of real band struc-\nture.\nIt is essential that the Gilbert damping coefficient, α,\nentersEq.(21). Therelaxationprocessofthe DWdynam-\nics is governed by the Boltzmann relaxation followed by\nthe Gilbert damping in hierarchical manner. As summa-\nrized in Figs.2(a) and (b), in our treatment, it is crucial\nto recognize that the OPZ coordinate ξ0acquires finite\nvalue (i.e., accumulation) only for the current flowing\nstate which is non-equilibrium but stationary. This is the\ncase where dynamical relaxation leads to finite accumula-\ntion of physical quantities which are zero in equilibrium.\nAlthough essential role of the sliding mode to describe\nlocalized spin dynamics was pointed out before[8, 12]\nand importance of out-of-plane canting of the local spins\nwas stressed[1, 8], the OPZA presented in this letter has\nnot been discussed before. For example, the sliding mo-\ntion in Ref.[12] does not contain internal deformation\nof the DW. The OPZA is an outcome of time-reversal-\nsymmetry breaking by electric current. This interpre-\ntation seems natural because current-flowing state is off\nequilibrium.\nJ. K. acknowledges Grant-in-Aid for Scientific Re-\nsearch (C) (No. 19540371) from the Ministry of Educa-\ntion, Culture, Sports, Science and Technology, Japan.\n[1] J.C. Slonczewski, J. Magn. Magn. Mat. 159L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] M. D. Stiles and A. Zangwill, Phys. Rev. B66,\n014407(2002).\n[4] Ya.B. Bazaliy, B.A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[5] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[6] J. Xiao, A. Zangwill and M. D. Stiles, Phys. Rev. B73,\n054428 (2006).\n[7] S. Zhang, P.M. Levy and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[8] G. Tatara, H.Kohno, Phys.Rev.Lett. 92, 086601 (2004).\n[9] G. Tatara, H. Kohno and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[10] A. Thiaville, et al.,Europhys. Lett. 69, 990 (2005).\n[11] S. Petit, et al.,Phys. Rev. Lett. 98, 077203 (2007); Z.\nLi,et al.,Phys. Rev. Lett. 100, 246602 (2008).\n[12] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[13] G. E. Volovik, J. Phys. Condens. Matter 20, 83(1987).\n[14] The out-of-plane zero-mode was discussed in the con-\ntext of chiral helimagnet by the present authors:\nI. G. Bostrem, J. Kishine, and A. S. Ovchinnikov, Phys.\nRev.B77, 132405 (2008); Phys. Rev. B78, 064425\n(2008); I.G. Bostrem, J. Kishine, R. V. Lavrov, A.S.\nOvchinnikov, Phys. Lett. A 373, 558(2009).\n[15] W. D¨ oring, Zeits. f. Naturforschung 3a, 374 (1948); R.\nBecker, Proceedings of the Grenoble Conference, July\n(1950); C. Kittel, Phys. Rev. 80, 918 (1950)."    },    {        "title": "0912.0229v1.Approximate_Sparse_Recovery__Optimizing_Time_and_Measurements.pdf",        "content": "APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND\nMEASUREMENTS\nA. C. GILBERT, Y. LI, E. PORAT, AND M. J. STRAUSS\nAbstract. Anapproximate sparse recovery system consists of parameters k;N, anm-by-Nmea-\nsurement matrix ,\b, and a decoding algorithm, D. Given a vector, x, the system approximates x\nbybx=D(\bx), which must satisfy kbx\u0000xk2\u0014Ckx\u0000xkk2, where xkdenotes the optimal k-term\napproximation to x. For each vector x, the system must succeed with probability at least 3/4.\nAmong the goals in designing such systems are minimizing the number mof measurements and the\nruntime of the decoding algorithm, D.\nIn this paper, we give a system with m=O(klog(N=k)) measurements|matching a lower\nbound, up to a constant factor|and decoding time O(klogcN), matching a lower bound up to\nlog(N) factors. We also consider the encode time ( i.e., the time to multiply \bbyx), the time to\nupdate measurements ( i.e., the time to multiply \bby a 1-sparse x), and the robustness and stability\nof the algorithm (adding noise before and after the measurements). Our encode and update times\nare optimal up to log( N) factors. The columns of \bhave at most O(log2(k) log(N=k)) non-zeros,\neach of which can be found in constant time. If xis an exact k-sparse signal and \u00171and\u00172are\narbitrary vectors (regarded as noise), then, setting bx=D(\b(x+\u00171) +\u00172), we get\nkbx\u0000xk2\u00142k\u00171k2+ log(k)k\u00172k2\nk\bk2 2;\nwherek\bk2 2is a natural scaling factor that makes our result comparable with previous results.\n(The log(k) factor above, improvable to log1=2+o(1)k, makes our result (slightly) suboptimal when\n\u001726= 0.) We also extend our recovery system to an FPRAS.\n1.Introduction\nTracking heavy hitters in high-volume, high-speed data streams [4], monitoring changes in data\nstreams [5], designing pooling schemes for biological tests [10] (e.g., high throughput sequencing,\ntesting for genetic markers), localizing sources in sensor networks [15, 14] are all quite di\u000berent\ntechnological challenges, yet they can all be expressed in the same mathematical formulation. We\nhave a signal xof lengthNthat is sparse or highly compressible; i.e., it consists of ksigni\fcant\nentries (\\heavy hitters\") which we denote by xkwhile the rest of the entries are essentially negligible.\nWe wish to acquire a small amount information (commensurate with the sparsity) about this signal\nin a linear, non-adaptive fashion and then use that information to quickly recover the signi\fcant\nentries. In a data stream setting, our signal is the distribution of items seen, while in biological group\ntesting, the signal is proportional to the binding a\u000enity of each drug compound (or the expression\nlevel of a gene in a particular organism). We want to recover the identities and values only of the\nheavy hitters which we denote by xk, as the rest of the signal is not of interest. Mathematically,\nwe have a signal xand anm-by-Nmeasurement matrix \bwith which we acquire measurements\ny=\bx, and, from these measurements y, we wish to recover bx, withO(k) entries, such that\nkx\u0000bxk2\u0014Ckx\u0000xkk2:\nGilbert is with the Department of Mathematics, The University of Michigan at Ann Arbor. E-mail: annacg@umich.\nedu. Li is with the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann\nArbor. E-mail: leeyi@umich.edu . Porat is with the Department of Computer Science, Bar-Ilan University. E-\nmail: porately@cs.biu.ac.il . Strauss is with the Department of Mathematics and the Department of Electrical\nEngineering and Computer Science, The University of Michigan at Ann Arbor. E-mail: martinjs@umich.edu .\n1arXiv:0912.0229v1  [cs.DS]  1 Dec 20092 GILBERT, LI, PORAT, STRAUSS\nPaper No. Measurements Encode time Column sparsity/ Decode time Approx. error\nUpdate time\n[8, 3] klog(N=k)Nklog(N=k)klog(N=k)\u0015N`2\u0014(1=p\nk)`1\n[4, 7] klogcN NlogcN logcN klogcN`2\u0014C`2\n[6] klogcN NlogcN logcN klogcN`1\u0014C`1\nThis paper klog(N=k)NlogcN logcN klogcN`2\u0014C`2\nFigure 1. Summary of the best previous results and the result obtained in this paper.\nOur goal, which we achieve up to constant or log factors in the various criteria, is to design\nthe measurement matrix \band the decoding algorithm in an optimal fashion: (i) we take as\nfew measurements as possible m=O(klog(N=k)), (ii) the decoding algorithm runs in sublin-\neartimeO(klog(N=k)), and (iii) the encoding and update times are optimal O(Nlog(N=k)) and\nO(klog(N=k)), respectively. In order to achieve this, our algorithm is a randomized algorithm; i.e.,\nwe specify a distribution on the measurement matrix \band we guarantee that, for each signal, the\nalgorithm recovers a good approximation with high probability over the choice of matrix.\nIn the above applications, it is important both to take as few measurements as possible and\nto recover the heavy hitters extremely e\u000eciently. Measurements correspond to physical resources\n(e.g., memory in data stream monitoring devices, number of screens in biological applications)\nand reducing the number of necessary measurements is critical these problems. In addition, these\napplications require e\u000ecient recovery of the heavy hitters|we test many biological compounds\nat once, we want to quickly identify the positions of entities in a sensor network, and we cannot\na\u000bord to spend computation time proportional to the size of the distribution in a data stream\napplication. Furthermore, Do Ba, et al. [2] give a lower bound on the number of measurements for\nsparse recovery \n( klog(N=k)). There are polynomial time algorithms [13, 3, 12] meet this lower\nbound, both with high probability for each signal and the stronger setting, with high probability for\nall signals1. Previous sublinear time algorithms, whether in the \\for each\" model [4, 7] or in the \\for\nall\" model [11], however, used several additional factors of log( N) measurements. We summarize\nthe previous sublinear algorithms in the \\for each\" signal model in Figure 1. The column sparsity\ndenotes how many 1s there are per column of the measurement matrix and determines both the\ndecoding and measurement update time and, for readability, we suppress O(\u0001). The approximation\nerror signi\fes the metric we use to evaluate the output; either the `2or`1metric. In this paper,\nwe focus on the `2metric.\nWe give a joint distribution over measurement matrices and sublinear time recovery algorithms\nthat meet this lower bound (up to constant factors) in terms of the number of measurements and\nare within log( k) factors of optimal in the running time and the sparsity of the measurement matrix.\nTheorem 1. There is a joint distribution on matrices and algorithms, with suitable instantiations\nof anonymous constant factors, such that, given measurements \bx=y, the algorithm returns bx\nand approximation error\nkx\u0000bxk2\u00142k\u00171k2\nwith probability 3=4. The algorithm runs in time O(klogc(N))and\bhasO(klog(N=k))rows.\nFurthermore, our algorithm is a fully polynomial randomized approximation scheme.\nTheorem 2. There is a joint distribution on matrices and algorithms, with suitable instantiations\nof anonymous constant factors (that may depend on \u000f, such that, given measurements \bx=y, the\nalgorithm returns bxand approximation error\nkx\u0000bxk2\u0014(1 +\u000f)k\u00171k2\n1albeit with di\u000berent error guarantees and di\u000berent column sparsity depending on the error metric.APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 3\nwith probability 3=4. The algorithm runs in time O((k=\u000f) logc(N))and\bhasO((k=\u000f) log(N=k))\nrows.\nFinally, our result is robust to corruption of the measurements by an arbitrary noise vector \u00172,\nwhich is an important feature for such applications as high throughput screening and other physical\nmeasurement systems. (It is less critical for digital measurement systems that monitor data streams\nin which measurement corruption is less likely.) When \u001726= 0, our error dependence is on \u00172is\nsuboptimal by the factor log( k) (improvable to log1=2+o(1)k). Equivalently, we can use log( k) times\nmore measurements to restore optimality.\nTheorem 3. There is a joint distribution on matrices and algorithms, with suitable instantiations\nof anonymous constant factors (that may depend on \u000f), such that, given measurements \bx+\u00172=\ny+\u00172, the algorithm returns bxand approximation error\nkx\u0000bxk2\u0014(1 +\u000f)k\u00171k2+\u000flog(k)k\u00172k2\nk\bk2 2\nwith probability 3=4. The algorithm runs in time O((k=\u000f) logc(N))and\bhasO(k=\u000flog(N=k))\nrows.\nPrevious sublinear algorithms begin with the observation that if a signal consists of a single\nheavy hitter, then the trivial encoding of the positions 1 through Nwith log(N) bits, referred to\nas a bit tester, can identify the position of the heavy hitter. The second observation is that a\nnumber of hash functions drawn at random from a hash family are su\u000ecient to isolate enough of\nthe heavy hitters, which can then be identi\fed by the bit tester. Depending on the type of error\nmetric desired, the hashing matrix is pre-multiplied by random \u00061 vectors (for the `2metric) in\norder to estimate the signal values. In this case, the measurements are referred to as the Count\nSketch in the data stream literature [4] and, without the premultiplication, the measurements are\nreferred to as Count Median [6, 7] and give `1\u0014C`1error guarantees. In addition, the sublinear\nalgorithms are typically greedy, iterative algorithms that recover portions of the heavy hitters with\neach iteration or that recover portions of the `2(or`1) energy of the residual signal.\nWe build upon the Count Sketch design but incorporate the following algorithmic innovations\nto ensure an optimal number of measurements:\n\u000fWith a random assignment of Nsignal positions to O(k) measurements, we need to encode\nonlyO(N=k) positions, rather than Nas in the previous approaches. So, we can reduce\nthe domain size which we encode.\n\u000fWe use a good error-correcting code (rather than the trivial identity code of the bit tester).\n\u000fOur algorithm is an iterative algorithm but maintains a compound invariant: the number of\nun-discovered heavy hitters decreases at each iteration while, simultaneously, the required\nerror tolerance and failure probability become more stringent. Because there are fewer\nheavy hitters to \fnd at each stage, we can use more measurements to meet more stringent\nguarantees.\nIn Section 2 we detail the matrix algebra we use to describe the measurement matrix distribution\nwhich we cover in Section 3, along with the decoding algorithm. In Section 4, we analyze the\nforegoing recovery system.\n2.Preliminaries\n2.1.Vectors. Letxdenote a vector of length N. For eachk\u0014N, letxkdenote either the usual\nk'th component of xor the signal of length Nconsisting of the jlargest-magnitude terms in x; it\nwill be clear from context. The signal xkis the best k-term representation of x. The energy of a\nsignal xiskxk2\n2=PN\ni=1jxij2.4 GILBERT, LI, PORAT, STRAUSS\noperator name input output dimensions and construction\n\brrow direct sum A:r1\u0002NM: (r1+r2)\u0002N\nB:r2\u0002NMi;j=(\nAi;j; 1\u0014i\u0014r1\nBi\u0000r1;j;1 +r1\u0014i\u0014r2\n\felement-wise product A:r\u0002NM:r\u0002N\nB:r\u0002NMi;j=Ai;jBi;j\nnrsemi-direct product A:r1\u0002NM: (r1r2)\u0002N\nB:r2\u0002hMi+(k\u00001)r2;`=(\n0;Ak;`= 0\nAk;`Bi;j;Ak;`=jth nonzero in row `\nFigure 2. Matrix algebra used in constructing an overall measurement matrix.\nThe last column contains both the output dimensions of the matrix operation and\nits construction formula.\n2.2.Matrices. In order to construct the overall measurement matrix, we form a number of di\u000ber-\nent types of combinations of constituent matrices and to facilitate our description, we summarize\nour matrix operations in Table 2. The matrices that result from all of our matrix operations have\nNcolumns and, with the exception of the semi-direct product of two matrices nr, all operations are\nperformed on matrices AandBwithNcolumns. A full description can be found in the Appendix.\n3.Sparse recovery system\nIn this section, we specify the measurement matrix and detail the decoding algorithm.\n3.1.Measurement matrix. The overall measurement matrix, \b, is a multi-layered matrix with\nentries inf\u00001;0;+1g. At the highest level, \bconsists of a random permutation matrix Pleft-\nmultiplying the row direct sum of (lg( k)) summands, \b(j), each of which is used in a separate\niteration of the decoding algorithm. Each summand \b(j)is the row direct sum of two separate\nmatrices, an identi\fcation matrix,D(j), and an estimation matrix,E(j).\n\b=P2\n6664\b(1)\n\b(2)\n...\n\b(lg(k))3\n7775where \b(j)=E(j)\brD(j):\nIn iteration j, the identi\fcation matrix D(j)consists of the row direct sum of O(j) matrices, all\nchosen independently from the same distribution. We construct the distribution ( C(j)nrB(j))\fS(j)\nas follows:\n\u000fForj= 1;2;:::; lg(k), the matrix B(j)is a Bernoulli matrix with dimensions kcj-by-N,\nwherecis an appropriate constant 1 =2<c< 1. Each entry is 1 with probability \u0002\u0000\n1=(kcj)\u0001\nand zero otherwise. Each row is a pairwise independent family and the set of row seeds is\nfully independent.\n\u000fThe matrixC(j)is an encoding of positions by an error-correcting code with constant\nrate and relative distance. That is, \fx an error-correcting code and encoding/decoding\nalgorithm that encodes messages of \u0002(log log N) bits into longer codewords, also of length\n\u0002(log logN), and can correct a constant fraction of errors. The i'th column of C(j)is\nthe direct sum of \u0002(log log N) copies of 1 with the direct sum of E(i1);E(i2);:::, where\ni1;i2;::: are blocks of O(log logN) bits each whose concatenation is the binary expansion\nofiandE(\u0001) is the encoding function for the error-correcting code. The number of columns\ninC(j)matches the maximum number of non-zeros in B(j), which is approximately theAPPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 5\nexpected number, \u0002\u0000\ncjN=k\u0001\n, wherec<1. The number of rows in C(j)is the logarithm of\nthe number of columns, since the process of breaking the binary expansion of index iinto\nblocks has rate 1 and encoding by E(\u0001) has constant rate.\n\u000fThe matrixS(j)is a pseudorandom sign-\rip matrix. Each row is a pairwise independent\nfamily of uniform \u00061-valued random variables. The sequence of seeds for the rows is a fully\nindependent family. The size of S(j)matches the size of C(j)nrB(j).\nNote that error correcting encoding often is accomplished by a matrix-vector product, but we are\nnotencoding a linear error-correcting code by the usual generator matrix process. Rather, our\nmatrix explicitly lists all the codewords. The code may be non-linear.\nThe identi\fcation matrix at iteration jis of the form\nD(j)=2\n664h\n(C(j)nrB(j))\fS(j)i\n1\n:::h\n(C(j)nrB(j))\fS(j)i\nO(j)3\n775:\nIn iteration j, the estimation matrix E(j)consists of the direct sum of O(j) matrices, all chosen\nindependently from the same distribution, B0(j)\fS0(j), so that the estimation matrix at iteration\njis of the form\nE(j)=2\n664h\nB0(j)\fS0(j)i\n1\n:::h\nB0(j)\fS0(j)i\nO(j)3\n775:\nThe construction of the distribution is similar to that of the identi\fcation matrix, but omits the\nerror-correcting code and uses di\u000berent constant factors, etc., for the number of rows compared\nwith the analogues in the identi\fcation matrix.\n\u000fThe matrixB0(j)is Bernoulli with dimensions O(kcj)-by-N, for appropriate c, 1=2<c< 1.\nEach entry is 1 with probability \u0002\u0000\n1=(kcj)\u0001\nand zero otherwise. Each row is a pairwise\nindependent family and the set of seeds is fully independent.\n\u000fThe matrixS0(j)is a pseudorandom sign-\rip matrix of the same dimension as B0(j). Each\nrow ofS0(j)is a pairwise independent family of uniform \u00061-valued random variables. The\nsequence of seeds for the rows is a fully independent family.\n3.2.Measurements. The overall form of the measurements mirrors the structure of the measure-\nment matrices. We do not, however, use all of the measurements in the same fashion. In iteration\njof the algorithm, we use the measurements y(j)=\b(j)x. As the matrix \b(j)=E(j)\brD(j),\nwe have a portion of the measurements w(j)=D(j)xthat we use for identi\fcation and a portion\nz(j)=E(j)xthat we use for estimation. The w(j)portion is further decomposed into measurements\n[v(j);u(j)] corresponding to the run of O(log logN) 1's inC(j)and measurements corresponding to\neach of the blocks in the error-correcting code. There are O(j) i.i.d. repetitions of everything at\niterationj.\n3.3.Decoding. The decoding algorithm is shown in Figure 3 in the Appendix.\n4.Analysis\nIn this section we analyze the decoding algorithm for correctness and e\u000eciency.6 GILBERT, LI, PORAT, STRAUSS\n4.1.Correctness. Letx=xk+\u00171where we assume xis normalized so that k\u00171k2= 1 and xkis\nthe vector xwith all but the largest-magnitude kentries zeroed out. Our goal is to guarantee an\napproximation bxwith approximation error kx\u0000bxk2\u0014(1 +\u000f)k\u00171k2+\u000fk\u00172k2. But observe that \u00172\nis a di\u000berent type of object from xorbx;\u00172is added to \bx. For the main theorem to make sense,\ntherefore, we need to normalize \b. We discuss this now.\nObserve that the matrix \bcan be scaled up by an arbitrary constant factor c>1 which can be\nundone by the decoding algorithm: Let D0be a new decoding algorithm that calls the old decoding\nalgorithm Das follows: D0(y) =D\u00001\ncy\u0001\n, so that D0(c\bx+\u00172) =D\u0000\n\bx+1\nc\u00172\u0001\n. Thus we can\nreduce the e\u000bect of \u00172by an arbitrary factor cand so citing performance in terms of k\u00172kalone is not\nsensible. Note also that \u00172andxare di\u000berent types of objects; \b, as an operator, takes an object\nof the type of xand produces an object of the type of \u00172. We will stipulate that the appropriate\nnorm of \bbe bounded by 1, in order to make our results quantitatively comparable with others.\nOur error guarantee is in `2norm, so we should use a 2-operator norm; i.e.,, max k\bxk2overx\nwithkxk2= 1. But our algorithm's guarantee is in the \\for each\" signal model, so we need to\nmodify the norm slightly.\nDe\fnition 4. Thek\bk2 2norm of a randomly-constructed matrix \bismaxxEhk\bxk2\nxi\n. the\nsmallestMsuch that, for all xwithkxk2= 1, we havek\bxk2<M except with probability 1/4.\nNow we boundk\bk2 2. Each row \u001aof a Bernoulli( p) matrix with sign \rips, B\fS, satis\fes\nE[j\u001axj2] =pkxk2\n2. So 1=psuch rows satisfy k(B\fS)xk2\n2\u0014O\u0010\nkxk2\n2\u0011\n. Our matrix \brepeats the\nabovejtimes in the j'th iteration, j\u0014log2(k), and combines it with an error-correcting code\nmatrix of \u0002(log( N=k)) dense rows. It follows that\nk\bk2\n2 2=O(log2(k) log(N=k)):\nWe are ready to state the main theorem.\nTheorem 3 Consider the matrices in Section 3.1 and the algorithms in Section 3.3 (that share\nrandomness with the matrices). The joint distribution on those matrices and algorithms, with\nsuitable instantiations of anonymous constant factors (that may depend on \u000f), are such that, given\nmeasurements \bx+\u00172=y+\u00172, the algorithm returns bxwith approximation error\nkx\u0000bxk2\u0014(1 +\u000f)k\u00171k2+\u000flog(k)k\u00172k2\nk\bk2 2\nwith probability 3=4. The algorithm runs in time klogcNand\bhasO(klog(N=k))rows.\nIn this extended abstract, we give the proof only for \u000f= 1. Our results generalize in a straightfor-\nward way for general \u000f>0 (roughly, by replacing kwithk=\u000fat the appropriate places in the proof)\nand the number of measurements is essentially optimal in \u000f. Because our approach builds upon\ntheCount Sketch approach in [4], we omit the proof of intermediary steps that have appeared\nearlier in the literature.\nWe maintain the following invariant. At the beginning of iteration j, the residual signal has the\nform\n(Loop Invariant )r(j)=x(j)+\u0017(j)\n1with\r\r\rx(j)\r\r\r\n0\u0014k\n2j, and\r\r\r\u0017(j)\n1\r\r\r\n2\u00142\u0000\u00103\n4\u0011j\nexcept with probability1\n4(1\u0000(1\n2)j), wherek\u0001k0is the number of non-zero entries. The vector x(j)\nconsists of residual elements of xk. Clearly, maintaining the invariant is su\u000ecient to prove the\noverall result. In order to show that the algorithm maintains the loop invariant, we demonstrate\nthe following claim.\nClaim 1. Letb(j)be the vector we recover at iteration j.APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 7\n\u000fThe vectorb(j)contains all but at most1\n4k\n2jresidual elements of x(j)\nk, with \\good\" estimates.\n\u000fThe vectorb(j)contains at most1\n4k\n2jresidual elements of xkwith \\bad\" estimates.\n\u000fThe total sum square error over all \\good\" estimates is at most\n\"\n2\u0000\u00123\n4\u0013j+1#\n\u0000\"\n2\u0000\u00123\n4\u0013j#\n=1\n4\u00123\n4\u0013j\n:\nProof. To simplify notation, let Tbe the set of un-recovered elements of xkat iteration j; i.e., the\nsupport of x(j). We know thatjTj\u0014k=2j. The proof proceeds in three steps.\nStep 1. Isolate heavy hitters with little noise. Consider the action of a Bernoulli sign-\rip\nmatrixB\fSwithO(k=2j) rows. From previous work [4, 1], it follows that, if constant factors\nparametrizing the matrices are chosen properly,\nLemma 5. For each row \u001aofB, the following holds with probability \n(1) :\n\u000fThere is exactly one element tofT\\hashed\" byB; i.e., there is exactly one t2Twith\n\u001at= 1.\n\u000fThere areO(N\u00012j=k)total positions (out of N) hashed byB.\n\u000fThe dot product (\u001a\fS)r(j)isStr(j)\nt\u0006O\u0010\n2j\nk\r\r\r\u0017(j)\n1\r\r\r\n2\u0011\n.\nProof. (Sketch.) For intuition, note that the estimator St(\u001a\fS)r(j)is a random variable with mean\nr(j)\ntand variance\r\r\r\u0017(j)\n1\r\r\r2\n2. Then the third claim and the \frst two claims assert that the expected\nbehavior happens with probability \n(1). \u0003\nIn our matrix B(j), the number of rows is not k=2jbutkcjfor somec, 1=2< c < 1. Take\nc= 2=3. We obtain a stronger conclusion to the lemma. The dot product ( \u001a\fS)r(j)is\nStr(j)\nt\u0006O\u00121\nk(2=3)j\r\r\r\u0017(j)\n1\r\r\r\n2\u0013\n=Str(j)\nt\u00061\n8\u0012\n(3=4)j2j\nk\r\r\r\u0017(j)\n1\r\r\r\n2\u0013\n;\nprovided constants are chosen properly. Our lone hashed heavy hitter twill dominate the dot\nproduct provided\n\f\f\fr(j)\nt\f\f\f\u00151\n8\u0012\n(3=4)j2j\nk\r\r\r\u0017(j)\n1\r\r\r\n2\u0013\n:\nWe show in the remaining steps that we can likely recover such heavy hitters; i.e., Identify\nidenti\fes them and Estimate returns a good estimate of their values. There are at most ( k=2j)\nheavy hitters of magnitude less than1\n8\u0010\n(3=4)j2j\nk\r\r\r\u0017(j)\n1\r\r\r\n2\u0011\nwhich we will not be able to identify nor\nto estimate but they contribute a total of1\n8\u0010\n(3=4)j\r\r\r\u0017(j)\n1\r\r\r\n2\u0011\nnoise energy to the residual for the\nnext round (which still meets our invariant).\nStep 2. Identify heavy hitters with little noise. Next, we show how to identify t. Since\nthere areN=k\u0002(1)positions hashed by B(j), we need to learn the O(log(N=k)) bits describing tin\nthis context. Previous sublinear algorithms [7, 11] used a trivial error correcting code, in which the\nt'th column was simply the binary expansion of tin direct sum with a single 1. Thus, if the signal\nconsists of xtin thet'th position and zeros elsewhere, we would learn xtandxttimes the binary\nexpansion of t(the latter interpreted as a string of 0's and 1's as real numbers). These algorithms\nrequire strict control on the failure probability of each measurement in order to use such a trivial\nencoding. In our case, each measurement succeeds only with probability \n(1) and, generally, fails\nwith probability \n(1). So we need to use a more powerful error correcting code and a more reliable\nestimate ofjxtj.8 GILBERT, LI, PORAT, STRAUSS\nTo get a reliable estimate of jxtj, we use the b= \u0002(log log N)-parallel repetition code of all 1s.\nThat is, we get bindependent measurements of jxtjand we decode by taking the median. Let p\ndenote the success probability of each individual measurement. Then we expect the fraction pto\nbe approximately correct estimates of jxtj, we achieve close to the expectation, and we can arrange\nthatp>1=2. It follows that the median is approximately correct. We use this value to threshold\nthe subsequent measurements (i.e., the bits in the encoding) to 0 =1 values.\nNow, let us consider these bit estimates. In a single error-correcting code block of b= \u0002(log log N)\nmeasurements, we will get close to the expected number, bp, of successful measurements, except\nwith probability 1 =log(N), using the Cherno\u000b bound. In the favorable case, we get a number of\nfailures less than the (properly chosen) distance of the error-correcting code and we can recover\nthe block using standard nearest-neighbor decoding. The number of error-correcting code blocks\nassociated with tisO(log(N=k)=log logN)\u0014O(logN), so we can take a union bound over all\nblocks and conclude that we recover twith probability \n(1). The invariant requires that the failure\nprobability decrease with j. Because the algorithm takes O(j) parallel independent repetitions, we\nguarantee that the failure probability decreases with jby taking the union over the repetitions.\nWe summarize these discussions in the following lemma. We refer to these heavy hitters in the\nlist \u0003 as the j-large heavy hitters.\nLemma 6. Identify returns a set \u0003of signal positions that contains at least 3=4of the heavy\nhitters inT,jTj\u0014k=2j, that have magnitude at least1\n8\u0010\n(3=4)j2j\nk\r\r\r\u0017(j)\n1\r\r\r\n2\u0011\n.\nWe also observe that our analysis is consistent with the bounds we give on the additional mea-\nsurement noise \u00172. The permutation matrix Pin\bis applied before \u00172is added and then P\u00001is\napplied after \u00172by the decoding algorithm. It follows that we can assume \u00172is permuted at random\nand, therefore, by Markov's inequality, each measurement gets at most an amount of noise energy\nproportional to its fair share of k\u00172k2\n2. Thus, If there are m= \u0002(klogN=k) measurements, each\nmeasurement getsk\u00172k2\n2\nmnoise energy and identi\fcation succeeds anyway provided the lone heavy\nhittertin that bucket has square magnitude at leastk\u00172k2\n2\nm, so the at most ksmaller heavy hitters,\nthat we may miss, together contribute energykk\u00172k2\n2\nm=O\u0010k\u00172k2\n2\nlog(N=k)\u0011\n. If we recall the de\fnition and\nvalue ofk\bk2 2, we see that this error meets our bound.\nStep 2. Estimate heavy hitters. Many of the details in this step are similar to those in Lemma 5\n(as well as to previous work as the function Estimate is essentially the same as Count Sketch ),\nso we give only a brief summary.\nFirst, we discuss the failure probability of the Estimate procedure. Each estimate is a complete\nfailure with probability 1 \u0000\n(1) and the total number of identi\fed positions is O\u0010\njk(2=3)j\u0011\n.\nBecause we perform jparallel repetitions in estimation, we can easily arrange to lower that failure\nprobability, so we assume that the failure probability is at most \u0002\u0010\n(3=4)j\u0011\n, and that we get\napproximately the expected number of (nearly) correct estimates. There are k(2=3)jheavy hitters\nin \u0003, so the expected number of failures is (1 =4)(k=2j). These, along with the at most 1 =4(k=2j)\nmissedj-large heavy hitters, will form x(j+1), the at-most- k=2j+1residual heavy hitters at the next\niteration.\nIn iteration j,Identity returns a list \u0003 with k(2=3)jheavy hitter position identi\fed. A group of\nk(2=3)jmeasurements in E(j)yields estimates for the positions in \u0003 with aggregate `2error\u0006O(1),\nadditively. An additional O\u0010\n(4=3)j\u0011\ntimes more measurements, O(k(8=9)j) in all, improves the\nestimation error to (1 =8) (3=4)j, additively. These errors, together with the omitted heavy hitters\nthat are not j-large and\u0017(j)form the new noise vector at the next iteration, \u0017(j+1).APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 9\nFinally, consider the e\u000bect of \u00172. We would like to argue that, as in the identi\fcation step, the\nnoise vector \u00172is permuted at random and each measurement is corrupted byk\u00172k2\n2\nm, wherem=\n\u0002(klog(N=k)) is the number of measurements, approximately its fair share of k\u00172k2\n2. Unfortunately,\nthe contributions of \u00172to the various measurements are not independent as \u00172is permuted, so we\ncannot use such a simple analysis. Nevertheless, they are negatively correlated and we can achieve\nthe result we want using [9]. The total `2squared error of the corruption over all O(k) estimates\nisk\u00172k2\n2=log(N=k), which will meet our bound. That is, since k\bk2\n2 2=O(log2(k) log(N=k)), the\n\u00172contribution to the error is\nO \nk\u00172k2p\nlogN=k!\n=O\u0012log(k)k\u00172k2\nk\bk2 2\u0013\n;\nas claimed, whence we read o\u000b the factor, log( k) (improvable to log1=2+o(1)k), which is directly\ncomparable to other results that scale \bproperly.\n\u0003\n4.2.E\u000eciency.\n4.2.1. Number of Measurements. The analysis of isolation and estimation matrices are similar; the\nnumber of measurements in isolation dominates.\nThe number of measurements in iteration jis computed as follows. There are O(j) parallel\nrepetitions in iteration j. They each consist of k(2=3)jmeasurements arising out of B(j)for\nidenti\fcation times O(log(N=k)) measurements for the error correcting code, plus k(2=3)jtimes\nO((4=3)j) for estimation. This gives\n\u0002 \njk\u00122\n3\u0013j\nlog(N=k) +jk\u00128\n9\u0013j!\n=klog(N=k)\u00128\n9+o(1)\u0013j\n:\nThus we have a sequence bounded by a geometric sequence with ratio less than 1. The sum, over\nallj, isO(klog(N=k)).\n4.2.2. Encoding and Update Time. The encoding time is bounded by Ntimes the number of non-\nzeros in each column of the measurement matrix. This was analyzed above in Section 4.1; there are\nlog2(k) log(N=k) non-zeros per column, which is suboptimal by the factor log2(k). By comparison,\nsome proposed methods use dense matrices, which are suboptimal by the exponentially-larger factor\nk. This can be improved slightly, as follows. Recall that we used jparallel repetitions in iteration\nj;j < log(k), to make the failure probability at iteration be; e.g., 2\u0000j, so the sum over jis bounded.\nWe could instead use failure probability 1 =j2, so that the sum is still bounded, but the number of\nparallel repetitions will be log( j), forj\u0014log(k). This results in log( k) log log(k) log(N=k) non-zeros\nper column and \u00172contribution to the noise equal top\nlog(k) log log(k)k\u00172k2\nk\bk2 2.\nWe can use a pseudorandom number generator such as i7!b(ai+bmodd)=Bcfor random a\nandb, whereBis the number of buckets. Then we can, in time O(1), determine into which bucket\nanyiis mapped and determined the i'th element in any bucket.\nAnother issue is the time to \fnd and to encode (and to decode) the error-correcting code. Observe\nthat the length of the code is O(log logN). We can a\u000bord time exponential in the length, i.e., time\nlogO(1)N, for \fnding and decoding the code. These tasks are straightforward in that much time.\n4.2.3. Decoding Time. As noted above, we can quickly map positions to buckets and \fnd the i'th\nelement in any bucket, and we can quickly decode the error-correcting code. The rest of the claimed\nruntime is straightforward.10 GILBERT, LI, PORAT, STRAUSS\n5.Conclusion\nIn this paper, we construct an approximate sparse recovery system that is essentially optimal:\nthe recovery algorithm is a sublinear algorithm (with near optimal running time), the number of\nmeasurements meets a lower bound, and the update time, encode time, and column sparsity are\neach within log factors of the lower bounds. We conjecture that with a few modi\fcations to the\ndistribution on measurement matrices, we can extend this result to the `1\u0014C`1error metric\nguarantee. We do not, however, think that this approach can be extended to the \\for all\" signal\nmodel (all current sublinear algorithms use at least one factor O(logN) additional measurements)\nand leave open the problem of designing a sublinear time recovery algorithm and a measurement\nmatrix with an optimal number of rows for this setting.\nReferences\n[1] N. Alon, Y. Matias, and M. Szegedy. The Space Complexity of Approximating the Frequency Moments. J.\nComput. System Sci. , 58(1):137{147, 1999.\n[2] K. Do Ba, P. Indyk, E. Price, and D. Woodru\u000b. Lower bounds for sparse recovery. In ACM SODA , page to\nappear, 2010.\n[3] E. J. Cand\u0012 es, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements.\nComm. Pure Appl. Math. , 59(8):1208{1223, 2006.\n[4] M. Charikar, K. Chen, and M. Farach-Colton. Finding frequent items in data streams. ICALP , 2002.\n[5] G. Cormode and S. Muthukrishnan. What's hot and what's not: Tracking most frequent items dynamically. In\nProc. ACM Principles of Database Systems , pages 296{306, 2003.\n[6] G. Cormode and S. Muthukrishnan. Improved data stream summaries: The count-min sketch and its applications.\nFSTTCS , 2004.\n[7] G. Cormode and S. Muthukrishnan. Combinatorial algorithms for Compressed Sensing. In Proc. 40th Ann. Conf.\nInformation Sciences and Systems , Princeton, Mar. 2006.\n[8] D. L. Donoho. Compressed Sensing. IEEE Trans. Info. Theory , 52(4):1289{1306, Apr. 2006.\n[9] Devdatt Dubhashi and Volker Priebe Desh Ranjan. Negative dependence through the fkg inequality. In Research\nReport MPI-I-96-1-020, Max-Planck-Institut fur Informatik, Saarbrucken , 1996.\n[10] Yaniv Erlich, Kenneth Chang, Assaf Gordon, Roy Ronen, Oron Navon, Michelle Rooks, and Gregory J. Han-\nnon. Dna sudoku|harnessing high-throughput sequencing for multiplexed specimen analysis. Genome Research ,\n19:1243|1253, 2009.\n[11] A. C. Gilbert, M. J. Strauss, J. A. Tropp, and R. Vershynin. One sketch for all: fast algorithms for compressed\nsensing. In ACM STOC 2007 , pages 237{246, 2007.\n[12] P. Indyk and M. Ruzic. Near-optimal sparse recovery in the l1norm. FOCS , 2008.\n[13] D. Needell and J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl.\nComp. Harmonic Anal. , 2008. To appear.\n[14] Y. H. Zheng, N. P. Pitsianis, and D. J. Brady. Nonadaptive group testing based \fber sensor deployment for\nmultiperson tracking. IEEE Sensors Journal , 6(2):490{494, 2006.\n[15] Y.H. Zheng, D. J. Brady, M. E. Sullivan, and B. D. Guenther. Fiber-optic localization by geometric space coding\nwith a two-dimensional gray code. Applied Optics , 44(20):4306{4314, 2005.\n6.Appendix\nWe have a full description of the matrix algebra de\fned in Table 2.\n\u000fRow direct sum. The row direct sum A\brBis a matrix with Ncolumns that is the\nvertical concatenation of AandB.\n\u000fElement-wise product. IfAandBare bothr\u0002Nmatrices, then A\fBis also anr\u0002N\nmatrix whose ( i;j) entry is given by the product of the ( i;j) entries inAandB.\n\u000fSemi-direct product. SupposeAis a matrix of r1rows (andNcolumns) in which each\nrow has exactly hnon-zeros and Bis a matrix of r2rows andhcolumns. Then BnrAis\nthe matrix with r1r2rows, in which each non-zero entry aofAis replaced by atimes the\nj'th column of B, whereais thej'th non-zero in its row. This matrix construction has\nthe following interpretation. Consider ( BnrA)xwhereAconsists of a single row, \u001a, with\nhnon-zeros and xis a vector of length N. Lety=\u001a\fxbe the element-wise product of \u001aAPPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 11\nandx. If\u001ais 0/1-valued, ypicks out a subset of x. We then remove all the positions in\nycorresponding to zeros in \u001a, leaving a vector y0of lengthh. Finally, (BnrA)xis simply\nthe matrix-vector product By0, which, in turn, can be interpreted as selecting subsets of\ny, and summing them up. Note that we can modify this de\fnition when Ahas fewer than\nhnon-zeros per row in a straightforward fashion.12 GILBERT, LI, PORAT, STRAUSS\nRecover (\b;y)\nOutput:bx=approximate representation of x\ny=P\u00001y\na(0)= 0\nForj= 0 toO(logk)f\ny=y\u0000P\u00001\ba(j)\nsplity(j)=w(j)\brz(j)\n\u0003 =Identify (D(j);w(j))\nb(j)=Estimate (E(j);z(j);\u0003)\na(j+1)=a(j)+b(j)\ng\nbx=a(j)\nIdentify (D(j);w(j))\nOutput: \u0003 = list of positions\n\u0003 =;\nDividew(j)into sections [v;u]of sizeO(log(cj(N=k)))\nFor each section f\nu= median(jv`j)\nFor each` // threshold measurements\nu`= \u0002(u`\u0000u=2) //\u0002(u) = 1 ifu>0,\u0002(u) = 0 otherwise\nDivideuinto blocks biof sizeO(log logN)\nFor eachbi\n\fi=Decode (bi) // using error-correcting code\n\u0015=Integer (\f1;\f2;:::) // integer rep'ed by bits \f1;\f2;:::\n\u0003 = \u0003[f\u0015g\ng\nEstimate (E(j);z(j);\u0003)\nOutput:b=vector of positions and values\nb=;\nFor each\u00152\u0003\nb\u0015= median`s:t:B(j)\n`;\u0015=1(z(j)\n`S(j)\n`;\u0015)\nFigure 3. Pseudocode for the overall decoding algorithm."    },    {        "title": "0912.3125v1.Toward_a_dynamical_shift_condition_for_unequal_mass_black_hole_binary_simulations.pdf",        "content": "arXiv:0912.3125v1  [gr-qc]  16 Dec 2009Toward a dynamical shift condition for unequal\nmass black hole binary simulations\nDoreen M¨ uller, Bernd Br¨ ugmann\nTheoretical Physics Institute, University of Jena, 07743 Jena, G ermany\nAbstract. Movingpuncturesimulationsofblackholebinariesrelyonaspecificgau ge\nchoice that leads to approximately stationary coordinates near ea ch black hole. Part\nof the shift condition is a damping parameter, which has to be proper ly chosen for\nstable evolutions. However, a constant damping parameter does n ot account for the\ndifference in mass in unequal mass binaries. We introduce a position de pendent shift\ndamping that addresses this problem. Although the coordinates ch ange, the changes\nin the extracted gravitational waves are small.\nPACS numbers: 04.25.D-, 04.25.dg,04.30.Db\n1. Introduction\nEarlynumericalrelativitysimulationsusinga3+1splitoftheEinsteineq uationssuffered\nfrom so-called slice stretching, an effect which occurs when using sin gularity avoiding\nslicing together with a vanishing shift. The slices become highly distort ed when time\nmarches on in the outer regions of the grid but slows down in the vicinit y of the black\nhole. It became clear that a non-vanishing, outwards pointing shift vector would be\nrequired in order to redistribute grid points and also to prevent grid points from falling\ninto the black hole. Inspired by Balakrishna et al.[1], Alcubierre et al.[2, 3] combined\nthe1+log slicing conditionwith a dynamical shift condition called gamma- driver. These\ngauge conditions successfully prevented slice stretching in black ho le simulations using\nexcision. It turned out that such gauge conditions could be used als o for fixed punctures\nwith slight modifications to keep the puncture from evolving [4]. The fix ed-puncture\nmodification was removed in [5, 6] when the moving puncture method was introduced.\n1+log slicing with gamma-driver shift succeeds in moving the puncture freely through\nthe grid while simultaneously avoiding slice-stretching. The basic reas on for the success\nofthisgaugeconditionisthatwhentheslicesstarttostretch, the shiftvectorcounteracts\nby pulling out grid points from the region near the black hole.\nIn this paper we focus on the dissipation or damping parameter in the gamma-\ndriver shift condition, which plays an important role in the success of this gauge. In\norder to reduce oscillations in the shift vector, the authors of [3] n oticed the necessity\nof a damping term in the shift condition. Adjusting the strength of t he damping via a\ndampingparameterwasfoundtoallowfreezingoftheevolutionatlat etimesin[4]andtoToward a dynamical shift condition for unequal mass black ho le binary simulations 2\navoid drifts in metric variables in [7]. Additionally, the value of the damp ing coefficient\nwas found to affect the coordinate location of the apparent horizo n and therefore the\nresolution ofthe black holeonthe numerical grid [7, 8]. The right cho ice ofthe damping\nvalue is therefore important if one wants to resolve the black hole pr operly while still\ndriving the coordinates to a frame where they are stationary when the physical situation\nis stationary and hence obtain a stable evolution.\nThe specific value of the damping parameter has to be adapted to th e black hole\nmass in order to obtain long term stable evolutions. If the damping pa rameter is\neither too small or too large, there are unwanted oscillations or a co ordinate instability,\nrespectively. In binary simulations, a typical choice is a constant va lue of roughly 2 /M,\nwhereMis the total mass of the system. However, using a constant dampin g parameter\nfor black hole binaries with unequal masses leads to a fundamental p roblem. With a\nconstant damping parameter, the effective damping near each blac k hole is asymmetric\nfor unequal black hole masses since the damping parameter has dime nsions 1/M. For\nlargemassratios, thisasymmetryinthegridcanbesolargethatsimu lationsfailbecause\nthe damping may become too large for one of the black holes. This is on e of the reasons\nwhy the highest mass ratio that has been successfully simulated up t o now is 10 : 1 [9].\nAdvantageous would be a position-dependent damping parameter t hat adapts to\nthe local mass, in particular such that in the vicinity of the ithpuncture with mass\nMiits value approaches 1 /Mi. It was noticed before [4, 9] that a damping coefficient\nadapted to the various parameters of the simulation would be benefi cial. In [10] a\nposition-dependent formula was introduced for head-on collisions o f black holes, which\nto our knowledge was only used in one other publication [11], prior to th e moving\npuncture framework. In this paper, we take first steps towards a position-dependent\ndamping parameter for moving punctures. As a consequence, the local coordinates\nchange compared to standard simulations, but this does not signific antly affect gauge\ninvariant quantities like the extracted waves as we discuss below.\n2. Dynamical damping in the shift equation\n2.1. Numerical setup\nWe focus on the gauge condition used in the 3+1 splitting of the Einste in equations, in\nparticular on the condition for the shift vector.\nThe slices are determined by the 1+log slicing condition [12] for the laps e function\nα,\n∂0α=−2αK, (1)\nwhereKis the trace of the extrinsic curvature. The coordinates of a given slice are\ngoverned by the gamma-driver shift condition introduced in [4] as\n∂2\n0βi=3\n4∂0˜Γi−ηs∂0βi, (2)Toward a dynamical shift condition for unequal mass black ho le binary simulations 3\nwhere˜Γiare the contracted Christoffel symbols of the conformal metric ˜ γij,βiis the\nshift vector and ηsis the damping coefficient we will discuss in this publication. In\nEqs. (1) and (2), ∂0is defined as ∂0=∂t−βi∂ias suggested by [13, 14, 15].\nExamining the physical dimensions, we see that [ βi] = 1 and [∂0] = 1/M, whereM\nis the mass (e.g. the total mass of the spacetime under considerat ion). For this reason,\nthe second term on the right hand side of equation (2) requires the damping parameter\nto carry units,\n[ηs] =1\nM. (3)\nIn simulations of a single Schwarzschild puncture of mass M1, we typically choose\na damping parameter of ηs≈1/M1for obtaining enough damping in the shift without\nproducing instabilities. In numerical experiments for a Schwarzsch ild puncture (to be\ndiscussed elsewhere), we find that 0 ≤ηs/lessorapproxeql3.5/M1is necessary for a stable and\nconvergent numerical evolution. Some minimal amount of damping is im portant to\nsuppress noise in the gauge when a puncture is moving. On the other hand, ifηsis too\nlargethen there aregaugeinstabilities, leading toa loss of converge nce and toinstability\nof the entire numerical evolution. Furthermore, early simulations f or fixed punctures\nalso found that ηsshould take values around 1 /M, whereMis the total mass, to avoid\nlong-term coordinate drifts at the outer boundary [4].\nIn simulations of black hole binaries with total mass M=M1+M2, we usually set\nηs= 2/Mwhich has been found to work well in equal mass binaries simulations. ( For\nequal masses, M1=M2=M/2, so near one of the punctures the value of ηsdiscussed\nabove for Schwarzschild becomes ηs= 1/M1= 2/M.) For unequal mass binaries, the\ndifferent black holes tolerate different ranges of ηsaccording to the above statement\nabout single black holes. Ideally, ηsshould be ≈1/Mi, which cannot be accomplished\nsimultaneously for unequal masses using a constant value of ηs= 2/M. In fact, for the\nmass ratio 1:10 in [9], the choice ηs= 2/Mfailed, but a smaller value for ηswas chosen\nsuch thatηs/lessorapproxeql3.5/Mifor bothi= 1 andi= 2.\nTo overcome the conflicts between punctures with different masse s in evolutions of\ntwo or more black holes, we suggest to construct a non-constant , position-dependent\ndamping parameter which knows about the position and mass of each puncture and\ntakes a suitable value at every grid point.\n2.2. Using ψ−2to determine the position of the punctures\nWe thus desire a definition of ηswhich respects the unit requirements found in Eq. (3)\nand which asymptotes to specifiable values at the location of the pun ctures and at\ninfinity. Typical values are ηs= 1/Miat theithblack hole and ηs= 2/Mat large\ndistances. Thiscanbeachievedbydetermining ηsthroughapositiondependent function\ndefined on the whole grid instead of using a constant as before. We d esire a smooth ηs\nwhich avoids modes which travel at superluminal speeds. Since we us e the Baumgarte–\nShapiro–Shibata–Nakamura (BSSN) system of Einstein’s equations [16, 17], we wantToward a dynamical shift condition for unequal mass black ho le binary simulations 4\nthe form of ηsto depend only on the BSSN variables in a way that does not change th e\nprincipal part of the differential operators.\nIn this paper, we choose to use the conformal factor ψ, which contains information\naboutthelocationsandmassesofthepunctures. Theformulawew illusefordetermining\nthe damping coefficient ηs(/vector r) is\nηs(/vector r) =ˆR0/radicalbig\n˜γij∂iψ−2∂jψ−2\n(1−ψ−2)2, (4)\nwith ˜γijthe inverse of the conformal 3–metric and ˆR0a dimensionless constant. While\nψ, ˜γij, andˆR0are dimensionless, the partial derivative introduces the appropria te\ndependence on the mass since [ ∂i] = 1/Mand hence [ ηs(/vector r)] = 1/M.\nFor a single Schwarzschild puncture of mass Mlocated atr= 0 the behavior of\nEq. (4) near the puncture and near infinity is as follows. According t o [18], for small\nradiir(near the puncture) the conformal factor asymptotically equals\nψ−2≃p1r (5)\nfor a known constant p1. The next to leading order behavior is less simple [19]. The\npointr= 0 corresponds to a sphere with finite areal radius R0,\nR0= lim\nr→0ψ2r=1\np1=ˆR0M. (6)\nNumerically, ˆR0≈1.31. The inverse of the conformal metric behaves like\n˜γij≃δij. (7)\nTherefore, we find for small r\n/radicalbig\n˜γij∂iψ−2∂jψ−2≃p1=1\nˆR0M(8)\nand\n(1−ψ−2)2≃(1−p1r)2≃1 (9)\nwhen keeping only leading order terms in r. Equations (8) and (9) combine according\nto (4) to give\nηs(r= 0) = 1/M. (10)\nFor largerwe can expand the conformal factor in powers of 1 /r,\nψ−2≃/parenleftbigg\n1+M\n2r/parenrightbigg−2\n≃1−M\nr, (11)\nresulting in\n/radicalbig\n˜γij∂iψ−2∂jψ−2=M\nr2. (12)\nand\nηs(r→ ∞)≃ˆR0M/r2\n(M/r)2=ˆR0\nM. (13)Toward a dynamical shift condition for unequal mass black ho le binary simulations 5\nIn summary, Eq. (4) leads to\nηs(r)→/braceleftBigg\n1/M, r →0\nˆR0/M, r→ ∞(14)\nfor a single puncture at r= 0. Note that using Eq. (4) in Eq. (2) does not affect the\nprincipal part of (2). Therefore, the system remains strongly hy perbolic, same as for\nηs= const. according to [14, 13].\n3. Results\nOur Eq. (4) analytically gives the desired 1 /Mbehavior near the puncture and near\ninfinity for a single, non–spinning and non–moving puncture. Now it re mains to be\ntested whether these properties persist in actual numerical simu lations, especially for\nunequal mass binaries.\nSimulations are performed with the BAM code described in [7, 20]. The c ode\nuses the BSSN formulation of Einstein’s equations and employs the mo ving puncture\nframework [5, 6]. Spatial derivatives are 6thorder accurate and time integration is\nperformed using the 4thorder Runge–Kutta scheme. The numerical grid is composed\nof nested boxes with increasing resolution, where the boxes of high est resolution are\ncentered around the black holes. These boxes are advanced in time with Berger–Oligar\ntime stepping [21]. We are using puncture initial data with Bowen–York extrinsic\ncurvature and solve the Hamiltonian constraint using a pseudospec tral collocation\nmethod described in [22]. The momentum parameter in the Bowen–Yo rk extrinsic\ncurvature is chosen such that we obtain quasi–circular orbits in our binary simulations\nusing the method of [23].\nFor binary simulations with unequal masses, we will use the mass ratio q=M2/M1\nto denote the runs, Mibeing the bare mass of the ithpuncture. The physical masses\nof the punctures (obtained after solving the constraints) differ b y less than 10% from\nthe bare masses for the orbits considered here, so the ηsvalues derived for a single\npuncture should remain valid. When comparing simulations run with ηs= 2/Mand\nηs(/vector r) following Eq. (4) we will refer to them as “standard” and “new” or “ dynamical”\ngauge, respectively, throughout this paper.\n3.1. Single Schwarzschild Black Hole\nIn order to test the 1 /M–behavior of (4) near the puncture and infinity, we first\nperformed a series of evolutions for a time of 100 Mof a single, non–spinning puncture\nwhile varying its mass. We then measured the value of ηs(/vector r) near the puncture and\nat the outer boundary of the grid and compared these values to th e limits (14). The\ndata points in Fig. 1 correspond to these measurements while the line s are fits to the\nnumerical data. The values of ηs(/vector r) near the puncture as a function of total mass M\nare fitted to ηs(M) = 1.05/Mwhich agrees well with the analytical limit r→0 of (14).\nFitting toηsmeasured near the physical boundary of the grid reveals ηs(M) = 1.311/MToward a dynamical shift condition for unequal mass black ho le binary simulations 6\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n0 1 2 3 40.51.01.52.02.53.03.5\nMΗs/LParen1r0/RParen1r0/EquΑl125 M\nr0/RArrow0\nFigure 1. Numerical test of the analytical limits (14) of ηs(/vector r) using single, non–\nspinning punctures with different masses. Shown are the values of ηs(r) near the\npuncture (gray squares) and at the outer boundary (black dots ). Note that here Mis\nidentifiedwith adimensionlessnumber, so ηs(r) isdimensionlessaswell. Thefits tothe\ndata points are consistent with the analytic prediction. Numerically, ηs(M) = 1.05/M\n(grayline) nearthe puncture and ηs(M) = 1.311/M(blackline) at the outerboundary.\n0 2 4 6 8 100.000.050.100.150.20\nr/LBracket1M/RBracket1ΒxΗs/LParen1r/OverRVector/RParen1\nΗs/EquΑl2/Slash1M\n0 2 4 6 8 101.001.051.101.151.201.25\nr/LBracket1M/RBracket1Γ/OverTilde\nxxΗs/LParen1r/OverRVector/RParen1\nΗs/EquΑl2/Slash1M\nFigure 2. x-component of the shift vector (upper panel) and xx-component of the\nconformal 3-metric (lower panel) of a single, non-spinning punctur e at time t= 100M,\nwhere the simulations have reached a stationary state. The dashe d black curves use\ndynamical damping, Eq. (4), the gray ones use ηs= 2.0/Min the shift condition\nEq. (2).\nand therefore fulfills the limit r→ ∞of (14) even though the outer boundary is situated\nonly at 130 M.\nUsing a modified shift condition, the shift itself will, of course, change . We compare\nthex-component of the shift vector for using ηs= 2.0/Mandηs(/vector r) in Fig. 2. A change\nin the shift implies a change of the coordinates and therefore, coor dinate dependent\nquantities willchange, too. Asanexample, the xx-component oftheconformal3-metric,\n˜γxx, is compared for ηs= 2/Mandηs(/vector r) in the lower panel of Fig. 2. The comparisons\nare made at time t= 100M, when the simulations have reached a stationary state.\nThe changes in the shift should only affect the coordinates and coor dinate\nindependent quantities should not change. This can be examined by lo oking at a scalar\nas a function of another scalar, e.g. the lapse αas a function of extrinsic curvature K,\nα=α(K). Both scalars should see the same coordinate drifts and therefo re, no changesToward a dynamical shift condition for unequal mass black ho le binary simulations 7\n0.00 0.05 0.10 0.15 0.20 0.25 0.300.00.20.40.60.8\nK/LBracket1M/Minus1/RBracket1Α0123450.00.20.40.60.8\nr/LBracket1M/RBracket1Α\n0123450.000.050.100.150.200.250.30\nr/LBracket1M/RBracket1K/LBracket1M/Minus1/RBracket1\nΗs/LParen1r/OverRVector/RParen1\nΗs/EquΑl2/Slash1M\nFigure 3. The lapse function αas a function of extrinsic curvature Kfor a single,\nnon-spinning and non-movingpuncture after a time t= 50.M.We compare using ηs(/vector r)\n(black, dashed line) and ηs= 2.0/M(gay line). The two curves lie perfectly on top\nof each other and are therefore indistinguishable. The insets show lapse (upper panel)\nand extrinsic curvature (lower panel) as functions of distance fro m the puncture.\nare expected in α(K). Figure 3 confirms this expectation. The two curves α(K) for\nηs= 2.0/Mandηs(/vector r) are lying perfectly on top of each other. We therefore believe\nthat using the dynamical damping introduces only coordinate chang es in our puncture\nsimulations.\n3.2. Black hole binary with equal masses\nWhile Eq. (4) has been introduced in order to allow for numerical simula tions of two\nblack holes with highly different masses, we first apply it to equal mass simulations in\norder to perform several consistency checks.\nThe (first order) coordinate independent quantity to look at in bina ry simulations\nis the Newman–Penrose scalar Ψ 4. We use Ψ 4for the extraction of gravitational waves\n(see [7] for details of the wave extraction algorithm), decomposed into modes using\nspin-weighted spherical harmonics Y−2\nlm. Since Ψ 4is only first-order gauge invariant and\nwe furthermore extract waves at a finite, fixed coordinate radius , it is a priori an open\nquestion how much the changes in the shift affect the wave forms.\nAs the most dominant mode of Ψ 4in an equal mass simulation is the l=|m|= 2\nmode, its real part multiplied by the extraction radius ( rex= 90Min this case) is\ndisplayed in Fig. 4. We look at amplitude and phase of this mode in Figs. 5 a nd 6. The\ninitial separation was chosen to be D= 7M. The black holes complete about 3 orbits.\nThree different resolutions are used corresponding to the three d ifferent colors in Figs. 4,\n5 and 6. We use the number of grid points in the inner boxes (centere d around the black\nholes) to denote the different resolutions. The grid configurations , in the terminology\nof [7], areφ[5×56 : 5×112 : 6],φ[5×64 : 5×128 : 6], and φ[5×72 : 5×144 : 6]Toward a dynamical shift condition for unequal mass black ho le binary simulations 8\n260 280 300 320 340 360/Minus0.06/Minus0.04/Minus0.020.000.020.040.06\nt/LBracket1M/RBracket1Re/LBrace1/CΑpPsi422/RBrace1rexN/EquΑl56\nN/EquΑl64\nN/EquΑl72\nFigure 4. Real part of the 22-mode of Ψ 4times extraction radius rexfor an equal\nmass binary with initial separation D= 7Musingηs= 2.0/M(solid lines) and ηs(/vector r)\nfollowing Eq. (4) (dashed lines) in three different resolutions (blue, r ed, green lines)\naccording to the grid configurations described in the text.\nwhich corresponds to resolutions on the finest grids of 3 M/112, 3M/128 andM/48,\nrespectively. These are the grid configurations used in [20].\nIn Fig. 5, we compare the amplitude A22in the standard gauge, ηs= 2.0/M,\ndisplayed as solid lines, to the new one, Eq. (4), plotted as dashed line s. We find that\nthe differences between standard and new gauge for a given grid re solution are much\nsmaller than differences due to using different resolutions. This stre ngthens the belief\nthat we only introduced coordinate changes to the system when us ing Eq. (4). The\nmaximum relative deviation between the amplitudes A22of old and new gauge amounts\nto about 3% for the lowest resolution ( N= 56) and decreases with increasing resolution,\nwhich can be seen in the inset of Fig. 5. For the phase, the absolute d ifferences are not\nvisible by eye and therefore, we only plot the relative deviations betw eenφ22in the\nstandard and new gauge for the three resolutions. For the lowest resolution ( N= 56),\nthe maximum deviation is only 0 .35%. As for the amplitude, this deviation decreases\nwith increases in resolution. This shows how the differences in the wav eforms disappear\nwith increasing resolution.\nThe fact that there actually aredifferences visible in the waves, though very small\nones, is not surprising when considering the way we extract gravita tional waves. We fix\na certain extraction radius and compute the Newman–Penrose sca lar on a sphere of this\nradius. The radius itself is coordinate dependent and we are compar ing Ψ4extracted\nat slightly different radii in the standard and new gauges. In future work we plan to\ncompare wave forms extrapolated in radius to infinity, although it is w orth noting how\nsmall the deviations are without additional processing.\n3.3. Black hole binary with mass ratio 4:1\nAfter having examined the influence of using a dynamical damping coe fficientηs(/vector r) for\nan equal mass binary, the next step is to look at its behavior for une qual masses. The\nfollowing results are obtained from a simulation of two black holes with m ass ratioq= 4\nand initial separation D= 7M. We used the grid configurations φ[5×N: 7×2N: 6]Toward a dynamical shift condition for unequal mass black ho le binary simulations 9\n260 280 300 320 340 3600.000.010.020.030.040.050.060.07\nt/LBracket1M/RBracket1A22\n260 280 300 320 340 360/Minus0.02/Minus0.010.000.010.020.03\nt/LBracket1M/RBracket1/CΑpDeltΑ/CΑpAlpΗΑ22/Slash1/CΑpAlpΗΑ22N/EquΑl56\nN/EquΑl64\nN/EquΑl72\nFigure 5. Amplitude of the 22-mode of Ψ 4for the same binary as in Fig. (4) using\nηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in three different resolutions (blue,\nred, green lines) according to the grid configurations described in t he text. The inset\nshows the relative deviation ∆ A22/A22between the amplitude in the standard and in\nthe new gauge, again for the three different resolutions.\n260 280 300 320 340 360/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.50.0\nt/LBracket1M/RBracket1/CΑpDeltΑΦ22/Slash1Φ22/LBracket1/Multiply10/Minus3/RBracket1N/EquΑl56\nN/EquΑl64\nN/EquΑl72\nFigure 6. Relative phase difference of the 22-mode of Ψ 4for the same binary as in\nFig. (4) using ηs= 2/Mandηs(/vector r). Compared are three different resolutions (blue, red\nand green lines) according to the grid configurations described in th e text.\nwithN= 72,80 which have also been used in [24] for mass ratio 4:1. An interesting\nquestion in this context is how Eq. (4) behaves for a simulation with tw o punctures\nwith different masses. The analytical behavior (14) was deduced fo r a single, non-\nmoving, stationary puncture but now we are using it for two moving p unctures, which\nare during most of the simulations far from having reached a station ary state globally,\nbut approximately stationary locally at the punctures.\nFigure 7 illustrates the distribution of ηs(/vector r) between the two punctures. For\nconvenience, the conformal factor φ= lnψis also plotted in order to indicate the\npositions of the punctures via its maxima (the divergences are not r esolved). The\nsnapshot is taken at a time during the simulation when the punctures are still well\nseparated. Similar to the simulations of a single puncture, according to (14) we expect\nto findηs(/vector r)≃2 near the puncture with mass M1= 0.5 andηs(/vector r)≃0.5 in the vicinity\nof the second puncture with M2= 2.0. Near the outer boundary, ηsis supposed to\ntake the value 1 .31/(M1+M2) = 0.52. Here the Miare chosen to be dimensionless, so\nηs(/vector r) is dimensionless as well. Figure 7 confirms that we do obtain the expec ted values,\nalthough they are not reached exactly. The latter is not a problem a s simulations workToward a dynamical shift condition for unequal mass black ho le binary simulations 10\n/Minus15/Minus10/Minus5 0 5 100.00.51.01.52.0\nd/LBracket1M/RBracket1Ηs,Φ\nFigure 7. ηs(/vector r) (black line) for two punctures of bare masses M1= 0.5 andM2= 2.0\nand the conformal factor φ(gray line) whose maxima show the current positions of\nthe punctures. In this plot the Miare dimensionless, so ηs(/vector r) is dimensionless as\nwell. The snapshot is taken right after the beginning of the q= 4 simulation with\ninitial separation D= 5Mdescribed in the text. The smaller black hole is at position\nd=−3.6M, the larger one at position d= 1.2M.\n100 110 120 130 140 150 160/Minus0.03/Minus0.02/Minus0.010.000.010.020.03\nt/LBracket1M/RBracket1Re/LBrace1/CΑpPsi422/RBrace1rexN/EquΑl72\nN/EquΑl80\nFigure 8. Real part of the 22-mode of Ψ 4multiplied by the extraction radius rex\nforq= 4 and initial separation D= 5Mruns. Compared are results for employing\nηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in two different resolutions (red and\ngreen lines) according to the grid configurations described in the te xt.\nnicely as long as ηs(/vector r) is in the right range for each black hole. For this reason, Eq. (4)\nalso seems to work rather nicely for two punctures with unequal ma sses.\nAs we did in the equal mass case in section 3.2, we compare the 22-mod e of Ψ 4\nin the new gauge with the standard gauge. To see how small the differ ences using ηs\norηs(/vector r) are, we plot its real part using two different resolutions which corr espond to\nthe two different colors in Fig. 8. Figure 9 shows the amplitude for the two different\nresolutions. The inset gives the relative differences between amplitu des in the standard\nand new gauge. The maximum relative deviation appears for the lower resolution and\namounts to about 3%. The high resolution gives 0 .5% relative difference. The phases\nin standard and new gauge are compared in Fig. 10. Again, we show on ly the relative\ndeviations as the absolute ones are too small to be seen. We find rela tive differences of\nup to 0.4% for the lower resolution and only 0 .1% for the high one. This confirms that\nwe are changing only the coordinates, as we found before in Section s 3.1 and 3.2.\nWhile the invariance of the waveforms is the most important feature of the newToward a dynamical shift condition for unequal mass black ho le binary simulations 11\n100 110 120 130 140 150 160 1700.0000.0050.0100.0150.0200.0250.0300.035\nt/LBracket1M/RBracket1A22\n80 100 120 140 160/Minus0.020.000.020.04\nt/LBracket1M/RBracket1/CΑpDeltΑ/CΑpAlpΗΑ22/Slash1/CΑpAlpΗΑ22N/EquΑl72\nN/EquΑl80\nFigure 9. Amplitude of the 22-mode of Ψ 4for the same runs as in Fig. 8. We\ncompare results using ηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in two different\nresolutions (red and green lines) according to the grid configuratio ns described in the\ntext. The inset shows the relative deviation ∆ A22/A22between the amplitude in the\nstandard and in the new gauge, again for the same two resolutions.\n80 100 120 140 160/Minus4/Minus3/Minus2/Minus1012\nt/LBracket1M/RBracket1/CΑpDeltΑΦ22/Slash1Φ22/LBracket1/Multiply10/Minus3/RBracket1N/EquΑl72\nN/EquΑl80\nFigure 10. Relative phase difference of the 22-mode of Ψ 4for the same run as in\nFig. 8ηs= 2.0/Mandηs(/vector r). Compared are two different resolutions according to the\ngrid configurations described in the text.\ngaugeηs(/vector r), it is illuminating to examine how the black holes are represented on th e\nnumerical grid. To this end, the apparent horizons (AH) are compu ted for both gauges\nin Fig. 11. We show the result in the ( x,y)–plane, in which the orbital plane lies. For\nclarity, the slices through the apparent horizons are only shown at 4 different times. In\nthe beginning of the simulations, the AH pertaining to the same black h ole are lying\non top of each other. With time, they separate as the coordinates become more and\nmore different in the two simulations. Two observations can be made. First, the ratio\nbetween the coordinate area ofthe AH of thelarger black hole andt he oneof the smaller\nblack hole is larger in the simulation using ηs= 2.0/M. This means the black holes are\nrepresented more equally on the grid in the simulation using Eq. (4). T his fact can be\nseen even more clearly in Fig. 12 where we plot the coordinate area of the apparent\nhorizons comparing the standard gauge (red lines) and the new one (black lines). While\nthe coordinate sizes of the smaller black hole (dashed lines) are near ly equal in both\ngauges, the sizes of the larger black hole (solid lines) differ by roughly 2M2. Second,\nthe shape of the horizon of the smaller black hole is more and more dist orted in the\nηs= 2.0/M-simulation when the black holes come closer together. This deforma tion isToward a dynamical shift condition for unequal mass black ho le binary simulations 12\n-4-2 0 2 4\n-4 -2  0  2  4PSfrag replacements\nx[M]y[M]\nt = 5.7M-4-2 0 2 4\n-4 -2  0  2  4PSfrag replacements\nx[M]y[M]\nt = 32 .8M\n-4-2 0 2 4\n-4 -2  0  2  4PSfrag replacements\nx[M]y[M]\nt = 54 .4M-4-2 0 2 4\n-4 -2  0  2  4PSfrag replacements\nx[M]y[M]\nt = 70 M\nFigure 11. Comparison of apparent horizons in the orbital plane using ηs= 2.0/M\n(red lines) and ηs(/vector r) (black lines) at different times during the evolution for a q= 4\nrun with initial separation D= 5M.\n 0 1 2 3 4 5 6 7 8 9 10\n 0  10  20  30  40  50  60PSfrag replacements\nAAH[M2]\nt [M]0\n1\n2\n3\n4\n5\n6\n7\n8\n9\nFigure 12. Comparison of the coordinate area of the apparent horizons using\nηs= 2.0/M(red lines) and ηs(/vector r) (black lines) over evolution time for a q= 4 run with\ninitial separation D= 5Muntil shortly before a common apparent horizon appears.\nThe dashed lines belong to the smaller black hole whereas the solid lines r epresent the\nlarger black hole.\nnot visible in the new coordinates. The progressive stretching of th e apparent horizon\nshapeandthereforethedistortionofthecoordinatesnear theb lack holescanbeasource\nof instabilities, e.g. [25]. Using Eq. (4) seems to be profitable in this reg ard.\n3.4. Behavior of ηs(/vector r)and influence on the shift vector\nDespite the encouraging results we have seen so far, there is a non –negligible concern\nusing Eq. (4) in the gamma-driver condition (2). Although we do not d etermine\nthe damping coefficient via a wave equation, we see wavy features in ηs(/vector r) traveling\noutwards. These distortions even leave remnants on the grid, esp ecially when they pass\nthrough a refinement boundary. The form of ηs(x) after different evolution times canToward a dynamical shift condition for unequal mass black ho le binary simulations 13\n 0 50 100 150 200 250 300 350 400 450\n800700600500400300200100 0 0 0.5 1 1.5 2 2.5\nPSfrag replacements\nx[M]t[M]ηsηs\nFigure 13. Form of ηs(/vector r) inx-direction at different times during an equal mass\nbinary simulation. Noise travels outwards and leaves strong distort ions on the grid.\n0 100 200 300 400 5000.0000.0020.0040.0060.0080.010\nx,y/LBracket1M/RBracket1Βx,ΒyΗs/LParen1r/OverRVector/RParen1\nΗs/EquΑl2/Slash1M\nFigure 14. x-component of the shift vector in x-direction (solid curves) and y-\ncomponent of the shift vector in y-direction (dashed curves) after the merger of two\nequal mass black holes at time t= 500Mwhen using either the standard gauge\nηs= 2.0/M(gray lines) or the dynamical one, ηs(/vector r), (black lines).\nbe seen in Fig. 13 for the equal mass binary described in Sec. 3.2. The result is similar\nin theq= 4 simulation and even in the Schwarzschild simulation, an outward tra veling\npulse is present, which however does not leave visible distortions on t he grid and the\nrelative amplitude of which decreases for higher mass. The effort we made before in\norder to achieve the correct value of ηs(/vector r) near the outer boundary seem to be canceled\nout by the disturbed shape we find now. As the peaks travel to a re gion of the grid\nwhere we have no punctures, we might take the point of view that th e exact value of\nηs(/vector r) and therefore the distortions are of no importance for our simula tions. Indeed the\noscillations do not translate to oscillations in the shift vector as one m ight think. In the\nshift, we find no gauge “waves” related to the ones in ηs(/vector r). Nevertheless, there is an\nunusual behavior. After merger, when going away from the punct ures the shift does not\nfall off to zero as fast as it does when using ηs= const. but keeps a shoulder (compare\nFig. 14) which might lead to an unphysical and unwanted drift of the c oordinate system.\nWe are planning to investigate these issues in more detail in the futur e.Toward a dynamical shift condition for unequal mass black ho le binary simulations 14\n4. Discussion\nWe presented a new approach to determine the coordinates in slices of spacetime for\nbinary black hole simulations where we take the distribution of mass ov er the grid into\naccount. We have shown that our approach of determining the dam ping parameter in\nthe gamma-driver conditiondynamically via Eq. (4) gives stable evolut ions and doesnot\nsignificantly change the gravitational waves extracted from binar y systems of equal or\nunequal masses. Furthermore, the use of Eq. (4) in an unequal m ass simulation resulted\nin a more regular shape of the apparent horizon of the smaller black h ole as the binary\nmerges. Thecoordinatesizeoftheapparenthorizonsbecamemor euniformwiththenew\ndamping coefficient which is a first step towards representing and re solving black holes\nwith different masses equally and hence removing the large asymmetr y which usually\ndistorts the numerical grid in unequal mass simulations. We found ga uge waves in our\ndamping coefficient which might affect the stability in very long-term sim ulations and\nlead to coordinate drifts after the merger of the binary. We will add ress these issues in\na future publication [26].\nAcknowledgments\nIt is a pleasure to thank Jason Grigsby for discussions and for his va luable comments\non this publication. We also thanks David Hilditch for discussions on the hyperbolicity\nof the BSSN system. This work was supported in part by DFG grant S FB/Transregio 7\n“Gravitational Wave Astronomy” and the DLR (Deutsches Zentru m f¨ ur Luft und\nRaumfahrt). Doreen M¨ uller was additionally supported by the DFG R esearch Training\nGroup 1523 “Quantum and Gravitational Fields”. Computations wer e performed on\nthe HLRB2 at LRZ Munich.\nReferences\n[1] J. Balakrishna, G. Daues, E. Seidel, W.-M. Suen, M. Tobias, and E. Wang. Coordinate conditions\nin three-dimensional numerical relativity. Class. Quantum Grav. , 13:L135–142, 1996.\n[2] Miguel Alcubierre and Bernd Br¨ ugmann. Simple excision of a black h ole in 3+1 numerical\nrelativity. Phys. Rev. D , 63:104006, 2001.\n[3] Miguel Alcubierre, Bernd Br¨ ugmann, Denis Pollney, Edward Seide l, and Ryoji Takahashi. Black\nhole excision for dynamic black holes. Phys. Rev. D , 64:061501(R), 2001.\n[4] Miguel Alcubierre, Bernd Br¨ ugmann, Peter Diener, Michael Kop pitz, Denis Pollney, Edward\nSeidel, and Ryoji Takahashi. 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Dynamic al damping for unequal mass black\nhole binary simulations. 2009. in preparation."    },    {        "title": "0912.5521v1.Spin_torque_and_critical_currents_for_magnetic_vortex_nano_oscillator_in_nanopillars.pdf",        "content": "1 \n  \nSpin torque and critical currents for magnet ic vortex nano-oscillator in nanopillars \n \nKonstantin Y. Guslienko1,2*, Gloria R. Aranda1, and Julian M. Gonzalez1 \n \n1Dpto. Fisica de Materiales, Universidad de l Pais Vasco, 20018 Donostia-San Sebastian, Spain  \n2IKERBASQUE, the Basque Foundation for Science, 48011 Bilbao, Spain \n \n \n   We calculated the main dynamic parameters of the spin polarized current  induced magnetic vortex \noscillations in nanopillars, such as the range of current density, wh ere a vortex steady oscillation state \nexists, the oscillation frequency and orbit radius. We accounted for both the non-linear vortex frequency \nand non-linear vortex damping. To describe the vortex excitations by the spin polarized current we used \na generalized Thiele approach to motion of the vortex  core as a collective coordinate. All the results are \nrepresented via the free layer sizes, saturation magnetiza tion, Gilbert damping and the degree of the spin \npolarization of the fixed layer. Pr edictions of the developed model can be checked experimentally.  \n  \nKey words: spin polarized current, magnetic nanopillar, nano-oscilla tors, magnetic vortex \n \n*Corresponding author. Electronic mail: sckguslk@ehu.es  2 \n      Now excitations of the microwave oscillatio ns in magnetic nanopilla rs, nanocontacts and tunnel \njunctions by spin polarized curren t as well as the current induced domain wall motions in nanowires are \nperspective applications of spintronics.1 A general theoretical approach to microwave generation in \nnanopillars/nanocontacts driven by spin-polarized current based on the universal model of an auto-\noscillator with negative damping and nonlinear frequency shift was de veloped recently by Slavin and \nTiberkevich [see Ref. 2 and references therein]. Th e model was applied to the case of a spin-torque \noscillator (STO) excited in a uniformly magnetized free layer  of nanopillar,  and explains the main \nexperimentally observed effects such as the power and frequency of the gene rated microwave signal. \nHowever, the low generated power ~ 1 nW of such STO prevents their practic al applications. Recently  \nextremely narrow linewidth of 0.3 MHz and relatively high generated power was detected for the \nmagnetic vortex (strongly non-uniform st ate) nano-oscillat ors in nanopillars.3 The considerable \nmicrowave power emission from a vortex STO in magnetic tunnel junctions was observed.4 It was \nestablished that the permanent perpendicular to the plane (CPP) spin polarized current I can excite \nvortex motion in free layer of th e nanopillar if the current intens ity exceeds some critical value, Ic1.5 \nThen, in the interval Ic1 <I <Ic2 (Ic2 is a second critical current) th ere is microwave generation at a \nfrequency which smoothly increase with the current increasing.6 This frequency corresponds to the \nvortex oscillations with a stationary orbit determ ined by the current value. If the current exceeds a \ncritical value Ic27,8 the vortex steady state is not stable anym ore, presumably because the vortex reaches a \ncritical velocity9 and reverses its core. The vortex with op posite core cannot be excited for the given \ncurrent sign I and the oscillations stop. Such  excitation scheme is different  from the current-in-plane \n(CIP) case, where one needs to apply a.c. CIP of about the resonance frequency to excite the vortex \nmotion.10 Low value of Ic1 and high value of Ic2 make the vortex CPP nano-os cillators attractive for \napplications as microwave devices. However, the calcu lations of the spin torque term (ST force) gave \ncontradictory results for the ST force and Ic1. The standard STO approach of Ref. 2 is not applicable to 3 \n the vortex dynamics due to specific damping term. The problem was reduced to  the problem of vortex \ncore reversal in the perpe ndicular to the layer plane  magnetic field in Ref. 7 but vortex steady state \ndynamics was not accounted. Using the Thiele approach  the ST force was calculated in Refs. 5, 8 which \ndiffers in 2 times from one calculate d form the energy dissipation balance.6 Non-linearity of the main \ngoverning parameters and the Oersted field of curre nt were not accounted or accounted incorrectly. The \ncritical current Ic2 has not yet been calculated.  \n     In this Letter we present a simple and effectiv e approach to calculate the ST force and the critical \ncurrents of the vortex STO in nanopillar. The appro ach is based on the Thiele formulation of the non-\nuniform magnetization dynamics11 and conception of the linear spin excitations.12 The nanopillar device \nconsists of two ferromagnetic layers , typically FeNi or Co and a nonma gnetic metallic spacer, typically \nCu, arranged in a vertical stack (Fig . 1). Magnetization of one layer is fixed (this layer is the so called \npolarizer), whereas the magnetization of  the second layer of the nanopillar ()t,rM  is free to rotate. The \ncurrent of spin polarized elec trons transfers some torque sτ from the polarizer, which excites \nmagnetization dynamics of the free layer. We st art from the Landau-Lifshi tz equation of motion \ns LLG eff τ mm Hm m γ α γ +× + × −= \u0005 \u0005 , where sM/Mm= , Mss  is the saturation magnetization, γ>0 is the \ngyromagnetic ratio,  Heff is the effective field, and LLGα  is the Gilbert damping. We use the ST term in \nthe form suggested by Slonczewski,13 () Pm mτ × × =Jsσ , where ()sLMe2/η σ== , η is the current \nspin polarization ( η=0.2 for FeNi), e is the electron charge, L is the free layer (dot) thickness, J is the \ncurrent density, and z PP=  is the unit vector of the polarizer magnetization ( P=+1/-1). We assume the \npositive vortex core polarization p=+1, P=+1 and define the current (flow of the positive charges) as \npositive I>0 when it flows from the polarizer to free layer.  The spin polarized curr ent can excite a vortex \nmotion in the free layer if only IpP > 0 (only the electrons bringing  a magnetic moment from the \npolarizer to free layer opposite to the core polarization can excite a vortex motion). Except p, the vortex 4 \n is described by its core position in the free layer, X=(X,Y), and chirality C=±1 .14 Let denote the \nSlonczewski´s energy density which correspond s to the spin polarized current as sw. Then, using the \nThiele approach and the ST field Pm m × =∂ ∂ a ws/ , the ST force acting on th e vortex in the free layer \ncan be written as \n                                          ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂× ⋅ =∂∂−= ∫ ∫\nα αα\nXd aL dVwXFs STmmρ P2,                                             (1) \nwhere J Masσ = , α=x, y, ()ϕρ, =ρ  is the in-plane radius vector, the derivative is taken with respect \nto the vortex core position X assuming an ansatz () ( ) [] t t Xρmρm , ,=  (m dependence on thickness \ncoordinate z is neglected). X has sense of the amplitude of the vortex gyrotropic eigenmode. \n     We use representation of m-components by the spherical angles ΦΘ, (Fig. 1) as  \n) cos, sin sin, cos (sin Θ Φ Θ Φ Θ =m  and find the expression for the ST force \n                                                          Xρ F∂Φ∂Θ =∫2 2sindaLST .                                                            (2) \n     In the main approximation we use the decompositions ( ) () () ( ) ρX Xρ ˆ cos ,0⋅ + =Θ = ρ ρ g m mz z , \n() ( ) [] ϕ ϕ ρ cos sin ,0 0 Y X m − +Φ= Φ Xρ , where ()ρ0\nzm , 0Φ are the static vortex core profile and phase, \n() () ( )22 2 2 2/ 1 4 ρ ρ ρ ρ + + = c pc g  is the excitation amplitude of the z-component of the vortex \nmagnetization ( RRcc/ = , cR is the vortex core radius, ρ, X are normalized to the dot radius R) and \n()() ρ ρ ρ / 12\n0 −= m  is the gyrotropic mode profile.12 One can conclude from Eq. (2) that only moving \nvortex core contribute to  the ST force because the contribu tion of the main dot area where 2/π=Θ  is \nequal to zero due to vanishing integrals on azimuthal angle φ from the gradient of the vortex phase Φ∂X 5 \n (it was checked accounting in ()Xρ,Φ  the terms up to cubic terms in X α-components). This is a reason \nwhy the ST contribution is relatively small bei ng comparable with the damping contribution.  \n     The integration in Eq. (2) yields the ST force ()Xz F × = ˆaLSTπ . This force contributes to the Thiele’s \nequation of motion ST D W FX XGX + + −∂=× \u0005 \u0005 ˆ , where γ π / 2ˆs pLMzG=  is the gyrovector, Dˆ is the \ndamping tensor. The vortex energy ()XW  and restoring force WR X F −∂=  can be calculated from an \nappropriate model14 (the force balance is shown in Fig. 2). For circular steady st ate vortex core motion \nthe XωX ×=\u0005  relation holds, which allows calculating Jc1. To calculate the vortex steady orbit radius \nX=sR  we need, however, to account non-linear on X α terms in the vortex damping and frequency (the \naccount only non-linear frequency as in Ref. 5 is not sufficient). The gyr ovector also depends on X, but \nthis dependence is essential only for the vortex core p reversal, where G changes its sign.  As we show \nbelow, the most important non-linearity co mes from the damping tensor defined as \n                                                   ()\nβ ααβγαX XdVMDs\nLLG∂∂⋅∂∂−=∫m mX ,                                                  (3)  \nor () [] Φ∂Φ∂Θ +Θ∂Θ∂ −= ∫ β α β α αβ γ α2 2sin /ρd LM Ds LLG  in ΦΘ, -representation Accounting \nαβ αβ δD D=  and introducing dimensionless damping parameter 0 /> −= GD d15 we can write the \nequation for a steady state vortex motion with the orbit radius X=sR : () ()ϕωST G F sRs Gsd =  from \nwhich RRss/ =  and the critical currents Jc1, Jc2 can be found. In the second order non-linear \napproximation ()2\n1 0 sd dsd + = , ()2\n1 0 s sG ω ω ω + =  and aLRs FSTπϕ= , whereGω is the vortex \nprecession frequency, 01>ω  is a function of the dot aspect ratio β=L/R  calculated from the vortex \nenergy decomposition ()sW  i n  s e r i e s  o f  R s /X= . It can be shown that () ( ) [] 3/41 9/200 β βγ βω − =sM  \nand () () βω βω0 1 4≈  for quite wide range of β= 0.01-0.2 of practi cal interest, whereas considerably larger 6 \n non-linearity () () 8.42 /0 1 =βωβω  was calculated in Ref. 5 due to in correct account of the magnetostatic \nenergy. We use the pole free model of the shifted vortex ()[] tXρm, , where the dynamic magnetization \nsatisfies the strong pinning boundary condition at the dot circumference16 R=ρ . The damping \nparameters are () () 2/ / ln8/50 c LLG RR d + =α , () 4/3/8 /2 2\n1 − =c LLG RR d α . We need also to account for \nthe Oersted field of the current, wh ich leads to contribution to the vo rtex frequency proportional to the \ncurrent density () ( ) J Jeω βω βω ω + = =0 0 0 , , where () ( ) CcRe ξ γ π ω / 15/8= , () R Rc8/ 2/12ln151 − +=ξ  \nis the correction for the finite core radius cR<<R. In the linear approximation we get the equality of the \ndamping force and the ST force (negative damp ing) as the condition to find the value of Jc1, with the \ncontribution of the Oersted field of the current accounted. The values of FST, Jc1 coincide with \ncalculation of Ref. 6 conducted by the method of the damping energy balance and differ by the \nmultiplayer 2 from Refs. 5, 8. The first critical current is ()e c d d J ω γσω0 00 1 2/ / − =  and the steady \nstate vortex orbit radius is \n \n                       () 1 /1− =cJJ Js λ ,     1cJJ> ,     ()[]10 1 011 2\n21\nω ωγσλd J dJ\ncc\n+=                      (4) \n     In this approximation the vortex trajectory radius ()Js  increases as square root of the current \novercriticality ()1 1/c cJ JJ−  (for the typical parameters and R=80-120 nm we get λ=0.25-0.30) and the \nvortex frequency () ( )1 12\n0 1 / ω λ ω βω ω − + + =c e G JJ J  increases linearly with J increasing. The vortex \nsteady orbit can exist until the m oving vortex crosses the dot border s=1 or its velocity X\u0005 reaches the \ncritical velocity cυ defined in Ref. 9. The later allows to write equation for the s econd critical current Jc2 \nas () ()c G RJsJ υ ω =. Substituting to this expression the equations for ()JGω  and ()Js  derived above 7 \n we get a cubic equation for Jc2 in the form () [ ] R xx J d Jc ce c λυ λω ω γσ / 2/2/1 2\n1 1 0 1 = + + , \n() 1 /1 2 − =c cJ J x . This equation has one positive root xc and the value of Jc2 can be easily calculated \n(Fig. 3). The former condition ( s=1) gives the second critical current ()12\n2 /11c c J J λ+=′ . More detailed \nanalysis shows that both the mechanisms of the hi gh current instability of the vortex motion are possible \ndepending on the dot sizes L, R, and the critical current is th e lower value of the currents Jc2, J’c2. The \nvortex core reversal inside the dot occurs for large enough R (> 100 nm) and L. For the typical sizes \nL=10 nm, R=120 nm and C=1, the critical currents are Jc1=6.3 106 A/cm2 (Ic1=2.9 mA), Jc2=1.13 108 \nA/cm2 (Ic2=51 mA), and for L= 5 nm, R=100 nm we get Jc1=1.8 106 A/cm2 (Ic1=0.56 mA), J’c2=2.7 107 \nA/cm2 (I’c2=8.4 mA). \n   In summary, we calculated the main physical para meters of the spin polar ized CPP current induced \nvortex oscillations in na nopillars, such as the cr itical current densities Jc1, Jc2, the vortex steady state \noscillations frequency and orbit radius. All the results are represented via the free layer sizes ( L, R), \nsaturation magnetization, Gilbert damp ing and the degree of the spin polarization of the fixed layer. \nThese parameters can be obtained from independent e xperiments. We demonstrated that the generalized \nThiele approach is applicable to the problem of the vortex STO excitations by the CPP spin polarized \ncurrent. The spin transfer torque force is related to the vortex core only. \n     The authors thank J. Grollier and A.K. Khvalkovsk iy for fruitful discussions. K.G. and G.R.A. \nacknowledge support by IKERBASQUE  (the Basque Science Foundation) and by the Program JAE-doc \nof the CSIC (Spain), respectively. The author s thank UPV/EHU (SGIker Arina) and DIPC for \ncomputation tools. The work was part ially supported by the SAIOTEK grant  S-PC09UN03. \n \n 8 \n References  \n1 G. Tatara, H. Kohno, and J. Shibata, Phys. Rep . 468, 213 (2008). \n2 A. Slavin and V. Tiberkevich, IEEE Trans. Magn.  45, 1875 (2009). \n3 V.S. Pribiag, I.N. Krivorotov, G.D. Fuchs et al., Nature Phys . 3, 498 (2007). \n4 A. Dussaux, B. Georges, J. Grollier et al. , submitted to Nature Phys.  (2009). \n5 B. A. Ivanov an d C. E. Zaspel, Phys. Rev. Lett.  99, 247208 (2007). \n6 A.V. Khvalkovskiy, J. Grollier, A. Dussaux, K.A. Zvezdin, and V. Cros, Phys. Rev . B 80, 140401 \n(2009).  \n7 J.-G. Caputo, Y. Gaididei, F.G. Mertens and D.D. Sheka, Phys. Rev. Lett.  98, 056604 (2007); D.D.  \n  Sheka, Y. Gaididei, and F.G. Mertens, Appl. Phys. Lett.  91, 082509 (2007). \n8 Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett.  91, 242501 (2007). \n9 K.Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett . 100, 027203 (2008); K.-S. Lee et al., Phys. \nRev. Lett.  101, 267206 (2008). \n10 S. Kasai, Y. Nakatani, K. Koba yashi, H. Kohno, and T. Ono, Phys. Rev. Lett.  97, 107204 (2006);  \n   K. Yamada, S. Kasai, Y. Naka tani, K. Kobayashi, and T. Ono, Appl. Phys. Lett.  93, 152502 (2008). \n11 A. A. Thiele, Phys. Rev. Lett . 30, 230 (1973).  \n12 K.Y. Guslienko, A.N. Slavin, V. Tiberkevich, S. Kim, Phys. Rev. Lett.  101, 247203 (2008).  \n13 J. Slonczewski, J. Magn. Magn. Mat . 159, L1 (1996); J. Magn. Magn. Mat . 247, 324 (2002).  \n14 K.Y. Guslienko, J. Nanosci. Nanotechn.  8, 2745 (2008). \n15 K.Y. Guslienko, Appl. Phys. Lett.  89, 022510 (2006).  \n16 K. Y. Guslienko et al., J. Appl. Phys . 91, 8037 (2002); V. Novosad et al. , Phys. Rev . B 72, 024455 \n(2005). 9 \n  \nCaptions to the Figures \n \nFig. 1. Sketch of the magnetic nan opillar with the coordinate system used. The upper (free) layer is in \nthe vortex state with non- uniform magnetization distribution. The polarizer layer (red color) is in \nuniform magnetization state w ith the magnetization along Oz axis. The positive current I (vertical arrow) \nflows from the polarizer to free layer.  \n Fig. 2. Top view of the free laye r with the moving vortex. The arrows  denote the force balance for the \nvortex core. The spin torque (\nFST), damping ( FD), restoring (RF) and gyro- ( FG) forces are defined in the \ntext. The vortex core steady trajectory Rs is marked by orange color. The vortex chirality is C=+1. \n \nFig. 3. Dependence of the critical currents Jc1 (solid red line), Jc2 (dashed green line) and J’c2 of the \nvortex motion instability on the radius R of the free layer. L= 10 nm, Ms =800 G, η =0.2, 01 .0=LLGα , \nγ/2 =2.95 MHz/Oe, Rc=12 nm.  The vortex STO motion is stable at Jc1 < J < min( Jc2, J’c2). \n \n \n     10 \n  \n  Fig. 1. \n \n  \n \n \n     \n \n     11 \n  \n  Fig. 2. \n \n \n \n \n   \n \n      \n 12 \n  \n  Fig. 3. \n \n \n60 80 100 120 14056789101112\nJc2\nJ'c2Current density, J (107 A/cm2)\nDot radius, R (nm)FeNi\nMs=800 G\nL=10 nm\nJc1x10\n "    },    {        "title": "1001.2845v1.Resonance_Damping_in_Ferromagnets_and_Ferroelectrics.pdf",        "content": "arXiv:1001.2845v1  [cond-mat.other]  16 Jan 2010Resonance Damping in Ferromagnets and Ferroelectrics\nA. Widom\nPhysics Department, Northeastern University, Boston, MA U SA\nS. Sivasubramanian\nNSF Nanoscale Science & Engineering Center for High-rate Na nomanufacturing,\nNortheastern University, Boston MA USA\nC. Vittoria and S. Yoon\nDepartment of Electrical and Computer Engineering, Northe astern University, Boston, MA USA\nY.N. Srivastava\nPhysics Department and & INFN, University of Perugia, Perug ia IT\nThe phenomenological equations of motion for the relaxatio n of ordered phases of magnetized\nand polarized crystal phases can be developed in close analo gy with one another. For the case of\nmagnetized systems, thedrivingmagnetic fieldintensityto ward relaxation was developedbyGilbert.\nFor the case of polarized systems, the driving electric field intensity toward relaxation was developed\nby Khalatnikov. The transport times for relaxation into the rmal equilibrium can be attributed to\nviscous sound wave damping via magnetostriction for the mag netic case and electrostriction for the\npolarization case.\nPACS numbers: 76.50.+g, 75.30.Sg\nI. INTRODUCTION\nIt has long been of interest to understand the close\nanalogiesbetween orderedelectric polarized systems, e.g.\nferroelectricity , and ordered magnetic systems, e.g. fer-\nromagnetism . Atthe microscopiclevel, the sourceofsuch\nordering must depend on the nature of the electronic en-\nergy spectra. The relaxation mechanism into thermal\nequilibrium state must be described by local electric field\nfluctuationsforthe electricpolarizationcaseandbymag-\nnetic intensity fluctuations for the magnetization case;\nSpecifically, the field fluctuations for each case\nGpol\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Ej(r′,−iλ)∆Ei(r,t)/an}bracketri}htdλ,\nGmag\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Hj(r′,−iλ)∆Hi(r,t)/an}bracketri}htdλ,\nwherein β=¯h\nkBT,(1)\ndetermine the relaxation time tensor for both cases via\nthe fluctuation-dissipation formula1–4\nτij=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVGij(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(2)\nWe have unified the theories of relaxation in ordered po-\nlarized systems and ordered magnetized systems via the\nKubo transport time tensor in Eqs.(1) and (2).\nThe transport describing the relaxation of or-\ndered magnetization is the Landau-Lifshitz-Gilbert\nequation5–7. This equation has been of considerable\nrecent interest8–10in describing ordered magnetic reso-\nnancephenomena11–14. The equationdescribingthe elec-tric relaxation of an ordered polarization is the Landau-\nKhalatinikov-Tani equation15–17. This equation can be\nsimplymodeled18–21witheffectiveelectricalcircuits22–25.\nInformation memory applications26–29of such polarized\nsystem are of considerable recent interest30–32.\nThe unification of the magnetic Gilbert-Landau-\nLifshitz equations and the electric Landau-Khalatnikov-\nTani equations via the relaxation time tensor depends\non the notion of a nonequilibrium driving field . For the\nmagnetic case, the driving magnetic intensity Hddeter-\nmines the relaxation of the magnetization via the torque\nequation\n˙M=γM×Hd, (3)\nwhereinγisthegyromagneticratio. Fortheelectriccase,\nthe driving electric field Eddetermines the relaxation of\nthe polarization via the equation of motion for an ion of\nchargeze\nm¨r=zeEd. (4)\nThe unification of both forms of relaxation lies in the\nclose analogy between the magnetic driving intensity Hd\nand the electric driving field Ed.\nIn Sec.II the thermodynamics of ordered magnetized\nand polarized systems is reviewed. The notions of mag-\nnetostrictionand electrostrictionaregiven aprecise ther-\nmodynamic definition. In Sec.III, the phenomenology of\nthe relaxation equations are presented. The magnetic\ndriving intensity Hdand the electric driving field Edare\ndefined in terms of the relaxation time tensor Eq.(2). In\nSec.IV, we introduce the crystal viscosity tensor. From a\nKubo formula viewpoint, the stress fluctuation correlax-2\nation\nFijkl(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆σkl(r′,−iλ)∆σij(r,t)/an}bracketri}htdλ,(5)\ndetermines the crystal viscosity\nηijkl=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVFijkl(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(6)\nFor models of magnetic relaxation wherein acoustic heat-\ning dominates via magnetostriction33and for models of\nelectric relaxation wherein acoustic heating dominates\nvia electrostriction, the relaxation time tensor in Eq.(2)\ncan be related to the viscosity tensor Eq.(6). An inde-\npendent microscopic derivation of viscosity induced re-\nlaxationisgiveninAppendixA. IntheconcludingSec.V,\nthe sound wave absorption physics of the viscous damp-\ning mechanism will be noted.\nII. THERMODYNAMICS\nOur purpose is to review the thermodynamic proper-\nties of both magnetically ordered crystals and polariza-\ntion ordered crystals. The former is characterized by\na remnant magnetization Mfor vanishing applied mag-\nnetic intensity H→0 while the latter is characterized by\na remnant polarization Pfor vanishing applied electric\nfieldE→0.\nA. Magnetically Ordered Crystals\nLetwbe the enthalpy per unit volume. The funda-\nmental thermodynamic law determining the equations of\nstate for magnetically ordered crystals is given by\ndw=Tds+H·dM−e:dσ, (7)\nwhereinsistheentropyperunitvolume, Tisthetemper-\nature,eis the crystal strain and σis the crystal stress.\nThe magnetic adiabatic susceptibility is defined by\nχ=/parenleftbigg∂M\n∂H/parenrightbigg\ns,σ. (8)\nIf\nN=M\nM⇒N·N= 1 (9)\ndenotes a unit vector in the direction of the magnetiza-\ntion, then the tensor Λ ijkldescribing adiabatic magne-\ntostriction coefficients may be defined as34\n2ΛijklNl=M/parenleftbigg∂eij\n∂Mk/parenrightbigg\ns,σ=−M/parenleftbigg∂Hk\n∂σij/parenrightbigg\ns,M.(10)When the system is out of thermal equilibrium, the driv-\ning magnetic intensity is\nHd=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−τ·/parenleftbigg∂M\n∂t/parenrightbigg\n,(11)\nwherein τare the relaxation time tensor transport co-\nefficients which determine the relaxation of the ordered\nmagnetic system into a state of thermal equilibrium.\nB. Ordered Polarized Crystals\nThe fundamental thermodynamic law determining the\nequations of state for ordered polarized crystals is given\nby\ndw=Tds+E·dP−e:dσ, (12)\nwherein wis the enthalpy per unit volume, sis the en-\ntropy per unit volume, Tis the temperature, eis the\ncrystal strain and σis the crystal stress. The electric\nadiabatic susceptibility is defined by\nχ=/parenleftbigg∂P\n∂E/parenrightbigg\ns,σ. (13)\nThe tensor βijkdescribing adiabatic electrostriction co-\nefficients may be defined as34\nβijk=/parenleftbigg∂eij\n∂Pk/parenrightbigg\ns,σ=−/parenleftbigg∂Ek\n∂σij/parenrightbigg\ns,P.(14)\nThe piezoelectric tensor is closely related to the elec-\ntrostriction tensor via\nγijk=/parenleftbigg∂eij\n∂Ek/parenrightbigg\ns,σ=/parenleftbigg∂Pk\n∂σij/parenrightbigg\ns,E=βijmχmk.(15)\nWhen the system is out of thermal equilibrium, the driv-\ning electric field is\nEd=E−/parenleftbigg∂w\n∂P/parenrightbigg\ns,σ−τ·/parenleftbigg∂P\n∂t/parenrightbigg\n,(16)\nwherein τis the relaxation time tensor transport coef-\nficients which determine the relaxation of the ordered\npolarized system into a state of thermal equilibrium.\nIII. RESONANCE DYNAMICS\nHere we shall show how the magnetic intensity Hd\ndrives the magnetic resonance equations of motion in\nmagnetically ordered systems. Similarly, we shall show\nhowtheelectricfield Eddrivesthe polarizationresonance\nequations of motion for polarized ordered systems.3\nA. Gilbert-Landau-Lifshitz Equations\nThe driving magnetic intensity determines the torque\non the magnetic moments according to\n∂M\n∂t=γM×Hd. (17)\nEmploying Eqs.(11) and (17), one finds the equations for\nmagnetic resonance in the Gilbert form\n∂M\n∂t=γM×/bracketleftBigg\nH−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−/parenleftbiggα\nγM/parenrightbigg\n·∂M\n∂t/bracketrightBigg\n,(18)\nwherein the Gilbert dimensionless damping tensor αis\ndefined as\nα= (γM)τ. (19)\nOnemaydirectlysolvetheGilbert equationsforthe driv-\ning magnetic intensity according to\nHd+α·/parenleftbig\nN×Hd/parenrightbig\n=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ.(20)\nEqs.(17) and (20) expressthe magneticresonancemotion\nin the Landau-Lifshitz form.\nB. Landau-Khalatnikov-Tani Equations\nThe driving electric field gives rise to a polarization\nresponse according to\n∂2P\n∂t2=/parenleftBigg\nω2\np\n4π/parenrightBigg\nEd, (21)\nwhereinωpis the plasma frequency. A simple derivation\nof Eq.(21) may be formulated as follows. In a large vol-\numeV, the polarization due to charges {zje}is given\nby\nP=/parenleftbigg/summationtext\njzjerj\nV/parenrightbigg\n. (22)\nIf the drivingelectric field acceleratesthe chargesaccord-\ning to\nmj¨rj=zjeEd, (23)\nthen Eq.(21) holds true with the plasma frequency\nω2\np= 4πe2lim\nV→∞/bracketleftBigg/summationtext\nj(z2\nj/mj)\nV/bracketrightBigg\n= 4πe2/summationdisplay\nanaz2\na\nma,(24)\nwhereinnais the density of charged particles of type a.\nThe polarization resonance equation of motion follows\nfrom Eqs.(16) and (21) as17\n/parenleftbigg4π\nω2p/parenrightbigg∂2P\n∂t2+τ·∂P\n∂t+∂w(P,s,σ)\n∂P=E.(25)The electric field Einduces the polarization Pat reso-\nnant frequencies which are eigenvalues of the tensor Ω\nfor which\nΩ2=ω2\npχ−1\n4π≡ω2\np(ǫ−1)−1. (26)\nThedecayratesforthepolarizationoscillationsareeigen-\nvalues of the tensor Γfor which\nΓ=ω2\npτ\n4π. (27)\nIfthedecayratesarelargeonthescaleofthetheresonant\nfrequencies, then the equation of motion is over damped\nso that\nmin\njΓj≫max\niΩiimplies\nτ·∂P\n∂t+∂w(P,s,σ)\n∂P=E. (28)\nEq.(28) represents the Landau-Khalatnikov equation for\npolarized systems.\nIV. HEATING RATE PER UNIT VOLUME\nLet us here consider the heating rate implicit in relax-\nation processes. Independently of the details of the mi-\ncroscopic mechanism for generating such heat, the rates\nof energy dissipation are entirely determined byτ. Ex-\nplicitly, the heating rates per unit volume for magnetiza-\ntion and polarization are given, respectively, by\n˙qM=∂M\n∂t·τ·∂M\n∂t, (29)\nand\n˙qP=∂P\n∂t·τ·∂P\n∂t. (30)\nFinally, the notion of crystal viscosity ηijklis introduced\ninto elasticity theory35via the heating rate per unit vol-\nume from rates of change in the strain ∂e/∂t; It is\n˙qe=∂eij\n∂tηijkl∂ekl\n∂t. (31)\nCrystal viscosity is employed to describe, among other\nthings, sound wave attenuation. Our purpose is to de-\nscribe how heating rates in Eqs.(29) and (30) can be re-\nlated to the heating rate in Eq.((31)). This allows us to\nexpressthetransportcoefficients τintermsofthecrystal\nviscosity.\nA. Relaxation via Magnetostriction\nFrom the magnetostriction Eq.(10), it follows that\nmagnetic relaxation gives rise to a strain\n∂eij\n∂t=2\nMΛijklNk∂Ml\n∂t, (32)4\nand thereby to the heating rate,\n˙q=4\nM2∂Mi\n∂t(ΛmnqiNq)ηmnrs(ΛrskjNk)∂Mj\n∂t,(33)\nin virtue of Eq.(31). Employing Eqs.(29) and (33), we\nfind that the magnetic relaxation transport coefficient in\nthe magnetostriction model\nτij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk).(34)\nThe Gilbert damping tensor follows from Eqs.(19) and\n(34) as\nαij=4γ\nM(ΛmnqiNq)ηmnrs(ΛrskjNk).(35)\nThe central relaxation tensor Eq.(35) describes the mag-\nnetic relaxation in terms of the magnetostriction coeffi-\ncients and the crystal viscosity.\nB. Relaxation via Electrostriction\nFromtheelectrostrictionEq.(14), it followsthatatime\nvarying polarization gives rise to a time varying strain\n∂eij\n∂t=βijk∂Pk\n∂t, (36)\nand thereby to the heating rate,\n˙q=∂Pi\n∂tβkliηklmnβmnj∂Pj\n∂t, (37)\nin virtue of Eq.(31). Employing Eqs.(30) and (37), we\nfind that the electric relaxation transport coefficient in\nthe electroostriction model\nτij=βkliηklmnβmnj. (38)\nThecentralrelaxationtensorEq.(38) describesthe polar-\nization relaxation time tensor coefficients in terms of the\nelectrostriction coefficients and the crystal viscosity. The\nimplications ofthe electrostrictionmodel forthe Landau-\nKhalatnikov equation is to the authors knowledge a new\nresult.\nV. CONCLUSIONS\nFor ordered polarized and magnetized systems, we\nhave developed phenomenological equations of motion inclose analogywith one another. For the magnetized case,\nthe relaxation is driven by the magnetic intensity Hd\nyielding the Gilbert equation of motion7. For the polar-\nized case, the relaxation is driven by the electric field Ed\nyielding the Tani equation of motion17. In both cases,\nthe relaxation time tensor τis determined by the crystal\nviscosity as derived in the Appendix A; i.e. in Eqs.(A3)\nand (A6). The viscosity can be measured independently\nfrom the magnetic or electrical relaxation by employing\nsound absorption techniques36.\nAppendix A: Kubo formulae\nFrom the thermodynamic Eq.(10), the fluctuations in\nthe magnetic intensity are given by magnetostriction, i.e.\n∆Hk(r,t) =−/parenleftbigg2ΛijklNl\nM/parenrightbigg\n∆σij(r,t).(A1)\nEqs.(A1), (1) and (5) imply\nGmag\nij(r,r′,t) =\n4\nM2(ΛmnqiNq)Fmnrs(r,r′,t)(ΛrskjNk).(A2)\nEmploying Eqs.(A2), (2) and (6), one finds the central\nresult for the magnetic relaxation time tensor; It is\nτmag\nij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk) =αij\nγM.(A3)\nFrom the thermodynamic Eq.(14), the fluctuations in\nthe electric intensity are given by electrostriction, i.e.\n∆Ek(r,t) =−βijk∆σij(r,t). (A4)\nEqs.(A4), (1) and (5) imply\nGpol\nij(r,r′,t) =βkliFklmn(r,r′,t)βmnj.(A5)\nEmploying Eqs.(A5), (2) and (6), one finds the central\nresult for the electric relaxation time tensor; It is\nτpol\nij=βkliηklmnβmnj. (A6)\n1S. Machlup and L. Onsager, Phys. Rev. 91, 1505 (1953).\n2S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953).\n3H.B. Callen, Fluctuation, Relaxation, and Resonance inMagnetic Systems , Editor D. ter Haar, p 176, Oliver &\nBoyd Ltd., London (1962).\n4R. Kubo, M. Toda and N. Hashitsume, Statistical Physics5\nII, Nonequilibrium Statistical Mechanics Springer, Berlin\n(1998).\n5L. Landau and L. Lifshitz, Phys. Zeit. Sowjetunion 8,153\n(1935).\n6T. L. Gilbert, Armor Research Foundation Rep. No. 11\nChicago, IL. (1955).\n7T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004).\n8M. Fahnle, D. Steiauf, and J. Seib, J. Phys. D: Appl. Phys.\n41, 164014 (2008).\n9E. Rossi, O.G. Heinonen, andA.H. MacDonald, Phys. Rev.\nB 72, 174412 (2005).\n10V. Kambersky, Phys. Rev. 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B: Microelectronics and Nanometer Structures 27,\n498 (2009).\n31N. Inoue and Y. Hayashi, IEEE Trans. on Elect. Dev. 48,\n2266 (2001).\n32T. Mikolajick, C. Dehm, W. Hartner, I. Kasko, M. Kast-\nner, N. Nagel, M. Moert and C. Mazure, Microelectronics\nReliability 41, 947 (2001).\n33C. Vittoria, S. D. Yoon and A. Widom, Phys. Rev. B 81,\n014412 (2010).\n34L.D. Landau and E.M. Lifshitz, “ Electrodynamics of Con-\ntinuos Media ”, Pergamon Press, Oxford (1960).\n35L.D. Landau and E.M. Lifshitz, “ Theory of Elasticity ”,\nSecs.10, 23 and 34, Pergamon Press, Oxford (1986).\n36A.B. Bhatia, Ultrasonic Absorption , Oxford University\nPress, Oxford (1967)."    },    {        "title": "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": "1001.4578v1.Strategies_and_tolerances_of_spin_transfer_torque_switching.pdf",        "content": " Page1\nStrategies and tolerances of spin transfer torque \nswitching \nDmitri E. Nikonov, George I. Bourianoff \nIntel Corp., Components Research, Santa Clara, California, 95052 \nGraham Rowlands, Ilya N. Krivorotov \nDepartment of Physics and Astronomy, Univers ity of California–Irvine, Irvine, California \n92697, USA \ndmitri.e.nikonov@intel.com \nAbstract \nSchemes of switching nanomagnetic memories via the effect of spin torque with various \npolarizations of injected electrons are studied. Simulations based on macrospin and \nmicromagnetic theories are performed and compared. We demonstrate that switching \nwith perpendicularly polarized current by short pulses and free precession requires \nsmaller time and energy than spin torque switching with collinear in plane spin \npolarization; it is also found to be superior to other ki nds of memories. We study the \ntolerances of switching to the magnitude of  current and pulse duration. An increased \nGilbert damping is found to improve tolera nces of perpendicular switching without \nincreasing the threshold current , unlike in plane switching.  Page2\n \n1. Introduction \nResearch in spintronics resulted in huge  technological impact  via development of \nextremely high capacity hard  drives and magnetic RAM1. The basic paradigm of such \ndevices is a stack of two ferromagnetic (FM)  layers separated by a non-magnetic layer, \nsee Fig. 1a. Giant magnetoresistance2,3 (GMR) – dependence of resi stance of such a stack \non the relative direction of magne tization in the FM layers – provides the crucial path to \ninterfacing magnetic and electroni c states in the device. If the non-magnetic layer is a \ndielectric, this effect is calle d tunneling magnetoresistance (TMR)4,5,6,7. \nA better way of switching ma gnetization in such devices  was first theoretically \npredicted8,9 followed by the demonstration10,11,12,13,14 of spin transfer torque effect. This \neffect consists of precession of magnetization of one of the FM layers as current flows \nacross the stack. As this happens, the angul ar momentum of spin-polarized current \noriginating in one FM layer is transferred to  the magnetization of another FM layer, see \nFig 1a. Since this original work, a tremendous  amount of research has been conducted in \nthe field (see review15). Good values of the performance metrics have been achieved: the \nmemory cell size has been reduced to a few square microns, the switching time to a few \nnanoseconds, and the switching current to a few milliamps. Spin transfer torque random \naccess memory (STTRAM) has been prototyped16 and is close to commercialization. \nStill to be competitive with the incumbent memory technologies, such as SRAM, DRAM, \nand flash, STTRAM has to surpass them in the majority of a set of metrics (size, speed,  Page3\nenergy). The fact that STTRAM is non-volatile is an important advantage, but would not \nalone ensure commercial success. For this r eason it is necessary to devise ways to \ndecrease STTRAM’s threshold switching curren t and time, and thus the overall switching \nenergy. One of the possible pathways to th is end is conducting sw itching by short pulses \nrather than quasi-steady currents. \nA previous theoretical treatment17 of pulsed currents envisioned two pulses of opposite \npolarity without a gap between them. Experi mental demonstration of switching with \npulses was performed with single pulses 18,19 or double pulses 20 of constant polarity.   \nIn the present work we attempt to give a comprehensive view of the pulsed switching of \nnanomagnetic memories. We consider a variet y of cases of spin polarization of the \ncurrent injected from the fixed FM layer to  the free FM layer. In considering various \ndynamical switching strategies, we first define  the energy landscape of the system and \nrepresent a specific switching strategy as a specific path across th e topological surface \nwhich defines the energy landscape   The anal ysis contained here identifies the optimal \nswitching strategy to be along the path of steep est accent (unlike the tr aditional, in-plane \ncollinear polarization switching which pro ceeds along the path of minimum accent). We \nperform simulations both in the macrospi n and micromagnetic approximations and \ncompare them side by side in order to make conclusions on applicabil ity of these methods \nin each of the cases. In these simulations we se parate the contribution of spin transfer and \nfield-like torques 21,22,23 to draw a conclusion on how they  affect switching in each of the \ncases. Finally we perform multiple simulations over a range of switching parameters in \norder to predict the tolerances of the memori es relative to the va riations in both the  Page4\nmagnitude of current and the pulse duration. Su ch variations of curre nt and pulse duration \nare bound to exist in elec tronic circuits due to fabricati on variability and the temperature \ndrift. Counter-intuitively, we find that Gilbert damping is be neficial for the tolerance of \nswitching and increases neither the thres hold current nor the switching duration.  \nThe paper is organized as follows. Section 2 co ntains the description of the mathematical \nmodels of the macrospin approximation and micromagnetic simulator OOMMF used in \nthis paper. Section 3 analyzes the energy dependence on the direction of magnetization \nand various strategies for switching based on it. Time dependence of magnetization in \nvarious strategies of switching is also ex emplified. Results of multiple simulation runs, \npresented in Section 4, establish tolerances of  switching relative to current magnitude and \npulse duration. In Section 5 we compare th e results of macrospin and micromagnetic \nsimulations side by side. The conclusions of this work ar e summarized in Section 6.  \n2. Mathematical models \nThe mathematical model of macrospin dynami cs is based on the Landau-Lifshitz-Gilbert \n(LLG) equation (see the review24). In addition it assumes that spatial variation of \nmagnetization can be neglected, and the w hole magnetic moment of the nanomagnet \n(“macrospin”) can be represented by a single average vector of magnetization M, or \ndimensionless magnetization /sM=mM , where the saturation magnetization of the \nmaterial is sM. Thus the LLG equation, containing the spin torque terms Γ, is   Page5\n 0 effdd\ndt dtγµ α⎡⎤⎡⎤=− + +⎣⎦ ⎢⎥⎣⎦mmm×H m× Γ, (1) \nwhere the gyromagnetic constant is Bgµγ==, the Bohr magneton is Bµ, and the Lande \nfactor is g, the permeability of vacuum is 0µ, and the Gilbert damping factor is α. The \neffective magnetic field originates from all contributions to the energy E per unit volume \nof the nanomagnet: \n 01\neffEδ\nµδ=−HM (2) \nThe energy of the nanomagnet includes the demagnetization term – coming from the \ninteraction of magnetic dipoles between them selves, the material anisotropy, and the \nZeeman energy due to an external magnetic fiel d. In this article we disregard the latter \ntwo contributions for the free FM layer, and focus on the former. The demagnetization \nterm (synonymous with the shape an isotropy of the nanomagnet) is \n 0\n2Eµ= MNM , (3) \nThe demagnetization tens or is diagonal and =1xx yy zzNNN++ , if the coordinate axes \ncoincide with the principal axes of the nanomagnet:  Page6\n 00\n0000xx\nyy\nzzN\nN\nN⎛⎞\n⎜⎟=⎜⎟⎜⎟⎝⎠N . (4) \nFor the shape of typical nanomagnets, ellipti cal cylinder, the demagnetization tensor is \ncalculated according to Ref. 25. A heuristic rule for demagneti zation energy is that it is \nlowest when magnetization points along the l ongest axis of a nanomagnet and highest \nwhen it points along the shortest axis. For exam ple, in the case considered here of the \nnanomagnet with dimensions of 120nm*60nm*3nm, the demagnetization tensor elements \nare =0.0279,  0.0731,  =0.8990xx yy zzNN N = . \nThe spin torque contribution is described by th e spin transfer (Slonczewski) and field like \nterms (in the brackets of the following equation) \n [][]() '\nsJ\nMe tγεε=+ ⎡⎤⎣⎦Γ m× p×m p×m=, (5) \nwhere t is the thickness of the nanomagnet,  e is the absolute value of the electron charge, \nJ is the density of current perpendicu lar to the plane of the nanomagnet, p is the unit \nvector of polarizati on of electrons , and P is the degree of polari zation of these electrons. \nIn this paper we disregard the angular depende nce in the prefactor fo r simplicity, so that \n 2Pε= (6) \nThe mathematical model of micromagnetics is realized in a widely used simulator \nOOMMF 26. It is based on the same LLG equation (1). The main difference from the  Page7\nmacrospin model is that now magnetization M is considered a function of spatial \ncoordinates (discretized on a gr id) and its spatial variation plays an important role. The \nexchange interaction of between spin s is included as the energy density \n ( )2 22\nxyz EAm m m= ∇ +∇ +∇  (7) \nand the demagnetization energy is calculat ed by explicit summation of dipole-dipole \ninteractions between the parts of the nanomagnet, see Ref. 26. In this paper we will use \nthe following set of typical parameters a nd hope that the reader  agrees that the \nconclusions of the paper do not depend on this  particular choice of numerical values: \nsaturation magnetization 1 /sM MA m= , Lande factor 2g=, spin polarization 0.8P= , \nand the exchange constant 1121 0 /A Jm−=⋅ . In the cases when we include the field-like \ntorque, we set its constant '0 . 3εε=  to be in approximate agr eement with the results of \nRefs. 21 and 22. \n3. Energy profile and strategies for switching \nWe are applying the above mathematical model to treat various cases of spin transfer \ntorque switching, Fig. 1. In these schemes of  nanopillars, the upper blue layer designates \na free nanomagnet. We introduce th e coordinate axes as follow s: x along the long axis of \nthe nanomagnet, in plane of the chip, y perp endicular to x in plane of the chip, and z \nperpendicular to the plane of the chip. We  also introduce the angles to specify the \nmagnetization direction: θ, the angle from the x-axis, and φ, the angle of the projection \non the yz-plane from the y axis. The free nanomagnet has two stable (lowest energy)  Page8\nstates when magnetization points al ong +x or –x directions, i.e. 0, θπ= . The goal of \nmemory engineering is to switch magnetiza tion between these two states. The bottom \nblue layer designates the fixed nanomagnet. Even though the spin torque acts on this \nlayer as well, one keeps its magnetization fr om switching by coupling it to an adjacent \n(“pinning”) antiferromagnetic la yer (not shown in the pictur e). The magnetization of the \nfixed nanomagnet can be set in various direc tions by fabricating it with the right shape \nand magnetocrystalline anisotropy. One cas e is when the fixed magnetization is \napproximately along the x-axis, Fig. 1a. If both magnetizations were exactly along the x-\naxis, the spin torque acting on the free laye r would be zero, and sw itching would not take \nplace. In fact, thermal fluctuations cause the angles of both free and fixed nanomagnets to \nhave random values around 0θ=. Therefore in the macrospi n simulations, we formally \nset the initial angle of the free nanomagnet to 0.1θ= radians, and set the angle of the \nfixed layer to 0θ=. Another case, Fig 1b, is that of th e fixed magnetization in plane of \nthe chip, with various θ and 0φ=. Finally, the case in Fig. 1c is that of perpendicular \nmagnetization of the fixed nanomagnet /2φπ=± . We will see further on that the \ndirections of magnetization of the fixed layer,  which we consider identical to directions \nof spin polarization of the el ectrons injected into the free layer, correspond to different \nstrategies of switching. \nIn order to gain an intuitive understandi ng of the process of switching, one needs to \nvisualize the “energy landscape” – the patter n of demagnetization energy in the phase \nspace of magnetization angles, according to Eq. (3). The map of this sphere of angles on \nthe plane is shown in Fig. 2. For a different sh ape of the nanomagnet, or in the presence \nof material anisotropy and exte rnal field, this dependence wi ll be quantitatively different,  Page9\nbut the same qualitative approach applies. The salient features of the energy landscape \nare: a) the stretched ellipse-shaped “basins” close to the poles – th e stable equilibrium \nstates; b) the two “valleys” stretching from one pole to the other – the states with in-plane \nmagnetization; c) they have “mount ain passes”, or saddle points at /2θπ=  and 0,φπ=   \nd) two peaks corresponding to magnetiza tion perpendicular to the plane, at /2θπ=  and \n/2φπ=± . The iso-energy lines are shown in the contour plot, Fig. 2. In the absence of \ndamping and spin torque, they would coincide with closed orbits of magnetization. There \nare orbits of low energy precessing around one of the poles, and orbits of high energy \noscillating between the two poles. In the presence of damping, the nanomagnet will \nevolve to one of the basins and eventually to  the pole inside it. Spin torque can cause the \nnanomagnet to gain or lose energy, under some  conditions moving between the basins, i.e. \nswitching. \nAt this point, let us agree on the definitions for switching time. We will consider \nswitching under the action of rectan gular pulse currents of magnitude I and duration puτ \n(marked in the following plots). These valu es are relevant for the switching charge \npuIτand energy swp uE UIτ=  dissipated in switching under the voltage bias U. Therefore \nthe pulse duration puτ is the time important in the technological sense for optimizing the \nenergy per writing a bit. From simulations  we obtain the switching time, which \ncharacterizes how fast the nanomagnet responds to the current pulse. We customarily \ndefine the switching time swτ as the time over which the magne tization is switched from   \n10% to 90% of its limiting values.  In our part icular case it is the interval between the \nfirst time the projection of magnetization xm goes below 0.8 till the last time it is over -  Page\n10\n0.8. The 10-90% time gives the lower bound of switching time. Its importance is to \ncharacterize the switchin g time pertinent to the strategy, ra ther than influence of initial \nconditions. The reason that we us e it instead of 0-100% time is  that, in the collinear in-\nplane case, the latter strongly depends on the choice of the initial angle of magnetization \n(see discussion below). The total write time needs to be longer than the largest of the two \ntime measures.  \nIn the following, we will plot the switching times swτ resulting from the simulation of \nmagnetization dynamics over certain time interv als, typically 1, 2.5, 3, or 4ns. When the \nswitching time reaches this constant value, it really means that switching does not occur \nover the simulation time, and, with high probabi lity, even after any duration of evolution. \nThus these limiting constant values on th e plot are just tokens for “no switching \noccurring”. \nCollinear polarization spin torque switching is done in a configuration of Fig 1a. Since \nthe injected spin polarization is in plane close to 0θ=, the resulting spin transfer torque \npushes the magnetization to rotate in plane, along the slope of sl owest accent in energy, \nwhich might seem at first glance like the optimal strategy of switching. An example of \nswitching dynamics for this case is shown in  Fig. 3. From the time evolution plot we see \nthat magnetization performs an oscillatory mo tion close to in plan e position with slowly \nincreasing amplitude. Torque increases with the angle from the x-axis, and this reinforces \nthe growth of this angle. At some point the projections on the x-axis abruptly switches to \na negative value and then the amplit ude of the oscillations is damped27,28. From the \ntrajectory in phase space of magnetization direction, we see that the switching happens by   Page\n11\nmoving along one of the valleys and crossing over a saddle point. The duration of the \npulse is sufficient if it is l onger than the time necessary to go to the other side of this \nsaddle point. \nAnother strategy is the in th e configuration of Fig. 1b. It is similar to the previous \nstrategy, with a few modifications. In case of th e spin polarization 90 degree in plane, i.e. \n/2θπ= , see Fig. 4, the torque is maximal in the initial instant when the magnetization is \nat 0θ=. At very large current values, when th e magnetization reaches the saddle point, \nthe torque turns to zero and the nanomagnet dwells in an unstable equilibrium until the \nend of the current pulse. At this point it falls towards one of the equilibriums; the choice \nof which is governed by the randomness of its  position at that moment. Overall, this \nlooks like an unreliable method of switching. Also it requires a much higher current than \ncollinear polarization switching a nd the switching time is longer29. The situation is \ndrastically improved for the case of injected spin polarization at a different angle, e.g. \n135 degree in plane, 3/ 4θπ= , shown in Fig. 5. The path of switching still goes along \nthe energy valley and over the saddle point. But in that case, torque is not zero at the \ninitial instant, and it does not turn to zero  at the position of th e energy saddle point. \nSwitching time proves to be shorter. But one im portant similarity is that the pulse time \nneeds to be relatively long in order to cross over the saddle point. The strategy of \nswitching with perpendicular spin polarization, as shown in Fig 1c, turns out to be very \ndifferent from collinear polari zation switching. It stems from  the fact that the spin \ntransfer torque acts in the direction perpe ndicular to the sample plane as well. In a \ncounter-intuitive manner, it pushes the nanomagne t along the path of steepest accent. The \nway to take advantage of this situation is to use a very short pulse, which will supply   Page\n12\nsufficient energy to the nanomagnet. The adva ntage of a short pulse is that it requires \nsmaller switching energy supplied by the curren t. After the initial short current pulse, the \nnanomagnet precesses due to the torque from the shape anisotropy at zero spin torque, see Fig 6 for an example of such evolu tion. Under the condition of a small Gilbert \ndamping, its trajectory will be close to the iso-energy line. A necessary condition for switching is that the nanomagnet has sufficient energy to be on the trajectory that crosses \nto the other basin, even with the account of  loss to damping. The sufficient condition of \nsuccessful switching is that by the time magne tization reaches the other basin, it loses \nenough energy so that it cannot cross back to the original basin. If this condition is \nsatisfied, the nanomagnet slowly loses energy to damping and approaches the equilibrium. \nDue to the small value of damping the switching time turns out to be long.   \nA variation on this strategy is to apply a nother pulse of curren t after the nanomagnet \ncrosses to the other basin, Fig. 7. Th is pulse can have the same duration \npuτ and start \nafter a delay time, gτ, after the trailing edge of the fi rst pulse but must have the opposite \npolarity of the current. Such a tw o-pulse sequence with the total time 2tp u gτττ=+ , will \nefficiently decrease the energy and avoid the process of slow energy damping. As a result \nthe switching time swτ becomes very short, ~0.2ns, comparable topuτ. The downside of \nthis strategy is that two pulses of course require twice the energy of one pulse with the \nsame current magnitude and duration. Also it re quires a more complicated circuit to time \nthe pulses of opposite polarity. We note that th e difference of the strategy considered here \nfrom the one of Kent et al.17 is that in their approach the two pulses of the opposite \npolarity did not have a gap between them, so  the stage of free precession was absent.   Page\n13\nWe make the following approximate estimate of the pulse parameters for the pulsed \nswitching. The energy that the nanomagnet gains in  the first pulse must be larger than the \nenergy necessary to cros s over the saddle point.  \n 2\nzz z y y x xmN N N=−  (8) \nOn the other hand, the out of plane projection at the end of the pulse is obtained from Eq.  \n 2Bp u\nz\nsgI PmMeVµτ= . (9) \nFor the parameters used in this paper, it amounts to the switching charge of 81puIf Cτ=  \nper bit. This is much smalle r than the best achieved values  for the collinear polarization \nswitching, ~5pC per bit. Moreover, if we  compare the switching time and energy \nprojected here with incumbent type s of memory, see Table 3 in Ref. 30, we see that \nSTTRAM with pulsed switching is superior  to all other types of memory. From \nprototypes of STTRAM 16 we also know that its density can be comparable to that of \nDRAM. Therefore the proposed improvement gives it the crucial performance boost to \npotentially be the universal memory and to replace all other kinds. \n4. Tolerances of switching and in fluence of field-like torque \nSpin transfer torque memories operate in el ectronic circuits. The ci rcuits naturally have \nvariability originating from fabrication imperfections. Also the state of the circuit is subject to electronic noise and temperature drif t. Obviously it is not  possible to guarantee \nprecise values of switching current and time fo r elements of memories. Therefore it is   Page\n14\nespecially important to study the tolerances  of memory operation relative to external \nparameters. We believe such studies have not been conducted up to now. Here we run \nmultiple simulations over a wide set of parameters to draw some conclusions about these \ntolerances.  \nAt the same time we study the effect of the field-like torque (FLT) contribution. \nExperiments on separate measurements of the spin transfer (Slonczewski) and field like \ntorques have been conducted 21,22. But the implications of th ese contributions to current \ninduced magnetization switching have not been  sufficiently clarified. The experiments \nshow that FLT increases with increasing appl ied voltage. To account for this we consider \ntwo cases – no field-like torque '0ε= and large field-like torque '0 . 3εε= . \nThe contour plots of switching time vs. current and pulse duration for collinear \npolarization switching are shown in Fig. 8. It is common to think of spin torque switching \nas having a threshold (or critical) current cI. However from this plot we see that, for \nsufficiently short pulse du ration, the switching current 1/cp uIIτ−∼  is much larger than \nthe critical current. Therefore it is the threshold charge, 1.2puI pCτ≈ , that determines \nwhether the memory state is sw itched.  Above this threshold,  switching can be quite fast, \n~0.2ns, but the total write time is limited by the pulse duration instead. One can notice \nthat the shapes of the swit ching time dependence with and without FLT are remarkably \nsimilar, but they appear to be shifted. For th at reason, one needs to be cautious of the fact \nthat for a specific values of current and pul se duration, the dynami cs may be different \nwith and without FLT. The reason for similarity  is that for this cas e FLT plays a role of \nan effective magnetic field in the z-direction,  in addition to a large effective field from   Page\n15\ndemagnetization. There are curious geometri cal features of the switching threshold \nborder. Even though they are persistent in simula tion, we believe that they are artifacts of \nthe choice of initial magnetization and of os cillations (“ringing”) of magnetization after \nswitching. In reality, thermal fluctuation wi ll vary the initial magnetization, and the \nfeatures on the plot will be washed away for the thermally averaged switching time. \nOverall, this strategy of switching gives an excellent tolerance when the switching \ncurrent and pulse duration are se t high enough above the threshold. \nThe switching time diagrams for the 90 degree in  plane polarization ar e shown in Fig. 9. \nIn this case, the threshold current is actua lly a good criterion of  switching; and this \nthreshold turns out to be very high, ~10mA. Without FLT, even above threshold there are \ntightly interlaced regions of successful and uns uccessful switching. This attests to the \nunstable nature of such switchi ng. It cannot be used for a pr actical device. The situation \nis different with FLT. There are large regions of small sw itching time, and therefore good \ntolerance to parameters, above th e threshold. The reason for this  stabilization is that even \nthough the Slonczewski torque vanishes at /2θπ= , FLT is still finite and it succeeds in \npushing the nanomagnet over the energy saddl e point. However at  excessively high \ncurrent we encounter the regions of unsucce ssful switching that memory designers need \nto avoid. \nThe switching diagrams for 135degree in plan e polarization are shown in Fig 10. The \nthreshold behavior is in be tween the collinear polarizati on and 90degree in plane cases. \nThe threshold current is not c onstant, and it is ~2-4mA, which is lower than that for 90 \ndegrees. But it is not inversely proportional to the pulse duration either. The threshold   Page\n16\ncharges are in the range 0.6 1.6puI pC τ≈÷ . Overall it has the same excellent tolerance to \ncurrent and pulse variation as the collinear polarization switc hing, but in the absence of \nFLT, regions of unsuccessful switchi ng are observed at higher current. \nThe switching diagrams for out of plane sp in polarization and one  switching pulse are \nshown in Fig. 11. The areas of successful sw itching are seen as na rrow strips across the \nplot. They are interlaced with stripes of unsuccessful switching. The reason for such \nbehavior is the precession character of  magnetization dynamics for this switching \nstrategy. If too much energy is transferre d from the current to the nanomagnet, it \novershoots and returns to  the basin around the initial ma gnetization state. The set of \nsuccessful switching correspond to 0.5, 1.5, 2.5 etc. full turns of magnetization. The \nlowest of these stripes corresponds  to the threshol d of switching, 100puIf C τ≈ . Though \nthe threshold is only approximately given by the product of the current and the pulse \nduration. This is in a very good agreemen t with the analytical estimate (9).  \nThis is the first case when we encounter the problem of tolerances in earnest. From the \nleft plot in Fig. 11 for Gilbert damping of 0. 01, we estimate the tolerances to be 1ps and \n0.1mA. This is likely too tight for a realistic memory circuit. The right plot is calculated \nfor Gilbert damping 0.03. We see that, c ontrary to what is known about collinear \npolarization switching, the thre shold is almost unchanged at a higher Gilbert damping. At \nthe same time, the tolerances are much rela xed, to 5ps and 0.7mA. The switching stripes \nbecame wider, and the next order of switc hing with 1.5 turns is moved to a higher \nswitching charge. This seems to be the fi rst occasion when increasing damping is \nbeneficial for the device performance. We note that these simulations are done with   Page\n17\ninclusion of FLT. The results with zero F LT (not included in this  paper) are almost \nindistinguishable from those. The reason for this is that the current pulses act when \nmagnetization is close to the poles, and FLT has projections mostly in-plane of the \nnanomagnet, which contribute only negligibly  to the precessional type of switching. \nThe switching diagrams for out of plane polariz ation and two pulses are shown in Fig. 12. \nThe total pulse time is fixed at 0.2t nsτ=    , while the pulse time puτ  is given on the \nhorizontal axis of the plot.  Their overall character is sim ilar to those for a single pulse. \nThe threshold condition is approximately the sa me. The first stripe is very narrow, with \nthe tolerances 0.3ps and 30uA. Surprisingly the second stripe  corresponding to 1.5 turns, \nhas a much larger and acceptable value of to lerance of 4ps and 0.4mA. We do not have \nan intuitive explanation for this difference between the two switching cases. For smaller \nGilbert damping 0.01 the switching time is quite short, ~0.2ps. This is due to the fact that the second pulse eliminates ringing of magnetizat ion, as it was discussed in the previous \nsection. With increase of Gilbert damping to 0.03, the tolerances in the first stripe improve to 2ps and 0.2mA. This is manifested  as appearances of several satellite strips \naround the first one, meaning that at highe r damping the condition of timing the gap \nbetween the pulses becomes less crucial fo r successful switchi ng. Conversely, the \nswitching time gets slower, ~0.5ps. This is because the magnetization oscillations are not \neliminated as efficiently is the pulses are not  finely timed.   Like for the single pulse, the \ninclusion of FLT changes the results very insignificantly.   Page\n18\n5. Comparison of macrospin an d micromagnetic simulations \nThe simulation of micromagnetic dynamics is a more rigorous model and presumably \ngives a better approximation to reality that the macrospin approximation. However it is \nalso much more computationally demanding. Fo r this reason it is us eful to compare the \nresults of macrospin and micromagnetic m odels. We compare the switching time at \nvarious pulse durations but fixe d values of current. Microma gnetic simulations start with \nan initial state obtained by rela xing its energy to a minimum. It has the general direction \nof 0θ=, and a “leaf state” pattern, i.e. magnetization is closer to being parallel to the \nlonger sides of the ellipse. \nThe comparison for the collinear polarization switching is shown in Fig. 13. Here we set \nthe direction of the sp in polarization of the fixed layer to be 0.1θ= . This value is an \nestimate of the r.m.s deviation of the angle of magnetization due to thermal distribution \nof energy. We make this choice of the initia l angle only in the case of collinear in-plane \nswitching, because spin torque would vanish for 0θ=. For other cases the spin torque \ndoes not vanish at the zero initial angle.  The agreement is surprisingly good. Both \nmodels give approximately the same value of  threshold charge. This may be because we \nare focusing on the evolu tion starting from angle 0.1θ= . The evolution around 0θ= \noccurs under a much smaller torque and carries more un certainty. Macrospin model \nexhibits more oscillations close to the thre shold. Micromagnetic model predicts shorter \nswitching time above threshold, probably due to more efficient damping of higher modes \nof magnetization.   Page\n19\nThe comparison in the case of 90degree in plan e polarization, Fig. 14, shows essentially \nthe same lower envelope of switching time. The micromagnetic model shows fewer areas \nof failed switching. We speculate that this  is due to stronger effect of FLT on non-\nuniform magnetization. \nThe disagreement is more pronounced in the cas e of single pulse out of plane polarization, \nFig. 15. The similarities are the same positi on on the time scale of  successful switching \nstripes for 0.5 and 3.5 turns. The regions of  unsuccessful switching for 1 and 3 turns are \nabsent in the micromagnetic model, probabl y because the barrier for crossing into the \nother basin is increased for a non-uniform ma gnetization pattern.   The switching times \nare generally predicted to be shor ter in the micromagnetic model.  \nA similar situation is observed in out of plane pol arization switching with two pulses, see \nFig. 16. The positions of successful switching on the time scale are shifted. This is \nprobably due to the fact that  the nanomagnet energy is di fferent with the account of \nexchange and dipole-dipole interaction, and thus the time of free precession of \nmagnetization is different too.  As before, micromagnetics pr edict shorter switching times, \nas well as giving better tolerance for 0.5 turn switching. In fact, micromagnetic \nsimulations for one and two pulses look quite si milar. This points to higher significance \nof damping than of the second pulse in bri nging the magnetization to  its final state.  \nThis paper includes just a few examples of co mparison. A more complete set of data is \navailable 31. All of this supports the conclusion th at macrospin simulati ons give generally \nthe same qualitative dependence of switchi ng time on the current a nd pulse duration as   Page\n20\nOOMMF simulations. The former can be used to  infer general trends . The latter should \nbe saved for obtaining quanti tatively precise results. \n6. Conclusions \nWe have compared various strategies of spin torque switching. The tolerances of \nswitching, performance limits of out-of plane pol arization devices, and the effect of field-\nlike torque have been comprehensively studied  for the first time. In summary, switching \nwith short pulses of out of plane polarization is the prefer red strategy. It has a much \nlower threshold charge than other strategies  of switching, and suffers less from low \ntolerance to current magnitude and pulse dura tion. This problem of low tolerance can be \nresolved by increasing Gilbert damping in the nanomagnet. Field-like torque significantly \nchanges the results for non-co llinear in-plane switching a nd happens to produce a minor \neffect for other switching strategies. Implem entation of this strategy would put STTRAM \nin a position of a technological leadership  among memories. We find that macrospin \ngives good qualitative prediction of the dyna mics, though micromagnetic models should \nbe used to get better quantitative precision. \n8. Acknowledgements \nG. R. and I. N. K. gratefully acknowle dge the support of DARPA, NSF (grants DMR-\n0748810 and ECCS-0701458) and the Nanoelectr onic Research Ini tiative through the \nWestern Institute of Nanoelectronics. Co mputations supporting this paper were \nperformed on the BDUC Compute Cluster donated by Broadcom, co-run by the UCI OIT \nand the Bren School of Inform ation and Computer Sciences.   Page\n21\nReferences \n                                                 \n1  A. Fert, “Nobel Lecture: Origin, developmen t, and future of spintronics”, Rev. Mod. \nPhys. 80, 1517-30 (2008).  \n2 M. N. Baibich, J. M. Broto, A. Fert, F. N.  Vandau, F. Petroff, P.  Eitenne, G. Creuzet, A. \nFriederich, J. Chazelas, “Giant Magneotres istance of (100) Fe / (100) Cr magnetic \nsuperlattices”, Phys. Rev. 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Page\n25\n \n \n \nFigure 1. The geometry of the spin tor que memory nanopillars. Red layer – tunneling \nbarrier oxide, orange – non-ma gnetic electrodes, blue with  double-sided arrow – free \nlayer with in-plane magnetization (easy x-axis),  blue with one sided arrow – fixed layer. \nPolarizations: a) in plane collin ear with x-axis (left), b) in plane at an angle to x-axis \n(middle), c) perpendicula r to the plane (right). \n a b c  Page\n26\n \n \nFigure 2. Map of the demagnetization ener gy of the nanomagnet (normalized) in the \nmacrospin model. θ – angle of magnetization from the x-axis (easy axis), φ – angle of \nprojection of magnetization within  the yz-plane (hard plane). \n   Page\n27\n \n \n  \n \nFigure 3. Magnetization projections vs. time and the trajectory of magnetization for \ncollinear polarization switching (0  degree in plane polarization). \n   Page\n28\n \n \n  \n \nFigure 4. Magnetization projections vs. time a nd the trajectory of magnetization for 90 \ndegree in plane polarization.    Page\n29\n \n \n  \n \nFigure 5. Magnetization projection vs. time a nd the trajectory of magnetization for 135 \ndegree in plane polarization.   Page\n30\n \n \n  \n \nFigure 6. Magnetization projections vs. time and the trajectory of magnetization for \nperpendicular out of plane polarization, one pulse.   Page\n31\n \n \n  \n \nFigure 7. Magnetization projections vs. time and the trajectory of magnetization for \nperpendicular out of plane polarization, two pulses with a total time of 0.2ns. \n   Page\n32\n \n \n  \n \nFigure 8. Contour maps of switching time, left  without the field like torque and right with \n0.3 factor of field like tor que, for collinear polarization switching (0 degree in plane \npolarization). Simulation time 2ns.    Page\n33\n \n \n  \n \nFigure 9. Contour maps of switching time, left  without the field like torque and right with \n0.3 factor of field like torque, for 90 degree in plane polarization. Simulation time 4ns. \n   Page\n34\n \n \n  \n \nFigure 10. Contour maps of switching time, le ft without the field like torque and right \nwith 0.3 factor of field like  torque, for 135 degree in plan e polarization. Simulation time \n4ns.    Page\n35\n \n \n  \n \nFigure 11. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 \nfactor of field like torque, for perpendicular out of pl ane polarization, one pulse. \nSimulation time 1ns.    Page\n36\n \n \n  \n \nFigure 12. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 \nfactor of field like torque, fo r perpendicular out of plane pol arization, two pulses with a \ntotal time of 0.2ns. Simulation time 1 ns.    Page\n37\n \n \n  \n \nFigure 13. Comparison of the switching time vs. pulse length simulated by a macrospin \nmodel (left) and OOMMF (right ) for collinear polarization switching at I=2.5mA, field-\nlike torque 0.3, and polarization angle in  plane of 10 degrees. Simulation time 3ns. \n   Page\n38\n \n \n  \n \nFigure 14. Same as Fig. 13 at I=20mA, field- like torque 0.3, and polarization angle in \nplane of 90 degrees. Simulation time 2.4ns.    Page\n39\n \n \n  \n \nFigure 15. Same as Fig. 13, single puls e at I=4mA, field-like torque 0.3, α=0.01, and \npolarization angle perpendi cular to plane of 90 degrees. Simulation time 1.8ns, \n   Page\n40\n \n \n  \n \nFigure 16. Same as Fig. 13 for two pulses with  total duration of 0. 2ns, at I=4mA, field-\nlike torque 0.3, α=0.01, and polarization angle perp endicular to plane of 90 degrees. \nSimulation time 1.8ns. \n "    },    {        "title": "1002.3295v1.Measurement_of_Gilbert_damping_parameters_in_nanoscale_CPP_GMR_spin_valves.pdf",        "content": "Measurement of Gilbert damping paramete rs in nanoscale CPP-GMR spin-valves  \n \nNeil Smith, Matthew J. Carey, and Jeffrey R. Childress. \nSan Jose Research Center \n Hitachi Global Storage Technologies \nSan Jose, CA 95120 \n \nabstract ⎯ In-situ, device level measurement of thermal mag-noise spectral linewidths in  60nm diameter CPP-GMR spin-valve stacks of \nIrMn/ref/Cu/free , with reference and free la yer of similar CoFe/CoFeGe alloy, are used  to simultaneously determine the intrins ic Gilbert damping for \nboth magnetic layers. It is shown that careful alignment at a \"magic-angle\" between free and reference layer static equilibrium  magnetization can \nallow direct measurement of the broadband intrinsic thermal spectr a in the virtual absence of spin-torque effects which otherwi se grossly distort the \nspectral line shapes and require linewidth extrapolations to zer o current (which are nonetheless al so shown to agree well with the direct method). The \nexperimental magic-angle spectra are shown to be in good qualit ative and quantitative agreement with both macros pin calculation s and \nmicromagnetic eigenmode analysis. Despite similar composition and thickness, it is repeatedly found that the IrMn exchange pinn ed reference layer \nhas ten times larger  intrinsic Gilbert damping  than that of the free-layer ) 1 . 0 ( ≈ α ) 01 . 0 ( ≈α .It is argued that the large reference layer damping \nresults from strong, off -resonant coupling to to lossy modes of an IrMn/ref couple,  rather than commonly invoked two-magnon pr ocesses.  \n \n \nI. INTRODUCTION \n     Spin-torque phenomena, in tunneling magnetoresistive (TMR) \nor giant-magnetoresistive (GMR) film stacks lithographically patterned into ~100 nm nanopillars and driven with dc electrical \ncurrents perpendicular to the plane (CPP) of the films have in \nrecent years been the topic of numerous theoretical and \nexperimental papers, both for their novel physics as well as \npotential applications for magnetic memory elements, microwave oscillators, and magnetic field sensors and/or \nmagnetic recording heads.\n1 In all cases, the electrical current \ndensity at which spin-torque instability or oscillation occurs in \nthe constituent magnetic film layers is closely related to the \nmagnetic damping  of these ferromagnetic (FM) films \n     This paper considers the electrical measurement of thermal \nmag-noise spectra to determine intrinsic damping at the device \nlevel  in CPP-GMR spin-valve stacks of sub-100nm dimensions \n(intended for read head applications), which allows simultaneous R-H and transport characterization on the same device. \nCompared to traditional ferromagnetic resonance (FMR) linewidth measurements at the bulk film level, the device-level approach naturally includes finite-size and spin-pumping\n2 effects \ncharacteristic of actual devices, as well as provide immunity to inhomogeneous and/or two-magnon linewidth broadening not relevant to nanoscale devices. Complimentary to spin-torque-\nFMR using ac excitation currents,\n3 broadband thermal excitation \nnaturally excites all modes of the system (with larger, more quantitatively modeled signal amplitudes) and allows \nsimultaneous damping measurement in both reference and free FM layers of the spin-valve, which will be shown to lead to \nsome new and unexpected conclusions. However, spin-torques at \nfinite dc currents can substantially alter the absolute linewidth, and so it is necessary to account for or eliminate this effect in \norder to determine the intrinsic damping. \n                                     1  \nII. PRELIMINARIES AND MAGIC-ANGLES \n \n     Fig. 1a illustrates the basic film stack structure of a \nprospective CPP-GMR spin-valve (SV) read sensor, which apart \nfrom the Cu spacer between free-layer (FL) and reference layer \n(RL), is identical in form to well-known, present day TMR sensors. In addition to the unidirectional exchange coupling \nbetween the IrMn and the pinned-layer (PL), the usual \n\"synthetic-antiferromagnet\" (SAF) structure PL/Ru/RL is meant to increase magnetostatic stability and immunity to field-induced \nrotation of the PL-RL couple, as well as strongly reduce its net \ndemagnetizing field on the FL which otherwise can rotate in \nresponse to signal fields. However, for simplicity in interpreting \nand modeling the spectral and transport data of Sec. III , the \npresent experiment restricts attention to devices with a single RL directly exchange-coupled to IrMn, as shown in Fig. 1b. \n     The simplest practical model for describing the physics of the device of Fig. 1b is a macrospin model that treats the RL unit \nmagnetization as fixed, with only the FL magnetization \nRLˆm\n) (ˆ ) ( ˆFL t tm m ↔  as possibly dynamic in time. As was described \npreviously,4 the linearized  Gilbert equations for small deviations \n) , (z ym m′ ′′ ′=′m  about equilibrium x m′ ↔ˆ ˆ0  can be expressed \nin the primed coordinates as a 2D tensor/matrix equation5: \nmm\nmH\nmmm Hhmmh mm\n′∂∂⋅∂∂⋅∂′∂−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛⋅ ≡ ′Δ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−\nγ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nγα ≡⋅∂′∂≡′=′⋅′+′⋅ +\nˆ\nˆ ˆ 1 00 1)ˆ () (,0 11 0,1 00 1) (ˆ) ( ) (\neff\n0effFL\nHmV MppGpDt p t HdtdG D\ns\ntt tt tt\n       (1) \ncap\nRL\nIrMnCu\nseedFL\nPL\nIrMn\nseedcap\nRuCu\nRL\n(a)(b)FL\nxz\nFIG. 1. (a) Cartoon of prospective CPP-GMR spin-valve sensor stack, \nanalogous to that used for contemporary TMR read head. (b) Cartoon of simplified spin-valve stack used for present experiments, patterned into ~60nm circular pillars using e-\nbeam lithography.  In (1),  is a 3D Cartesian tensor, m Hˆ/eff∂ ∂ m m′∂ ∂/ˆ  is a 2 3× \ntransformation matrix between 3D unprimed and 2D primed \nvectors (with  its transpose) which depends only on \n, and  is a 3D perturbation field supposed as the origin \nof the deviations . The magnetic moment m mˆ/∂′∂\n0ˆm ) (th\n) (tm′ mΔ is an \narbitrary fixed  value, but  is a natural choice for \nSec. II. Using an explicit Slonczewski6 type expression for the \nspin-torque contribution, the general form for  is  FL) (V M ms → Δ\n)ˆ(effm H\n \nm J P e HHE\nm\neΔ ≡ ⋅ ≡ θ× θ η −∂∂\nΔ−=\n/ ) 2 / ( and , ˆ ˆ cos),ˆ ˆ ( ) (cosˆ1\neffeff\nST FL RLFL RL ST\nh m mm mmH            (2) \n \nfor any free energy function . A positive electron current \ndensity  implies electron flow from the RL to the FL.  is \nthe net spin polarization of th e current inside the Cu spacer. \nOersted-field contributions to )ˆ(mE\neJeffP\neffH  will be neglected here.  \n                                     2      With  in (1), nontrivial solutions  \nrequire s satisfy 0 ) (=thste t−′=′ m m) (\n0 | ) ( | det= + −G D s Httt\n. The value  \nwhen defines the critical onset of spin-torque instability. \nUsing (1), the general criticality  condition is expressible as  crit\ne eJ J≡\n0 Re=s\n \n0 ) (\nt independen=′−′+′+′ α\n∝′ ′ ′ ′ ′ ′ ′ ′4434421 4434421\ne e Jy z z y\nJz z y y H H H H\n-                 (3a) \n) cos ( ) ( 2 ) 1 (2\nSTθ ≡ η −η− ≅′−′′ ′ ′ ′q q qdqdqHH Hy z z y       (3b) \n) ( 2 / ) 1 () (\n) 2 / (2effcrit\nq q dq d qH H\nP emJz z y y\neη − η −′+′ α Δ= ⇒′ ′ ′ ′\nh            (3c) \n \nwhere  is the Gilbert damping. The -scaling of the terms in \nin (3a) follows just from the form of (2). The result in (3b) was \nderived earlier4 in the present approximation of rigid . αeJ\nRLˆm\n     With θ the angle between  and  (at equilibrium), it \nfollows from (3c) that at a \"magic-angle\"  where the \ndenominator vanishes,  and spin-torque effects are \neffectively eliminated from the system at finite . To pursue this point further, explicit results for  will be used from \nthe prototypical case where th e CPP-GMR stack (Fig. 1b) is \napproximately symmetric about th e Cu spacer, which is roughly \nequivalent to the less restrictiv e situation where the RL and FL \nare similar materials with thicknesses that are not small \ncompared the spin-diffusion length. For this quasi-symmetric \ncase, both quasi-ballistic6 and fully diffusive7 transport models \nyield the following simple functional forms:  \nRLˆmFLˆm\nmagicθ\n∞ →crit\neJ\neJ) (cosθ η\n \n) cos 1 () (cos ) (cos) (cos] cos ) 1 ( 1 [ / ) (cos\nmin maxminθ −Γθ η=− ≡ Δ− θ ≡ δ≡ θθ − Γ + + Γ Γ=θ η\nR R RR R Rr     (4) \n \nwhich also relates η to the normalized resistance r 1 0 (≤≤r ) \nwhich is directly measurable experimentally. The transport \nparameter Γ is theoretically related to the Sharvin resistance6,8 \nor mixing conductance8 at the Cu/FL interface, but will be \nestimated via  measurement in Sec III. Using crit\neJ ) (qη  from (4) \nin (3), magicθ  and ) (magicθr vs.  curves are shown in Fig. 2.  Γ\n     The \"magic-angle\" concept also applies to mag-noise power \nspectral density (PSD)  at bias \ncurrent , arising from thermal fluctuations in θ θ Δ = S d dr R I SV2\nbias bias ] ) / ( [\nbiasI θabout \nequilibrium bias angle biasθ . Assuming  in/near the \nfilm plane (FL RL 0ˆ, - m\n=≅′z zˆ ˆplane-normal), and requiring  | , \nit can be shown5 from fluctuation-dissipation arguments that  | | |crit\nbias eI I<\n \n] ) ( [ andwhere) ( ) () (4) ()] ( [ ) ( , ) (4) (\n02 2 2\n022 2 2 21\nz y y z y y z zy z z y z z y yz y z z By yB\nH H H HH H H HH H\nmT kf SG D i H DmT kf S\n′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′\nθ−\n′ ′ θ\n′−′+′+′ α γ = ω Δ′ ′ −′ ′ γ = ωω Δ ω + ω − ωω + ′+′ γ\nΔα γ≅ ⇒+ ω − ′ = ω χ χ ⋅ ⋅ χΔ γ≅tt t t t tt @\n    (5) \n \nComparing (5) with (3), it is se en that the spectral linewidth \nωΔis predicted to be a linear function of  but with ,eJ\n0 /→ ωΔedJ d  when magic bias θ → θ . Since y y z zH H ′ ′ ′ ′′> > ′  \n(due to ~10 kOe out-of-plane demag fields) and z y y yH H ′ ′ ′ ′′> > ′  \n(e.g., for the  measurements in Sec. III), it is only in \nthe linewidth crit\ne eJ J<\nωΔ that the off-diagonal terms y z z yH H ′ ′ ′ ′′ ′,  can \nbe expected to influence . Therefore, measurement of \n with ) (f Sθ\n) (f SV magic bias θ≅ θ  ideally allows direct measurement \nof the natural thermal-equ ilibrium mag-noise spectrum , \nfrom which can be extracted the intrinsic (i.e., -independent) \nGilbert damping constant) (f Sθ\neJ\nα. This is the subject of Sec. III.   9095100105110115120\n  \n 0.20.250.30.350.40.450.5rmagic\n1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6θmagic\n(deg)\nΓFL0 1112m2\nm = +− Γ+ Γ+ c c\nFIG. 2.  Graph of θmagic(blue) and rmagic= rbias(θma g i c) (red) vs. ΓFLas \ndescribed by (4). The equation for cm= cos( θmagic) follows from (3) and \n(4). The red solid squares are measured ( ΓFL, rmagic) from Figs. 3,4 and 6. III. EXPERIMENTAL RESULTS \n \n     The results to be shown below were measured on CPP-GMR-\nspin-valves of stack structure: seed-layers/IrMn (60A)/RL/Cu (30A)/FL/cap layers. The films were fabricated by magnetron \nsputtering onto AlTiC substrates  at room temperature, with \n2mTorr of Ar sputter gas. The bottom contact was a ~1-\nμm thick \nNiFe layer, planarized using ch emical-mechanical polishing. To \nincrease ΔR/R, both the RL and FL were made from \n(CoFe) 70Ge30 magnetic alloys.9 The RL includes a thin CoFe \nbetween IrMn and CoFeGe to help maximize the exchange coupling strength, and both RL and FL include very thin CoFe at \nthe Cu interface.  The resultant  product for the RL and FL \nwere about 0.64 emu/cm\n2. After deposition, SV films were \nannealed for 5hours at 245C in 13kOe applied field to set the \nexchange pinning direction. The IrMn/RL exchange pinning \nstrength of ≈0.75 erg/cm2\n was measured by vibrating sample \nmagnetometry. After annealing, patterned devices with ≈ 60 nm \ndiameter (measured at the FL) were fabricated using e-beam lithography and Ar ion milling. A 0.2\nμm-thick Au layer was \nused as the top contact to devices.  t Ms\n     Fig. 3 illustrates a full measurement sequence. Devices are \nfirst pre-screened to find samp les with approximate ideal in-\nplane δR-H loops (Fig. 3a) for circul ar pillars: non-hysteretic, \nunidirectionally-square loops with  parallel with the \nRL's exchange pinning direction ||H H=\n),ˆ(x+ along with symmetric \nloops about when  is transverse The right-shift in the \n0=H⊥=H H axis).ˆ(-y||H R-δ  loop indicates a large demagnetizing \nfield of ~500 Oe from the RL on the FL.  \n     As shown previously,4 narrow-band \"low\"-frequency  \nmeasurements (eI N-\nMHz) 100 ( = ≡ f PSD N , 1MHz bandwidth) \ncan reveal spin-torque criticality as the  very rapid onset  of \nexcess (1/ f-like) noise when exceeds .  loops \nare measured with  sourced from a continuous sawtooth \ngenerator (2-Hz) which also triggers 1/2 sec sweeps of an \nAgilent-E4440 spectrum analyzer (i n zero-span, averaging mode) \nfor ≈50 cycles. With high sweep repeatability and virtually no \n-hysteresis, this averaging is  sufficient so that after \n(quadratically) subtracting the mean | |eI | |crit\neIeI N-\neI\neI\nHz nV/ 1 ) 0 ( ≈ ≈eI N  \nelectronics noise, the resultant  loops (Fig. 3b) indicate \nstochastic uncertainty eI N-\n. Hz nV/ 1 . 0 < <  \n      With 1 cos±=θ , it readily follows from (3c) and (4) that \n \ncrit critcrit crit\nPAP\n) 0 () (\nI II I\nee\n≡ = θ≡ π = θ− = Γ                           (6) \n \nHence, to estimate Γ,  are measured with applied fields eI N-\nkOe2 . 1 , 45 . 0|| + −≈H  (Fig. 3b),which more than sufficient to \nalign  antiparallel (AP), or parallel (P) to ,respectively \n(see Fig. 3a), thereby reducing possible sensitivity to Oersted \nfield and/or thermal effects. (Reducing  by ~200-300 Oe \ndid not significantly change either  curve.) With  \ndenoting electron  flow from RL  to FL, it is readily found from \n(3) that  and  for the FL. By symmetry, it \nmust follow that and  for spin-torque \ninduced instability of the RL. This sign convention readily \nidentifies these four critical points by inspection of the  \ndata. To account for possible small (thermal) spread in critical \nonset, specific values for the  (excluding ) are defined \nby where the  curves cross the FLˆmRLˆm\n| |||H\neI N- 0>eI\n0crit\nFL AP>-I 0crit\nFL P<-I\n0crit\nRL AP<-I 0crit\nRL P>-I\neI N-\ncrit\neIcrit\nRL P-I\neI N- Hz nV/ 2 . 0 line, which is \neasily distinguished from the mA / Hz nV/ 05 . 0 ~  residual \nmagnetic/thermal background. is estimated in Fig. 3b (and \nrepeatedly in Figs. 4-7) to be ≈ +4.5 mA. Arbitrariness in the \nvalue of  from using the crit\nRL P-I\ncrit\neI Hz nV/ 2 . 0 criterion is thought to \nonly be of minor significance for , due to the rounded \nshape of the AP  curves near this particular critical point, \nwhich may in part explain why  estimated from  is \nfound to be systematically  somewhat larger than crit\nRL AP-I\neI N-\nRL/CuΓeI N-\nCu/FLΓ . \n                                     3      However, the key results here are the 0.1-18 GHz broad-band \n(rms)  spectra (Fig. 3c). They are measured at \ndiscrete dc bias currents with the same Miteq preamp (and in-\nseries bias-T) used for the data, the latter being insitu  gain-) ; PSD(eI f\neI N-H (kOe)-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.2Ω\n(a)\nFIG. 3. Measurement set for 60nm device. (a) δR-H ||(black) and δR-H ⊥\n(gray) loops at -5mV bias. (b) P-state  N-Ieloops at H| |≈+1.2 kOe (re d), \nand  AP-state N-Ieloops at H ||≈-0.45 kOe ( blue); FL critical currents to \ndeter mine ΓFL(via (6)) enclosed by oval. (c) rms PSD (f, Ie) (normalized to \n1 mA) with Ieas indicated by color. Thin black curves are least-squares fits \nvia (7), fitted values for αFL, αRLlisted on top of graph. M easured rbiasand \napplied field Hlisted inside graph. Field strength and direction (see Fig. 9) \nadjusted to achieve \"magic-angle\".  ±1.5 mA spectra shown, but not fit.02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nnV\nHzPSD\nfrequency (GHz)I  = +0.4, -0.4, +0.6, +0.8, -0.8  mA -0.6, \nH l +750 Oeα  = 0.12, 0.13, 0.11, 0.12, 0.10 R L 0.10, α = 0.011, 0.011 , 0.011 , 0.012, 0.010 F L 0.010, \nrbiasj 0.36normalized \nto 1 mA\n-1.5 mA+1.5 mA\n(c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nnV\nHzPSD H  l -0.45 kOe  | |\nΓ  l 4(100 MHz) RL\nΓ  l 3.1FL\n(b)calibrated vs. frequency (with ≈50Ω preamp input impedance \nand additionally compensating the present ≈0.7 pF device \ncapacitance) to yield quantitative ly absolute values for these \n (each averaged over ~100 sweeps, with \nsubtracted post-process) . To confirm the real \nexistence of an effective \"magic-angle\", the applied field H was \ncarefully adjusted (by repeated  trial and error) in both amplitude \nand direction  to eliminate as much as possible any real-time \nobserved dependence of the raw  near the FL FMR \npeak (~ 6 GHz) on the polarity  as well as amplitude of  over a \nsufficient range. This procedure was somewhat tedious and \ndelicate, and initial attempts us ing a nominally transverse field \n were empirically found inferior to additionally adjusting the \ndirection of the field, here rotated somewhat toward the pinning direction for the RL. Using a mechanically-positioned permanent magnet as a field source, this field rotation was only \ncrudely estimated at the time to be ~20-30\no (see also Sec. IV). \nWith both H and bias-point \"optimized\" as such, an -\nseries of  were measured, after which the bias-\nresistance , and finally  and  were measured at \na common (low) bias of −10 mV to determine  (as in (4)). ) ; PSD(eI f\n) 0 ; PSD(=eI f\n) ; PSD(eI f\neI\n⊥H\nbiasθeI\n) ; PSD(eI f\nbiasRminRmaxR\nbiasr\n     The key feature of the rms  in Fig. 3c is that \nthese measured spectra (excluding  appear \nessentially independent of both the polarity and magnitude of  \n(after 1mA-normalization), de fining a \"universal\" spectrum \ncurve over the entire 18GHz bandwidth, including the \nunexpectedly wide, low amplitude  RL-FMR peak near 14 GHz \n(more on this below). Because of the relatively large ) ; PSD(eI f\nmA) 5 . 1 + =eI\neI\nHz nV/ 1 ~ ) 0 ; PSD( =eI f  background, these RL peaks were \nnot well discernible during ra w spectrum measurements, and were practically revealed only after electronics background noise subtraction. As suggested in Fig. 3c, eventual breakdown of the \nmagic-angle condition was genera lly found to first occur from \nspin-torque instability of the FL at larger positive .  \neI\n     The spectra Fig. 4 shows the equivalent set of measurements \non a physically different (tho ugh nominally identical) 60-nm \ndevice. They are found to be remarkably alike in all properties to those of Fig. 3, providing additional confirmation that the \"magic-angle\" method can work on real nanoscale structures to \ndirectly obtain the intrinsic   in the absence of of \nspin-torque effects. This appears further confirmed by the close \nagreement of measured  pairs (from data of Figs. 3,4, \nand 6) and the macrospin model predictions described in Fig. 2. ) 0 ; (=θ eI f S\n) , (Cu/FLΓbr\n     To obtain values for linewidth and then damping ω Δ α from \nthe measured , regions of spectra several-GHz wide, \nsurrounding the FL and RL FMR peaks are each nonlinear least-sqaures fitted to the functional form for) ; PSD(eI f\n) 0 ; (=θ eI f S  in (5). In \nparticular, the fitting function is taken to be  \n \nz z z z y yy y z z z z y yz z y y\nV\nH H HH H H HH H\nS f S\n′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′\nπω\n′ ′ α + γ ω → ′′+′ α γ = ω Δ ′ ′ γ = ωω Δ ω + ω − ωω′ ′ + ω ω\n= =\n/ ] 2 / ) ( ) / [( and) ( , with,\n) ( ) (] ) / ( [\n) (\n2\nfit2\npeakfit 02 2 2\n022 2\n02\n0\n0 2\n     (7) \n \nThree fitting parameters are used: ) 0 (0 = = f S SV , fitα, and \npeakω , the latter being already well defined by the data itself. \nThe substitution for y yH ′ ′′ is accurate to order , leaving 2α\nz zH ′ ′′ as yet unknown. With  dominated by out-\nof-plane demagnetizing fields,  depends mostly on the \nproduct y y z zH H ′ ′ ′ ′′> >′\n) (f SV\nz zH ′ ′′ αfit . For simplicity, fixed values  \nand  were used here, based on macrospin \ncalculations that approximately account for device geometry and net  product for FL and RL films. The fitted  \ncurves, and the values obtained for  and  are also \nincluded in Figs. 3c and 4c. These values are notably independent  of (or show no significant trend with) .   kOe 8FL=′′ ′z zH\nkOe, 10RL=′′ ′z zH\nt Ms ) ; PSD(eI f\nFL\nfitαRL\nfitα\neI\n                                     4      Although the  repeatedly found from these data is \na quite typical magnitude for Gilbert damping in CoFe alloys, \nthe extremely large, 10× greater value of  is quite \nnoteworthy, since the RL and FL are not too dissimilar in \nthickness and composition. Although the small amplitude of the \nRL-FMR peaks in Figs. 3-4 (everywhere below the raw 01 . 0FL\nfit≈ α\n1 . 0RL\nfit≈ α\nHz nV/ 1 electronics noise), may suggest a basic unreliability \nin this fitt ed value for , this concern is seemingly dismissed \nby the data of Fig. 5. Measured on a third (nominally identical) \ndevice, an alternative \"extrapolation-method\" was used, in which RL\nfitα-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.0Ω\nH (kOe)(a)\nFIG. 4. Analogous measurement set for a different (but nominally identical) \n60nm device. as that shown in Fig. 3.  0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.02.5\nfrequency (GHz)I  = +0.5, -0.5, +1.0, -1.5 mA α  = 0.12, 0.13, 0.10, 0.12 R L\n-1.0, 0.12, α = 0.012, 0.011 , 0.013 , 0.013 F L 0.012 , \nrbiasj 0.39nV\nHzPSD\nnormalized \nto 1 mAH l +600 Oe\n(c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nΓ  l 2.7FLΓ  l 4RLnV\nHzPSD H  l -0.45 kOe  | |\n(100 MHz)\n(b)                                     5 the applied field was purposefully reduced in magnitude (and \nmore transversely aligned than for magic-angle measurements) \nto increase  and thus align  to be more antiparallel to \n. As a result, spin-torque effects at larger negative - will \ndecrease  and concomitantly enhance  RL-FMR peak \namplitude (and visa-versa for the FL), bringing this part of the \nmeasured spectrum above the raw el ectronics noise background.  biasrFLˆm\nRLˆmeI\nω Δ\n     Using the same fitting function from (7), it is now necessary \nto extrapolate the to  (Fig. 5d) in order to \nobtain the intrinsic damping. This method works well in the case \nof the RL since  and the extrapolated ) (RL\nfit eI α 0→eI\n0 | | /RL\nfit< αeI d d 0=eI  \nintercept value of is necessarily larger  than the measured \n, and hence will be (proportionately) less sensitive to \nuncertainty in the estim ated extrapolation slope.  As can be seen \nfrom Fig. 5d, the extrapolated values for intrinsic RLα\n) (RL\nfit eI α\nRLα  are \nvirtually identical to those obtain ed from the data of Figs. 3,4. \nThe extrapolated  is also quite consistent as well. The \nextrapolation data also confirm the expectation (noted earlier \nfollowing (5)) that linewidth  will vary linearly with . FLα\nω ΔeI\n     Comparing with Figs. 3c,4c, the spectra in Fig. 5c illustrate \nthe profound effect of spin-torque on altering the linewidth and peak-height of both FL and RL FMR peaks  even if the system is \nonly moderately misaligned from  the magic-angle condition. By \ncontrast, for other frequencies (where the ωΔ term in the \ndenominator of (5) is unimportant), the 1mA-normalized spectra \nare independent of . Being consistent with (5), this appears to \nverify that this 2nd form of fluctuation-dissipation theorem \nremains valid despite that the system of (1) is not in thermal equilibrium\n10 at nonzero . (Alternatively stated, spin-torques \nlead to an asymmetric  eI\neI\nHt\n, but do not alter the damping tensor \nDt\n in (1)). The α-proportionality in the prefactor of  in \n(5) relatedly shows that the effect of spin-torque on ) (f Sθ\nωΔ is not \nequivalent to additional dampin g (positive or negative) as may \nbe commonly misconstrued. It fu rther indicates that Oersted-\nfield effects, or other -dependent terms in eI Ht\n not contributing \nto ωΔ, are insignificant in this experiment.  \n     Analogous to Figs. 4,5, the data of Figs. 6,7 are measured on \nCPP-GMR-SV stacks differing only by an additional 1-nm thick \nDy cap layer deposited directly on top of the FL. The use of Dy \nin this context (presumed spin-pumping from FL to Dy, but possibly including Dy intermixing near the FL/Dy interface\n11) \nwas found in previous work12 to result in an ~3 × increase in FL-\ndamping, then inferred from the ~3 × increase in measured . \nHere, a more direct measure from the FL FMR linewidth \nindicates a roughly similar,  increase in | |crit\nFLI\n× ≈3 . 2FLα(now using \nsomewhat thicker FL films). This ratio is closely consistent with \nthat inferred from  data measured in this experiment over \na population of devices (see Table 1). Notably, the values found \nfor | |crit\nFLI\nRLα remain virtually the same as before.  \n     Finally, Fig.8 shows results for a \"synthetic-ferrimagnet\" (SF) \nfree-layer of the form FL1/Ru(8A)/FL2. The Ru spacer provides -1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.5Ω\nH (kOe)(a)\nI  (mA)e- 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81\nΓ  l 3.3FLΓ  l 3.7RLnV\nHzPSD\n(100 MHz)\n(b)H  l -0.45 kOe  | |H  l +1.2 kOe     | |\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5\nfrequency (GHz)I  =  -1.0, -1.5, -2.5, -3.0 mA -2.0,  \nrbiasj 0.53 H l +500 Oe\nnV\nHzPSD\nnormalized \nto 1 mA\n(c)\nFIG. 5. Measurement set for a different (but nominally identical) 60nm \ndevice as that shown in Figs. 3-4. (c) rms spectra (with least-sqaures fits) \nmeasured at larger r biasand θbi as> θmagic. (d) Ie-dependent values of αfi t(Ie) \nfor FL (red) and RL (blue), with suggested Ie→0 extrapolation lines.0.0 0.5 1.0 1.5 2.0 2.5 3.00.000.050.100.15\nα fitRL\n(d)\n|I  | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810\nH (kOe)(%)δR\nRRj19.9Ω\n(a)\n-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nH  l -0.45 kOe  \nΓ  l 3.2FLΓ  l 4.2RLnVPSD\nHz| |\n(100 MHz)\n(b)\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5\nfrequency (GHz)I  = +0.7, -0.7, +1.0 , +1.4, -1.4  mA α  = 0.11, 0.11, 0.11 , 0.11, 0.10 R Lα = 0.027 , 0.026 , 0.027 , 0.027, 0.026 F L\n-1.0, 0.10, 0.026, \nH l +700 Oe\nrnVPSD\nbiasj 0.36 Hz\nnormalized \nto 1 mA\n(c)\nFIG. 6. Analogous measurement set as in Figs.3-4, for (an otherwise \nidentical) device with a 10A Dy cap layer in direct contact with the FL                                     6 an interfacial antiferromagnetic  coupling of . Here, \nFL1 has a thicker CoFeGe layer than used for prior FL films, \nand FL2 is a relatively thin CoFe layer chosen so that \n ≈ 0.64 erg/cm2. Although \nhaving similar static M-H or R-H characteristics to that of the \nsimple FL (of similar net  product) used in earlier \nmeasurements, the transport of the SF-FL in regard to spin-\ntorque effects in particular is fundamentally distinct. The basic \nphysics of this phenomenon was described in detail previously.13 \nIn summary, a spin-torque induced quasi-coresonance between \nthe two natural oscillation modes of the FL1/FL2 couple in the \ncase of negative   and  , can act to transfer \nenergy out of the mode that is destabilized by spin-torque, \nthereby delaying the onset of criticality and substantially \nincreasing . Indeed, the side-by-side comparison of   loops provided in Fig. 8b indicate a nearly 5 × increase  in \n, despite that  remains virtually unchanged.  2erg/cm 0 . 1 ≅\nFL 2 1 FL ) ( ) ( ) (FL t M t M t Ms s s ≅ −\nt Ms\neI 0 ˆ ˆRL 1 FL> ⋅ m m\n| |crit\nFL P-IeI N-\n| |crit\nFL P-I | |crit\nFL AP-I\n     For the SF-FL devices, attempts at finding the magic-angle \nunder similar measurement conditions as used for Figs. 3c,4c, and 6c were not successful, and so  the extrapolation method at \nsimilar \n4 . 0bias≈ r was used instead. To improve accuracy for \nextrapolated-FLα , the data of Fig. 8c include measurements \nfor mA 3 . 0 | | ≤eI  (so that ) for which electronics noise \noverwhelms the signal from the RF FMR peaks. Showing \nexcellent linearity of over a wide -range, the \nextrapolated intrinsiccrit\nFLI Ie<\neI. vsFL\nfitαeI\n01 . 0FL≈ α  is, as expected, unchanged \nfrom before. The same is true for the extrapolated RLα as well. \n     Table 1 summarizes the mean critical voltages   (less \nsensitive to lithographic variations in actual device area) from a \nlarger set of  measurements. The crit\nFL P-I R−\neI- PSD ×≈3 . 2 increase in \n with the use of the Dy-cap is in good agreement with \nthat of the ratio of measured .  | |crit\nFL P-I R\nFLα\n -1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.5Ω\nH (kOe)(a)\n- 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81\nΓ  l 3.3FLΓ  l 4.4RL\nI  (mA)enV\nHzPSD\n(100 MHz)\n(b)H  l -0.45 kOe  | |H  l +1.2 kOe     | |\n0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.0\nfrequency (GHz)I  = -0.7, -1.0,  -1.3,  -1.8, -2.0mA -1.6 , \nnormalized \nto 1 mArbiasj 0.66H l +400 OenV\nHzPSD\n(c)\nFIG. 7. Analogous measurement set as in Fig. 5 for a different (but \nnominally identical) device as that in Fig. 6 with a 10A Dy cap layer..0.0 0.5 1.0 1.5 2.00.000.050.100.15\nα fitRL\n(d)\n|I  | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj18.2Ω\nH (kOe)(a)\n-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nH  l -0.45 kOe  \nnVPSD\nHz| |\n(100 MHz)\n(b)\n0.01.02.03.04.05.06.0I  = +0.15 , -0.15 , +0.3, -0.3,  -1., -2., -2.5 mA \n0 2 4 6 8 10 14 16 18 12-1.5., \nrbiasj 0.41H l +600 OenVPSD\nHz\nnormalized \nto 1 mA\n0 2 4 6 8 1 01 21 41 61 80.00.20.40.60.81.0\nfrequency (GHz)I  = +0.15 , -0.15 , +0.3 , -0.3,  -1., -2., -2.5 mA -1.5., \n(c)\n0.0 0.5 1.0 1.5 2.0 2.50.000.011 FL\n0.026 /  \n0.011 / 0.011 /  αFL0.050.100.15\nα fit\n|I  | (mA)eRL\n(d)\nFIG.  8.  Analogous m ea surem ent set as in Figs.5, for (an otherwise identica l) \ndevic e with a synthetic-ferrimagnet FL (SF-FL) as described in text. (b)  \nincludes for comparison N-Ieloops (in lighter color) from  Fig. 3b ; arrows\nshow SF-FL Icr itfor P-state (red) and AP-state (blue). (c) spectral data and \nfits are repeatedly shown (for clarity) using two different ordinate scales. \n0.12 44.9 !2.0 SF-FL0.11  10.4 !0.1 Control/ αRLstack \n0.11 24.5 !0.5 Dy cap 0.026 /  \n0.011 / 0.011 /  αFL\n0.11 24.5 !0.5 Dy cap\n0.12 44.9 !2.0 SF-FL0.11  10.4 !0.1 Control/ αRLstack (mV)crit\nFLR I −\nTable. 1. Summary of critical voltages (measured over ≈ 8 devices each) \nand damping parameter values α for the present experiment. Estimated \nstatistical uncertainty in the α-values is ~10%.   IV. MICROMAGNETIC MODELLING  \n \n     For more quantitative comparison with experiment than \nafforded by the 1-macrospin model of Sec. II, a 2-macrospin \nmodel equally treating both  and  is now considered \nhere as a simpler, special case of a more general micromagnetic \nmodel to be discussed below. The values , \n, , and will be used \nas simplified, combined representations (of similar thickness and \n) to the actual CoFe/CoFeGe multilayer films used for the \nRL and FL. The magnetic films are geometrically modeled as 60 \nnm squares which (in the macrospin approximation) have zero \nshape anisotropy (like circles), but allow analytical calculation \nof all magnetostatic interactions. The effect of IrMn exchange \npinning on the RL is simply included as a uniform  field \n with measured . \nFirstly, Fig. 9b shows simulated  and  curves \ncomputed assuming , roughly the mean value found \nfrom the data of Sec. III. The agreement with the shape of \nthe measured  is very good (e.g., Figs. 6,7 in particular), \nwhich reflects how remarkably closely these actual devices \nresemble idealized (macrospin) behavior.  RLˆmFLˆm\nemu/cc 950FL=sM\nnm 7FL=t emu/cc 1250RL=sM nm 5RL=t\nt Ms\nx H ˆ] ) /( [RL pin pin t M Js =2erg/cm 75 . 0pin≅ J\n|| bias H r-⊥H r-bias\n2 . 3= Γ\ncrit\nFLI\nH R-\n     Next, Fig. 9d shows simulated PSD curves  computed \n(see Appendix) in the absence of spin-torque  (i.e., ) (f SV\n) 0ST= H , \nbut otherwise assuming typical experimental values R=19Ω, ΔR/R=9%, and T=300K, as well as  and 01 . 0FL= α 1 . 0RL=α , \nso to be compared with the magic-angle spectra of Figs. 3,4. Since (as stated in Sec. III) th e experimental field angle was not \naccurately known, the field angle  was varied systematically \nfor the simulations, and in each case the field-magnitude H was \niterated until Hφ\n37 . 0bias≅ r , approximately matching the mean \nmeasured value. In terms of both absolute values and the ratio of \nFL to RL FMR peak amplitudes, the location of  \n(particularly for the FL), and the magnitude of H (on average \n650-700 Oe from the three magic-angle data in Sec. III), the best \nmatch with experiment clearly occurs with . \nThe agreement, both qualitatively and quantitatively, is again \nremarkable given the simplicit y of the 2-macrospin model. peakf\no o40 30 ≤ φ ≤H\n     Finally, results from a di scretized micromagnetic model are \nshown in Fig. 10. Based on Fig.9, the value  was fixed, \nand H = 685 Oe was determined by iteration until o35= θH\n37 . 0bias≅ r . \nThe equilibrium bias-point magneti zation distribution is shown \n60 nm\nRL FL(a)\n60 nm\nRL FL(a)\nFIG. 10. Micromagnetic model results. (a) cell discretizations with arrow-\nheads showing magnetization orie nta tion when | H|=685 Oe and φH=35o\n(see Fig. 9c). (b) simulated partial rms PSD for first 7 eigenmodes (as \nlabeled) computed individually with αFL=0.01 andαRL=0. 01, other \nparameter values indicated. ( c) simulated total rms PSD with αFL=0.01 and \nαRL=0.01 (green) or αRL=0.1(red or blue); blue curve excludes \ncontribution from 5th(FL) eigenmode at 16 GHz.                                      7 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nfrequency (GHz)nV\nHzPSD rbiasj 0.37\nΓ =3.2\nα   =0.01FLH   =0STφ   =35HoH=685 Oe\nexclude #5 include # 1-7\nα    =0.01RL\n(c)α    =0.1RL02468 1 0 1 2 1 4 1 6 1 8 2 00.00.51.01.52.02.5\nfrequency (GHz)nV\nHzPSDI=1mA R=19 ΩΔR/R=9%\nΓ =3.2\nα  =0.01  RLT=300K\nH   =0ST(#1)\n(#2)\n(#3)(#4)\n(#5)\n(#6)\n(#7)rbiasj 0.37\nφ   =35HoH=685 Oe\n\"FL\"-mode \"RL\"-mode\nα =0.01  FL\n(b)-1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81\nH (kOe)δR\nΔRΓ = 3.2\n(b)5 nm3nm7 nm\n60 nm60 nm\nRLFL\n(c)(a)\nH\nx\nzymRL\nmFLφH\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nfrequency (GHz)ΔR/R=9% R=19 Ω I=1mA \nφ  =10 , H=802  Oeo\nHφ  =0 , H=895  Oeo\nH\nφ  =20 , H=723  Oeo\nH\nφ  =30 , H=657  Oeo\nH\nφ  =40 , H=605  Oeo\nHα   =0.10RLT=300K\nr H   =0ST j 0.37bias\nnVPSD\nα   =0.01Γ =3.2Hz\nFL\n(d)\nFIG. 9. Two-macrospin model results. (a) cartoon of model geometry.  (b) \nsimulated δR-H loops analogous to data of Figs 3-8c. (c )  cartoon defining \nvector orientations (RL exchange pinned along + x direction). (d) simulated \nrms PSD assuming parameter values indicated, with variable | H| to maintain \na fixed rbiasat each φH( as indicated by color).  in Fig. 10a for this 416 cell model. Estimated values for \nexchange stiffness,  and erg/cm 4 . 1FL μ = A erg/cm 2RL μ=A  \nwere assumed. The simulated spectra in Fig. 10b are shown one \neigenmode at a time  (see Appendix), for the 7 eigenmodes with \npredicted FMR frequencies below 20 GHz (the 8th mode is at \n22.9 GHz). The 1st, 2nd, 5th, and 7th modes involve mostly FL \nmotion, the nearly degenerate 3rd and 4th modes (and the 6th) \nmostly that of the RL. (The amplitudes of  from the 6th or \n7th mode are negligible.). For illustration purposes only, Fig. 10b \nassumed identical damping in each film. ) (f SV\n01 . 0FL FL= α = α\n     For Fig. 10c, the computation of  is more properly \ncomputed using either 6 or all 7 eigenmodes simultaneously, \nwhich includes damping-induced coupling between the modes. \nIncluding higher order modes makes negligible change to \n (but rapidly increases computation time). As \nwas observed earlier, the agreement between simulated and \nmeasured spectra in Figs. 3c,4c is good (with ) (f SV\nGHz) 20 ( <f SV\n1 . 0RL=α ), and \nthe simulations now include the small, secondary FL-peak near  \n8 GHz clearly seen in the measured data (including that of Fig. \n6c), though it is somewhat more pronounced in the model results. \nNotably, the computed spectrum near the RL FMR peak more resembles the measurements after removing the 16-GHz 5\nth \nmode from the calculation, as this (FL) mode does not appear to be physically present in the Figs. 3c,4c spectra. \n     While it is perhaps expected that higher order modes in a \nmicromagnetic simulation assuming perfectly homogeneous \nmagnetic films would show deviations from real devices with finite grain-size, edge-roughness/damage, etc., the situation is \nactually more interesting. Fig. 11 shows measured spectra on yet \nanother device (again, nominally identical to that of Figs. 3-5) in which the experiment was perhaps slightly off from the optimum \nmagic-angle condition, as evidenced by the very small shift in \nthe 6-GHz FL FMR peak position with polarity of . More \nnoteworthy, however, is the clear polarity asymmetry and nonlinear-in-  peak-amplitude (for  in particular) of \nboth the 8-GHz secondary FL mode and a higher order mode \nclose to 15 GHz. (Both resemble typical spin-torque effect at \nangles more antiparallel than the magic-angle condition.) This \nsimilarity in behavior indicates with near certainty that this 15-\nGHz mode is also FL-like in origin, and is thus a demonstration of the \"missing\" 5\nth mode predicted in Fig. 10. (In hindsight, \nthere is now discernible a small but similar 15-GHz peak in the spectra of Fig. 4c). It is worth remembering that the \"magic-\nangle\" argument was based on a simple 1-macrospin model, and so remarkably there appears to be circumstances where this \n\"spin-torque null\" actually does apply simultaneously to both the \nFL and RL, as well as to higher order modes.  \neI\neI 0>eI \nV. DISCUSSION \n \n     In addition to the direct  evidence from the measured spectral \nlinewidth in Figs. 3-8, evidence for large Gilbert damping \nFL RL α> > α  for the RL is also seen in the  data. As ratios \n and  are (from Figs. 3-5 data) both \nroughly ~7, this conclusion is semi-quantitatively consistent with \nthe basic scaling (from (3c)) that . This, as well as the \nsubstantial, 2-3 × variation of  with  in Figs. 5d, and 7d, \nappears to rather conclusively (and expectedly) confirm that \ninhomogeneous broadening is not a factor in the large linewidth-\ninferred values of crit\neI\ncrit crit\nFL P RL P/- -I Icrit crit\nFL AP RL AP /- - I I\nα ∝crit\neI\nRL\nfitαeI\nRLα found in these nanoscale  spin-valves.  \n     Large increases in effective damping of \"bulk\" samples of \nferromagnetic (FM) films in cont act with antiferromagnet (AF) \nexchange pinning layers has been reported previously.14-16 The \nexcess damping was generally attributed to two-magnon scattering processes\n17 arising from an inhomogeneous AF/FM \ninterface. However, the two-magnon description applies to the \ncase where the uniform, ( ,  mode is pumped by a \nexternal rf source to a high excitation ( magnon) level, which \nthen transfers energy via two-magnon scattering into a large  \n(quasi-continuum) number of degenerate 0=k )0ω ≡ ω\n) , 0 (0ω=ω≠k k  \nspin-wave modes, all with low (thermal) excitation levels and \nmutually coupled by the same two-magnon process. In this \ncircumstance, the probability of en ergy transfer back to the \nuniform mode (just one among the degenerate continuum) is \nnegligible, and the resultant one-way flow of energy out of the \nuniform mode resembles that of intrinsic damping to the lattice. \nBy contrast, for the nanoscale  spin-valve device, the relevant \neigenmodes (Fig. 10) are discrete and  generally nondegenerate.  \nin frequency. Even for a coincide ntal case of a quasi-degenerate \npair of modes (e.g., RL modes #3 and #4 in Fig. 10), both modes \nare equally excited to thermal equilibrium levels (as are all \nmodes), and have similar intrinsi c damping rates to the lattice. \nAny additional energy transfer via a two-magnon process should \nflow both ways, making impossible a large (e.g., ~10× ) increase \nin the effective net damping of either mode.  \n0.00.51.01.52.0\nI  = +0.7, -0.7 , +1.0 , +1.4, -1.4 mA \n                                     8     Two alternative hypotheses for large RLα  which are \nessentially independent of device size are 1) large spin-pumping \neffect at the IrMn/RL interface, or 2) strong interfacial exchange coupling at the IrMn/RL resulting in non-resonant  coupling to \nhigh frequency modes in either the RL and/or or the IrMn film.  \nHowever, these two alternatives can be distinguished since the \nexchange coupling strength can be greatly altered without \nnecessarily changing the spin-pumping effect. In particular, \nRLα was very recently measured by conventional FMR methods -1.0, \nnVPSD\nHz\n0 2 4 6 8 1 01 21 41 61 8\nfrequency (GHz)r(normalized \nto 1 mA) j 0.35biasH l +700 Oe\nFIG. 11. The rms PSD measured on a physically diffe rent (but nominally \nidentical) device as that generating the analogous \"magic-angle\" spectra shown in Figs. 3c and 4c. Table. 2. Summary of bulk film FMR measurements18 for reduced film \nstack structure: seed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap.  Removal of IrMn, \nor alternatively a lack of proper seed layer and/or use of a sufficiently thick \ntCu≈30A can each effectively eliminate exchange pinning strength to RL. 0.013 tAF= tCu= 0          \n(out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A ,  tCu= 0                  \n(no seed layer for IrMn)0.011 tAF= tCu= 0sample type\n0.013 tAF= tCu= 0          \n(out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A ,  tCu= 0                  \n(no seed layer for IrMn)0.011 tAF= tCu= 0sample type ) / (23\nRL ω Δ γ = α d H d\nby Mewes18 on bulk film samples (grown by us with the same \nRL films and IrMn annealing procedure as that of the CPP-\nGMR-SV devices reported herein) of the reduced stack structure: \nseed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap. For all four cases \ndescribed in Table 2, the exchange coupling was deliberately \nreduced to zero, and the measured  was found to be \nnearly identical to that found here for the FL of similar CoFeGe \ncomposition. However, for the two cases with tAF = 60A, excess \ndamping due to spin-pumping of electrons from RL into IrMn \nshould not have been diminished (e.g., the spin diffusion length \nin Cu is ~100 × greater than tCu ≈ 30A). This would appear to rule \nout the spin-pumping hypothesis. 012 . 0RL≈ α\n    The second hypothesis emphasizes the possibility that the \nenergy loss takes place inside the IrMn, from oscillations excited \nfar off resonance by locally strong interfacial exchange coupling \nto a fluctuating . This local interfacial exchange coupling \n can be much greater than , since the latter reflects a \nsurface average over inhomogeneous spin-alignment (grain-to-\ngrain and/or from atomic roughness) within the IrMn sub-lattice that couples to the RL. Further, though such strong but \ninhomogeneities coupling cannot truly be represented by a \nuniform  acting on the RL, the similarity between \nmeasured and modeled values of ~14 GHz for the \"uniform\" RL \neigenmode has clearly been demonstrated here. Whatever are the natural eigenmodes of the real device, the magic-angle spectrum \nmeasurements of Sec. III reflect the thermal excitation of all \neigenmodes for which \"one-way\" intermodal energy transfer should be precluded by the condition of thermal equilibrium and \nthe orthogonality\n19 of the modes themselves. Hence, without an \nadditional energy sink exclusive  of the RL/FL spin-lattice system, \nthe linewidth of all modes should arguably reflect the intrinsic Gilbert damping of the FL or RL films, which the data of Sec. III \nand Table 2 indicate are roughly equal with . Inclusion \nof IrMn as a combined AF/RL system, would potentially provide \nthat extra energy loss channel for the RL modes. RLˆm\nexJpinJ\npinH\n01 . 0 ~ α\n                                     9      A rough plausibility argument for the latter may be made with a crude AF/FM model in which a 2-sublattice AF film is \ntreated as two ferromagnetic layers (#1 and #2) occupying the same  physical location. Excluding magnetostatic contributions, \nthe free energy/area for this 3-macrospin system is taken to be  \n x m m mx m x m m m\nˆ ˆ ] [ ˆ ˆ] )ˆ ˆ ( ) ˆ ˆ [( ) ( ˆ ˆ ) (\nFM FMAF AF AF\npin 0 2 ex2\n22\n1 212 1\n⋅ − + ⋅ −⋅ + ⋅ − ⋅\nJ J Jt K H t Ms   (8) \n \nFor IrMn with Neel temperature of , the internal AF \nexchange field .20 With K T700N≈\nOe 10 ~ / ~7\nB B AFμNT k H A, 60AF=t  \nAF uniaxial anisotropy is estimated to be .21 A \nrough estimate  for strong  interfacial exchange \nis obtained by equating interface energy erg/cc 10 ~6\nAFK\nFM) / ( 8 ~ex t A J\n2 /2\nexφJ  to the bulk \nexchange energy t A/ 42φ  of a hypothetical, small angle Bloch \nwall ) 2 0 (φ ≤ φ ≤ twisting through the FM film thickness. \nTaking nm 5≈t  and A ~ 10-6 erg/cm yields . \nThe value of  in the last \"field-like\" \nterm  in (8) is more precisely chosen to maintain a constant \neigenfrequency for the FM layer independent  of  or , \nthus accounting for the weaker inhomogeneous coupling averaged over an actual AF/FM interface.  2\nex erg/cm 15 ~ J\n1\nex 0 ] ) ( / 1 / 1 [ ~AF−+ t K J J\nexJAFK\n     As shown in Fig. 12, this crude model can explain a ~10 × \nincrease in the FM linewidth provided  á 5-  and  exJ2erg/cm 10\n1 . 0 05 . 0 ~AF - α . It is worthily noted20 that for the 2-sublattice \nAF, the linewidth ) ) / ( /( 2 /AF AF AF 0 sM K H HK≡ α ≈ ω ω Δ  is \nlarger by a factor of 100 ~ / 2AF KH H  compared to high order \nFM spin-wave modes in cases of comparable α and 0ω (with \nHz 10 ~ 212\n0 AF KH H γ ≈ ω  for the AF). Since the lossy  part \nof the \"low\" frequency susceptibility for FM or AF modes scales \nwith ωΔ, it is suggested that the IrMn layer can effectively sink \nenergy from the ~14 GHz RL mode despite the ~100 × disparity \nin their respective resonant frequencies. S ize-independent  \ndamping mechanisms for FM films exchange-coupled to AF \nlayers such as IrMn are worthy of further, detailed study.  \n0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05\nfrequency (GHz)T=300K\n(GHz)-1/2α   =0.01  FMSθFMJ   =0α   = 0.02,\n          0.05, 0.10  0.01, exAF\nerg\ncm2 J   =10ex\n0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05\nfrequency (GHz)T=300K\nα   =0.01  FMJ   =0exerg 1, 3, 10, 100cm2 J   =ex\nS α   =0.10  θFM\nAF\n(GHz)-1/2(a)\n(b)\nFIG.  12. Simulated rms PSD SθFM(f) for a 3-macrospin model of an AF/FM \ncouple as described via (8) and in the text. The FM film parametrics are the \nsame as used for macrospin RL model in Fig. 9, with  αFM=0.01 and       \nJpin= 0.75 erg/cm2.  (a) varied α AF(denoted by color) with  Jex=10erg/cm2.  \n(b) varied  Jex(denoted by color) with αAF=0.1. The black curve in (a) or \nin (b) corresponds to Jex=0. For AF, Msis taken to be 500 emu/cc. \nACKNOWLEDGMENTS \n \nThe authors wish to acknowledge Jordan Katine for the e-beam \nlithography used to make all the measured devices, and Stefan \nMaat for film growth of alternative CPP spin-valve stacks useful \nfor measurements not included here. The authors wish to thank Tim Mewes (and his student Zachary Burell) for making the \nbulk film FMR measurements on rather short notice. One author \n(NS) would like to thank Thomas Schrefl for a useful suggestion for micromagnetic modeling of an AF film. \n \nAPPENDIX \n \n     As was described in detail elsewhere,22 the generalization of  \n(1) or (5) from a single macrospin to that for an N-cell \nmicromagnetic model takes the form \n \n1)] ( [ ) ( ,2) () ( ) (\n−+ ω − ′ = ω ⋅ ⋅Δ γ≡ ω′=′⋅′+′⋅ +\nG D H D Sh m HmG D\ntt t t ttt trrtrtt\nimT ktdtd\nBχ χ χ@    (A1) \n \nwhere  m′r) (orh′r\n is an  column vector built from the N \n2D vectors , and 1 2×N\nN j... 1=′m H G Dttt\nand , ,  are  matrices \nformed from the  array of 2D tensors N N2 2×\nN N× ,jkDt\n ,jkGt\nand \n Here, and , though .jkHt\njk jk D Dδ =t t\njk jkG Gδ =t t\n.jkHt\nis \nnonlocal  in cell indices j,k  due to the magnetost atic interaction.  \n      The PSD  for any scalar quantity is22 ) (f SQ })ˆ({j Qm\n \nj jj\njN\nk jk jk j QQS f Sm mm\nd d dˆ ˆ, ) ( 2 ) (\n1 , ∂∂⋅∂′∂\n≡′ ′ ⋅ ω ⋅ ′ =∑\n=t\n            (A2) \nThe computations for the PSD of Figs.9, 10 took ) (mrQ  to be \n \n∑\n= ⋅ − Γ + + Γ⋅ − Δ=iN\ni i ii i\niR I\nNQ\n1bias\nFL RLFL RL\nˆ ˆ ) 1 ( 1)ˆ ˆ 1 ( 1\nm mm m \n \naveraged over the  cell pairs at the RL-FL interface. 2 /N Ni=\n     For a symmetric  Ht\n (e.g., the set of eigenvectors ), 0ST= H\nm err←  of the system (A1) can be defined from the following \neigenvalue matrix equation \n \nn n N n ie e H Gr rtt\nω = ⋅ ⋅=−\n2 ... 11) (                       (A3) \n \nThe eigenvectors come in N complex conjugate pairs − +e err, \nwith real eigenfrequencies . With suitably normalized ω ± ,ner \nmatrices  and  are diagonal in the \neigenmode basis .22. The analogue to (A1) becomes  mn mn H δ =n mn mni G ω δ =/\n ∑∑\n′⋅ ≡ ′ ′ ω ′ =ω χ ω χΔ γ≡ ω⋅ ⋅ ≡ ω − δ ω ω − = ω χ\n∗ ∗∗\n′\n′ ′′ ′ ′∗ −\nn mn n n mn m Qn n\nn mn m m mB\nmnn m mn mn n mn\nd d S d f SDmT kSD i\n,,1\n, ) ( 2 ) () ( ) (2) () ( ) / 1 ( )] ( [\nd ee D e\nrrrtr\n     (A4) \n \nThe utility of eigenmodes for computing PSD, e.g, in the \ncomputations of Fig. 10, is that only a small  fraction (e.g., 7 \nrather than 416 eigenvector pairs) need be kept in (A4) (with all \nthe rest simply ignored ) in order to obtain accurate results in \npractical frequency ranges (e.g., GHz). Despite that  \nis (in principle) a full matrix, the reduction in matrix size for the \nmatrix inversion to obtain  at each frequency  more than \nmakes up for the cost of computing the 20<mnD\n) (ω χ\n) , (n nerω  which need be \ndone only once independent of frequency or α-values.  \n \n \nREFERENCES  \n1 Volume 320 (2008) of the Journal of Magnetism and Magnetic \nMaterials offers a series of review articles (with many additional \nreferences therein) covering this field. Some examples include: \n   J. A. Katine and E. E. Fullerton, pp. 1217-1226, \n   J. Z. Sun and D. C. Ralph, pp. 1227-1237, \n   T. J. Silva  and W. H. Rippard, pp. 1260-1271. \n2Y. Tserkovnyak, A. Brataas, and G.  E. W. Bauer, Phys. Rev. B \n  67, 140404(R) (2003). \n3G. D. Fuchs et al. , Appl. Phys. Lett. 91, 062505 (2007). \n4 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, \n   IEEE Trans. Magn. 41, 2935 (2005). \n5 As discussed in N. Smith, Phys. Rev. B 80, 064412 (2009), the spin-\npumping effect described in Ref. 2 l eads to additional contributions to \nthe damping tensor-matrix Dt\n in (1) or (A1) which are anisotropic, \nangle-dependent, and nonlocal betw een RL and FL. For simplicity, \nthese are here simply lumped into the Gilbert damping parameter α \nfor either RL or FL. In this c ontext, \"intrinsic \" damping refers \napproximately to that in the linit , but also in the presence of \nthe complete stack structure of the spin-valve device of interest. 0→eI\n6  J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002). \n7 N. Smith, J. Appl. Phys. 99, 08Q703 (2006). \n8 Y. Tserkovnyak, A. Brataas, G. E. W.  Bauer,  and B. I. Halperin,  \n   Rev. Mod. Phys. 77, 1375 (2005). \n9 S. Maat, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. \n   93, 143505 (2008). \n10 R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald, \n    Phys. Rev. B 75, 214420 (2007). \n11 G. Woltersdorf, M. Kiessling, G. Meye r, J. U. Thiele, and C. H. Back, \n    Phys. Rev. Lett. 102, 257602 (2009). \n12 S. Maat, N. Smith, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. \n    93 , 103506 (2008). \n13 N. Smith, S. Maat, M. J. Carey, and J. R. Childress, Phys. Rev. Lett. \n    101 , 247205, (2008). \n14 R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., \n    J. Appl. Phys.,  83, 7037 (1998). \n15 S. M. Rezende, A. Azevedo, M. A. Lucena, and F. M. de Aguiar, \n    Phys. Rev. B 63, 214418 (2001). \n16 M. C. Weber, H. Nembach, B. Hillebrands, M. J. Carey, and \n    J. Fassbender, J. Appl. Phys. 99, 08J308 (2006). \n17  R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). \n                                     10 18 T. Mewes, MINT  Center, U. Alabama (Tuscaloosa). These bulk film \nFMR measurements were made in-plane, except for the last entry in \nTable 2 which was measured in th e out-of-plane configuration which \nshould exclude two-magnon contributions  (Ref. 17). The similarity in \nα -values suggests two-magnon is unim portant here, which in turn \nthen suggests that RL-FL spin-pumping contributions in the present \nCPP-GMR nanopillars spin-valves (footnote (5)) that would not be \npresent in the RL-only bulk samples are also small. Further \ncomparisons of present measuremen ts with additional bulk film FMR \nresults are expected to be addressed in a future publication.  19 A nonzero damping matrix Dt\n can technically cause a coupling of \nnondegenerate eigenmodes if , though this does not \nchange the argument citing this f ootnote. This small effect can be \nseen in Fig. 10c as the slight change in the 6-GHz FL FMR peak \nheight and linewidth when 0≠ ⋅ ⋅∗≠ n n me D ertr\nRLαis grossly varied from 0.01 to 0.1.  \n20 Magnetic Oscillations and Waves , A. G. Gurevich and G. A. Melkov, \nCRC Press (Florida, USA) 1996. Chap. 3. \n21 K. O'Grady, L. E. Ferndandez- Outon, G. Vallejo-Fernandez,  \n    J. Magn. Magn. Mater. 322, 883 (2010). \n22 Appendix A of N. Smith, J. Magn. Magn. Mater.  321, 531 (2009). \n \n \n                                     11 "    },    {        "title": "1002.4958v1.Correlation_Effects_in_the_Stochastic_Landau_Lifshitz_Gilbert_Equation.pdf",        "content": "arXiv:1002.4958v1  [cond-mat.mes-hall]  26 Feb 2010Correlation Effects in Stochastic Ferromagnetic Systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: June 16, 2018)\nAbstract\nWe analyze the Landau-Lifshitz-Gilbert equation when the p recession motion of the magnetic\nmoments is additionally subjected to an uniaxial anisotrop y and is driven by a multiplicative cou-\npled stochastic field with a finite correlation time τ. The mean value for the spin wave components\noffers that the spin-wave dispersion relation and its damping is strongly influenced by the deter-\nministic Gilbert damping parameter α, the strength of the stochastic forces Dand its temporal\nrangeτ. The spin-spin-correlation function can be calculated in t he low correlation time limit by\nderiving an evolution equation for the joint probability fu nction. The stability analysis enables us\nto find the phase diagram within the α−Dplane for different values of τwhere damped spin wave\nsolutions are stable. Even for zero deterministic Gilbert d amping the magnons offer a finite life-\ntime. We detect a parameter range wherethe deterministic an d the stochastic damping mechanism\nare able to compensate each other leading to undamped spin-w aves. The onset is characterized by\na critical value of the correlation time. An enhancement of τleads to an increase of the oscillations\nof the correlation function.\nPACS numbers: 75.10.Hk, 05.40.-a, 75.30.Ds,72.70.+m,76.60.Es\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nMagnetism can be generally characterized and analyzed on different length and time scales.\nThe description of fluctuations of the magnetization, the occurre nce of damped spin waves\nand the influence of additional stochastic forces are successfully performed on a mesoscopic\nscale where the spin variables are represented by a continuous spa tio-temporal variable [1].\nIn this case a well established approach isbased uponthe Landau-L ifshitz equation [2] which\ndescribes the precession motion of the magnetization in an effective magnetic field. This\nfield consists of a superposition of an external field and internal fie lds, produced by the in-\nteracting magnetic moments. The latter one is strongly influenced b y the isotropic exchange\ninteraction and the magnetocrystalline anisotropy, for a recent r eview see [3]. The studies\nusing this frame are concentrated on different dynamical aspects as the switching behav-\nior of magnetic nanoparticles which can be controlled by external tim e-dependent magnetic\nfields [4] and spin-polarized electric currents [5, 6]. Such a current- induced spin transfer\nallows the manipulation of magnetic nanodevices. Recently, it has bee n demonstrated that\nan electric current, flowing through a magnetic bilayer, can induce a coupling between the\nlayers [7]. Likewise, such a current can also cause the motion of magn etic domain walls in\na nanowire [8]. Another aspect is the dynamical response of ferrom agnetic nanoparticles\nas probed by ferromagnetic resonance, studied in [9]. In describing all this more complex\nbehavior of magnetic systems, the Landau-Lifshitz equation has t o be extended by the in-\nclusion of dissipative processes. A damping term is introduced pheno menologically in such a\nmanner, that the magnitude of the magnetization /vectorSis preserved at any time. Furthermore,\nthe magnetization should align with the effective field in the long time limit. A realization\nis given by [2]\n∂S\n∂t=−γ[S×Beff]−ε[S×(S×Beff)]. (1)\nThe quantities γandεare the gyromagnetic ratio and the damping parameter, respectiv ely.\nAn alternative equation for the magnetization dynamics had been pr oposed by Gilbert [10].\nThe Gilbert equation yields an implicit form of the evolution of the magne tization. A com-\nbination of both equations, called Landau-Lifshitz-Gilbert equation (LLG) will be used as\nthe basic relation for our studies, see Eq. (2). The origin of the dam ping term as a non-\nrelativistic expansion of the Dirac equation has been discussed in [11 ] and a generalization\nof the LLG for conducting ferromagnetics is offered in [12]. The form of the damping seems\n2to be quite general as it has been demonstrated in [13] using symmet ry arguments for fer-\nroelectric systems.\nAs a new aspect let us focus on the influence of stochastic fields. Th e interplay between\ncurrent and magnetic fluctuations and dissipation has been studied recently in [14]. Via the\nspin-transfer torque, spin-current noise causes a significant en hancement of the magnetiza-\ntion fluctuations. Such a spin polarized current may transfer mome ntum to a magnet which\nleads to a spin-torque phenomenon. The shot noise associated with the current gives rise to\na stochastic force [15]. In our paper we discuss the interplay betwe en different dissipation\nmechanism, namely the inherent deterministic damping in Eq. (1) and t he stochastic mag-\nnetic field originated for instance by defect configurations giving ris e to a different coupling\nstrength between the magnetic moments. Assuming further, tha t the stochastic magnetic\nfield is characterized by a finite correlation time, the system offers m emory effects which\nmight lead to a decoherent spin precession. To that aim we analyze a f erromagnet in the\nclassical limit, i.e., the magnetic order is referred to single magnetic at oms which occupy\nequivalent crystal positions, and the mean values of their spins exh ibit a parallel orientation.\nThe last one is caused by the isotropic exchange interaction which will be here supplemented\nby a magneto-crystalline anisotropy that defines the direction of t he preferred orientation.\nEspecially, we discuss the influence of an uniaxial anisotropy. The co upling between differ-\nent dissipation mechanisms, mentioned above, leads to pronounced correlations, which are\ndiscussed below. Due to the multiplicative coupling of the stochastic fi eld and the finite\ncorrelation time the calculation of the spin-spin correlation function is more complicated.\nTo that aim we have to derive an equivalent evolution equation for the joint probability\ndistribution function. Within the small correlation time limit this approa ch can be fulfilled\nin an analytical manner. Our analysis is related to a recent paper [16] in which likewise the\nstochastic dynamics of the magnetization in ferromagnetic nanopa rticles has been studied.\nFurther, we refer also to a recent paper [17] where the mean first passage time and the\nrelaxation of magnetic moments has been analyzed. Different to tho se papers our approach\nis concentrated on the correlation effects in stochastic system wit h colored noise.\nOur paper is organized as follows: In Sec.II we discuss the LLG and ch aracterize the ad-\nditional stochastic field. The equations for the single and the two pa rticle joint probability\ndistribution are derived in Sec.III. Using these functions we obtain t he mean value of the\nspin wave variable and the spin-spin correlation function. The phase diagram, based on the\n3stability analysis, is presented in Sec.IV. In Sec.V we finish with some co nclusions.\nII. MODEL\nIn order to develop a stochastic model for the spin dynamics in ferr omagnetic systems let\nus first consider the deterministic part of the equation of motion. W e focus on a description\nbased upon the level of Landau-Lifshitz phenomenology [2], for a r ecent review see [3]. To\nfollow this line we consider a high spin systems in a ferromagnet sufficien tly below the\nCurie temperature. In that regime the dynamics of the magnet are dominated by transverse\nfluctuationsofthespatio-temporalvaryinglocalmagnetization. Theweakexcitations, called\nspin waves or magnons, are determined by a dispersion relation, the wavelength of which\nshould be large compared to the lattice constant a, i.e., the relation q·a≪1 is presumed to\nbe satisfied, where qis the wavenumber. In this limit the direction of the spin varies slowly\nwhile its magnitude |S|=msremains constant in time. A proper description for such a\nsituation is achieved by applying the Landau-Lifshitz-Gilbert equatio n (LLG) [4, 10, 18].\nThe spin variable is represented by S=msˆ n, whereˆ n(r,t) is a continuous variable which\ncharacterizes the local orientation of the magnetic moment. The e volution equation for that\nlocal orientation reads\n∂ˆ n\n∂t=−γ\n1+α2ˆ n×[Beff+α[ˆ n×Beff]]. (2)\nThe quantities γandαare the gyromagnetic ratio and the dimensionless Gilbert damping\nparameter, respectively, where αis related to εintroduced in Eq. (1). Beffis the effective\nmagnetic field that drives the motion of the spin density. Generally, it consists of an internal\npart originated by the interaction of the spins and an external field . This effective field is\nrelated to the Hamiltonian of the system by functional variation with respect to ˆ n\nBeff=−m−1\nsδH\nδˆ n. (3)\nIn absence of an external field the Hamiltonian can be expressed as [19, 20]\nH=/integraldisplay\nd3r{wex+wan},with\nwex=1\n2msκ(∇ˆ n)2andwan=1\n2msΓ sin2θ.(4)\nThereby, the constants κand Γ denote the exchange energy density and the magneto-\ncrystalline anisotropy energy density. To be more precise, κ∝Ja2,Jbeing the coupling\n4strength that measures theinteraction between nearest neighb ors inthe isotropic Heisenberg\nmodel [21]. Once again ais thelattice constant. Notice that the formof theexchange ener gy\nin the Hamiltonian (4) arises from the Heisenberg model in the classica l limit. The quantity\nθrepresents the angle between ˆ nand the anisotropy axis ˆν= (0,0,1), where ˆνpoints\nin the direction of the easy axis in the ground state in the case of zer o applied external\nfield. Thus, the constant Γ >0 characterizes anisotropy as a consequence of relativistic\ninteractions (spin-orbital and dipole-dipole ones [20]). In deriving Eq . (4) we have used\nˆ n2= 1. Although it is more conventional to introduce the angular coord inates (θ,Φ) [2, 4],\nwe find it more appropriate to use Cartesian coordinates. To proce ed, we divide the vector\nˆ ninto a static and a dynamic part designated by µandϕ, respectively. In the linearized\nspin wave approach let us make the ansatz\nˆ n(r,t) =µ(r)+ϕ(r,t) =µˆν+ϕ, µ= const., (5)\nwhereˆ n2= 1 is still valid. The effective field can now be obtained from Eqs. (3) an d (4).\nThis yields\nBeff=κ∇2ϕ−Γϕ′;ϕ′= (ϕ1,ϕ2,0). (6)\nEq. (2) together with Eqs. (3) and (4) represent the determinist ic model for a classical\nferromagnet. In order to extent the model let us supplement the effective magnetic field in\nEq. (6) by a stochastic component yielding an effective random field Beff=Beff+η(t). The\nstochastic process η(t) is assumed to be Gaussian distributed with zero mean and obeying\na colored correlation function\n˜χij(t,t′) =∝an}b∇acketle{tηi(t)ηj(t′)∝an}b∇acket∇i}ht=˜Dij\n˜τijexp/bracketleftbigg\n−|t−t′|\n˜τij/bracketrightbigg\n. (7)\nHere,˜Dijand ˜τijare the noise strength and the finite correlation time of the noise η.\nDue to the coupling of the effective field to the spin orientation ˆ nthe stochastic process\nis a multiplicative one. Microscopically, such a random process might be originated by\na fluctuating coupling strength for instance. The situation associa ted with our model is\nillustrated in Fig. 1 and can be understood as follows: The stochastic vector fieldη(t) is able\nto change the orientation of the localized moment at different times. Therefore, fixed phase\nrelations between adjacent spins might be destroyed. Moreover, theη(tk) are interrelated\ndue to the finite correlation time τ. The anisotropy axis defines the preferred orientation of\nthe mean value of magnetization. Due to the inclusion of η(t) the deterministic Eq. (2) is\n5xyz\nanisotropy axis ˆν\nexchange ∝Jaη(t1)η(t2)η(t3)random field at\ndifferent times ti\nFIG. 1. Part of a ferromagnetic domain influenced by stochast ic forces for the example of cubic\nsymmetry with lattice constant a. The black spin in the center only interacts with its nearest\nneighbors (green), where Jis a measure for the exchange integral.\ntransformed into the stochastic LLG. Using Eq. (5) it follows\n∂ϕ\n∂t=−γ\n1+α2(µ+ϕ)×[Beff+α[(µ+ϕ)×Beff]]. (8)\nThe random magnetic field is defined by\nBeff=κ∇2ϕ−Γϕ′+η(t), (9)\nwhereϕ′is given in Eq. (6). With regard to the following procedure we suppose the random\nfield to be solely generated dynamically, i.e., ˆ n×η(t) =ϕ×η(t). So far, the dynamics\nof our model (Eqs. (8) and (9)) are reflected by a nonlinear, stoc hastic partial differential\nequation (PDE). Using Fourier transformation, i.e., ψ(q,t) =F{ϕ(r,t)}and introducing\nthe following dimensionless quantities\nβ= (l0q)2+1, l2\n0=κ\nΓ, ω=γΓ,¯t=ωt ,λ(t) =η(t)\nΓ,(10)\nthe components ψi(q,t) fulfill the equation\nd\ndtψi(q,t) = Ωi(ψ(q,t))+Λ ij(ψ(q,t))λj(t). (11)\n6The quantity l0is the characteristic magnetic length [22]. The vector Ωand the matrix Λ\nare given by\nΩ=ξµβ\n−(αµψ1+ψ2)\nψ1−αµψ2\n0\n, ξ=1\n1+α2, (12)\nand\nΛ =ξ\nαµψ3ψ3−(ψ2+αµψ1)\n−ψ3αµψ3ψ1−αµψ2\nψ2−ψ1 0\n. (13)\nFor convenience we have substituted ¯t→tagain. The statistical properties of λ(t) are\nexpressed as ∝an}b∇acketle{tλ(t)∝an}b∇acket∇i}ht= 0 and\nχkl(t,t′) =∝an}b∇acketle{tλk(t)λl(t′)∝an}b∇acket∇i}ht=Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\nτkl→0− −− →2Dklδklδ(t−t′).(14)\nIncidentally, in the limit τ→0 the usual white noise properties are recovered. We empha-\nsize that although we regard the long-wavelength limit ( a·q≪1), wave vectors for which\nl0·q≫1 (in Eq. (10)) can also occur [22]. But this case is not discussed in the present\npaper and will be the content of future work. Whereas, in what follo ws we restrict our\nconsiderations to the case q→0 so that, actually, l0·q≪1 is fulfilled. Hence, we can set\nβ= 1 approximately in Eq. (10). Due to the anisotropy the spin wave dis persion relation\noffers a gap at q= 0. Owing to this fact ψis studied at zero wave vector. For this situation\nthe assumption of a space-independent stochastic force ηi(t), compare Eq. (7), is reasonable.\nFor non-zero wave vector the noise field should be a spatiotempora l fieldηi((r,t). Because\nour model is based on a short range interaction we expect that the corresponding noise\ncorrelation function is δ-correlated, i.e. instead of (14) we have\nχkl(r,t;r′,t′) =Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\n2Mδ(r−r′),\nwhereMis the strength of the spatial correlation. Using this relation we are able to study\nalso the case of small qwhich satisfies l0·q≪1. In the present paper we concentrate on\nthe case of zero wave vector q= 0.\n7III. CORRELATION FUNCTIONS\nIn the present section let us discuss the statistical behavior of th e basic Eqs. (11)-(14). They\ndescribe a non-stationary, non-Markovian process attributed t o the finite correlation time.\nDue to their common origin both characteristics can not be analyzed separately. In the\nlimitτ→0, Eq. (11) defines a Markovian process which provides also station arity by an\nappropriate choice of initial conditions [23]. However, the present s tudy is focused on the\neffectofnonzerocorrelationtimes. Tothatpurposeweneedapro perprobabilitydistribution\nfunction which reflects the stochastic process defined by Eqs. (1 1)-(14). In deriving the\nrelevant joint probability distribution function we follow the line given in [24], where the\ndetailedcalculationshadbeencarriedout, seealsothereferences citedtherein. Inparticular,\nit has been underlined in those papers that in order to calculate corr elation functions of type\n∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hta single probability distribution function P(ψ,t) is not sufficient. Instead of\nthat one needs a joint probability distribution of the form P(ψ,t;ψ′,t′). Before proceeding\nlet us shortly summarize the main steps to get the joint probability dis tribution function.\nTo simplify the calculation we assume τkl=τδklandDkl=Dδkl. Notice that our system\nhas no ergodic properties what would directly allow us to relate the st ochastic interferences\nwith temperature fluctuations by means of a fluctuation-dissipatio n theorem. Based on\nEq. (11) the appropriate joint probability distribution is defined by [2 4, 25], for a more\ngeneral discussion compare also [26]:\nP(ψ,t;ψ′,t′) =∝an}b∇acketle{tδ(ψ(t)−ψ)δ(ψ(t′)−ψ′)∝an}b∇acket∇i}ht. (15)\nHere the average is performed over all realizations of the stochas tic process. In defining the\njoint probability distribution function we follow the convention to indic ate the stochastic\nprocess by the function ψ(t) whereas the quantity without arguments ψstands for the\nspecial values of the stochastic variable. These values are even re lalized with the probaility\nP(ψ,t;ψ′,t′). The equation of motion for this probability distribution reads acco rding to\n8[24]\n∂\n∂tP(ψ,t;ψ′,t′)\n=−∂\n∂ψit/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t)\nδλk(t1)/bracketrightbigg\nψ(t)=ψ·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1\n−∂\n∂ψ′it′/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t′)\nδλk(t1)/bracketrightbigg\nψ(t′)=ψ′·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1,(16)\nwhere Novikov’s theorem [27] has been applied. Expressions for the response functions\nδψi(t)/δλk(t1) andδψi(t′)/δλk(t1) can be found by formal integration of Eq. (11) and\niterating the formal solution. After a tedious but straightforwar d calculation including the\ncomputation of the response functions to lowest order in ( t−t1) and (t′−t1) and the\nevaluation of several correlation integrals referring to χklfrom Eq. (14), Eq. (16) can be\nrewritten in the limit of small correlation time τas\n∂\n∂tPs(ψ,t;ψ′,t′) =/braceleftbig\nL0(ψ,τ)\n+exp[−(t−t′)/τ]D∂\n∂ψiΛik(ψ)∂\n∂ψ′\nnΛnk(ψ′)/bracerightbigg\nPs(ψ,t;ψ′,t′).(17)\nThereby, transient terms and terms of the form ∝τexp[−(t−t′)/τ] (these terms would lead\nto terms of order τ2in Eq. (22)) have been neglected. The result is valid in the stationary\ncase characterized by t→ ∞andt′→ ∞but finites=t−t′. In Eq. (17) L0is the operator\nappearing in the equation for the single probability density. Following [2 4, 28] the operator\nreads\nL0(ψ,τ) =−∂\n∂ψiΩi(ψ)+∂\n∂ψiΛik(ψ)∂\n∂ψn/braceleftigg\nD/bracketleftbig\nΛnk(ψ)−τMnk(ψ)/bracketrightbig\n+D2τ/bracketleftbigg\nKnkm(ψ)∂\n∂ψlΛlm(ψ)+1\n2Λnm(ψ)∂\n∂ψlKlkm(ψ)/bracketrightbigg/bracerightigg\n,(18)\nwith\nMnk= Ωr∂Λnk\n∂ψr−Λrk∂Ωn\n∂ψr\nKnlk= Λrk∂Λnl\n∂ψr−∂Λnk\n∂ψrΛrl.(19)\nThe equation of motion for the expectation value ∝an}b∇acketle{tψi∝an}b∇acket∇i}htscan be evaluated from the single\nprobability distribution in the stationary state\n∂\n∂tPs(ψ,t) =L0Ps(ψ,t). (20)\n9One finds\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi∝an}b∇acket∇i}hts+D/angbracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/angbracketrightbigg\ns−D2τ/braceleftigg/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/angbracketrightbigg\ns/bracerightigg\n.(21)\nThe knowledge of the evolution equation of the joint probability distr ibutionP(ψ,t;ψ′,t′)\ndue to Eqs. (17) and (18) allows us to get the corresponding equat ion for the correlation\nfunctions. Following again [24], it results\nd\ndt∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi(ψ(t))ψj(t′)∝an}b∇acket∇i}hts+D/angbracketleftbigg/bracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n−D2τ/braceleftigg/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns/bracerightigg\n+Dexp/bracketleftbigg\n−t−t′\nτ/bracketrightbigg\n∝an}b∇acketle{tΛik(ψ(t))Λjk(ψ(t′))∝an}b∇acket∇i}hts,(22)\nwhere the symbol [ ...]tdenotes the quantity [ ...] at timet. As mentioned above the result\nis valid for t, t′→ ∞whiles=t−t′>0 remains finite. The quantities MnkandKklmare\ndefined in Eq. (19). The components Ω iand Λ ijare given in Eqs. (12) and (13). Performing\nthe summation over double-indices according to Eqs. (21) and (22) we obtain the evolution\nequations for the mean value and the correlation function\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t)∝an}b∇acket∇i}hts, (23)\nand\nd\ndsCij(s) =d\nds∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t′+s)ψj(t′)∝an}b∇acket∇i}hts\n+Dexp/bracketleftig\n−s\nτ/bracketrightig\n∝an}b∇acketle{tΛik(ψ(t′+s))Λjk(ψ(t′))∝an}b∇acket∇i}hts.(24)\nNotice, that in the steady state one gets Cij(t,t′) =Cij(s) withs=t−t′. The matrix\ncomponents of Gikare given by\nGik=\n−A1A20\n−A2−A10\n0 0 −A3\n, (25)\n10where\nA1=−D2τ(6µ2α2−1)ξ4+2µ2αDτξ3−D(µ2α2−2)ξ2+µ2αξ\nA2=1\n2µαD2τ/parenleftbig\n11−3µ2α2/parenrightbig\nξ4+µDτ/parenleftbig\nµ2α2−1/parenrightbig\nξ3+3µDαξ2−µξ\nA3= +D2τ/parenleftbig\n3µ2α2+1/parenrightbig\nξ4−4µ2αDτξ3+2Dξ2,(26)\nandξis defined in Eq. (12). At this point let us stress that in the case t′= 0 the term\n∝exp[−(t−t′)/τ] on the rhs. in Eqs. (22) and (24), respectively, would vanish in the steady\nstate, i.e.\n∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts∝ne}ationslash=∝an}b∇acketle{tψi(s)ψj(0)∝an}b∇acket∇i}hts.\nTheoccurrenceofsuchatermisastrongindicationforthenon-st ationarityofourmodel. An\nexplicit calculation shows, that in general this inequality holds for non -stationary processes\n[23].\nIV. RESULTS\nThe solution of Eq. (23) can be found by standard Greens function methods and Laplace\ntransformation. As the result we find\n∝an}b∇acketle{tψ(t)∝an}b∇acket∇i}hts=\ne−A1tcos(A2t)e−A1tsin(A2t) 0\n−e−A1tsin(A2t)e−A1tcos(A2t) 0\n0 0 e−A3t\n·∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts, (27)\nwhere∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts=∝an}b∇acketle{tψ(t= 0)∝an}b∇acket∇i}htsare the initial conditions. The parameters A1,A3andA2defined\nin Eqs. (26) play the roles of the magnon lifetime and the frequency o f the spin wave at\nzero wave vector, respectively. As can be seen in Eq. (26) all of th ese three parameters are\naffected by the correlation time τand the strength Dof the random force. Moreover, the\nGilbert damping parameter αinfluences the system as well. The solution of Eq. (24) for\nthe correlation function in case of t′= 0 is formal identical to that of Eq. (27). The more\ngeneral situation t′∝ne}ationslash= 0 allows no simple analytic solution and hence the behavior of the\ncorrelation function C(s) is studied numerically. In order to analyze the mean values and\nthe correlation function let us first examine the parameter range w here physical accessible\nsolutions exist. In the following we assume ∝an}b∇acketle{tψ1(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ2(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ0∝an}b∇acket∇i}htand∝an}b∇acketle{tψ3(0)∝an}b∇acket∇i}ht= 0, since\nthe solutions for ψ1(t) andψ2(t) on the one hand and ψ3(t) on the other hand are decoupled\n11in Eq. (27). Therefore, spin wave solutions only exists for non-zer o averages ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htand\n∝an}b∇acketle{tψ2(t)∝an}b∇acket∇i}ht. The existence of such non-trivial solutions are determined in depe ndence on the\nnoise parameters Dandτand the deterministic damping parameter α. Notice, that the\ndimensionless quantity D=˜D/Γ, i.e.,Dis the ratio between the strength of the correlation\nfunction (Eq. (7)) and the anisotropy field in the original units. The stability of spin wave\nsolutions is guaranteed for positive parameters A1andA3. According to Eqs. (26) the\nphase diagrams are depicted in Fig. 2 within the α−Dplane for different values of the\ncorrelation time τ. The separatrix between stable and unstable regions is determined by\nthe condition A1= 0. The second condition A3= 0 is irrelevant due to the imposed initial\nconditions. As the result of the stability analysis the phase space dia gram is subdivided into\nfour regions where region IV does not exist in case of τ= 0, see Fig. 2(a). For generality,\nwe take into account both positive and negative values of Dindicating correlations and\nanti-correlations of the stochastic field. Damped spin waves are ob served in the areas I and\nIV, whereas the sectors II and III reveal non-accessible solutio ns. In those regions the spin\nwave amplitude, proportional to exp[ −A1t], tends to infinity which should not be realized,\ncompare Figs. 2(b)-2(d). Actually, a reasonable behavior is obser ved in regions I and IV. As\nvisible from Fig. 2 damped spin waves will always emerge for D>0 even in the limit of zero\ndamping parameter αand vanishing correlation time τ. This behavior is shown in Fig. 3,\nwhere theevolution of ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htis depicted for different valuesof α. As canbeseen in Fig.2(a)\nthesolutionfor D<0isunlimitedandconsequently, itshouldbeexcludedfurther. Contr ary\nto this situation, additional solutions will be developed in region IV in ca se ofτ >0 and\nsimultaneously α= 0, see Figs. 2(b)-2(d). Thereby the size of area IV grows with inc reasing\nτ. Likewise, the extent of region I decreases for an enhanced τ. However, in the limit of\nD= 0 and consequently for τ= 0, too, only damped spin waves are observed. Immediately\non the separations line undamped periodic solutions will evolve, compa re the sub-figures in\nFig. 2. This remarkable effect can be traced back to the interplay be tween the deterministic\ndamping and the stochastic forces. Both damping mechanism are co mpensated mutually\nwhich reminds of a kind of resonance phenomenon. The difference to conventional resonance\nbehavior consists of the compensation of the inherent determinist ic Gilbert damping and the\nstochastic one originated from the random field. This statement is e mphasized by the fact\nthat undamped periodic solutions do not develop in the absence of st ochastic interferences,\ni.e.,D= 0. The situation might be interpreted physically as follows: the requ ired energy\n12(a)τ= 0 (b)τ= 0.1\n(c)τ= 1 (d)τ= 10\nFIG. 2.α−Dplane for fixed magnetization µ= 0.9 and different values of τ.\nthat enables the system to sustain the deterministic damping mecha nisms is delivered by\nthe stochastic influences due to the interaction with the environme nt. To be more precise, in\ngeneral, the Gilbert damping enforces the coherent alignment of th e spin density along the\nprecession axis. Contrary, the random field supports the dephas ing of the orientation of the\nclassical spins. Surprisingly, the model predicts the existence of a critical value τ=τc≥0\n13FIG. 3. Evolution of the mean value ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}ht, withµ= 0.9,D= 0.1 andτ= 0.αvaries from 0\n(dash-dotted line), 0 .05 (solid line), 0 .5 (dotted line) and 1 (dashed line).\ndepending on αandDwhich determines the onset of undamped periodic solutions. Notice,\nthat negative values of τcare excluded. The critical value is\nτc=−[µ2(α3−Dα2+α)+2D](1+α2)2\n2Dµ2(α3−3Dα2+α)+D2. (28)\nHence, this result could imply the possibility of the cancellation of both damping processes.\nExamples according to the damped and the periodic case are displaye d in Fig. 4. An increas-\ningτfavors the damping process as it is visible in Fig. 4(a). Based on estima tions obtained\nfor ferromagnetic materials [29] and references therein, the Gilbe rt damping parameter can\nrange between 0 .04<α<0.22 in thin magnetic films, whereas the bulk value for Co takes\nαb≈0.005. The phase space diagram in Fig. 2 offers periodic solutions only fo r values of\nαlarger than those known from experiments. Therefore such perio dic solutions seem to be\nhard to see experimentally. We proceed further by analyzing the be havior of the correlation\nfunction by numerical computation of the solution of Eq. (24) with E qs. (25) and (26). As\ninitial values we choose Cik(t=t′,t′) =Cik(s= 0) =C0for every combination i,k={1,2,3}.\nThe results are depicted in Figs. 5 and 6. Inspecting Figs. 5(a)-5(c ) one recognizes that an\nenhancement of the correlation time τleads to an increase of the oscillations within the\ncorrelation functions C1k,k={1,2,3}. Moreover, Fig. 5(d) reveals that the oscillatory\n14(a) (b)\nFIG. 4. Evolution of the mean values ∝an}b∇acketle{tψ1,2(t)∝an}b∇acket∇i}ht, withµ= 0.9. (a):D= 0.1,α= 0.005 andτvaries\nfrom 10 (solid line), 1 (dotted line) and 0 (dash-dotted line ). (b):D= 2,α= 1 andτ=τc≈1.79\n(Eq. (28)). The solid line represents ∝an}b∇acketle{tψ1∝an}b∇acket∇i}htand the dash-dotted line is ∝an}b∇acketle{tψ2∝an}b∇acket∇i}ht.\nbehavior of C31seems to be suppressed. Obviously, the decay of the correlation f unction is\nenhanced if τgrowths up. The pure periodic case for τ=τc, corresponding to Fig. 4(b),\nis depicted in Fig. 6. Exemplary, C12andC31are illustrated. The behavior of the latter is\nsimilar to the damped case, displayed in Fig. 5(d), unless slight oscillatio ns occur. However,\nif one compares the form of C12in Fig. 5(b) and Fig. 6 the differences are obvious. The am-\nplitude of the correlation function for the undamped case grows to the fourfold magnitude\nin comparison with C0, whereas the damped correlation function approaches zero. Fur ther,\na periodic behavior is shown in Fig. 6, and therefore the correlation w ill oscillate about zero\nbut never vanish for all s=t−t′>0.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamics of a classical spin model with uniaxial\nanisotropy. Aside from the deterministic damping due to the Landau -Lifshitz-Gilbert\nequation the system is subjected to an additional dissipation proce ss by the inclusion of a\nstochastic field with colored noise. Both dissipation processes are a ble to compete leading to\n15(a) (b)\n(c) (d)\nFIG. 5. Correlation functions Cik(s) forµ= 0.9,D= 0.1 andα= 0.005.τtakes 0 (dotted line),\n1 (solid line) and 10 (dash-dotted line).\na more complex behavior. To study this one we derive an equation for the joint probability\ndistribution which allows us to find the corresponding spin-spin-corr elation function. This\nprogram can be fulfilled analytically and numerically in the spin wave appr oach and the\nsmall correlation time limit. Based on the mean value for the spin wave c omponent and\n16FIG. 6. Correlation functions Cik(s) forτ=τc≈1.79 (Eq. (28)), µ= 0.9,D= 2 andα= 1. The\ndotted line represents C12and the solid line is C31.\nthe correlation function we discuss the stability of the system in ter ms of the stochastic\nparameters, namely the strength of the correlated noise Dand the finite correlation time\nτ, as well as the deterministic Gilbert damping parameter α. The phase diagram in the\nα−Dplane offers that the system develops stable and unstable spin wave solutions due to\nthe interplay between the stochastic and the deterministic damping mechanism. So stable\nsolutions evolve for arbitrary positive Dand moderate values of the Gilbert damping α.\nFurther, we find that also the finite correlation time of the stochas tic field influences the\nevolution of the spin waves. In particular, the model reveals for fix edDandαa critical\nvalueτcwhich characterizes the occurrence of undamped spin waves. The different situa-\ntions are depicted in Fig. 2. Moreover, the correlation time τaffects the damped spin wave\nwhich can be observed in regions I and IV in the phase diagram. If the parameters Dand\nαchanges within these regions, an increasing τleads to an enhancement of the spin wave\ndamping, cf. Fig. 4(a). The influence of τon the correlation functions is similar as shown\nin Figs. 5(a)-5(c). The study could be extended by the inclusion of fi nite wave vectors and\nusing an approach beyond the spin wave approximation.\n17ACKNOWLEDGMENTS\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is\nsupported by the Saxony-Anhalt State, Germany.\n18[1] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media (Pergamon\nPress, Oxford, 1989).\n[2] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halpe rin, Rev. Mod. Phys. 77, 1375\n(2005).\n[4] A. Sukhov and J. Berakdar, J. Phys. - Cond. 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Bauer, Phys. Rev . Lett.88, 117601 (2002).\n20"    },    {        "title": "1003.1868v1.Damping_of_Nanomechanical_Resonators.pdf",        "content": "arXiv:1003.1868v1  [cond-mat.mes-hall]  9 Mar 2010Damping of Nanomechanical Resonators\nQuirin P. Unterreithmeier,1,∗Thomas Faust,1and J¨ org P. Kotthaus1\n1Fakult¨ at f¨ ur Physik and Center for NanoScience (CeNS),\nLudwig-Maximilians-Universit¨ at, Geschwister-Scholl- Platz 1, D-80539 M¨ unchen, Germany\n(Dated: May 28, 2018)\nWe study the transverse oscillatory modes of nanomechanica l silicon nitride strings under high\ntensile stress as a function of geometry and mode index m≤9. Reproducing all observed resonance\nfrequencies with classical elastic theory we extract the re levant elastic constants. Based on the\noscillatory local strain we successfully predict the obser ved mode-dependent damping with a single\nfrequency independent fit parameter. Our model clarifies the role of tensile stress on damping and\nhints at the underlying microscopic mechanisms.\nThe resonant motion of nanoelectromechanical sys-\ntems receives a lot of recent attention. Their large fre-\nquencies, low damping i.e. high mechanical quality fac-\ntors, and small masses make them equally important as\nsensors[1–4] and for fundamental studies[3–9]. In either\ncase, low damping of the resonant motion is very desir-\nable. Despite significant experimental progress[10, 11],\na satisfactory understanding of the microscopic causes of\ndamping is not yet achieved. Here we present a system-\natic study of the damping of doubly-clamped resonators\nfabricated out of prestressed silicon nitride leading to\nhighmechanicalqualityfactors[10]. Reproducingtheob-\nserved mode frequencies applying continuum mechanics,\nwe are able to quantitatively model their quality factors\nby assuming that damping is caused by the local strain\ninduced by the resonator’s displacement. Considering\nvarious microscopic mechanisms, we conclude that the\nobserved damping is most likely dominated by dissipa-\ntion via localized defects uniformly distributed through-\nout the resonator.\nWe study the oscillatory response of nanomechanical\nbeams fabricated from high stress silicon nitride (SiN).\nA released doubly-clamped beam of such a material is\ntherefore under high tensile stress, which leads to high\nmechanical stability[12] and high mechanical quality fac-\ntors[10]. Such resonators are therefore widely used in\nrecent experiments[6, 9]. Our sample material consists\nof a silicon substrate covered with 400nm thick silicon\ndioxide serving as sacrificial layer and a h= 100nm\nthick SiN device layer. Using standard electron beam\nlithography and a sequence of reactive ion etch and wet-\netch steps, we fabricate a series of resonators having\nlengths of 35 /nµm,n={1,...,7}and a cross section\nof 100·200nm as displayed in Fig.1a and b. Since the\nrespective resonance frequency is dominated by the large\ntensile stress[10, 13], this configuration leads to reso-\nnances of the fundamental modes that are approximately\nequally spaced in frequency. Suitably biased gold elec-\ntrodes processed beneath the released SiN strings actu-\nate the resonators via dielectric gradient forces to per-\nform out-of-plane oscillations, as explained in greater de-\ntail elsewhere [12]. The length and location of the gold\nxzxz\ny(a) (b)\n(c)\n5□µm\nFIG.1.Setup and Geometrya SEM-pictureofoursample;\nthe lengths of the investigated nanomechanical silicon nit ride\nstrings are 35 /nµm,n={1,...,7}; their widths and heights\nare 200nm and 100nm, respectively. bZoom-in of a: the res-\nonator (highlighted in green) is dielectrically actuated b y the\nnearby gold electrodes (yellow); its displacement is recor ded\nwith an interferometric setup. cMode profile and absolute\nvalue of the resulting strain distribution (color-coded) o f the\nlongest beam’s 4th harmonic as calculated by elastic theory .\nelectrodesis properlychosen to be able to also excite sev-\neral higher order modes of the beams. The experiment\nis carried out at room temperature in a vacuum below\n10−3mbar to avoid gas friction.\nThe displacement is measured using an interferometric\nsetup that records the oscillatory component of the re-\nflected light intensity with a photodetector and network\nanalyser[12, 14]. The measured mechanical response\naround each resonance can be fitted using a Lorentzian\nlineshape as exemplarily seen in the inset of Fig.2. The\nthereby obtained values for the resonance frequency f\nand quality factor Qfor all studied resonators and ob-\nserved modes are displayed in Fig.2 (filled circles). In\norder to reproduce the measured frequency spectrum,\nwe apply standard beam theory (see e.g.[15]). With-\noutdamping, the differentialequationdescribingthespa-\ntial dependence of the displacement for a specific mode\nmof beam n un,m[x] at frequency fn,mwrites (with\nρ= 2800kg /m3being the material density[16]; E1,σ02\n50\n40\n30\n20\n10\n0Signal Amplitude[□µV]\n24.136 24.132 24.128\nFrequency[MHz]160\n140\n120\n100\n80\n60\n40\n20\n0Quality□Factor□[103]\n80 70 60 50 40 30 20 10 0\nFrequency□[MHz]Beam□lenght□[µm]\n35\n35/2 35/5\n35/3 35/6\n35/4 35/7\nFIG. 2. Resonance Frequency and Mechanical Qual-\nity Factor The harmonics of the nanomechanical resonator\nshow a Lorentzian response (exemplary in the inset). Fittin g\nyields the respective frequency and mechanical quality fac tor.\nThe main figure displays these values for several harmonics\n(same color) ofdifferentbeams as indicted bythecolor. Tore -\nproduce the resonance frequencies, we fit a continuum model\nto the measured frequencies. We thereby retrieve the elas-\ntic constants of our (processed) material, namely the built -in\nstressσ0= 830MPa and Young’s modulus E= 160GPa.\nFrom the displacement-induced, mode-dependent strain dis -\ntribution, we calculate (except for an overall scaling) the me-\nchanical quality factors. Calculated frequencies and qual ity\nfactors are shown as hollow squares, the responses of the dif -\nferent harmonics of the same string are connected.\nare the (unknown) real Young’s modulus and built in\nstress, respectively)\n1\n12E1h2∂4\n∂x4un,m[x]−σ0∂2\n∂x2un,m[x]−ρ(2πfn,m)2un,m[x] = 0\n(1)\nSolutions of this equation have to satisfy the bound-\nary conditions of a doubly-clamped beam (displacement\nand its slope vanish at the supports ( un,m[±l/(2n)] =\n(∂/∂x)un,m[±l/(2n)] = 0,l/n: beam length). These con-\nditions lead to a transcendental equation that is numeri-\ncally solved to obtain the frequencies fn,mof the different\nmodes.\nThe results arefitted to excellently reproducethe mea-\nsured frequencies, as seen in Fig.2 (hollow squares).\nOne thereby obtains as fit parameters the elastic con-\nstants of the micro-processed material E1= 160GPa,\nσ0= 830MPa, in good agreement with previously pub-\nlished measurements [13].\nFor each harmonic, we now are able to calculate the\nstrain distribution within the resonator induced by the\ndisplacement u[x] and exemplarily shown in Fig.1c. The\nmeasured dissipation is closely connected to this induced\nstrainǫ[x,z,t] =ǫ[x,z]exp[i2πft]. As in the case of a\nZener model[17] we now assume that the displacement-\ninduced strain and the accompanying oscillating stressσ[x,z,t] =σ[x,z]exp[i2πft] are not perfectly in phase;\nthis can be expressed by a Young’s modulus E=E1+\niE2having an imaginary part. The relation reads again\nσ[x,z] = (E1+iE2)ǫ[x,z]. During one cycle of oscillation\nT= 1/f, a small volume δVof length sand cross section\nAthereby dissipates the energy ∆ UδV=AsπE2ǫ2. The\ntotal loss is obtained by integrating over the volume of\nthe resonator.\n∆Un,m=/integraldisplay\nVdV∆UδV=πE2/integraldisplay\nVdVǫn,m[x,z]2(2)\nThe strain variation and its accompanying energy\nloss can be separated into contributions arising from\noverall elongation of the beam and its local bending.\nIt turns out that here the former is negligible, de-\nspite the fact that the elastic energy is dominated by\nthe elongation of the string, as discussed below. To\nvery high accuracy we obtain for the dissipated energy\n∆Un,m≈π/12E2wh3/integraltext\nldx(∂2/(∂x)2un,m)2. A more rig-\norous derivation can be found in the Supplementary In-\nformation.\nThe total energy depends on the spatial mode (through\nǫn,m, see exemplary Fig.1c) and therefore strongly dif-\nfers for the various resonances. To obtain the qual-\nity factor, one has to calculate the stored energy\ne. g. by integrating the kinetic energy Un,m=/integraltext\nldxAρ(2πfn,m)2un,m[x]2. The overall mechanical qual-\nityfactoris Q= 2πUn,m/∆Un,m. Amoredetailed deriva-\ntion can be found in the Supplementary Information.\nAssuming that the unknown value of the imaginary\npartE2of the elastic modulus is independent of res-\nonatorlength andharmonicmode, weareleft with onefit\nparameter E2to reproduce all measured quality factors\nand find excellent agreement(Fig.2, hollowsquares). We\ntherefore successfully model the damping of our nanores-\nonators by postulating a frequency independent mecha-\nnism caused by local strain variation. Allowing E2to\ndepend on frequency, the accordance gets even better, as\ndiscussed in detail in the Supplement.\nWe now discuss the possible implications of our find-\nings, considering at first the cause ofthe high quality fac-\ntors in overall pre-stressed resonators and then the com-\npatibility of our model with different microscopic damp-\ning mechanisms. In a relaxed beam, the elastic energy\nis stored in the flexural deformation and becomes for a\nsmall test volume UδV= 1/2AsE1ǫ2. In the framework\nof a Zener model, as employed here, this result is pro-\nportional to the energy loss (see eq.3) and thus yields\na frequency-independent quality factor Q=E1/E2for\nthe unstressed beam. In accordance with this finding,\nRef.[10] reports a much weaker dependence of quality\nfactor on resonance frequency, in strong contrast with\nthe behavior of their stressed beams.\nSimilar as in the damping model, the total stored elas-\ntic energy in a beam can be very accurately separated3\ninto a part connected to the bending and a part coming\nfrom the overall elongation. The latter is proportional\nto the pre-stress σ0and vanishes for relaxed beams, refer\nto the Supplement for details. Assuming a constant E=\nE1+iE2, Fig.3 displays the calculation of the elastic en-\nergy and the quality factor for the fundamental mode of\nourlongest( l= 35µm)beamasafunctionofoverallbuilt\nin stress σ0. The total elastic energy is increasingly dom-\ninated by the displacement-induced elongation Uelong=\n1/2σ0wh/integraltext\nldx(∂/(∂x)u[x])2. In contrast the bending en-\nergyUbend= 1/24E1wh3/integraltext\nldx(∂2/(∂x)2u[x])2, which in\nour model is proportional to the energy loss, is found to\nincrease much slower with σ0. Thus one expects Qto in-\ncreasewith σ0, afindingalreadydiscussedbySchmidand\nHierold for micromechanical beams[18]. However, their\nmodel assumes the simplified mode profile of a stretched\nstring and can not explain the larger quality factors of\nhigher harmonics when compared to a fundamental res-\nonance of same frequency. Including beam stiffness, our\nmodel can fully explain the dependence of frequency and\ndamping on length and mode index, as reflected in Fig.2.\nIt also explains the initially surprising finding[19] that\namorphous silicon nitride resonators exhibit high quality\nfactors when stretched whereas having Q-factors in the\nrelaxed state that reflect the typical magnitude of inter-\nnal friction found to be rather universal in glassy materi-\nals[20]. More generally we conclude that the increase in\nmechanical quality factors with increasing tensile stress\nis not bound to any specific material.\nSince the resonance frequency is typically easier to\naccess in an experiment, we plot the quality factor vs.\ncorresponding resonance frequency in Fig.3b; with both\nnumbers being a function of stress. The resulting rela-\ntion of quality factor on resonance frequency is (except\nfor verylow stress) almost linear; experimental results by\nanother group can be seen to agree well with this find-\ning[21]. In addition, we show in the Supplement that\nalthough the energy loss per oscillation increases with\napplied stress, the linewidth of the mechanical resonance\ndecreases.\nWe willnowconsidertheintrinsicphysicalmechanisms\nthat could possibly contribute to the observed damp-\ning. As explained in greater detail in the Supplement,\nwe can safely neglect clamping losses[1, 2], thermoelastic\ndamping[4, 6] and Akhiezer damping[5, 6] since they all\npredict damping constants significantly smaller than the\nones observed.\nInstead, we would like to discuss the influence of lo-\ncalized defect states. Similar to the Akhiezer effect, it\nis assumed that the energy spectra of defects are modu-\nlated by strain[27], which thereby drives the occupancy\nout of thermal equilibrium. In Ref.[27], this effect is cal-\nculated for a broad spectrum of two-level systems. The\nenergy difference of two uncoupled levels as well as their\nseparating tunnel barrier height are assumed to be uni-\nformly distributed, leading to a broad yet not flat dis-10-2410-2310-2210-2110-20\nEnergy□[J]\n105106107108109\nStress□[Pa]Quality□factor□103\nResonance□frequency□[MHz]2.0\n1.5\n1.0\n0.5\n0.0Stress□□[GPa]elongation\nbending\ntotal□energyExperiment250\n200\n150\n100\n50\n0\n12 108 6 4 2Experiment(a) (b)\nFIG. 3.Elastic Energy and Mechanical Quality Factor\nof the Beam in Dependence of Stress a The elastic en-\nergies of the fundamental mode of the beam with l= 35µm\nare displayed vs. applied overall stress separated into the con-\ntributions resulting from the overall elongation and the lo cal\nbending. The dashed line marks the strain of the experimen-\ntally studied resonator σ0≈830MPa, there the elongation\nterm dominates noticeably. bQuality factor and frequency\nare calculated for varying stress σ0. In order to compare the\ncalculation with other published results quality factor an d\nstress are displayed vs. resulting frequency.\ntribution of relaxation rates. In the high temperature\nlimit the thus derived energy loss per oscillation becomes\nfrequency-independent as assumed in our model. In ad-\ndition, published quality factors of relaxed silicon nitride\nnano resonators[19] cooled down to liquid helium tem-\nperature display quality factors that are well within the\ntypical range of amorphous bulk materials[20]. More-\nover, on a different sample chip we measured a set of\nresonators showing quality factors that are uniformly de-\ncreased by a factor of approximately 1.4 compared to the\ndatapresentedinFig.2. Theirresponsecanstillbequan-\ntitativelymodeledusingnowanincreasedimaginarypart\nof Young’s modulus E2. We attribute this reduction to\na non-optimized RIE-etch step, leading to an increased\ndensity of defect states. The corresponding data is pre-\nsented in the Supplementary Information. These three\narguments clearly favor the concept of damping via de-\nfect states as the dominant mechanism.\nWe wish to point out that such a model calculation\nbased on two-level systems cannot be rigourouslyapplied\nat elevated temperatures, as the concept of two-level\nsystems should be replaced by local excitable systems.\nHowever, it seems plausible that such a system still ex-\nhibits abroadrangeofrelaxationrates, crucialto explain\nfrequency-independent damping. In contrast, a mecha-\nnism with discrete relaxation rates will exhibit damp-\ning maxima whenever the oscillation frequency matches\nthe relaxation rate[4, 6, 17]. We further notice that in\nourexperimentsbeamswithlargerwidthsexhibitslightly\nhigher quality factors. This indicates an increased defect\ndensity near the surface being either intrinsic or caused\nby the micro-fabrication[28] (RIE etch). For a fixed\ncross-section however, the applicability of our model is\nnot affected.\nIn conclusion, we systematically studied the transverse4\nmode frequencies and quality factors of prestressed SiN\nnanoscale beams. Implementing continuum theory, we\nreproduce the measured frequencies varying with beam\nlength and mode index over an order of magnitude. As-\nsuming that damping is caused by local strain variations\ninduced by the oscillation, independent of frequency, en-\nables us to calculate the observed quality factors with a\nsingle interaction strength as free parameter. We thus\nidentify the unusually high quality factors of pre-stressed\nbeams as being primarily caused by the increased elastic\nenergy rather than a decrease in damping rate. Several\npossible damping mechanisms are discussed; because of\ntheobservednearlyfrequency-independenceofthedamp-\ning parameter E2, we attribute the mechanism to inter-\naction of the strain with local defects of not yet identified\norigin. One therefore expects that defect-free resonators\nexhibit even larger quality factors, as recently demon-\nstrated for ultra-clean carbon nanotubes[11].\nFinancial support by the Deutsche Forschungsgemein-\nschaft via project Ko 416/18,the German Excellence Ini-\ntiative viathe NanosystemsInitiative Munich (NIM) and\nLMUexcellent as well as the Future and Emerging Tech-\nnologies programme of the European Commission, under\nthe FET-Openproject QNEMS (233992)is gratefullyac-\nknowledged.\nWe would like to thank FlorianMarquardtand Ignacio\nWilson-Rae for stimulating discussions.\n∗quirin.unterreithmeier@physik.uni-muenchen.de\n[1] K. Jensen, K. Kim, and A. Zettl, Nat Nano 3, 533 (2008).\n[2] B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and\nA. 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P.\nKouwenhoven, and H. S. J. van der Zant, Nano Letters 9,\n2547–2552 (2009).\n[12] Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus,\nNature458, 1001–1004 (2009).\n[13] Q. P. Unterreithmeier, S. Manus, and J. P. Kotthaus,Appl. Phys. Lett. 94, 263104–3 (2009).\n[14] N. O. Azak, M. Y. Shagam, D. M. Karabacak, K. L. Ek-\ninci, D. H. Kim, and D. Y. Jang, Applied Physics Letters\n91, 093112 (2007).\n[15] W. Weaver, S. P. Timoshenko, and D. H. Young, Vibra-\ntion Problems in Engineering (Wiley, New York, 1990).\n[16] M. G. el Hak, The MEMS Handbook (CRC Press, 2001).\n[17] A. N. Cleland, Foundations of Nanomechanics (Springer,\n2003).\n[18] S. Schmid and C. Hierold, J. Appl. Phys. 104, 093516\n(2008).\n[19] D. R. Southworth, R. A. Barton, S. S. Verbridge, B. Ilic,\nA.D. Fefferman, H.G. Craighead, and J. M. Parpia, Phys.\nRev. Lett. 102, 225503 (2009).\n[20] R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys.\n74, 991 (2002).\n[21] S. Verbridge, D. Shapiro, H. Craighead, and J. 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Esashi, Journal of Microelec-\ntromechanical Systems 11, 775–783 (2002).5\nSUPPLEMENTARY INFORMATION\nDAMPING MODEL\nIn a Zener model, an oscillating strain ǫ(t) =ℜ[ǫ[ω]exp[iωt]] and its accompanying stress σ[t] =ℜ[σ[ω]exp[iωt]] are\nout-of phase, described by a frequency-dependent, complex ela stic modulus σ(ω) =E[ω]ǫ[ω] = (E1[ω]+iE2[ω])ǫ[ω].\nThis leads to an energy loss per oscillation in a test volume δV=δA·δsof cross-section δAand length δs.\n∆UδV=/integraldisplay\nTdtEAǫ[t]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nforce·∂\n∂t(sǫ[t])\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nvelocity=πδAδsE 2ǫ2(3)\nWe now employ this model for our case, namely a pre-stressed, rec tangular beam of length l, widthwand height\nh, corresponding here to the x,y,z-direction, respectively. The orig in of the coordinate system is centered in the\nbeam. The resonator performs oscillations in the z-direction and, a s we consider a continuum elastic model, there\nwill be no dependence on the y-direction. For a beam of high aspect r atiol≫hand small oscillation amplitude,\nthe displacement of the m-th mode can be approximately written um[x,y,z] =um[x]. During oscillation, a small test\nvolume within the beam undergoes oscillating strain ǫm[x,z,t].\nThis strain arises because of the bending of the beam as well as its elo ngation as it is displaced. The stress caused by\nthe overall elongation is quadratic in displacement, therefore it occ urs at twice the oscillating frequency.\nǫm[x,z,t] =1\n2/parenleftbigg∂\n∂xum[x]ℜ[exp[iωt]]/parenrightbigg2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nelongation+z∂2\n∂x2um[x]ℜ[exp[iωt]]\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbending\n=1\n2/parenleftbigg∂\n∂xum[x]/parenrightbigg21\n2(1+ℜ[exp[2iωt]])+z∂2\n∂x2um[x]ℜ[exp[iωt]] (4)\nInserting this into eq.3 and integrating over the cross-section w·h, the accompanying energy losses can be seen to\nseparate into elongation and displacement caused terms.\n∆Uw·h=πsE2[2ω]wh\n8/parenleftbigg∂\n∂xum[x]/parenrightbigg4\n+πsE2[ω]wh3\n12/parenleftbigg∂2\n∂x2um[x]/parenrightbigg2\n(5)\nIntegrating over the length yields the total energy loss of a partic ular mode ∆ U=/integraltextl/2\n−l/2dx∆Uw·h. In the case that\nE2is only weakly frequency-dependent, it turns out that for our geo metries the elongation term is several orders of\nmagnitude (105−107) smaller than the term arising from the bending. The energy loss the refore may be simplified\nand writes\n∆U≈∆Ubending=πE2wh3\n12/integraldisplayl/2\n−l/2dx/parenleftbigg∂2\n∂x2um[x]/parenrightbigg2\n(6)\nELASTIC ENERGY OF A PRE-STRESSED BEAM\nA volume δVsubject to a longitudinal pre-stress σ0stores the energy UδVwhen strained; E1is assumed to be\nfrequency independent in the experimental range (5-100MHz)\nUδV=sA/parenleftbigg\nσ0ǫ+1\n2E1ǫ2/parenrightbigg\n(7)\nTo apply this formula to the case of an oscillating pre-stressed beam , we insert eq.4 |t=0(maximum displacement) and\nintegrate over the cross-section to obtain\nUw·h=1\n2E1/parenleftBigg\n1\n4wh/parenleftbigg∂\n∂xum[x]/parenrightbigg4\n+1\n12wh3/parenleftbigg∂2\n∂x2um[x]/parenrightbigg2/parenrightBigg\n+1\n2swhσ0/parenleftbigg∂\n∂xum[x]/parenrightbigg2\n(8)6\nAnalog to eq.5 we can omit the first term in the brackets; integratin g over the length yields\nU≈/integraldisplayl/2\n−l/2dx/parenleftBig1\n2whσ0/parenleftbigg∂\n∂xum[x]/parenrightbigg2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nelongation+1\n24E1wh3/parenleftbigg∂2\n∂x2um[x]/parenrightbigg2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbending/parenrightBig\n(9)\nWecanthereforedividethetotalenergyintopartsarisingfromth eelongationandthebendingofthebeam. Depending\non the magnitude of the pre-stress, either of the two energies ca n dominate as seen in Fig.3a of the main text. We\nhave checked that the kinetic energy Ukin= 1/2ρ(ωm)2/integraltextl/2\nl/2dx(um[x])2; (ωm/(2π),ρ: resonance frequency, material\ndensity, respectively) equals the total elastic energy, as expect ed.\nFREQUENCY-DEPENDENT LOSS MODULUS\nThere is no obvious reason that the imaginary part of Young’s modulu sE2should be completely frequency-\nindependent. We therefore assume that E2obeys a (weak) power-law and chose the ansatz:\nE2[f] =E2(f/f0)a(10)\nFitting our data with the thus extended theory, we achieve a very p recise agreement of measured and calculated\nquality factors, as seen in Fig.S1. The resulting exponent is a= 0.075;E2varies therefore by 20% when fchanges\nby one order of magnitude.\n160\n140\n120\n100\n80\n60\n40\n20\n0Quality Factor [103]\n80706050403020100\nFrequency [MHz] Beam lenght [µm]\n 35  35/5\n 35/2  35/5\n 35/3  35/6\n 35/4\n \nFIG. 4. Resonance frequencies and quality factors of the resonator saMeasured quality factor and resonance frequency of\nseveral harmonics of beams with different lengths (color-co ded) are displayed as filled circles (same data as in Fig.2 of t he main\ntext). The resonance frequencies are reproduced by a contin uum model; we calculate the quality factors using a model bas ed\non the strain caused by the displacement. In contrast to Fig. 2 of the main text and Fig.S2 we here allow E2to be (weakly)\nfrequency-dependent.\nLINEWIDTH OF THE MECHANICAL RESONANCE\nThe elastic energy of a harmonic oscillator is given by U= 1/2meffω2\n0x2\n0withmeff,ω0,x0being effective mass,\nresonance frequency and displacement, respectively. If we assu me the effective mass to be energy-independent, it\nappliesω0∝√\nU. Recalling the definition of the quality factor Q= 2πU/∆U∝U, one obtains for the for the Full\nWidth at Half Max (FWHM) of the resonance\n∆ω=ω0\nQ∝√\nU\nU/∆U=∆U√\nU(11)\nAs in the main text, the energy depends on the applied overall tensile stress. Figure6 shows a numerical calculation\nof the resulting linewidth vs. applied stress; one can see that increa se in energy loss per oscillation is dominated by7\nthe increase in energy, resulting in a decreased linewidth. The exact effective mass is included in this calculation; as\nit changes by less than 20%, the above assumption is justified.\n160\n140\n120\n100\n80\n60FWHM /(2 )Δω π\n103104105106107108109\nPrestress□[Pa]\nFIG. 5. Linewidth of the mechanical resonance The calculated linewidths (FWHM) for the fundamental mode o f the beam\nwithl= 35µm are displayed vs. applied overall stress.\nMICROSCOPIC DAMPING MECHANISMS\nWe start with clamping losses as discussed, e.g., in ref.[1, 2], i. e. the r adiation of acoustic waves into the bulk\ncaused by inertial forces exerted by the oscillating beam. With a sou nd velocity in silicon of vSi≈8km/s, the\nwavelength of the acoustic waves radiated at a frequency of 10MH z from the clamps into the bulk will be greater\nthan 500 µm, and thus substantially larger than the length of our resonators . Considering each clamping point as\na source of an identical wave propagating into the substrate, one would expect that mostly constructive/destructive\ninterference would occur for in-/out-of-phaseshear forces ex erted by the clamping points, respectively. With clamping\nlosses being important, one would therefore expect that spatially a symmetric modes with no moving center of mass\nexhibit significant higher quality factors than symmetric ones[3]. Ano ther way to illuminate this difference is that\nsymmetric modes give rise to a net force on the substrate, wherea s asymmetric modes yield a torque. Since the\nmeasurement (Fig. 2) does not display such an alternating behavior of the quality factors with mode index m(best\nseen for the longest beam), clamping losses are likely to be of minor imp ortance.\nThe next damping mechanism we consider are phonon-assisted losse s within the beam. At elevated temperatures,\nat least two effects arise, the first being thermoelastic damping: be cause of the oscillatory bending, the beam is com-\npressed and stretched at opposite sides. Since such volume chang es are accompanied by work, the local temperature\nin the beam will deviate from the mean. For large aspect ratios as in ou r case, the most prominent gradient is in the\nz direction. The resulting thermal flow leads to mechanical dissipatio n. We extend existing model calculations[4] to\ninclude the tensile stress of our beams. Using relevant macroscopic material parameters such as thermal conductivity,\nexpansion coefficient and heat capacity we derive Q-values that are typically three to four orders of magnitudes larger\nthan found in the experiment. Therefore, heat flow can be safely n eglected as the dominant damping mechanism.\nIn addition, the calculated thermal relaxation rate corresponds t o approximately 2GHz, so the experiment is in the\nso-called adiabatic regime. Consequently, one would expect the ene rgy loss to be proportional to the oscillation\nfrequency, in contrast to the assumption of a frequency indepen dentE2and our experimental findings.\nAnother local phonon-based damping effect is the Akhiezer-effect [5]; it is a consequence of the fact that phonon\nfrequencies are modulated by strain, parameterized by the Gr¨ un eisen tensor. If different phonon modes (characterized\nbytheirwavevectorandphononbranch)areaffecteddifferently, the occupancyofeachmodecorrespondstoadifferent\ntemperature. This imbalance relaxes towards a local equilibrium temp erature, giving rise to mechanical damping.\nIn a model calculation applying this concept to the oscillatory motion o f nanobeams[6], the authors find in the case\nof large aspect ratios length/height that the thermal heat flow re sponsible for thermoelastic damping dominates the\nenergy loss by the Akhiezer effect. We thus can safely assume this m echanism to be also negligible in our experiment.8\nREDUCED QUALITY FACTOR\nWe fabricated a set of resonators, shown in Fig.S1a, that showed lower quality factors than the ones presented in\nthe main text (Fig.2); we attribute this reduction to a non-optimize d RIE-etch step. As in the main article, it is\npossible to reproduce the quality factors using a single fit paramete r, namely the imaginary part of Young’s modulus\nE2. The ratio of the two sets of quality factors is displayed in Fig.S1 b an d can be seen to be around 1.4 with\nno obvious dependence on resonance frequency, mode index or len gth. A non-optimized etch step causes additional\nsurface roughnessand the addition of impurities, thereby increas ingthe density of defect states. As there is no obvious\nreason why another damping mechanism should be thereby influence d, we interpret this as another strong indication\nthat the dominant microscopic damping mechanism is caused by localize d defect states.\n100\n80\n60\n40\n20\n0Quality□Factor□[103]\n70 60 50 40 30 20 10 0\nFrequency□[MHz]Beam□length□[µm]\n35 35/5\n35/2 35/6\n35/3 35/7\n35/4\nQuality□Factor□Ratio\nFrequency□[MHz]1.5\n1.0\n0.5\n0.0\n70 60 50 40 30 20 10 0Beam□length□[µm]\n35 35/5\n35/2 35/6\n35/3 35/7\n35/4 (b) (a)\nFIG. 6. Comparison of the resonance frequencies and quality factor s of the sets of resonators aMeasured quality factor and\nresonance frequency of several harmonics of beams with diffe rent lengths (color-coded) are displayed as filled circles. The\nresonance frequencies are reproduced by a continuum model; a model based on the strain caused by the displacement allows\nus to calculate the quality factors, shown as hollow squares . The uniform reduction of the Q-factors is attributed to an n on-\noptimized RIE-etch. bThe ratio of the quality factors of the two sets resonators (F ig.2 main article and Fig.S2a) are displayed\nvs. frequency, being approximately constant.\n∗quirin.unterreithmeier@physik.uni-muenchen.de\n[1] Z. Hao, A. Erbil, and F. Ayazi, Sensors and Actuators A: Ph ysical109, 156–164 (2003).\n[2] I. Wilson-Rae, Phys. Rev. B 77, 245418 (2008).\n[3] I. Wilson-Rae, private communication\n[4] R. Lifshitz and M. L. Roukes, Phys. Rev. B 61, 5600–5609 (2000).\n[5] A. Akhieser, Journal of Physics-ussr 1, 277–287 (1939).\n[6] A. A. Kiselev and G. J. Iafrate, Phys. Rev. B 77, 205436 (2008)."    },    {        "title": "1003.3769v1.Dynamics_of_magnetization_on_the_topological_surface.pdf",        "content": "arXiv:1003.3769v1  [cond-mat.mes-hall]  19 Mar 2010Dynamics of magnetization on the topological surface\nTakehito Yokoyama1, Jiadong Zang2,3, and Naoto Nagaosa2,4\n1Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan\n2Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan\n3Department of Physics, Fudan University, Shanghai 200433, China\n4Cross Correlated Materials Research Group (CMRG), ASI, RIK EN, WAKO 351-0198, Japan\n(Dated: October 17, 2018)\nWe investigate theoretically the dynamics of magnetizatio n coupled to the surface Dirac fermions\nof athree dimensional topological insulator, byderiving t heLandau-Lifshitz-Gilbert (LLG) equation\nin the presence of charge current. Both the inverse spin-Gal vanic effect and the Gilbert damping\ncoefficient αare related to the two-dimensional diagonal conductivity σxxof the Dirac fermion,\nwhile the Berry phase of the ferromagnetic moment to the Hall conductivity σxy. The spin transfer\ntorque and the so-called β-terms are shown to be negligibly small. Anomalous behavior s in various\nphenomena including the ferromagnetic resonance are predi cted in terms of this LLG equation.\nPACS numbers: 73.43.Nq, 72.25.Dc, 85.75.-d\nTopologicalinsulator(TI) providesa new state of mat-\nter topologically distinct from the conventional band in-\nsulator[1]. In particular,the edge channelsorthe surface\nstates are described by Dirac fermions and protected by\nthe band gap in the bulk states, and backward scatter-\ning is forbidden by the time-reversal symmetry. From\nthe viewpoint of the spintronics, it offers a unique op-\nportunity to pursue novel functions since the relativistic\nspin-orbit interaction plays an essential role there. Actu-\nally, several proposals have been made such as the quan-\ntized magneto-electric effect [2], giant spin rotation [3],\nmagneto-transport phenomena [4], and superconducting\nproximity effect including Majorana fermions [5–7].\nAlso, a recent study focuses on the inverse spin-\nGalvanic effect in a TI/ferromagnet interface, predicting\nthe current-induced magnetization reversal due to the\nHall current on the TI [8]. In Ref. [8], the Fermi energy is\nassumed to be in the gap of the Dirac dispersion opened\nby the exchange coupling. In this case, the quantized\nHall liquid is realized, and there occurs no dissipation\ncoming from the surface Dirac fermions.\nHowever, in realistic systems, it is rather difficult to\ntune the Fermi energy in the gap since the proximity-\ninducedexchangefieldisexpectedtobearound5-50meV.\nTherefore, it is important to consider the generic case\nwhere the Fermi energy is at the finite density of states\nof Dirac fermions, where the diagonal conductivity is\nmuch larger than the transverse one, and the damping of\nthe magnetizationbecomes appreciable. Related systems\nare semiconductors and metals with Rashba spin-orbit\ninteraction, where the spin-Galvanic effect and current\ninduced magnetization reversal have been predicted [9]\nand experimentally observed [10, 11]. Compared with\nthese systems where the Rashba coupling constant is a\nkey parameter, the spin and momentum in TI is tightly\nrelated to each other corresponding to the strong cou-\npling limit of spin-orbit interaction, and hence the gigan-\ntic spin-Galvanic effect is expected.\n\u0001\u0000 \u0002 \u0003\u0004 \u0005 \u0006\n\u0007 \b \t\n\n\u000b \f \r \u000e\u000f\n\u0010\u0011 \u0012 \u0013 \u0014 \u0015\u0016 \u0017\u0018\u0019\u001a \u001b\u001c \u001d\n\u001e\u001f\nFIG. 1: (Color online) (a) Illustration of the Dirac dispers ion\non top of TI. The Fermi level εFis far above the surface\ngap opened by magnetization in the ferromagnetic layer. (b)\nSketch of FMR experiment in the soft magnetic layer. The\nsubstrate in the figure is TI, which is capped by a layer of\nsoft ferromagnet. The magnetization precesses around the\nexternal magnetic field Heff.\nIn this letter, we study the dynamics of the magnetiza-\ntion coupled to the surface Dirac fermion of TI. Landau-\nLifshitz-Gilbert (LLG) equationin the presenceofcharge\ncurrent is derived microscopically, and (i) inverse spin-\nGalvanic effect, (ii) Gilbert damping coefficient α, (iii)\ntheso-called β-terms, and(iv)thecorrectiontotheBerry\nphase, are derived in a unified fashion. It is found that\nthese are expressed by relatively small number of param-\neters, i.e., the velocity vF, Fermi wave number kF, ex-\nchange coupling M, and the transport lifetime τof the\nDirac fermions. It is also clarified that the terms re-\nlated to the spatial gradient are negligibly small when\nthe surface state is a good metal. With this LLG equa-\ntion, we propose a ferromagnetic resonance (FMR) ex-\nperiment, wheremodificationsoftheresonancefrequency\nand Gilbert damping are predicted. Combined with the\ntransport measurement of the Hall conductivity, FMR\nprovide several tests of our theory.2\nDerivation of LLG equation. — By attaching a ferro-\nmagnet on the TI as shown in Fig. 1, we can consider a\ntopological surface state where conducting electrons in-\nteract with localized spins, S, through the exchange field\nHex=−M/integraldisplay\ndrn(r)·ˆσ(r). (1)\nHere, we set S=Snwith a unit vector npointing in the\ndirection of spin, ˆσ(r) =c†(r)σc(r) represents (twice)\nthe electronspindensity, with c†= (c†\n↑,c†\n↓) beingelectron\ncreation operators, σthe Pauli spin-matrix vector, and\nMbeing the exchange coupling energy. The total Hamil-\ntonian of the system is given by Htot=HS+Hel+Hex,\nwhereHSandHelare those for localized spins and con-\nducting electrons, respectively.\nThe dynamics of magnetization can be described by\nthe LLG equation\n˙n=γ0Heff×n+α0˙n×n+t′\nel, (2)\nwhereγ0Heffandα0are an effective field and a Gilbert\ndamping constant, respectively, both coming from HS.\nEffects of conducting electrons are contained in the spin\ntorque\ntel(r)≡s0t′\nel(r) =Mn(r)×∝angbracketleftˆσ(r)∝angbracketrightne,(3)\nwhich arises from Hex. Here, s0≡S/a2is the local-\nized spin per area a2. In the following, we thus calculate\nspin polarization of conducting electrons perpendicular\nton,∝angbracketleftˆσ⊥(r)∝angbracketrightne, in such nonequilibrium states with cur-\nrent flow and spatially varying magnetization to derive\ntheβ-term, or with time-dependent magnetization for\nGilbert damping. Here and hereafter, ∝angbracketleft···∝angbracketrightnerepresents\nstatistical average in such nonequilibrium states.\nFollowing Refs. [12–14] we consider a small transverse\nfluctuation, u= (ux,uy,0),|u| ≪1, around a uniformly\nmagnetized state, n= ˆz, such that n= ˆz+u. In the\n‘unperturbed’ state, n= ˆz, the electrons are described\nby the Hamiltonian\nH0=/summationdisplay\nkvF(kyσx−kxσy)−Mσz−εF+Vimp(4)\nwhereVimpis the impurity potential given by Vimp=\nu/summationtext\niδ(r−Ri) in the first-quantization form. We take\na quenched average for the impurity positions Ri. The\nelectron damping rate is then given by γ= 1/(2τ) =\nπniu2νFin the first Born approximation. Here, niis the\nconcentration of impurities, and νF=εF/(2πv2\nF) is the\ndensity of states at εF. We assume that γ≪vFkF=/radicalbig\nε2\nF−M2,M, and calculate spin transfer torque in the\nlowest non-trivial order.\nIn the presence of u(r,t) =u(q,ω)ei(q·r−ωt), the con-\nducting electrons feel a perturbation (note that Hel+\nHex=H0+H1)\nH1=−M/summationdisplay\nkσc†\nk+qσck·u(q,ω)e−iωt,(5)and acquires a transverse component\n∝angbracketleftˆσ′α\n⊥(q,ω)∝angbracketrightne=Mχαβ\n⊥(q,ω+i0)uβ(q,ω) (6)\nin the first order in uin the momentum and frequency\nrepresentation. Here, χαβ\n⊥is the transverse spin suscep-\ntibility in a uniformly magnetized state with α,β=x,y,\nand summing over βis implied.\nNow, we study the ω-linear terms in the uniform ( q=\n0) part of the transverse spin susceptibility, χαβ\n⊥(q=\n0,ω+i0). We make the following transformation of the\noperator:\nc=U˜c=1/radicalbig\n2ε(ε+M)/parenleftbigg\nvF(ky+ikx)\nε+M/parenrightbigg\n˜c(7)\nwithε=/radicalbig\n(vFk)2+M2. Note U†U= 1,U†σxU=\nvFky/ε,andU†σyU=−vFkx/ε. This transformation\nmaps two component operator cinto one component op-\nerator on the upper Dirac cone ˜ c. With this new op-\nerator, we calculate the transverse spin susceptibility in\nMatsubara form\nχαβ\n⊥(0,iωλ) =/integraldisplayβ\n0dτeiωλτ/angbracketleftbig\nTτσα(0,τ)σβ(0,0)/angbracketrightbig\n=−T/summationdisplay\nk,nU†σαU˜G(k,iεn+iωλ)U†σβU˜G(k,iεn) (8)\nwith˜G(k,iεn) = (iεn−ε+εF+iγsgn(εn))−1. By sym-\nmetry consideration of the integrand in k-integral, we\nfindχαβ\n⊥(0,iωλ)∝δαβ. After some calculations, we ob-\ntain the torque stemming from the time evolution:\ntα\nel=M2iω\n2π1\n2v2\nF/parenleftbiggvFkF\nεF/parenrightbigg2\nεFτn×u (9)\n=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτ˙ n×n.(10)\nThis result fits the conventional Gilbert damping with\nα=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτa2\n¯hS. (11)\nWe next examine the case of finite current by applying\na d.c. electric field E, and calculate a linear response of\nσα\n⊥toE, i.e.,< σα\n⊥(q)>ne=Kα\ni(q)Ei. First, it is clear\nthatKα\ni(q=0) =−εiασxx/(evF) where εiαandσxx\nare the anti-symmetric tensor and diagonal conductivity,\nrespectively, because electron’s spin is ”attached” to its\nmomentum. This representsthe inversespin-Galvanicef-\nfect, i.e., chargecurrentinduces magneticmoment. Since\nwe assume that Fermi level is far away from the surface\ngap,σxx≫σxywhereσxyis the Hall conductivity. The\ndominant term in χis thusχxy∝σxx. This is quite\ndifferent from the case studied in Ref. [8], where Fermi\nlevel lies inside the surface gap and therefore σxxis van-\nishing. Hence, the only contribution to the inverse spin-\nGalvanic effect is χxx∝σxy, which is much smaller than3\nthe effect proposed in this letter. Compared with the in-\nverse spin-Galvanic effect in Rashba system [9–11], this\neffect is much stronger since the small Rashba coupling\nconstant, i.e., the small factor αRkF/EFin Eq. (16) of\nRef. [9], does not appear in the present case. Taking into\naccount the realistic numbers with α= 10−11eVmandvF= 3×105m/s, onefindsthat theinversespin-Galvanic\neffect in the present system is ∼50times largerthan that\nin Rashba systems.\nThe next leading order terms of the expansion in uβ\nandqjcan be obtained by considering the four-point ver-\ntices [12] as\n∝angbracketleftˆσα\n⊥(q)∝angbracketrightne=−eMπ\n45i\n8πε2\nFεikεjl[δαβδkl+δαkδβl+δαlδβk]qjuβEi (12)\n=−eM5i\n32ε2\nF[q·Euα−q·(u׈z)(E׈z)α+u·(E׈z)(q׈z)α]. (13)\nTherefore, the spin torque steming from the spatial gradient has the form:\ntβ\nel=−β1\n2e[n×(j·∇)n−(j−(j·n)ˆz)∇·(n׈z)+(∇−(n·∇)ˆz)n·(j׈z)] (14)\nwherej=σCEwith charge current jand conductivity\nσC=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. and\nβ=5π\n4εFτ/parenleftbiggM\nvFkF/parenrightbigg2\n. (15)\nFrom Eq.(14), one can find the followings: (i) The spin\ntransfer torque of the form ( j·∇)nis missing since\nwe consider the upper Dirac cone only. (ii) The β-\nterm has a form essentially different from that in the\nconventioal one.[12, 15, 16] In contrast to the conven-\ntionalferromagnet,[12] thisconstantcomesfromthe non-\nmagnetic impurity. Considering vFkF∼=εF, we get\nα/β∼(εFτ)2from Eqs. (11) and (15). Therefore,\ntheβ-terms are negligible for a good surface metal, i.e.,\nεFτ≫1.\nUp to now, we consider only one branch of the band\nwhere the Fermi energy is sitting. When we consider the\n2-band structure, i.e., the 2 ×2 matrix Hamiltonian H=\nvF[(ky+Mnx\nvF)σx−(kx−Mny\nvF)σy], we have the correctionto the Berry phase term. In analogy with the minimal\ncoupling of electromagnetic field, A=−M\nevF(−ny,nx)\nplays the same role as the U(1) gauge. By integrating\nthe fermions out, one can get a Chern-Simons term in\ntermsofthemagnetization LCS=σxyǫµνρAµ∂νAρwhere\nµ,ν,ρ=t,x,y. When the gradient of magnetization van-\nishes, it can be rewritten as\nLCS=σxy(M\nevF)2(nx˙ny−ny˙nx).(16)\nThis additional term can be interpreted as an additional\nBerryphase for the magnetization. In fact, as nzremains\nconstant in the present case, we have [ nx,ny] =inz.\nTherefore, nxandnybecome conjugate variables up\nto a factor, which naturally leads to a Berry phase:\nnx˙ny−ny˙nx. This term is exactly equivalent to the\nChern-Simons term.\nIncluding all the terms derived above, we finally arrive\nat a modified LLG equation:\n˙n−2σxy(M\nevF)2˙n/(s0N) =γ0Heff×n+/parenleftbiggM\nevFs0N/parenrightbigg\n(−j+(n·j)ˆz)+(α0+α/N)˙n×n+tβ\nel/(s0N) (17)\nwhereNisthenumberofferromagneticlayers. Notethat\nα-,β- andBerryphaseterms originatefromthe interplay\nbetween Dirac fermions and local magnetization which\npersists over a few layers of the ferromagnet. Therefore,\nthe overall coefficients are divided by the number of fer-\nromagnetic layers N.Ferromagnetic resonance. —Observingthe smallvalue\nofβ, the spatial gradient of magnetization can be ne-\nglected for the time being. Only one uniform domain in\nthe absence of current is taken into account for simplic-\nity. Without loss of generality, assume that an external\nmagnetic field is applied along zdirection, and consider4\nthe ferromagnet precession around that field. ˙ nz= 0\nis kept in the first order approximation, namely nzis a\nconstant in the time evolution. By inserting the ansatz\nnx(y)(t) =nx(y)e−iωtinto the modified LLG equation,\none obtains\nℜω=ξ\nξ2+η2ω0,ℑω=−η\nξ2+η2ω0(18)\nwhereη= (α0+α/N),ω0=γ0Heffandξ= 1−\n2σxy(M\nevF)2/(s0N). Expanding up to the first order in\nσxyandη, one gets ℜω=ω0+ 2σxy(M\nevF)2ω0/(s0N)\nandℑω=ηω0. Therefore, the precession frequency ac-\nquires a shift proportional to σxyin the presence of in-\nterplay between Dirac fermions and the ferromagnetic\nlayer. The relative shift of ℜωis 2σxy(M\nevF)2ω0/(s0N) =\n1\nπSNM\nεF(Ma\nvF)2∼1\nN(M\nεF)3[17]. By tuning the Fermi level,\nthis shift can be accessible experimentally.\nMeanwhile, the Gilbert damping constant αcan be\nmeasured directly without referring to the theoretical\nexpression in Eq. (11). One can investigate the fer-\nromagnetic layer thickness dependence of FMR line-\nwidth. While increasing the thickness Nof ferromagnet,\nthe Gilbert damping constant stemming from the Dirac\nfermions decreases inversely proportional to the thick-\nness. Taking into account the realistic estimation with\nεFτ∼100 and M/εF∼0.3, one has α/s0∼1, while\nα0∼0.001 usually. Therefore, even for a hundred of lay-\ners of ferromagnet, the contribution from the proximity\neffect is still significant compared to the one coming from\ntheferromagnetitself. Observingthattheimaginarypart\nof resonance frequency in Eq. (18) is proportional to η,\none may plot the relation between the FMR peak broad-\nening, namely ℑω, and 1/N. The broadening is a linear\nfunction of 1 /N, and approaches the value of the ferro-\nmagnet at large thickness limit. We can find the value of\nαfrom the slope of the plot.\nOn the other hand, the real part of FMR frequency\nprovides rich physics as well. Since in the presence of\nadditional Berry phase, the frequency shift is propor-\ntional to the Hall conductivity on the surface of TI, it\nleads to a new method to measure the Hall conductiv-\nity without four-terminal probe. In an ideal case when\nthe Fermi level lies inside the surface gap, this quantity\nis quantized as σ0\nxy=e2\n2h. However, in realistic case,\nFermi level is away from the surface gap, and therefore\nthe Hall conductivity is reduced to σxy=e2\n2hMnz\nεF[17]. As\na result, the shift of resonance frequency is proportional\nton2\nz∝cos2θ, and the FMR isotropy is broken. Here,\nθis the angle between effective magnetic field and the\nnormal to the surface of TI. One can perform an angle\nresolved FMR measurement. The signal proportional to\ncos2θcomes from additional Berry phase.\nSince parameters αandβdepend on Mandτ, it\nis quite important to measure these quantities directly.Molecular-beam epitaxy method can be applied to grow\nTI coated by a thin layer of soft ferromagnet. As is re-\nquired in the above calculation, Fermi level of TI should\nlie inside the bulk band gap. Also, the soft ferromag-\nnet should be an insulator or a metal with proper work\nfunction. One may employ angular resolved photoemis-\nsion spectroscopy(ARPES) or scanning tunneling micro-\nscope techniques to measure the surface gap ∆ opened\nby the ferromagnet, which is given by ∆ = Mnz. As the\neasy axis nzcan be found experimentally, Mcan be fixed\nas well. On the other hand, the lifetime τis indirectly\ndetermined by measuring the diagonal conductivity σxx\nviaσxx=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. Finally, Fermi surface can\nbe determined by ARPES, and all parameters in LLG\nequation Eq.(17) can be obtained.\nIn summary, we have investigated theoretically the\ndynamics of magnetization on the surface of a three\ndimensional topological insulator. We have derived\nthe Landau-Lifshitz-Gilbert equation in the presence of\ncharge current, and analyzed the inverse spin-Galvanic\neffect and ferromegnetic resonance predicting anomalous\nfeatures of these phenomena.\nThis work is supported by Grant-in-Aid for Scientific\nResearch (Grants No. 17071007, 17071005, 19048008\n19048015, and 21244053) from the Ministry of Educa-\ntion, Culture, Sports, Science and Technology of Japan.\n[1] M. Z. Hasan and C. L. Kane, arXiv:1002.3895; X. L.\nQi and S. C. Zhang, Physics Today, 63, 33 (2010) and\nreferences therein.\n[2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n78, 195424 (2008).\n[3] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nLett.102, 166801 (2009).\n[4] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B\n81, 121401(R) (2010).\n[5] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407\n(2008).\n[6] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.\nLett.103, 107002 (2009).\n[7] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N.\nNagaosa, Phys. Rev. Lett. 104, 067001 (2010).\n[8] I. Garate and M. Franz, arXiv:0911.0106.\n[9] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[10] A. Chernyshov et al., Nature Phys. 5, 656 (2009).\n[11] I.M. Miron et al., Nature Materials 9, 230 (2010).\n[12] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[13] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W.\nBauer, Phys. Rev. B 74, 144405 (2006); Y. Tserkovnyak,\nA. Brataas, and G. E. Bauer, J. Magn. Magn. Mater.\n320, 1282 (2008).\n[14] Y. Tserkovnyak, G.A. Fiete and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[15] Clement H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,5\n184411 (2009).\n[16] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).[17] J. Zang, and N. Nagaosa, arXiv:1001.1578"    },    {        "title": "1003.4681v1.Dynamical_shift_condition_for_unequal_mass_black_hole_binaries.pdf",        "content": "arXiv:1003.4681v1  [gr-qc]  24 Mar 2010Dynamical shift condition for unequal mass black hole binar ies\nDoreen M¨ uller, Jason Grigsby, Bernd Br¨ ugmann\nTheoretical Physics Institute, University of Jena, 07743 J ena, Germany\n(Dated: August 19, 2021)\nCertain numerical frameworks used for the evolution of bina ry black holes make use of a gamma\ndriver, whichincludesadampingfactor. Suchsimulations t ypicallyuseaconstantvaluefor damping.\nHowever, it has been found that very specific values of the dam ping factor are needed for the\ncalculation of unequal mass binaries. We examine carefully the role this damping plays, and provide\ntwo explicit, non-constant forms for the damping to be used w ith mass-ratios further from one. Our\nanalysis of the resultant waveforms compares well against t he constant damping case.\nPACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx\nI. INTRODUCTION\nThe ability to simulate the final inspiral, merger, and\nring-down of black hole binaries with numerical relativ-\nity [1–3] plays a key role in understanding a source of\ngravitational waves that may one day be observed with\ngravitational wave detectors. While initial simulations\nfocused on binaries of equal-mass, zero spin, and quasi-\ncircular inspirals, there currently is a large effort to ex-\nplore the parameter space of binaries, e.g. [4–7]. A key\npart of studying the parameter space is to simulate bina-\nries with intermediate mass-ratios.\nTodate, themassratiofurthestfromequalmassesthat\nhas been numerically simulated is 10:1 [8, 9]. These sim-\nulations use the Baumgarte-Shapiro-Shibata-Nakamura\n(BSSN) formulation [10–12] with 1+log slicing, and the\n˜Γ driver condition for the shift [13, 14]. In [8], it was\nnoted that the stability of the simulation is sensitive to\nthe damping factor, η, used in the ˜Γ driver condition,\n∂2\n0βi=3\n4∂0˜Γi−η∂0βi. (1)\nHere,βiistheshiftvectordescribinghowthecoordinates\nmove inside the spatial slices, ∂0≡∂t−βi∂i, and˜Γiis\nthe contraction of the Christoffel symbol, ˜Γi\njk, with the\nconformal metric, ˜ γjk.\nThe standard choice for ηis to set it to a constant\nvalue, which works well even for the most demanding\nsimulations as long as the mass ratio is sufficiently close\nto unity. In binary simulations, a typical choice is a con-\nstant value of about 2 /M, withMthe total mass of the\nsystem. This choice, however, leads to instabilities for\nthe mass ratio 10:1 simulation [8], although stability was\nobtained for η= 1.375/M. The value of ηis chosen to\ndamp an outgoing change in the shift while still yield-\ning stable evolutions. As we will show, if ηis too small,\nthere are unwanted oscillations, and values that are too\nlarge lead to instabilities. By itself, this observation is\nnot new, see e.g. [15–18]. The key issue for unequal\nmasses is that, as evident from (1), the damping factor η\nhas units of inverse mass, 1 /M. Therefore, the interval\nof suitable values for ηdepends on the mass of the blackholes. For unequal masses, a constant ηcannot equally\nwell accommodateboth black holes. Aconstantdamping\nparameter implies that the effective damping near each\nblack hole is asymmetric since the damping parameter\nhas dimensions 1 /M. For large mass ratios, this asym-\nmetry in the grid can be large enough to lead to a failure\nof the simulations because the damping may become too\nlarge or too small for one of the black holes. To cure this\nproblem, we need a position-dependent damping param-\neter that adapts to the local mass. In particular, we want\nit to vary such that, in the vicinity of the ithpuncture\nwith massMi, its value approaches 1 /Mi.\nA position-dependent ηwas already considered when\nthe˜Γdriverconditionwasintroduced[8,19–22], but such\nconstructions were not pursued further because for mod-\nerate mass ratios a constant ηworks well. Recently, we\nrevived the idea of a non-constant ηfor moving punc-\nture evolutions in order to remove the limitations of a\nconstantηfor large mass ratios. In [23], we constructed\na position-dependent ηusing the the conformal factor,\nψ, which carries information both about the location of\nthe black holes, and about the local puncture mass. The\nform ofηwas chosen to have proper fall-off rates both\nat the punctures and at large distance from the binary.\nIn [9], this approach was used successfully for mass ratio\n10:1. (We note in passing that damping is useful in other\ngauges as well, e.g. in [24] the modified harmonic gauge\ncondition includes position-dependent damping by use of\nthe lapse function.)\nIn the present work, we examine one potential short-\ncoming of the choice of [23], which leads us to suggest\nan alternative type of position-dependent η. Using [23],\nwe find large fluctuations in the values of that η, and\nthis might lead to instabilities in the simulation of larger\nmass-ratio binary black holes. To address this, we have\ntested two new explicit formulas for the damping factor\ndesignedto havepredictablebehaviorthroughoutthe do-\nmain of computation. We find the new formulas to pro-\nduce only small changes in the waveforms that diminish\nwithresolution,andthereisagreatdealoffreedominthe\nimplementation. Independently of our discussion here, in\n[16] the stability issues for large ηare explained, and a\nnon-constant ηis suggested (although not yet explored2\nin actual simulations), that, in its explicit coordinate de-\npendence, is similar to one of our suggestions.\nThe paperisorganizedasfollows. We firstdescribethe\nreasons for the damping factor and some of the reasons\nforlimitingitsvalueinSecII. InSec.III, wediscusssome\nprevious forms of ηthat have been used. We also present\ntwo new definitions and why we investigated them. In\nSec. IV, we find that these new definitions agree well\nwith the use of constant ηin the extracted gravitational\nwaves for mass ratios up to 4:1. Finally, in Sec. V, we\ndiscuss further implications of this work.\nII. MOTIVATION\nIn order to define a position-dependent form for η, it\nis important to determine what this damping parameter\naccomplishes in numerical simulations. For this reason,\nwe examine the effects of running different simulations\nwhile varying ηbetween runs. First we use evolutions of\nsingle non-spinning black-holes to identify the key phys-\nical changes. Then we examine equal-mass binaries to\ndetermine specific values desired in ηat both large and\nsmall radial coordinates.\nA. Numerics\nFor all the work in this paper, we have used the BAM\ncomputer codedescribed in [17, 25, 26]. It uses the BSSN\nformalism with 1+log slicing and ˜Γ driver condition in\nthe moving puncture framework [2, 27]. Puncture ini-\ntial data [28] with Bowen-York extrinsic curvature [29]\nhave been used throughout this work, solving the Hamil-\ntonian constraint with the spectral solver described in\n[30]. For binaries, parameters were chosen using [31] to\nobtain quasi-circularorbits, while the parametersfor sin-\ngle black holes were chosen directly. We extract waves\nvia the Newman-Penrose scalar Ψ 4. The wave extraction\nprocedure is described in detail in [17]. We perform a\nmode decomposition using spin-weighted spherical har-\nmonics with spin weight −2,Y−2\nlm, as basis functions and\ncalculate the scalar product\nΨlm\n4=/parenleftbig\nY−2\nlm,Ψ4/parenrightbig\n=/integraldisplay2π\n0/integraldisplayπ\n0sinθdθdϕY−2\nlmΨ4.(2)\nWe furthersplit Ψlm\n4intomode amplitude Almandphase\nφlmin order to cleanly separate effects in these compo-\nnents,rex·Ψlm\n4=Almeiφlm. In this paper, we focus on\none of the most dominant modes, the l=m= 2 mode,\nand report results for this mode unless stated otherwise.\nThe extraction radius used here is rex= 90M.B. Single, non-spinning puncture with constant\ndamping\nThedamping factor, η, in Eq.(1), isincluded to reduce\ndynamics in the gauge during the evolution. To examine\nthe problem brought up in the introduction, we compare\nresults of a single, non-spinning puncture with mass M.\nWe use a Courant factor of 0 .5 and 9 refinement levels\ncentered around the puncture. The resolution on the\nfinest grid is 0 .025M, and the outer boundary is situated\nat 256M. Varying the damping constant between 0 .0/M\nand4.5/M,twomainobservationscanbemade. First, as\ndesigned, a non-zero ηattenuates emerging gauge waves\nefficiently. Second, an instability develops for values of η\nthat are too large.\nFigs. 1, and 2 illustrate the first observation. Both\nfigures show the x-component of the shift along the x-\ncoordinate using η∈ {0.0/M,1.5/M,3.5/M}. Apart\n0 10 20 30 40\nx/M00.050.10.150.2βxη = 0.0/M\nη = 1.5/M\nη = 3.5/M\nFIG. 1: The x-component of the shift, βx, for a single non-\nspinning puncture of mass Mat timet= 15.2M. The three\nlines were taken for different values of the damping factor η.\nThe solid line (black) is for η= 0.0/M. The dashed line (red)\nis forη= 1.5/Mand the dotted-dashed line (green) is for\nη= 3.5/M. This shows the beginning of a pulse in βxfor\nsmaller values of η.\nfrom the usual shift profile, Fig. 1 shows the beginnings\nof a pulse in the η= 0.0/Mcase (solid line) at x≈10M\nafter 15.2Mof evolution. Examining Fig. 2, where we\nzoom in at a later time, t= 30.4M, one can see that\nthe pulse has started to travel further out (solid line).\nLooking carefully, one can also see a much smaller pulse\nin theη= 1.5/Mline (dashed). Lastly, by examination,\none can find almost no traveling pulse in the η= 3.5/M\ncurve (dotted-dashed line). The observed pulse in the\nshift travels to regions far away from the black hole and\neffects the gauge of distant observers. This might have3\n0 10 20 30 40 50\nx/M-0.0100.010.020.030.04βxη = 0.0/M\nη = 1.5/M\nη = 3.5/M\nFIG. 2: The x-component of the shift, βx, for a single non-\nspinning puncture of mass Mat timet= 30.4M. The three\nlines were taken for different values of the damping factor η\nwith the same line type and color scheme as in Fig. 1. Here\nit is clear a pulse radiates outward in the shift with smaller\nvalues of η.\nundesirable implications for the value of such numerical\ndata when trying to understand astrophysical sources.\nFor values of ηlarger than 3 .5/M, an instability\narises in the shift at larger radius. Fig. 3 shows the x-\ncomponent of the shift vector using damping constants\nη= 3.5/M(solid line), η= 4.0/M(dashed line) and\nη= 4.5/M(dotted-dashed line). The plots show an in-\nstability in simulations with η >3.5/Mdeveloping in\nβi, which eventually leads to a failure of the simulations.\nContrary to this, the simulation using η= 3.5/Mdoes\nnot show this shift related instability. In test runs we\nfound that by decreasing the Courant factor used, we\ncould increase the value of the damping factor and still\nget stable evolutions. This agrees with [16] where it was\nshown that the gamma driver possesses the stiff prop-\nerty, which limits the size of the time-step in numerical\nintegration based on the value of the damping.\nFigures 1, 2, and 3 make clear how the choice of the\ndamping factor affects the behavior of the simulations.\nThe value we choose for ηshould be non-zero and not\nlargerthan 3 .5/Mto allow for effective damping and sta-\nble simulations. The exact cutoff value between stable\nand unstable simulations is not relevant here since the\nposition dependent form we develop in Sec. III gives us\nthe flexibility we need to obtain stable simulations.0 50 100 15000.511.520 50 100 15000.511.52\nPSfrag replacementsx/M\nx/Mη= 3.5/M\nη= 3.5/Mη= 4.0/M\nη= 4.0/Mη= 4.5/Mβxβxt= 16M\nt= 30.4M\nFIG. 3: The x-componentof theshift vectorin the x-direction\nfor a single non-spinning puncture of mass Mat times t=\n16.0Mandt= 30.4M. The three different lines mark three\nvalues of the damping constant η. The solid line (black) is\nforη= 3.5/M, the dashed line (red) for η= 4.0/Mand the\ndotted-dashed line (green) for η= 4.5/M. Att= 16M, the\nsimulation using η= 4.5/Mdevelops an instability in the\nshift vector and fails soon afterward, the same happens for\nη= 4.0/Matt= 30.4M. In the simulation using η= 3.5,\nno such instability develops (not shown).\nC. Equal mass binary with constant damping\nTo examine the effect of ηon the extraction of gravi-\ntational waves, we compare the results from simulations\nof an equal mass binary with total mass Min quasi-\ncircular orbits with initial separation D= 10M, using\nη∈ {0.0/M,0.5/M,2.0/M}. Again, the Courant factor\nis chosen to be 0 .5 and we use, in the terminology of [17],\nthe grid configuration χ[6×56 : 5×112 : 6] with a finest\nresolution of 0 .013M. Here, the extraction radius rexis\nchosen to be 90 M.\nFor vanishing η, we find a lot of noise in the the real\npart of the 22-mode of rexΨ4, shown in the solid curve of\nFig. 4. A small, but non-vanishing ηsuffices to suppress\nthis noise, as seen in the dashed curve of this figure. The\ndotted-dashedcurvein this plot is the result forusing the\nvalueη= 2.0/M. We see a difference in time between\npeak amplitudes of the three curves due to the change\nof coordinates that the alternation of ηintroduces. We\ndid, however, find that by decreasing the Courant factor\nthosedifferencesbetween peakamplitudes summarilyde-\ncreased.\nTo understand the noise in the waves for η= 0.0/M,\nwe look at the shift vector at different times. The first\npanel of Fig. 5 shows the x-component of the shift over4\n0 200 400 600 800-0.008-0.006-0.004-0.00200.0020.004\nPSfrag replacementsη= 0.0/M\nη= 0.5/M\nη= 2.0/MRe{Ψ224}rex·M\nt/M0 200 400 600 800 1000 1200-0.06-0.04-0.0200.020.040.06\nPSfrag replacements\nηs= 0.0M\nηs= 0.5M\nηs= 2.0M\nRe{Ψ224}rex·M\nt/M\nFIG. 4: Real part of the 22-mode of Ψ 4over time for equal\nmass simulations using different values for η. The inset shows\nthe full waveform until ringdown. The solid curves (black)\nare forη= 0, the dashed curve (red) mark η= 0.5/Mand\nthe dotted-dashed curves (green) are for η= 2.0/M. Without\ndamping in the shift, the extracted waves are noisy at times\nwhen the amplitude is still small (black, solid curve).\nx, againforη∈ {0.0/M,0.5/M,2.0/M},shortlyafter the\nbeginning of the simulation. The fourth panel shows the\nsame at a time shortly before the merger, and the two\npanels in the middle represent intermediate times. We\nsee clear gauge pulses in the earliest time panel for all\nthree curves. We also observe the amplitude of this pulse\ndecreasing with increasing η. As time goes on, the gauge\npulse travelsoutwardsas in the case for a single puncture\nin section IIB. For vanishing η(solid line), the shift\nbecomes more and more distorted, and the distortions\ndo not leave the grid. For non-zero η, the amplitude of\nthe gauge pulse decreases when traveling outwards, and\nthe shape of βxis not distorted. There is, compared to\nη= 2.0/M(dotted-dashed line), only a small bump left\nin theη= 0.5/Mcase (dashed line), that changes its\nshape slightly during the simulation, but does not travel\nto large distances from the punctures. The coordinates\nare disturbed in the case where no damping is used, and\nthus the noise in rexRe{Ψ22\n4}is not surprising.\nIn this series, using a Courant factor of 0.5, we only\nobtained stable evolutions for η <3.5/Mwhich agrees\nwith thelimits foundin sectionIIB. Ifwechosethe value\nofηtoo large, the same kind of instability in the shift\nvectorwefoundtheredevelopsintheequalmasscaseand\nthe simulations fail. The failure occurs relatively early,\nbefore 50Mof evolution time, whereas the stable runs\nlasted about 1200 M,including mergerand ringdown(we\nstopped the runs after ringdown).\nIII. POSITION-DEPENDENT FORMS OF η\nInsectionIIB, wesawthatasufficientlevelofdamping\nis needed to limit gauge dynamics, and too much damp--200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004\n-200 -100 0 100 200-0.004-0.00200.0020.004PSfrag replacementsη= 0.0/M\nη= 0.5/M\nη= 2.0/Mt= 64M\nt= 170 M\nt= 456 M\nt= 751 Mβxβxβxβx\nx/Mx/Mx/Mx/M\nFIG. 5: x-component of the shift vector, βx, for three differ-\nent choices of ηat four different times during the simulation.\nThe physical system is the same as in Fig. 4. The merger\ntakes place at approximately t= 1000M. In the η= 0/M\ncase (black, solid curve), the shift vector is not damped and\ntherefore, a pulse travels outwards and distorts the shift o ver\nthe whole grid. The amplitude of this pulse is considerably\ndamped when using a non-vanishing ηand therefore the dis-\ntortions are reduced. For η= 0.5/M(red, dashed curve),\nthere are still small bumps traveling out which are reduced\nby using η= 2/M(green, dotted-dashed curve).\ning can lead to numerical instabilities. In section IIC,\nwe saw the positive effect that sufficient damping has on\nthe resultantwaveformforequalmassbinaries. While we\nstill need damping in the gamma driver in the unequal\nmass case, a constant value may not fulfill the require-\nments of limiting gauge dynamics and permitting stable\nevolutions. Rather, we need a definition for the damping\nthat adjusts the value to the local mass-scale.\nWe will examine definitions, that naturally track the\nposition, and mass of the individual black holes. The\nchoice ofηshould provide a reasonable value both near\nthe individual black holes, and at large distance from\nthe binary. We will start by examining some previous5\nwork, that has used non-constant forms of the damping\nparameter, and why it may be necessary to use other\nformulas. We will then present the two new formulas for\nη, which we designed for this work.\nA. Previous dynamic damping parameters\nA position dependent damping was introduced some\nyears ago by the authors of [20], and was later used in\n[21]. That formula reads\nη=ηpunc−ηpunc−η∞\n1+(ψ−1)2(3)\nwithηpunc,η∞being constants, and assuming ψ=\n1 +M1/(2r1) +M2/(2r2) (riis the distance to the ith\npuncture). Thisformulawasusedtodampgaugedynam-\nics while using excision for equal-mass head-on collisions.\nIt has since been found that using the moving puncture\nframework allows for constant damping in the approxi-\nmately equal mass case. We are looking for a formula\nwhich is suitable for the quasicircular inspiral of inter-\nmediate mass-ratio binaries.\nPreviously [23], we used the formula\nη(/vector r) =ˆR0/radicalbig\n˜γij∂iψ−2∂jψ−2\n(1−ψ−2)2, (4)\nfor determining a position dependent damping coeffi-\ncient instead of using a constant η. WithˆR0taken to be\na unitless constant, it can be seen that Eq. (4) has units\nof inverse mass. The dependence on the BSSN variable,\nψ, naturally tracks the position, and mass of the black\nholes. The application of Eq. (4) gave good values for\nthe damping both at the punctures, and at the outer\nboundary, and was even found to somewhat decrease the\ngrid-size of the larger black hole. The latter point could\nhave positive effects on how the individual black holes\nare resolved on the numerical grid. It even had the addi-\ntional effect of keeping the horizon shapes roughly circu-\nlar, even close to merger - something that doesn’t hold in\nthe constant ηcase. Most importantly, the simulations\nremained stable, without significantly changing the grav-\nitational waves. The formula was later used successfully\nfor the 10:1 mass-ratio in [9].\nDespite all this, Eq. (4) provides reason for concern.\nFig.6showstheformof ηusingEq.(4)foranon-spinning\nbinary of equal mass in quasicircular orbits starting at a\nseparation of D= 10Mat four different times in the\nsimulation. As can be seen, noise travels out from the\norigin as time progresses. This leaves steady features on\nthe form of ηwhich could spike to higher and lower val-\nues than the range determined in Sec. IIB. Additionally,\nthese sharp features may lead to unpredictable coordi-\nnate drifts, and could, in some cases, affect the long-term\nstability of the simulation.\nTo illuminate the origin of the disturbances in η(/vector r),\nwe looked at the development of η(/vector r) in simulations of0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\n0 200 400 600 80000.511.522.53\nPSfrag replacements\nx/Mη(x)·M η (x)·M η (x)·M η (x)·Mt= 0M\nt= 79M\nt= 165 M\nt= 300 M\nFIG. 6: Damping factor, η, along the x-axis using Eq. (4).\nThe simulated configuration is an equal mass binary with ini-\ntial separation D= 10Mand orbits lying in the ( x,y)-plane.\nShown are four different times during the simulation.\na single, non-spinning puncture, and a single, spinning\npuncture (Sz/M2= 0.25). The result for the spinning\ncase is plotted in Fig. 7 at two different times over the\nx-axis. Again, we see a pulse traveling outwards. Only\nthis time, it does not leave much noise on the grid. The\nfact that this pulse travels at a speed which is roughly\n1.39 (in our geometric units where c=G= 1) in both\nthe spinning and non-spinning scenario indicates that it\nis related to the gauge modes traveling at speed√\n2 in\nthe asymptotic regionwhere α≃1 (see [32] and [19] for a\ndiscussion of gauge speeds). In contrast to gauge pulses\nin the lapse, α, or shift vector, βi, the pulse in η(x) is\namplified as it walks out. We found the same result in\nthe single puncture simulation without spin. We believe\nthe reason for this behavior is that as the distance to the\npuncture increases,the conformalfactor, ψ, getscloserto\nunity. Therefore, the denominator in Eq. (4) approaches\nzero, and the gauge disturbances in the derivatives of ψ\nare magnified. We further observed reflections at the re-6\nfinement boundaries as this pulse passes through them.\nThismayexplainthe fluctuationsin η(x) shownin Fig.6.\nWhile one could continue to fine-tune a formula depen-\ndent on the conformalfactorto deal with these problems,\nwe looked in a different direction to determine the form\nof the damping parameter.\n-200 -100 0 100 20000.511.522.53\nPSfrag replacements\nx/Mη(x)·Mt= 50 .56M\nt= 101 .25M\nFIG. 7: Form of η(/vector r) for a single spinning puncture sitting\natx= 0 using Eq. (4) after simulation time t= 50.56M\n(solid black line) and t= 101.25M(dashed red line) over\nx−direction.\nB. Formulas for ηwith explicit dependence on the\nposition and mass of the punctures\nSince we always know the location of a puncture, and\nwe know what its associated mass, we chose a form\nof damping that uses this local information throughout\nthe domain. To address the demands and concerns dis-\ncussed in Section II and IIIA, we designed two position-\ndependent forms of η. The two forms we tested are\nη(/vector r) =A+C1\n1+w1(ˆr2\n1)n+C2\n1+w2(ˆr2\n2)n,(5)\nand\nη(/vector r) =A+C1e−w1(ˆr2\n1)n+C2e−w2(ˆr2\n2)n.(6)\nInEqs.(5) and(6), w1andw2arerequiredtobe positive,\nunitless parameters which can be chosen to change the\nwidth of the functions. The power nis taken to be a\npositive integer which determines the fall-off rate. The\nconstantsA,C1, andC2are then chosen to provide the\ndesired values of ηat the punctures, and at at infinity.\nLastly, ˆr1and ˆr2are defined as ˆ ri=|/vector ri−/vector r|\n|/vector r1−/vector r2|, whereiis\neither one or two, and /vector riis the position of the i’th black\nhole.\nThe definition of ˆ riis chosen to naturally scale the fall-\nofftothe separationofthe blackholes. w1,w2, andncan\nbe chosen to change the overallfall-off. Our work focuseson the choice w1=w2=wandn= 1. Following [23],\nwe construct the damping factor to have units of inverse\nmass. We choose A= 2/Mtot, whereMtot≡M1+M2\nis defined as the sum of the irreducible masses. We then\ntakeCi= 1/Mi−A. It is then evident that both Eqs. (5)\nand (6) will give a constant value of η= 2/Mtotin the\nequal mass case.\nWe designed the two formulas for ηin order to test\nthe value of using fundamentally different functions. In\nour simulations, we found little noticeable difference in\nthe application of one compared to the other. In the\nabsence of such a difference, it becomes more beneficial\nto use Eq. (5), as Gaussians are computationally more\nexpensive. It should be pointed out that Eq. (5) is very\nsimilar to Eq. (13) suggested in [16], and we believe the\nfollowing results are very similar to what would be found\nusing that form for the damping. Going into the present\nwork, we haveno ansatz which might suggestthese forms\nof damping yield wave forms which are any better than\nthe use of any previous form of η. However, as will be\nseen in the results sections, the waveforms we get from\nunequal mass binaries show noticeable improvement over\nthe constant ηcase.\nIV. RESULTS\nFor data analysis purposes, we are mainly interested in\nthe properties of the emitted gravitational waves of the\nblack hole binary systems under study. Hence, it is im-\nportant to check how the changes in the gauge alter the\nextracted waves. In the context of gravitational wave ex-\ntraction, Ψ 4is only first orderinvariant under coordinate\ntransformations. In addition, we haveto chosean extrac-\ntion radius rexfor the computation of modes, which is\nalso coordinate dependent. Although the last point can\nbe partly addressed by extrapolation of rex→ ∞, it is\na priori not clear how much a change of coordinates af-\nfects the gravitational waves. Furthermore, a change of\ncoordinates implies an effective change of the numerical\nresolution, and for practical purposes we have to ask how\nmuch waveforms differ at a given finite resolution.\nA. Waveform comparison using formula (5)\nThe results in the following section refers to the use of\nEq. (5). We compare numerical simulations using three\ndifferent grid configurations, which correspond to three\ndifferent resolutions. In the terminology of [17], the grid\nset-upsareφ[5×64 : 7×128 : 5],φ[5×72 : 7×144: 5], and\nφ[5×80 : 7×160: 5], which correspondsto resolutionson\nthe finest grids of 3 M/320 (N= 64),M/120 (N= 72)\nand 3M/400 (N= 80), respectively. When referring\nto results from different resolutions, we will from here\non use the number of grid points on the finest grid, N,\nto distinguish between them. In this subsection, we use\nw1=w2= 12 andn= 1 in Eq. (5). As test system we7\nuse an unequal mass black hole binary with mass ratio\nm2/m1= 4 andan initial separationof D= 5Mwithout\nspins in quasi-circular orbits.\nFor orientation, Fig. 8 shows the amplitude of the 22-\nmode,A22, computed with the standard gauge η= 2/M\n(displayed as solid lines) and with the new η(/vector r) using\nEq. (5) (displayed as non-solid lines). The three differ-\nent colors correspond to the three resolutions. The inset\nshows a larger time range of the simulation, while the\nmain plot concentrateson the time frame around merger.\nThe plotgivesacourseviewofthe closenessofthe results\nwe obtain with standard and new gauges.\nIn Fig. 9, we plot the relative differences between\nthe amplitudes at low and medium (solid lines), and\nmedium and high resolution (non-solid lines) obtained\nwithη= 2/M(light gray lines) as well as η(/vector r) (Eq. (5))\n(black lines). Here, we find the maximum error be-\ntween the low and medium resolution of the series using\nη= 2/Mamounts to about 12% (solid gray curve). Be-\ntween medium and high resolution (dashed gray curve),\nwe find a smaller relative error, but it still goes up to\n7% at the end of the simulation. Employing Eq. (5),\nthe maximum amplitude error between low and medium\nresolution (solid black line) is only about 4%, and there-\nfore even smaller than the error between medium and\nhigh resolution for the constant damping case. Between\nmedium and high resolution, the relative amplitude dif-\nferences for Eq. (5) are in general smaller than the ones\nbetween low and medium resolution, although the maxi-\nmum error is comparable to it (dot-dashed black line).\nWe repeat the previous analysis for the phase of the\n22-mode,φ22. Again, we comparethe errorsbetween res-\nolutions in a fixed gauge. Figure 10 shows that the error\nbetween lowest and medium resolution using η= 2/M\n(solid gray line) grows up to about 0.31 radians. For the\ndifferences between medium and high resolution (dashed\nline) we find a maximal error of 0.2 radians for η= 2/M.\nForη(/vector r) following Eq. (5), the phase error between low\nand medium resolution is only about 0.19 radians (solid\nblack line) and decreases to 0.1 radians between medium\nand high resolution (dot-dashed line). Again, employ-\ning the position dependent form of η, Eq. (5), the error\nbetween lowest and medium resolution is lower than the\none we obtain for constant ηbetween medium and high\nresolution. The results for amplitude and phase error\nsuggest that we can achieve the same accuracy with less\ncomputationalresourcesusingaposition-dependent η(/vector r).\nB. Waveform comparison using formula (6)\nWe repeated the analysis of Sec. IVA with the wave-\nforms we obtain using Eq. (6) (with w1=w2= 12 and\nn= 1). We use the same initial conditions (mass ratio\n4 : 1,D= 5M, no spins), and compare the amplitudes\nand phases of the 22-mode of Ψ 4with the results of the\nη= 2/M-runs. The grid configurations remain the same.\nThe results are very similar to the ones we obtained in160 165 170 175 180 185 1900.0150.0200.0250.0300.035\nt/Slash1MA22/DotMatΗM\n100 120 140 160 180 2000.0000.0050.0100.0150.0200.0250.0300.035\nt/Slash1MA22/DotMatΗMN/Equal64\nN/Equal72\nN/Equal80\nFIG. 8: Amplitude of the 22-mode of Ψ 4of a binary with\nmass ratio 4:1 and initial separation D= 5M. The different\ncolors correspond to three different resolutions according to\nthe grid setup described in the text. The solid lines are resu lts\nforη= 2/M, the dashed, dotted and dot-dashed ones are for\nη(/vector r) (Eq. (5)). The inset shows the simulation from shortly\nafter the junk radiation passed, in the main plot we zoom into\nthe region of highest amplitude (near the merger).\n100 120 140 160 180 200/Minus0.050.000.050.100.15\nt/Slash1M/CapDelta/CapAlpΗa22/Slash1/CapAlpΗa22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen164/Minus72/RParen1,Η/Equal2/Slash1M\n/LParen180/Minus72/RParen1,Η/Equal2/Slash1M\nFIG. 9: Relative differences of the amplitude of the 22-mode\nof Ψ4between resolutions N= 64 and N= 72 (gray solid\ncurve) as well as N= 72 and N= 80 (gray dashed curve)\nwhen using η= 2/M. The same for η(/vector r) (Eq. (5)) between\nN= 64 and N= 72 (black solid curve) and N= 72 and\nN= 80 (black dot-dashed curve). The physical situation is\nthe same as in Fig. 8. The maximum differences are above\n10%, comparing low and medium resolution of the constant\nηsimulations (gray solid line).\nFigs. 9 and 10, and we therefore do not show them here.\nAlthough Eqs. (5) and (6) result in different shapes for\nη(/vector r), Ψ22\n4is very similar. Therefore, the comparison to\nη= 2/Mnaturally gives very similar results, too. The\nphase differences between results from Eqs. (5) and (6)\nat a given resolution are shown in Fig. 11. These are,\nwith a maximum phase error of 0.004 radians, very small\ncomparedto the phase errorsbetween resolutions, which,\nat minimum, are about 0.1 radian (see Fig. 10). Fig. 12\ncomparesthe phase errorbetween low and medium (solid8\n100 120 140 160 180 2000.00.10.20.3\nt/Slash1M/CapDeltaΦ22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1\n/LParen164/Minus72/RParen1,Η/Equal2/Slash1M\n/LParen180/Minus72/RParen1,Η/Equal2/Slash1M\nFIG. 10: Phase differences between lowest and medium reso-\nlution for the series using η= 2/M(solid gray line) and η(/vector r)\n(Eq. (5)) (solid black line) as well as between medium and\nhigh resolution for η= 2/M(dashed gray line) and for η(/vector r)\n(Eq. (5)) (dot-dashed black line). The physical situation i s\nthe one of Fig. 8.\n100 120 140 160 180 200/Minus0.004/Minus0.0020.0000.0020.004\nt/Slash1M/CapDeltaΦ22N/Equal64,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nN/Equal72,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nN/Equal80,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1\nFIG. 11: Phase differences between waveforms obtained\nwith Eq. (5) and Eq. (6) in three different resolutions (solid ,\ndashed, dotted-dashed lines) for mass ratio 4:1, D= 5M.\nlines), and medium and high resolution (dotted-dashed\nand dashed line) of Eq. (6) (gray) to the ones of Eq. (5)\n(black). For comparison, the error between medium and\nhigh resolution is also plotted for Eq. (4) in this figure\n(dotted line). The plot indicates that the errors between\nresolutions are in good agreement for the different posi-\ntion dependent formulas of η.\nC. Behavior of the shift vector\nIn [23], we found an unusual behavior of the shift vec-\ntor. This is illustrated in Fig. 13, where we plot the\nx-component of the shift, βx, in thex-direction after\n160Mof evolution (this means approximately 80 Maf-\nter merger) for all four versions of the damping constant\nwe used for comparison in this paper before, and for the\nsame binary configuration as the one used in Secs. IVA100 120 140 160 180 2000.000.050.100.150.20\nt/Slash1M/CapDeltaΦ22/LParen180/Minus72/RParen1,ΗEq./LParen14/RParen1\n/LParen164/Minus72/RParen1,ΗEq./LParen15/RParen1\n/LParen180/Minus72/RParen1,ΗEq./LParen15/RParen1\n/LParen164/Minus72/RParen1,ΗEq./LParen16/RParen1\n/LParen180/Minus72/RParen1,ΗEq./LParen16/RParen1\nFIG. 12: Phase difference between waveforms at low and\nmedium resolution (solid lines) and medium and high reso-\nlution (dotted-dashed and dashed line) using either Eq. (5)\n(black lines) or Eq. (6) (gray lines) for mass ratio 4:1, D=\n5M. For comparison, we also show the phase difference ob-\ntainedwith Eq. (4) between mediumandhigh resolution (dot-\nted line).\n0 100 200 300 400 5000.0000.0010.0020.0030.004\nx/Slash1MΒxΗ/Equal2/Slash1M\nEq./LParen14/RParen1\nEq./LParen15/RParen1\nEq./LParen16/RParen1\nFIG. 13: x-component of the shift vector in x-direction after\n160Mof evolution of the system with mass ratio 4 : 1 and\nD= 10M. The black, dot-dashed line refers to the use of a\nconstant damping η, while the black, solid line uses Eq. (4).\nThe gray, dashed line is for the use of Eq. (6) and the gray,\ndotted one for Eq. (5). Except for the constant η(black, dot-\ndashed line), the results in this plot are indistinguishabl e.\nand IVB. Like in [23], we find that using Eq. (4) results\nin a shift which falls off to zero too slowly towards the\nouter boundary, and which develops a “bump” (black,\nsolid line), while the constant damping case (black, dot-\ndashed line) falls off to zero quickly. Employing Eqs. (5)\nor (6) avoids this undesirable feature. After merger, the\nshift falls off to zero when going awayfrom the punctures\nas it does in the constant damping case (gray dashed and\ndotted lines). Using Eq. (5) or (6) prevents unwanted co-\nordinate drifts at the end of the simulations.9\n0 10 20 30 40 500102030405060\nPSfrag replacements\nt/MAAH,1/AAH,2\nn= 1, w= 0.1\nn= 1, w= 0.5 n= 1, w= 6.0\nn= 1, w= 12.0\nn= 1, w= 16.0n= 1, w= 200 .0\nn= 3, w1= 0.01,\nw2= 0.001\nη= 2/M\nηfrom Eq. (4)\nFIG. 14: Shown is the time dependence of the ratio between\nthe coordinate areas of the apparent horizons of both black\nholes in a simulation with mass ratio 4 : 1 with initial separa -\ntionD= 5M. The black, blue and red lines use η(/vector r), Eq. (5)\nwithvaryingvaluesofthewidthparameter w. Theorange line\n(dash-dot-dot) uses the constant damping η= 2/Mand the\ngreen (dash-dot) one refers to the result of [23] with Eq. (4) .\nUsing Eq. (5), the coordinate areas can be varied with re-\nspect to each other depending on the choice of w. A ratio of\n1 means the black holes have the same size on the numerical\ngrid.\nV. DISCUSSION\nIn this work, we examined the role that the damping\nfactor,η, plays in the evolution of the shift when using\nthe gamma driver. In particular, we examined the range\nof values allowed in various evolutions, and what effects\nshowed up because of the value chosen. We then de-\nsigned a form of ηfor the evolution of binary black holes\nwhich provides appropriate values both near the individ-\nual punctures and far away from them with a smooth\ntransition in between.\nIn Sec. IV, we directly examined the waveformsfor the\ncaseusingEq.(5), where w1=w2= 12andn= 1. While\nthe form of ηis predictable, and can be easily adjusted\nfor stability, we also saw that the waveforms produced\nusing this definition showed less deviation with increas-\ning resolution than using a constant η. When examining\nthe waveforms produced using Eq. (6), we found simi-\nlar results. In the absence of a noticeable difference in\nthe quality of the waveforms, Eq. (5) is computationally\ncheaper, and, as such, is our preferred definition for the\ndamping.\nWe have already pointed out a certain freedom to\npick parameters in Eqs. (5) and (6). We did perform\nsome experimentation along this line where we varied\nw=w1=w2to see if we could get a useful effect of the\ncoordinate size of the apparent horizons on the numer-\nical grid. In [17, 33], it was noticed that the damping\ncoefficient affects the coordinate location of the apparent\nhorizon, and therefore the resolution of the black hole onthe numerical grid. Fig. 14 plots the ratio of the grid-\narea of larger apparent horizon to the smaller apparent\nhorizon as a function of time for w-values of 0 .1, 0.5 and\nfor 200, all with n= 1. Also plotted is the relative co-\nordinate size for the same binaries using a constant η\nin dashed, double-dotted line, and for using Eq. (4) in\na blue dashed-dotted line. All the evolutions show an\nimmediate dip, and then increase in the grid-area ratio\nduring the course of the evolution. While a very low ra-\ntio was found using Eq. (4), the orange dotted line was\nlater found for the choices of n= 3 withw1= 0.01 and\nw2= 0.0001 with Eq. (5). Due to this freedom in the\nimplementation of our explicit formula for the damping,\nit may be possible to further reduce the relative grid size\nof the black holes. This effect could be important in eas-\ning the computational difficulty of running a numerical\nsimulation for unequal mass binaries.\nHaving a form of ηthat leads to stable evolutions for\nany mass-ratio is an important step towards the numer-\nical evolution of binary black holes in the intermediate\nmass-ratio. We believe the form given in Eq. (5) pro-\nvides such a damping factor at a low computational cost,\nalthough the test results presented are limited to mass\nratio 4 : 1. We plan to examine larger mass ratios in\nfuture work. The new method should allow binary sim-\nulations for mass ratio 10 : 1, or even 100 : 1. It remains\nto be seen whether other issues than the gauge are now\nthe limiting factor for simulations at large mass ratios.\nAcknowledgments\nItisapleasuretothankZhoujianCaoandErikSchnet-\nter for discussions. This work was supported in part\nby DFG grant SFB/Transregio 7 “Gravitational Wave\nAstronomy” and the DLR (Deutsches Zentrum f¨ ur Luft\nund Raumfahrt). D. M. was additionally supported by\nthe DFG Research Training Group 1523 “Quantum and\nGravitational Fields”. Computations were performed on\nthe HLRB2 at LRZ Munich.10\n[1] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-\nqc/0507014.\n[2] M. Campanelli, C. O. Lousto, P. Marronetti, and\nY. Zlochower, Phys. Rev. Lett. 96, 111101 (2006), gr-\nqc/0511048.\n[3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and\nJ. van Meter, Phys. Rev. Lett. 96, 111102 (2006), gr-\nqc/0511103.\n[4] F. 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Gundlach and J. M. Martin-Garcia, Phys. Rev. D 74,\n024016 (2006), gr-qc/0604035.\n[33] F. Herrmann, D. Shoemaker, and P. Laguna (2006), gr-\nqc/0601026."    },    {        "title": "1003.5375v1.Damped_wave_dynamics_for_a_complex_Ginzburg_Landau_equation_with_low_dissipation.pdf",        "content": "arXiv:1003.5375v1  [math.AP]  28 Mar 2010DAMPED WAVE DYNAMICS FOR A COMPLEX GINZBURG-LANDAU\nEQUATION WITH LOW DISSIPATION\nEVELYNE MIOT\nAbstract. We consider a complex Ginzburg-Landau equation on RN, corresponding to a\nGross-Pitaevskii equation with a small dissipation term. W e study an asymptotic regime for\nlong-wave perturbations of constant maps of modulus one. We show that such solutions never\nvanish on RNand we derive a damped wave dynamics for the perturbation. Ou r results are\nobtained in the same spirit as those by Bethuel, Danchin and S mets for the Gross-Pitaevskii\nequation [2].\n1.Introduction\nWe consider a complex Ginzburg-Landau equation\n∂tΨ = (κ+i)[∆Ψ+Ψ(1 −|Ψ|2)], (C)\nwhereΨ = Ψ( t,x) :R+×RN→C, withN≥1, isacomplex-valued mapandwhere0 < κ <1.\nEquation (C) admits elementary non-vanishing solutions, w hich are given by all constant\nmaps of modulus equal to one. The aim of this paper is to study t he dynamics for (C) near\nsuch states. We focus on a regime in which the solutions Ψ do no t vanish on RN, so that we\nmay write them into the form\nΨ =rexp(iφ).\nSecondly, we assume that ( r2,∇φ) is a long-wave perturbation of (1 ,0). More precisely, we\nintroduce a small parameter ε >0 and we define ( r2,∇φ) through the change of variables\n\n\nr2(t,x) = 1+ε√\n2aε(εt,εx)\n2∇φ(t,x) =εuε(εt,εx),(1.1)\nwhere (aε,uε) belongs to C(R+,Hs+1×Hs), withs≥2, and satisfies suitable bounds.\nOur objective is two-fold. First, to define ( aε,uε) we wish to determine how long a solution\ninitially given by (1.1) does not vanish on RN. Our second purpose is to investigate the\ndynamics of ( aε,uε) whenεvanishes and κis small. This asymptotic dynamics depends on\nthe balance between the amount κof dissipation in Eq. (C) and the size εof the perturbation;\nto characterize this balance we introduce the ratio\nνε=κ\nε.\nAccording to (C) we obtain the equations for the perturbatio n (aε,uε)\n/braceleftigg\n∂taε+√\n2divuε+2νε−κε∆aε= fε(aε,uε)\n∂tuε+√\n2∇aε−κε∆uε= gε(aε,uε),(1.2)\nDate: November 23, 2018.\n12 EVELYNE MIOT\nwhere f εand gεare given by\n\n\nfε(aε,uε) =√\n2κ/parenleftig\n−2|∇ρa|2−ρ2\na|uε|2\n2−a2\nε/parenrightig\n−εdiv(aεuε)\ngε(aε,uε) =κε∇/parenleftbigg∇ρ2\na\nρ2a·uε/parenrightbigg\n+2ε∇∆ρa\nρa−εuε·∇uε,(1.3)\nwith\nρ2\na(t,x) = 1+ε√\n2aε(t,x).\nOur first result establishes that if the initial perturbatio n is not too large, the solution Ψ\nnever exhibits a zero so that (1.1) does hold for all time.\nTheorem 1.1. Letsbe an integer such that s >1 +N/2. There exist positive numbers\nK1(s,N),K2(s,N)and0< κ0(s,N)<1, depending only on sandN, satisfying the following\nproperty.\nLet0< κ≤κ0(s,N). For0< ε≤1, let(a0\nε,ϕ0\nε)∈Hs+1(RN)2such that\nM0:=/ba∇dbl(a0\nε,u0\nε)/ba∇dblHs+ε/ba∇dbla0\nε/ba∇dblHs+1+/ba∇dblϕ0\nε/ba∇dblL2≤min(νε,κ−1,ε−1)\nK1(s,N),\nwhereu0\nε= 2∇ϕ0\nε.\nThen Eq. (1.2)-(1.3)has a unique global solution (aε,uε)inC(R+,Hs+1×Hs)such that\n(aε,uε)(0) = (a0\nε,u0\nε). Moreover\n/ba∇dbl(aε,uε)/ba∇dblL∞(Hs)+ε/ba∇dblaε/ba∇dblL∞(Hs+1)≤K2(s,N)M0.\nFinally, if Ψdenotes the corresponding solution to Eq. (C), we have for all t≥0\n/vextenddouble/vextenddouble|Ψ(t)|2−1/vextenddouble/vextenddouble\n∞<1\n2.\nRemark 1.1. Fixingκ=κ0andε=ε0, Theorem 1.1 entails that for initial data\nΨ0(x) =/parenleftbig\n1+˜a0(x)/parenrightbig1/2exp(i˜ϕ0(x)),\nwith/ba∇dbl(˜a0,˜ϕ0)/ba∇dblHs+1≤C, whereConly depends on sandN, the corresponding solution1Ψ\nto Eq.(C)remains bounded and bounded away from zero for all time.\nRemark 1.2. For all0< ε≤ε0and0< κ≤κ0satisfying ε≤κ, so that νε≥1, Theorem\n1.1 allows to handle initial data\nΨ0\nε(x) =/parenleftbigg\n1+ε√\n2a0(εx)/parenrightbigg1/2\nexp(iϕ0(εx)), (1.4)\nwhere(a0,ϕ0)∈Hs+1(RN)2does not depend on ε, so that M0is constant, and where M0is\nsmaller than a number depending only on sandN.\nOnce the question of existence for ( aε,uε) has been settled, our next task is to determine\na simplified system of equations to describe its asymptotic d ynamics. From now on we focus\non a regime with low dissipation, namely we further assume th at\nκ=κ(ε) and lim\nε→0κ(ε) = 0.\n1Given by Theorem 3.1 below.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 3\nIn view of (1.3), this is a natural ansatz in order to treat the second members f εand gεas\nperturbations in the limit ε→0. Eq. (1.2) then formally reduces to a damped wave equation\n/braceleftigg\n∂ta+√\n2divu+2νεa= 0\n∂tu+√\n2∇a= 0,(1.5)\nwith propagation speed equal to√\n2 and damping coefficient equal to 2 νε.\nAs a consequence of Theorem 1.1 we can compare the solution ( aε,uε) to the one of the\nlinear damped wave equation (1.5) with loss of three derivat ives.\nTheorem 1.2. Letsbe an integer such that s >1+N/2. Let(a0\nε,ϕ0\nε)∈Hs+1(RN)2satisfy\nthe assumptions of Theorem 1.1. Let u0\nε= 2∇ϕ0\nε.\nWe denote by (aℓ,uℓ)∈C(R+,Hs+1×Hs)the solution of Eq. (1.5)with initial datum\n(a0\nε,u0\nε).\nThere exists a constant K3(s,N)depending only on sandNsuch that for all t≥0\n/ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−2≤K3(s,N)(εκt)1/2max(1,ν−1\nε)(M2\n0+M0),\nwhereM0is defined in Theorem 1.1.\nIn particular, for initial data given by (1.4), the approxim ation by the damped wave equa-\ntion is optimal when κandεare comparable. Moreover, Theorem 1.2 yields a correct appr ox-\nimation up to times of order C(κε)−1. In order to handle larger times, it is helpful to take\ninto account the linear parabolic terms in (1.2):\n/braceleftigg\n∂ta+√\n2divu+2νεa−κε∆a= 0\n∂tu+√\n2∇a−κε∆u= 0.(1.6)\nOur next result presents uniform in time comparison estimat es with the solution of Eq. (1.6)\nfor high order derivatives.\nTheorem 1.3. Letsbe an integer such that s >1+N/2. Let(a0\nε,ϕ0\nε)∈Hs+1(RN)2satisfy\nthe assumptions of Theorem 1.1.\nWe denote by (aℓ,uℓ)∈C(R+,Hs+1×Hs)the solution of Eq. (1.6)with initial datum\n(a0\nε,u0\nε).\nThere exists a constant K4(s,N)depending only on sandNsuch that\n•/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs−2)≤K4(s,N)/parenleftbig\nκmax(1,ν−1\nε)2M2\n0+εmax(1,ν−1\nε)M0/parenrightbig\n,\n•/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs−1)≤K4(s,N)/parenleftig\nmax(1,ν−1\nε)/parenleftbig\nmax(κ,ε)+ν−1\nε/parenrightbig\nM2\n0+ν−1\nεM0/parenrightig\n,\n•/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs)≤K4(s,N)/parenleftig\n(ν−1\nεmax(1,ν−1\nε)+κ−1)M2\n0+κ−1M0/parenrightig\n.\nFinally, for all t≥0\n•/ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−2≤K4(s,N)(εκt)1/2/parenleftbig\nmax(1,ν−1\nε)M2\n0+ν−1\nεM0/parenrightbig\n,\n•/ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−1≤K4(s,N)(εκ−1t)1/2M0.\nWe come back to initial data given by (1.4). Since κ−1diverges when ε→0, Theorem\n1.3 does not provide a correct approximation for s-order derivatives. However, Eq. (1.6)\nyields a satisfactory large in time approximation for the de rivatives of order s−1 ifν−1\nε\nvanishes with ε. In fact, the corresponding comparison estimate is optimal whenever κand√εare proportional. This is due to the fact that the regularizi ng properties of the parabolic\ncontributions in (1.6) become less efficient when κis small. On the other hand, as in Theorem4 EVELYNE MIOT\n1.2, the global in time comparison estimates involving the l ower (s−2)-order derivatives are\nmore efficient when κandεare proportional.\nThe complex Ginzburg-Landau equations are widely used in th e physical literature as a\nmodel for various phenomena such as superfluidity, Bose-Ein stein condensation or supercon-\nductivity, see [1]. In the specific form considered here, Eq. (C) corresponds to a dissipative\nextension of the purely dispersive Gross-Pitaevskii equat ion\n∂tΨ =i[∆Ψ+Ψ(1 −|Ψ|2)]. (GP)\nA similar asymptotic regime for (GP) has been recently inves tigated by Bethuel, Danchin and\nSmets [2]. The analysis of [2] exhibits a lower bound for the fi rst time Tεwhere the solution\nvanishes and shows that ( aε,uε) essentially behaves according to the free wave equation ( νε≡\n0), or to a similar version, until then.\nIn the two-dimensional case N= 2, there exists a formal analogy between Eq. (C) and\nthe Landau-Lifschitz-Gilbert equation for sphere-valued magnetizations in three-dimensional\nferromagnetics, see [3, 7]. We mention that a thin-film regim e leading to a damped wave\ndynamics for the in-plane components of the magnetization h as been studied by Capella,\nMelcher and Otto [3].\nFinally, still in the two-dimensional case N= 2, Eq. (C) presents another remarkable\nregime in which the solutions exhibit zeros (vortices). Thi s regime has been investigated by\nKurtzke, Melcher, Moser and Spirn [6] and the author [9] when κis proportional to |lnε|−1.\nIn this setting, Eq. (C) is considered under the form\n∂tΨε= (κ+i)[∆Ψε+1\nε2Ψε(1−|Ψε|2)], (Cε)\nwhich is obtained from the original equation via the parabol ic scaling\nΨε(t,x) = Ψ/parenleftbiggt\nε2,x\nε/parenrightbigg\n. (1.7)\nA natural extension of the results in [6, 9] would consist in a llowing for superpositions of\nvortices and oscillating phases in the initial data. This di fficult issue was a strong motivation\nto analyze the behavior of the phase in the regime (1.1), excl uding vortices, as a first attempt\nto tackle the general situation where it is coupled with vort ices.\n2.General strategy\nWe now present our approach for proving Theorems 1.1, 1.2 and 1.3, which will be partly\nborrowed from the analysis in [2] for the Gross-Pitaevskii e quation.\nFirst, we handle Eq. (C) in its parabolic scaling (1.7) yield ing Eq. (C ε). We define the\nvariables\n\nbε(t,x) =aε/parenleftbiggt\nε,x/parenrightbigg\nvε(t,x) =uε/parenleftbiggt\nε,x/parenrightbigg\n,\nso that in the regime (1.1) we have\nΨε(t,x) =ρε(t,x)exp(iϕε(t,x)) on R+×RN, (2.1)DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 5\nwhere \n\nρ2\nε(t,x) = 1+ε√\n2bε(t,x)\n2∇ϕε(t,x) =εvε(t,x).(2.2)\nThe system for ( bε,vε) translates into\n\n\n∂tbε+√\n2\nεdivvε+2νε\nεbε−κ∆bε=˜fε(bε,vε)\n∂tvε+√\n2\nε∇bε−κ∆vε= ˜gε(bε,vε),(2.3)\nwhere \n\n˜fε(bε,vε) =√\n2νε/parenleftbigg\n−2|∇ρε|2−ρ2\nε|vε|2\n2−b2\nε/parenrightbigg\n−div(bεvε)\n˜gε(bε,vε) =κ∇/parenleftbigg∇ρ2\nε\nρ2ε·vε/parenrightbigg\n+2∇/parenleftbigg∆ρε\nρε/parenrightbigg\n−vε·∇vε.(2.4)\nFor a map Ψ ∈H1\nloc, the Ginzburg-Landau energy of Ψ is defined by\nEε(Ψ) =/integraldisplay\nRN/parenleftig|∇Ψ|2\n2+(1−|Ψ|2)2\n4ε2/parenrightig\ndx,\nandEdenotesthecorrespondingspaceoffiniteenergyfields. Fort heGross-Pitaevskiiequation\nthe Ginzburg-Landau energy is an Hamiltonian, whereas for s olutions to Eq. (C ε) it decreases\nin time. Note that, in the regime (2.1)-(2.2), the solution Ψ εbelongs to Esince (bε,vε)∈\nH1×L2. In fact, one has\nEε(Ψε)≃C(/ba∇dbl(bε,vε)/ba∇dbl2\nL2+ε2/ba∇dbl∇bε/ba∇dbl2\nL2)\nprovided that /ba∇dbl|Ψε|−1/ba∇dbl∞<1.\nOur first issue is to solve the Cauchy problem for (C ε) so that ( bε,vε) being defined by\n(2.2), as long as Ψ εdoes not vanish, does belong to C(Hs+1×Hs). As mentioned, the initial\nfield Ψ0\nεhas finite Ginzburg-Landau energy. In [4] (see also [5]) it ha s been shown that\nE ⊂ W+H1(RN).\nHere the space W, which will be defined in Section 3 below, contains in particu lar all constant\nmaps of modulus one. It is therefore natural to determine the solution Ψ εinC(W+Hs+1).\nThis is done in Section 3.\nIn Theorems 1.1, 1.2 and 1.3 one assumes that /ba∇dblb0\nε/ba∇dbl∞is bounded in such a way that |Ψ0\nε|is\nbounded and bounded away from zero. More precisely, the cons tantK1(s,N) can be adjusted\nso that\nc(s,N)ε√\n2/ba∇dblb0\nε/ba∇dblHs<1\n2. (2.5)\nHere the constant c(s,N) corresponds to the Sobolev embedding Hs(RN)⊂L∞(RN) for\ns > N/2. Hence (2.5) guarantees that /ba∇dbl|Ψ0\nε|2−1/ba∇dbl∞<1/2.\nAs long as infRN|Ψε(t)|>0, one may define ( bε,vε)(t) explicitely as a function of Ψ ε(t). In\nfact, to prove that Ψ εand (bε,vε) are globally defined, and to establish Theorems 1.2 and 1.3\nit suffices to show that /ba∇dbl(bε,vε)/ba∇dblHs+1×Hsremains bounded. Moreover, to obtain the bound\n/ba∇dbl|Ψε(t)|2−1/ba∇dbl∞<1/2, it suffices to show that (2.5) holds as long as bεis defined.\nDue to the presence of higher order derivatives in the right- hand sides in (2.3), controlling\n/ba∇dbl(bε,vε)/ba∇dblHs+1×Hsis however a difficult issue. As in [2], this control will be car ried out by6 EVELYNE MIOT\nincorporating the equation satisfied by ∇ln(ρ2\nε). More precisely, we focus on the new variable\n(bε,zε), where\nzε=vε−i∇ln(ρ2\nε) =∇/parenleftbig\n2ϕε−iln(ρ2\nε)/parenrightbig\n∈CN.\nWe remark that ( bε,zε) is well-suited to our analysis since\nEε(Ψε) =1\n8/parenleftig\n/ba∇dblbε/ba∇dbl2\nL2+/ba∇dblzε/ba∇dbl2\nL2((1+εb/√\n2)dx)/parenrightig\n.\nMoreover, there exists a constant C=C(s,N) such that2\nC−1/ba∇dbl(bε,zε)/ba∇dblHs≤ /ba∇dbl(bε,vε)/ba∇dblHs+ε/ba∇dblbε/ba∇dblHs+1≤C/ba∇dbl(bε,zε)/ba∇dblHs.\nFrom now on we will sometimes omit the subscript εfor more clarity in the notations.\nThe equations for ( b,z) are given in the following\nProposition 2.1. Lets≥2,T0>0andΨbe a solution to (Cε)on[0,T0]satisfying\ninf\n(t,x)∈[0,T0]×RN|Ψ(t,x)| ≥m >0\nand such that (b,v)∈C1([0,T0],Hs+1×Hs). Then3\n\n\n∂tb+√\n2\nεdivRez=κ/parenleftig\n−(√\n2\nε+b)div(Imz)−1\n2(√\n2\nε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht\n−√\n2\nε(√\n2\nε+b)b/parenrightig\n−div(bRez)\n∂tz+√\n2\nε∇b= (κ+i)∆z+−1+κi\n2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+κ√\n2\nεi∇b.\nDealing with ( b,z) instead of ( b,v) presents many advantages when computing energy\nestimates. Indeed, in contrast with System (2.3) for ( b,v), the equations for ( b,z) involve\nonly non linear first-order quadratic terms and a linear seco nd-order operator ( κ+i)∆z. This\nis due to the identityε√\n2∇b=−(1+ε√\n2b)Imz,\nwhich enables to save one derivative.\nFor the Gross-Pitaevskii equation (GP), the energy estimat es performed in [2] for ( b,z)\ninvolve a family of semi-norms with a suitable weight\nΓk(b,z) :=/integraldisplay\nRN|Dkb|2+/integraldisplay\nRN(1+ε√\n2b)|Dkz|2, k= 0,...,s.\nIn particular, we have the remarkable identity\nΓ0(b,z) = 8Eε(Ψ),\nwhich in fact was the principal motivation to add the imagina ry part of z. Moreover we\nremark that Γk(b,z) and/ba∇dbl(Dkb,Dkz)/ba∇dbl2\nL2are comparable as long as |Ψ|is close to one.\nForthecomplexGinzburg-Landauequation(C ε)wewillpartlyrelyontheestimates already\nstated in [2] to establish the following\n2See (5.4) below.\n3Here/angbracketleftz,z/angbracketright=N/summationdisplay\ni=1z2\ni, wherez= (z1,...,z N)∈CN.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 7\nProposition 2.2. Lets > N/2andT0>0. LetΨbe a solution to (Cε)on[0,T0]such that\n/ba∇dbl|Ψ|2−1/ba∇dblL∞([0,T0]×RN)<1\n2\nand such that (b,z)∈C1([0,T0],Hs+1). There exists a constant K=K(s,N)depending only\nonsandNsuch that for 1≤k≤sandt∈[0,T0]\nd\ndt/parenleftbig\nΓk(b,z)+Eε(Ψ)/parenrightbig\n+κ\n2/parenleftbig\nΓk+1(b,z)+1\nε2Γk(b,0)/parenrightbig\n≤K/parenleftbig\nνε/ba∇dblb/ba∇dbl∞+κ/ba∇dbl(b,z)/ba∇dbl2\n∞+/ba∇dbl(Db,Dz)/ba∇dbl∞/parenrightbig/parenleftbig\nΓk(b,z)+Eε(Ψ)/parenrightbig\n.\nWe further assume that s >1+N/2. Combining Proposition 2.2 and Sobolev embedding\nwe readily find\n/ba∇dbl(b,z)(t)/ba∇dblHs≤C/ba∇dbl(b,z)(0)/ba∇dblHs+C(ε)/integraldisplayt\n0/ba∇dbl(b,z)(τ)/ba∇dbl3\nHsdτ.\nThisprovidesafirstcontrol of thenorm /ba∇dbl(b,z)(t)/ba∇dblHsuptotimes of order C(ε)−1/ba∇dbl(b,z)(0)/ba∇dbl−2\nHs.\nHowever, we need to refine this control since C(ε) diverges as εtends to zero. In fact, one\nmay also apply Cauchy-Schwarz inequality and Sobolev imbed ding together with Proposition\n2.2 to infer an estimate for /ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)in terms of the norms /ba∇dbl(b,z)/ba∇dblL2\nt(Hs)and/ba∇dblb/ba∇dblL2\nt(L∞).\nProposition 2.3. Under the assumptions of Proposition 2.2, we assume moreove r thats >\n1 +N/2. There exists a constant K=K(s,N)depending only on sandNsuch that for\n[0,T0]\nK−1/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)≤ /ba∇dbl(b,z)(0)/ba∇dblHs\n+νε/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)/ba∇dblb/ba∇dblL2\nt(L∞)+/parenleftbig\nκ/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)+1/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)\nand\nK−1κ/ba∇dbl(Db,Dz)/ba∇dbl2\nL2\nt(Hs)≤ /ba∇dbl(b,z)(0)/ba∇dbl2\nHs\n+/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)/parenleftbig\nνε/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)/ba∇dblb/ba∇dblL2\nt(L∞)+/parenleftbig\nκ/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)+1/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)/parenrightbig\n.\nIn the second step of the proofs, we will exploit the decreasi ng properties of the semi-\ngroup operator associated to System (2.3) to derive estimat es for the norms /ba∇dbl(b,z)/ba∇dblL2\nt(Hs)\nand/ba∇dblb/ba∇dblL2\nt(L∞)in terms of /ba∇dbl(b,z)/ba∇dblL∞\nt(Hs). These estimates are summarized in the following\nProposition 2.4. Under the assumptions of Proposition 2.3, there exists a con stantK=\nK(s,N)depending only on sandNsuch that for t∈[0,T0]\nK−1/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)≤κ1/2max(1,ν−1\nε)M0\n+/parenleftbig\n1+ε/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)/parenrightbig\n/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)/parenleftbig\nκ1/2/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)+(ε+ν−1\nε)/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)/parenrightbig\nand\nK−1/ba∇dblb/ba∇dblL2\nt(L∞)≤(εν−1\nε)1/2M0\n+/parenleftbig\n1+ε/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)/parenrightbig\n/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)εmax(1,ν−1\nε)/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs),\nwhereM0is defined in Theorem 1.1.\nCombining Propositions 2.3 and 2.4 yields an improved estim ate for/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)which,\nin turn, leads to Theorems 1.1, 1.2 and 1.3.\nThe remainder of this work is organized in the following way. In Section 3 we study the\nCauchy problem for (C ε) and prove local well-posedness for ( b,z). Propositions 2.1, 2.2 and8 EVELYNE MIOT\n2.3 are established in Section 4. Section 5 is devoted to the p roof of Proposition 2.4 by means\nof a Fourier analysis. We finally turn to the proof of Theorems 1.1 and 1.3 in Section 6. We\nomit the proof of Theorem 1.2, which can be obtained with some minor modifications. At\nsome places, we will rely on helpful estimates that are recal led or established in the appendix.\n3.The Cauchy problem for the complex Ginzburg-Landau equatio n\nIn this section, we address the Cauchy problem for (C ε) in a space including the fields\nΨ = (1+ a)1/2exp(iϕ), where ( a,ϕ)∈Hs+1(RN)2ands+1≥N/2. We consider the set\nW=/braceleftbig\nU∈L∞(RN),∇U∈H∞(RN) and 1 −|U|2∈L2(RN)/bracerightbig\n.\nApplying a standard fixed point argument (see, e.g., the proo f of Theorem 1 in [9]) and\nusing the Sobolev embedding Hs+1⊂L∞ifs+1> N/2, it can be shown the following\nTheorem 3.1. Lets+ 1> N/2andU0∈ W. For any ω0∈Hs+1(RN)there exists T∗=\nT(U0,ω0)>0and a unique maximal solution\nΨ∈ {U0}+C([0,T∗),Hs+1(RN))\nto Eq.(Cε)such that Ψ(0) =U0+ω0.\nThe Ginzburg-Landau energy of Ψis finite and satisfies\nEε(Ψ(t))≤Eε(Ψ(0)),∀t∈[0,T∗).\nMoreover, there exists a number Cdepending only on Eε(Ψ(0))such that\n/ba∇dblΨ(t)−Ψ(0)/ba∇dblL2(RN)≤Cexp(Ct),∀t∈[0,T∗).\nFinally, either T∗= +∞orlimsup\nt→T∗/ba∇dbl∇Ψ(t)/ba∇dblHs= +∞.\nWe recall that Edenotes the space of finite energy fields. Thanks to the alread y mentioned\ninclusion (see [4])\nE ⊂ W+H1(RN),\na consequence of Theorem 3.1 is the\nCorollary 3.1. Lets+1> N/2. Let(a0,ϕ0)∈Hs+1(RN)2. We assume that\nε√\n2/ba∇dbla0/ba∇dbl∞<1.\nThere exists T0>0and a unique solution (b,v)∈C([0,T0],Hs+1×Hs)to System (2.3)with\ninitial datum (a0,u0= 2∇ϕ0). Moreover, there exists ϕ∈C([0,T0],H1\nloc)such that v= 2∇ϕ.\nProof.Set\nΨ0(x) =/parenleftbig\n1+ε√\n2a0(x)/parenrightbig1/2exp(iϕ0(x)).\nBy assumption on ( a0,ϕ0), Ψ0belongs to Eand\n/ba∇dbl|Ψ0|2−1/ba∇dbl∞<1. (3.1)\nSinceE ⊂ W+H1(RN), wehave Ψ0∈ {U0}+H1(RN) for some U0∈ W. Usingtheembedding\nHs+1(RN)⊂L∞(RN), we check that\n/ba∇dbl∇Ψ0/ba∇dblHs≤C(1+/ba∇dbl(a0,u0)/ba∇dbl2\nHs+1×Hs).\nThis shows that actually Ψ0∈ {U0}+Hs+1(RN). Hence, by virtue of Theorem 3.1 there\nexistsT∗>0 and a unique maximal solution Ψ ∈ {U0}+C([0,T∗),Hs+1) to (C ε) such that\nΨ(0) = Ψ0.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 9\nNext, thanks to (3.1) and to the inclusion Hs+1(RN)⊂L∞(RN), there exists by time\ncontinuity a non trivial interval [0 ,T0]⊂[0,T∗) for which\ninf\n(t,x)∈[0,T0]×RN|Ψ(t,x)| ≥m >0.\nConsequently, we may find a lifting for Ψ on [0 ,T0] :\nΨ(t,x) =/parenleftbig\n1+ε√\n2b(t,x)/parenrightbig1/2exp(iϕ(t,x)),whereϕ∈L2\nloc.\nSetting then v= 2∇ϕ, we determine bandvin a unique way through the identities\nb=√\n2\nε(|Ψ|2−1) and v=2\n|Ψ|2(Ψ×∇Ψ).\nIn view of the regularity of Ψ we have ( b,v)∈C([0,T0],Hs+1×Hs). In addition, ( b,v) is a\nsolution to System (2.3) on [0 ,T0], and the conclusion follows. /square\n4.Proofs of Propositions 2.1, 2.2 and 2.3.\n4.1.Notations. We use this paragraph to fix some notations. The notation a·bdenotes the\nstandard scalar product on RNorR2N, which we extend to complex vectors by setting\nz·ζ= (Rez,Imz)·(Reζ,Imζ)∈R,∀z,ζ∈CN.\nWe define the complex product of z= (z1,...,zN) andζ= (ζ1,...,ζN)∈CNby\n/an}b∇acketle{tz,ζ/an}b∇acket∇i}ht=N/summationdisplay\nj=1zjζj∈C.\nTherefore when z=a+ib∈CNandζ=x+iy∈CNwitha,b,x,y∈RNwe have\n/an}b∇acketle{tz,ζ/an}b∇acket∇i}ht=a·x−b·y+i(a·y+b·x) and z·ζ=a·x+b·y.\nWith the same notations as above we finally introduce\n∇z=∇a+i∇b∈CN×N\nand\n∇z:∇ζ=∇a:∇x+∇b:∇y∈R,\nwhere for A,B∈RN×Nwe have set A:B= tr(AtB).\n4.2.Proof of Proposition 2.1. Since Ψ = ρexp(iϕ) is a solution to (C ε), we have, with\nv= 2∇ϕ,\n\n∂tρ2\nρ2= 2κ/parenleftbigg∆ρ\nρ−|v|2\n4+1−ρ2\nε2/parenrightbigg\n−div(ρ2v)\nρ2\n∂t(2ϕ) = 2/parenleftbigg∆ρ\nρ−|v|2\n4+1−ρ2\nε2/parenrightbigg\n+κdiv(ρ2v)\nρ2.\nTaking the gradient in both equations we obtain\n\n\n∇∂tρ2\nρ2= 2κ∇∆ρ\nρ−κ∇|v|2\n2+2κ∇1−ρ2\nε2−∇div(ρ2v)\nρ2\n∂tv= 2∇∆ρ\nρ−∇|v|2\n2+2∇1−ρ2\nε2+κ∇div(ρ2v)\nρ2.10 EVELYNE MIOT\nSince∂tz=∂tv−i∇∂tρ2\nρ2, we have\n∂tz= (1−κi)2∇∆ρ\nρ−(1−κi)∇|v|2\n2+2(1−κi)∇1−ρ2\nε2+(κ+i)∇div(ρ2v)\nρ2.\nNext, expanding\n∆lnρ=∆ρ\nρ−|∇ρ|2\nρ2,\nwe obtain\n2∇∆ρ\nρ=∇∆lnρ2+2∇|∇lnρ|2=−∆Imz+1\n2∇|Imz|2.\nOn the other hand, since vis a gradient we have\n∇div(ρ2v)\nρ2=∇divv+∇/parenleftig\nv·∇ρ2\nρ2/parenrightig\n= ∆Rez−∇/parenleftig\nImz·v/parenrightig\n.\nFinally, using the fact that\n2∇1−ρ2\nε2=−√\n2\nε∇b,\nwe are led to the equation for z\n∂tz= (κ+i)∆z−1−κi\n2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√\n2\nε(1−κi)∇b.\nWe next turn to the equation for b, recalling that ρ2verifies\n∂tρ2=κ/parenleftbigg\n2ρ∆ρ−ρ2|v|2\n2+2ρ2(1−ρ2)\nε2/parenrightbigg\n−div(ρ2v).\nExpanding the expression\n2ρ∆ρ=ρ2∆lnρ2+ρ2\n2|Imz|2=−ρ2divImz+ρ2\n2|Imz|2,\nwe find\n∂tρ2=κ/parenleftig\n−(1+ε√\n2b)divImz−1\n2(1+ε√\n2b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht−2(1+ε√\n2b)ε√\n2\nε2b/parenrightig\n−div/parenleftbig\n(1+ε√\n2b)Rez/parenrightbig\n,\nas we wanted. /square\n4.3.Proof of Proposition 2.2. We present now the proof of Proposition 2.2. In all this\nparagraph, Cstands for a number depending only on sandN, which possibly changes from\na line to another. We will make use of the identity\nε√\n2∇b=−(1+ε√\n2b)Imz. (4.1)\nAs we want to rely on the estimates already performed for the G ross-Pitaevskii equation\nin [2], it is convenient to write the equations for ( b,z) as follows\n/braceleftigg\n∂tb=κfd(b,z)+fs(b,z)\n∂tz=κgd(b,z)+gs(b,z),DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 1\nwhere we have introduced the dissipative part\n\n\nfd(b,z) =−(√\n2\nε+b)div(Imz)−1\n2(√\n2\nε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√\n2\nε(√\n2\nε+b)b,\ngd(b,z) = ∆z+i\n2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+i√\n2\nε∇b\nand the dispersive part\n\n\nfs(b,z) =−div/parenleftbig\n(√\n2\nε+b)Rez/parenrightbig\n,\ngs(b,z) =i∆z−1\n2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√\n2\nε∇b.\nLetk∈N∗. We compute\nd\ndtΓk(b,z) =d\ndt/integraldisplay\nRN(1+ε√\n2b)Dkz·Dkz+DkbDkb\n= 2/integraldisplay\nRN(1+ε√\n2b)Dkz·Dk∂tz+DkbDk∂tb+/integraldisplay\nRNε∂tb√\n2Dkz·Dkz\n=Is+Id,\nwhere\nIs= 2/integraldisplay\nRN(1+ε√\n2b)Dkz·Dkgs+DkbDkfs+/integraldisplay\nRNεfs√\n2Dkz·Dkz\nand\nκ−1Id= 2/integraldisplay\nRN(1+ε√\n2b)Dkz·Dkgd+DkbDkfd+/integraldisplay\nRNεfd√\n2Dkz·Dkz.\nTo estimate the first term Iswe invoke Proposition 1 in [2] :\n|Is| ≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dbl(Db,Dz)/ba∇dblL∞/parenleftig\nΓk(b,z)+Eε(Ψε)/parenrightig\n,\nso we only need to estimate the term Id. Inserting the expressions of fdandgdwe find\nId=κ(2I+2J+K),\nwhere\nI=/integraldisplay\nRN(1+ε√\n2b)/parenleftig\nDkz·Dk∆z+1\n2Dkz·iDk∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+√\n2\nεDkz·iDk∇b/parenrightig\n=I1+I2+I3,\nJ=/integraldisplay\nRN−DkbDk/parenleftig\n(√\n2\nε+b)div(Imz)/parenrightig\n−1\n2DkbDk/parenleftig\n(√\n2\nε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht/parenrightig\n−DkbDk/parenleftig√\n2\nε(√\n2\nε+b)b/parenrightig\n=J1+J2+J3,\nand\nK=−/integraldisplay\nRN(1+ε√\n2b)/parenleftig\ndiv(Imz)+1\n2Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht+√\n2\nεb/parenrightig\nDkz·Dkz.12 EVELYNE MIOT\nStep 1: estimate for I1.\nIntegrating by parts in I1, then inserting (4.1) we find\nI1=−/integraldisplay\nRN(1+ε√\n2b)∇Dkz:∇Dkz−ε√\n2∇b·(Dkz·∇Dkz)\n=−/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+/integraldisplay\nRN(1+ε√\n2b)Imz·(Dkz·∇Dkz)\n≤ −/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+/integraldisplay\nRN(1+ε√\n2b)1/2|Imz||Dkz|(1+ε√\n2b)1/2|∇Dkz|.\nApplying Young inequality to the second term in the right-ha nd side, we obtain\nI1≤ −1\n2/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+1\n2/integraldisplay\nRN(1+ε√\n2b)|Imz|2|Dkz|2,\nso finally\nI1≤ −1\n2/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblImz/ba∇dbl2\n∞/ba∇dblz/ba∇dbl2\nHk.\nStep 2: estimate for I2.\nExpanding I2thanks to Leibniz formula, we obtain\nI2=/integraldisplay\nRN(1+ε√\n2b)Dkz·Dk(i/an}b∇acketle{tz,∇z/an}b∇acket∇i}ht)\n=/integraldisplay\nRN(1+ε√\n2b)Dkz·i/an}b∇acketle{tz,∇Dkz/an}b∇acket∇i}ht+k−1/summationdisplay\nj=0Cj\nk/integraldisplay\nRN(1+ε√\n2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht.\nApplying then Young inequality to the first term in the right- hand side, we infer that\nI2≤1\n4/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblz/ba∇dbl2\n∞/ba∇dblz/ba∇dbl2\nHk\n+Ck−1/summationdisplay\nj=0/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRN(1+ε√\n2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle.\nFor each 0 ≤j≤k−1, we apply first Cauchy-Schwarz, then Gagliardo-Nirenberg (see Lemma\n7.4 in the appendix) inequalities. This yields\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRN(1+ε√\n2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblDkz/ba∇dblL2/ba∇dbl|Dk−jz||Dj(∇z)|/ba∇dblL2\n≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblDkz/ba∇dblL2/ba∇dblDz/ba∇dbl∞/ba∇dblz/ba∇dblHk,\nand we are led to\nI2≤1\n4/integraldisplay\nRN(1+ε√\n2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)(/ba∇dblz/ba∇dbl2\n∞+/ba∇dblDz/ba∇dbl∞)/ba∇dblz/ba∇dbl2\nHk.\nStep 3: estimate for I3.\nSinceDk∇b∈RNwe have by definition of the complex product\nI3=/integraldisplay\nRN(1+ε√\n2b)√\n2\nεDkz·iDk∇b=/integraldisplay\nRN(1+ε√\n2b)√\n2\nεDkImz·Dk∇b.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 3\nInserting first (4.1) and using then Leibniz formula we get\nI3=−2\nε2/integraldisplay\nRN(1+ε√\n2b)DkImz·Dk/parenleftbig\n(1+ε√\n2b)Imz/parenrightbig\n=−2\nε2/integraldisplay\nRN(1+ε√\n2b)2|DkImz|2−2\nε2k/summationdisplay\nj=1Cj\nk/integraldisplay\nRN(1+ε√\n2b)DkImz·/parenleftbig\nDj(1+ε√\n2b)Dk−jImz/parenrightbig\n.\nNow, we observe that for each j≥1, we have\nDj(1+ε√\n2b) =ε√\n2Djb.\nConsequently, applying Young inequality to each term of the sum we find\nI3≤ −1\nε2/integraldisplay\nRN(1+ε√\n2b)2|DkImz|2+Ck/summationdisplay\nj=1/integraldisplay\nRN|DjbDk−jImz|2,\nand we finally infer from Gagliardo-Nirenberg inequality th at\nI3≤C/parenleftbig\n/ba∇dblb/ba∇dbl2\n∞+/ba∇dblImz/ba∇dbl2\n∞/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nHk.\nStep 4: estimate for J1.\nA short calculation using (4.1) yields\nJ1=−/integraldisplay\nRNDkbDk/parenleftig\n(√\n2\nε+b)div(Imz)/parenrightig\n=−/integraldisplay\nRNDkbDkdiv/parenleftig\n(√\n2\nε+b)Imz/parenrightig\n+/integraldisplay\nRNDkbDk(∇b·Imz)\n=/integraldisplay\nRNDkbDkdiv(∇b)+/integraldisplay\nRNDkbDk(∇b·Imz).\nAfter integrating by parts in the first term in the right-hand side and expanding the second\nterm by means of Leibniz formula we obtain\nJ1=−/integraldisplay\nRN|∇Dkb|2+/integraldisplay\nRNDkb(Dk∇b)·Imz+k/summationdisplay\nj=1Cj\nk/integraldisplay\nRNDkb(Dk−j∇b)·DjImz.\nNext, combining Young, Cauchy-Schwarz and Gagliardo-Nire nberg inequalities we find\nJ1≤ −1\n2/integraldisplay\nRN|∇Dkb|2+C/ba∇dblImz/ba∇dbl2\n∞/ba∇dblb/ba∇dbl2\nHk+C/ba∇dblb/ba∇dblHk(/ba∇dbl∇b/ba∇dbl∞+/ba∇dblDz/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dblHk,\nso that\nJ1≤ −1\n2/integraldisplay\nRN|∇Dkb|2+C/parenleftbig\n/ba∇dblImz/ba∇dbl2\n∞+/ba∇dbl(∇b,Dz)/ba∇dbl∞/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nHk.14 EVELYNE MIOT\nStep 5: estimate for J2.\nSimilarly, we compute thanks to Leibniz formula\nJ2=−1\n2/integraldisplay\nRNDkbDk/parenleftig\n(√\n2\nε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht/parenrightig\n=−1\n2/integraldisplay\nRNDkb(√\n2\nε+b)Dk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)+1\n2k/summationdisplay\nj=1Cj\nk/integraldisplay\nRNDkbDjbDk−j(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)\n=−1\nε√\n2/integraldisplay\nRNDkbDk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)−1\n2/integraldisplay\nRNbDkbDk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)\n+1\n2k/summationdisplay\nj=1Cj\nk/integraldisplay\nRNDkbDjbDk−j(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht).\nInvoking Young and Cauchy-Schwarz inequalities, we obtain\nJ2≤1\nε2/integraldisplay\nRN|Dkb|2+C/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dbl2\nHk\n+C/parenleftbig\n/ba∇dblb/ba∇dbl∞/ba∇dblb/ba∇dblHk/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblHk+/ba∇dblb/ba∇dblHkk/summationdisplay\nj=1/ba∇dblDjbDk−j/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblL2/parenrightbig\n,\nso that by virtue of Lemma 7.4,\nJ2≤1\nε2/integraldisplay\nRN|Dkb|2+C/ba∇dbl(b,z)/ba∇dbl2\n∞/ba∇dbl(b,z)/ba∇dbl2\nHk.\nStep 6: estimate for J3.\nWe have\nJ3=−√\n2\nε/integraldisplay\nRNDkbDk/parenleftig\nb(√\n2\nε+b)/parenrightig\n=−2\nε2/integraldisplay\nRN|Dkb|2−√\n2\nε/integraldisplay\nRNDkbDk(b2),\nso, thanks to Cauchy-Schwarz inequality and Lemma 7.4,\nJ3≤ −2\nε2/integraldisplay\nRN|Dkb|2+C\nε/ba∇dblb/ba∇dbl∞/ba∇dblb/ba∇dbl2\nHk.\nStep 7: estimate for K.\nWe readily obtain\n|K| ≤C(1+ε/ba∇dblb/ba∇dbl∞)/parenleftig/ba∇dblb/ba∇dbl∞\nε+/ba∇dblDz/ba∇dbl∞+/ba∇dblz/ba∇dbl2\n∞/parenrightig\n/ba∇dblz/ba∇dbl2\nHk.\nGathering the previous steps we obtain\nd\ndtΓk(b,z)+κ\n2Γk+1(b,z)+2κ\nε2Γk(b,0)\n≤C(1+ε/ba∇dblb/ba∇dbl∞)/parenleftig\nκ/parenleftbig\n/ba∇dbl(b,z)/ba∇dbl2\n∞+ε−1/ba∇dblb/ba∇dbl∞/parenrightbig\n+/ba∇dbl(∇b,Dz)/ba∇dbl∞/parenrightig\n/ba∇dbl(b,z)/ba∇dbl2\nHk,DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 5\nholding for any 1 ≤k≤s. Following step by step the previous computations we readil y check\nthat it also holds for k= 0. Finally, we have by assumption\n1\n2≤1+εb√\n2≤3\n2on [0,T0]×RN,\nfrom which we infer that /ba∇dbl(b,z)/ba∇dbl2\nHk≤CΓk(b,z) for all 0 ≤k≤s. Therefore the proof of\nProposition 2.2 is complete. /square\n4.4.Proof of Proposition 2.3. Toshow thefirstinequality weaddtheinequalities obtained\nin Proposition 2.2 for kvarying from 1 to s. Since 1 /2≤1+εb/√\n2≤3/2, this yields\nd\ndt/ba∇dbl(b,z)/ba∇dbl2\nHs≤C(νε/ba∇dblb/ba∇dbl∞+κ/ba∇dbl(b,z)/ba∇dbl2\n∞+/ba∇dbl(Db,Dz)/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dbl2\nHs\n≤C/parenleftbig\nνε/ba∇dblb/ba∇dbl∞+(κ/ba∇dbl(b,z)/ba∇dblHs+1)/ba∇dbl(b,z)/ba∇dblHs/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nHs.\nAfter integrating on [0 ,T] and using Cauchy-Schwarz inequality this leads to\n/ba∇dbl(b,z)(T)/ba∇dbl2\nHs≤ /ba∇dbl(b,z)(0)/ba∇dbl2\nHs\n+C/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)/parenleftig\nνε/ba∇dblb/ba∇dblL2\nT(L∞)/ba∇dbl(b,z)/ba∇dblL2\nT(Hs)+(κ/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)+1)/ba∇dbl(b,z)/ba∇dbl2\nL2\nT(Hs)/parenrightig\n,\nfor allT∈[0,T0]. Considering the supremum over T∈[0,t] and applying Young inequality\nin the right-hand-side we find the result.\nFinally the second inequality in Proposition 2.3 is obtaine d by integrating on [0 ,t] and\nusing Sobolev and Cauchy-Schwarz inequalities. /square\n5.Proof of Proposition 2.4.\nIn this paragraph again, Crefers to a constant depending only on sandNand possibly\nchanging from a line to another.\nFirst, we formulate System (2.3)-(2.4) with second members involving only bandz. By the\nsame computations as those in Paragraph 4.2 we find\n\n∂tb+√\n2\nεdivv+2νε\nεb−κ∆b=f(b,z)\n∂tv+√\n2\nε∇b−κ∆v−ε√\n2∇∆b=g(b,z),(5.1)\nwheref=˜fandg= ˜g−ε√\n2∇∆bare defined by\n\n\nf(b,z) =νε/parenleftbigg\n−1√\n2(1+ε√\n2b)|z|2−√\n2b2/parenrightbigg\n−div(bRez)\ng(b,z) =−κ∇(Rez·Imz)+ε√\n2∇div(bImz)−1\n2∇Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht.(5.2)\n5.1.Some notations and preliminary results. As in [2], we symmetrize System (5.1) by\nintroducing the new functions\nc= (1−ε2\n2∆)1/2b, d= (−∆)−1/2divv,\nand\nF= (1−ε2\n2∆)1/2f, G= (−∆)−1/2divg.16 EVELYNE MIOT\nWe remark that, knowing d, one can retrieve vsincevis a gradient. We have\n\n\n∂tc+2νε\nεc−κ∆c+√\n2\nε(−∆)1/2(1−ε2\n2∆)1/2d=F\n∂td−κ∆d−√\n2\nε(−∆)1/2(1−ε2\n2∆)1/2c=G.(5.3)\nIn the following, we denote by ξ∈RNthe Fourier variable, by ˆfthe Fourier transform of\nfand byF−1the inverse Fourier transform.\nIn view of the definition of ( c,d), it is useful to introduce the frequency threshold |ξ| ∼ε−1.\nMore precisely, let us fix some R >0 and let χdenote the characteristic function on B(0,R).\nForf∈L2(RN), we define the low and high frequencies parts of f\nfl=F−1/parenleftbig\nχ(εξ)ˆf/parenrightbig\nandfh=F−1/parenleftbig\n(1−χ(εξ))ˆf/parenrightbig\n,\nso that/hatwidefland/hatwidefhare supported in {|ξ| ≤Rε−1}and{|ξ| ≥Rε−1}respectively.\nLemma 5.1. There exists C=C(s,N,R)>0such that the following holds for all 0≤m≤s\nandt∈[0,T0]:\n/ba∇dblg(t)/ba∇dblHm≈ /ba∇dblG(t)/ba∇dblHm,/ba∇dblfl(t)/ba∇dblHm≈ /ba∇dblFl(t)/ba∇dblHmand/ba∇dbl(ε∇f)h(t)/ba∇dblHm≈ /ba∇dblFh(t)/ba∇dblHm.\nIn addition,\n/ba∇dblv(t)/ba∇dblHm≈ /ba∇dbld(t)/ba∇dblHm,/ba∇dblbl(t)/ba∇dblHm≈ /ba∇dblcl(t)/ba∇dblHmand/ba∇dbl(ε∇b)h(t)/ba∇dblHm≈ /ba∇dblch(t)/ba∇dblHm.\nFinally,\n/ba∇dbl(b,z)(t)/ba∇dblHm≈ /ba∇dbl(b,v)l(t)/ba∇dblHm+/ba∇dbl(ε∇b,v)h(t)/ba∇dblHm.\nHere we have set for f1,f2∈Hm\n/ba∇dblf1/ba∇dblHm≈ /ba∇dblf2/ba∇dblHmif and only if C−1/ba∇dblf1/ba∇dblHm≤ /ba∇dblf2/ba∇dblHm≤C/ba∇dblf1/ba∇dblHm.\nProof.For the first two statements it suffices to consider the Fourier transforms of the func-\ntions and to use their support properties. The last statemen t is already established in [2],\nLemma 1. /square\nLemma 5.1 guarantees that for 0 ≤m≤s,\n/ba∇dbl(b,v)(t)/ba∇dblHm+ε/ba∇dblb(t)/ba∇dblHm+1≈ /ba∇dbl(b,z)(t)/ba∇dblHmand/ba∇dbl(b,z)(t)/ba∇dblHm≈ /ba∇dbl(c,d)(t)/ba∇dblHm,(5.4)\ntherefore we have /ba∇dbl(c,d)(0)/ba∇dblHs≤CM0, whereM0is defined in Theorem 1.1.\nOn the other side, when s−1> N/2, Sobolev embedding yields\n/ba∇dblbl(t)/ba∇dbl∞≤C/ba∇dblbl(t)/ba∇dblHs−1≤C/ba∇dblcl(t)/ba∇dblHs−1\nand\n/ba∇dblbh(t)/ba∇dbl∞≤C/ba∇dblbh(t)/ba∇dblHs−1≤C/ba∇dbl(ε∇b)h(t)/ba∇dblHs−1≤C/ba∇dblch(t)/ba∇dblHs−1.\nTherefore it suffices to establish the first inequality of Prop osition 2.4 for /ba∇dbl(c,d)/ba∇dblL2\nt(Hs)and\nthe second inequality for /ba∇dblc/ba∇dblL2\nt(Hs−1).\nNext, we have\nd\ndt/parenleftbiggˆc\nˆd/parenrightbigg\n+M(ξ)/parenleftbiggˆc\nˆd/parenrightbigg\n=/parenleftbiggˆF\nˆG/parenrightbigg\n,DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 7\nwhere\nM(ξ) =νε\nε\n2+ε2|ξ|2|ξ|\nνε(2+ε2|ξ|2)1/2\n−|ξ|\nνε(2+ε2|ξ|2)1/2ε2|ξ|2\n.\nBy Duhamel formula we have\n/hatwide(c,d)(t,ξ) =e−tM(ξ)/hatwide(c,d)(0,ξ)+/integraldisplayt\n0e−(t−τ)M(ξ)/hatwider(F,G)(τ,ξ)dτ.\nOurnextresult, whichisprovedintheappendix,establishe spointwiseestimates for e−tM(ξ).\nLemma 5.2. There exist positive numbers κ0,r,candCsuch that for all (a,b)∈C2, we\nhave for 0< ε≤1,κ < κ0andt≥0\n(1)If|ξ| ≤rνεthen\n/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle≤Cexp(−νεε|ξ|2t)/bracketleftbigg\nexp/parenleftig\n−νε\nεt/parenrightig\n(|a|+|b|)+exp/parenleftbigg\n−c|ξ|2\nνεεt/parenrightbigg\n(ν−1\nε|ξ||a|+|b|)/bracketrightbigg\n.\n(2)If|ξ| ≥rνεthen\n/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle≤Cexp/parenleftbigg\n−νε(1+ε2|ξ|2)\n2εt/parenrightbigg\n(|a|+|b|).\nHere for A= (a,b)∈C2we have set |A|=|a|+|b|.\nLemma 5.2 reveals the new frequency threshold |ξ| ∼νε. We may choose R > r, so that\nrνε< Rε−1. We are therefore led to split the frequency space into three regions\nRN=R1∪R2∪R3,\nwhere\n• R1={|ξ| ≤rνε}denotesthelow frequenciesregion, inwhichthesemi-group iscomposed\nof a parabolic part (exp( −(νεε)−1|ξ|2t)), and a damping part (exp( −νεε−1t)).\n• R2={rνε≤ |ξ| ≤Rε−1}denotestheintermediatefrequenciesregion, inwhichthed amp-\ning effect exp( −νεε−1t) is prevalent with respect to the parabolic contribution ex p(−νεε|ξ|2t).\n• R3={|ξ| ≥Rε−1}denotes the high frequencies region, in which the parabolic contribu-\ntion is strong and dominates the damping.\nWith respect to this decomposition we introduce the small, i ntermediate and high frequen-\ncies parts of f∈L2(RN) as follows\nfs=F−1/parenleftbig\nχ|ξ|≤rνεˆf/parenrightbig\n, fm=F−1/parenleftbig\nχrνε≤|ξ|≤Rε−1ˆf/parenrightbig\nandfh=F−1/parenleftbig\nχ|ξ|≥Rε−1ˆf/parenrightbig\n,\nwhereχEdenotes the characteristic function on the set E. Note that we have\nf=fs+fm+fh=fl+fh.\n5.2.Proof of Proposition 2.4. We first introduce some notations. Let\nL(b,z)(t) =/ba∇dbl(1+εb(t))|z(t)|2/ba∇dblHs+/ba∇dblb2(t)/ba∇dblHs+/ba∇dblb(t)z(t)/ba∇dblHs+/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht(t)/ba∇dblHs.\nNext, we sort the terms in the definitions of f(b,z) andg(b,z) in System (5.2) as follows. We\nset\nf(b,z) =νεf0(b,z)+f1(b,z)\nand\ng(b,z) =g1(b,z)+εg2(b,z) =∇h0(b,z)+ε∇h1(b,z),18 EVELYNE MIOT\nwhere the subscript j= 0,1,2 denotes the order of the derivative, so that\n\n\nf0(b,z) =−1√\n2(1+ε√\n2b)|z|2−√\n2b2\nf1(b,z) =−div(bRez)\nand\n\ng1(b,z) =−κ∇(Rez·Imz)−1/2∇Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht=∇h0(b,z)\ng2(b,z) =1√\n2∇div(bImz) =∇h1(b,z).\nThe proof of Proposition 2.4 relies on several lemmas which w e present now separately.\nLemma 5.3. Under the assumptions of Proposition 2.4 we have for T∈[0,T0]\nC−1/ba∇dbl(c,d)s/ba∇dblL2\nT(Hs)≤κ1/2max(1,ν−1\nε)M0+ε/ba∇dblL(b,z)/ba∇dblL2\nT+κ1/2/ba∇dblL(b,z)/ba∇dblL1\nT.\nProof.By virtue of Lemma 5.2 we have\n|/hatwide(c,d)s(t,ξ)| ≤C(I(t,ξ)+J(t,ξ)),\nwhere\nI(t,ξ) =e−νε\nεt|/hatwide(c,d)s(0,ξ)|+/integraldisplayt\n0e−νε\nε(t−τ)|/hatwider(F,G)s(τ,ξ)|dτ\nand\nJ(t,ξ) =e−c|ξ|2\nνεεt/vextendsingle/vextendsingle(|ξ|ν−1\nε/hatwidecs(0),/hatwideds(0))/vextendsingle/vextendsingle+/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)/vextendsingle/vextendsingle(|ξ|ν−1\nε/hatwiderFs,/hatwiderGs)/vextendsingle/vextendsingledτ\n=JL(t,ξ)+JNL(t,ξ).\nWe setˇI=F−1IandˇJ=F−1J, so that /ba∇dbl(c,d)s/ba∇dblL2\nT(Hs)≤C(/ba∇dblˇI/ba∇dblL2\nT(Hs)+/ba∇dblˇJ/ba∇dblL2\nT(Hs)).\nFirst step : estimate for /ba∇dblˇI/ba∇dblL2\nT(Hs).\nInvoking Lemma 7.3 we obtain\n/ba∇dblˇI/ba∇dblL2\nT(Hs)≤C/parenleftbig\n(εν−1\nε)1/2/ba∇dbl(c,d)s(0)/ba∇dblHs+εν−1\nε/ba∇dbl(f,g)s/ba∇dblL2\nT(Hs)/parenrightbig\n.\nLeth∈Hs. We observe that thanks to the support properties of /hatwidehs, we have\n/ba∇dblDkhs/ba∇dblHs≤Cνk\nε/ba∇dblhs/ba∇dblHs, k∈N.\nApplying this inequality to the higher order derivatives f1,g1andg2, we see that\n/ba∇dbl(f,g)s(t)/ba∇dblHs≤C(νε+εν2\nε)L(b,z)(t)≤CνεL(b,z)(t),\nand we conclude that\n/ba∇dblˇI/ba∇dblL2\nT(Hs)≤C/parenleftbig\n(εν−1\nε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2\nT/parenrightbig\n. (5.5)\nSecond step : estimate for /ba∇dblˇJ/ba∇dblL2\nT(Hs).\nWe have\n/ba∇dblˇJ/ba∇dblL2\nT(Hs)≤C(/ba∇dblˇJL/ba∇dblL2\nT(Hs)+/ba∇dblˇJNL/ba∇dblL2\nT(Hs)).DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 9\nFor the linear term we obtain\n/ba∇dblˇJL/ba∇dblL2\nT(Hs)≤/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2\nνεεt(|ξ|ν−1\nε|/hatwidecs(0)|+|/hatwideds(0)|)/vextenddouble/vextenddouble\nL2\nT(L2)\n≤C/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2\nνεεt|ξ|(ν−1\nε|/hatwidecs(0)|+|ξ|−1|/hatwideds(0)|)/vextenddouble/vextenddouble\nL2\nT(L2)\n≤Cmax(1,ν−1\nε)/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2\nνεεt|ξ|(|/hatwidecs(0)|+|/hatwiderϕs(0)|)/vextenddouble/vextenddouble\nL2\nT(L2),\nbecaused(0) =−2(−∆)1/2ϕ(0). By virtue of Lemma 7.1 in the appendix, this yields\n/ba∇dblˇJL/ba∇dblL2\nT(Hs)≤Cmax(1,ν−1\nε)(ενε)1/2/parenleftbig\n/ba∇dblcs(0)/ba∇dblHs+/ba∇dblϕs(0)/ba∇dblHs/parenrightbig\n≤Cmax(1,ν−1\nε)κ1/2M0.\nOn the other side, Lemma 5.1 yields\n/ba∇dblˇJNL/ba∇dblL2\nT(Hs)≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0(1+|ξ|s)e−c|ξ|2\nνεε(t−τ)/parenleftbig\n|ξ|ν−1\nε|/hatwiderFs|+|/hatwiderGs|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0(1+|ξ|)se−c|ξ|2\nνεε(t−τ)/parenleftbig\n|ξ|ν−1\nε|/hatwidefs|+|/hatwidegs|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2).\nInserting the expressions f=νεf0+f1andg=∇h0+ε∇h1we obtain\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)(1+|ξ|s)/parenleftbig\n|ξ|ν−1\nε|/hatwidefs|+|/hatwidegs|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|2(1+|ξ|s)/parenleftbig\nν−1\nε|ξ|−1|/hatwidef1|+ε|ξ|−1|/hatwiderh1|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n+/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|(1+|ξ|s)/parenleftbig\n|/hatwidef0|+|/hatwiderh0|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2).\nFirst, invoking Lemma 7.1, we find\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|2(1+|ξ|s)/parenleftbig\nν−1\nε|ξ|−1|/hatwidef1|+ε|ξ|−1|/hatwiderh1|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n≤Cενε/ba∇dbl(1+|ξ|s)(ν−1\nε|ξ|−1/hatwidef1,ε|ξ|−1/hatwiderh1)/ba∇dblL2\nT(L2)\n≤Cενε(ν−1\nε+ε)/ba∇dbl(1+|ξ|s)(|/hatwidebRez|+|/hatwidebImz|)/ba∇dblL2\nT(L2)\n≤Cενε(ν−1\nε+ε)/ba∇dblb·z/ba∇dblL2\nT(Hs)\n≤Cε/ba∇dblL(b,z)/ba∇dblL2\nT.\nNext, we infer from Lemma 7.2 in the appendix that\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|(1+|ξ|s)/parenleftbig\n|/hatwidef0|+|/hatwiderh0|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n≤C(ενε)1/2/ba∇dbl(1+|ξ|s)(|/hatwidef0|+|/hatwiderh0|)/ba∇dblL1\nT(L2)\n≤Cκ1/2/ba∇dblL(b,z)/ba∇dblL1\nT.\nGathering the previous steps and noticing that ( εν−1\nε)1/2≤κ1/2max(1,ν−1\nε), we conclude\nthe proof of the lemma. /square\nLemma 5.4. Under the assumptions of Proposition 2.4 we have for T∈[0,T0]\nC−1/parenleftig\n/ba∇dbl(c,d)m/ba∇dblL2\nT(Hs)+/ba∇dbl(c,d)h/ba∇dblL2\nT(Hs)/parenrightig\n≤(εν−1\nε)1/2M0+(ε+ν−1\nε)/ba∇dblL(b,z)/ba∇dblL2\nT.20 EVELYNE MIOT\nProof.We divide the proof into several steps.\nFirst step : intermediate frequencies rνε≤ |ξ| ≤Rε−1.\nAnother application of Lemma 5.2 yields\n|/hatwide(c,d)m(t,ξ)| ≤Ce−νε\n2εt|/hatwide(c,d)m(0,ξ)|+C/integraldisplayt\n0e−νε\n2ε(t−τ)|/hatwider(F,G)m(τ,ξ)|dτ,\nwhence, according to Lemma 7.3,\n/ba∇dbl(c,d)m/ba∇dblL2\nT(Hs)≤C(εν−1\nε)1/2/ba∇dbl(c,d)(0)/ba∇dblHs+Cεν−1\nε/ba∇dbl(F,G)m/ba∇dblL2\nT(Hs).\nLet us set\n(F,G)m=Am+Bm,\nwhereAmandBm∈L2\nT(Hs×Hs), to be determined later on, are such that /hatwidestAm(t,·) and\n/hatwiderBm(t,·) are compactly supported in/parenleftbig\nR1∪ R2={|ξ| ≤Rε−1}/parenrightbig2. Owing to these support\nproperties we find\n/ba∇dbl(F,G)m/ba∇dblL2\nT(Hs)≤ /ba∇dblAm/ba∇dblL2\nT(Hs)+/ba∇dblBm/ba∇dblL2\nT(Hs)≤C(ε−1/ba∇dblAm/ba∇dblL2\nT(Hs−1)+ε−2/ba∇dblBm/ba∇dblL2\nT(Hs−2)),\nso finally\nC−1/ba∇dbl(c,d)m/ba∇dblL2\nT(Hs)≤(εν−1\nε)1/2M0+ν−1\nε/parenleftbig\n/ba∇dblAm/ba∇dblL2\nT(Hs−1)+ε−1/ba∇dblBm/ba∇dblL2\nT(Hs−2)/parenrightbig\n.(5.6)\nSecond step : high frequencies |ξ| ≥Rε−1.\nFor the high frequencies we neglect the contribution of the d ampinge−νε\n2εtand only take the\ncontribution of e−νεε|ξ|2tinto account. Exploiting again Lemma 5.2 we have\n|/hatwide(c,d)h(t,ξ)| ≤Ce−νεε|ξ|2t|/hatwide(c,d)h(0,ξ)|+C/integraldisplayt\n0e−νεε|ξ|2(t−τ)|/hatwider(F,G)h(τ,ξ)|dτ\n≤Cε|ξ|e−νεε|ξ|2t|/hatwide(c,d)h(0,ξ)|+C/integraldisplayt\n0e−νεε|ξ|2(t−τ)|/hatwider(F,G)h(τ,ξ)|dτ,\nwhere the second inequality is due to the fact that 1 ≤Cε|ξ|on the support of /hatwide(c,d)h. By\nvirtue of Lemma 7.1 we obtain\n/ba∇dbl(c,d)h/ba∇dblL2\nT(Hs)≤C/parenleftbig\n(εν−1\nε)1/2/ba∇dbl(c,d)h(0)/ba∇dblHs+(νεε)−1/ba∇dbl(F,G)h/ba∇dblL2\nT(Hs−2)/parenrightbig\n.(5.7)\nAs in the first step, we set\n(F,G)h=Ah+Bh,\nwhereAhandBh∈L2\nT(Hs−1×Hs−1) will be set in such a way that /hatwiderAh(t,·) and/hatwiderBh(t,·) are\nsupported in the region/parenleftbig\nR3={|ξ| ≥Rε−1}/parenrightbig2. Thanks to these support properties we can\nsave one factor εto the detriment of one derivative :\n/ba∇dbl(F,G)h/ba∇dblL2\nT(Hs−2)≤ /ba∇dblAh/ba∇dblL2\nT(Hs−2)+/ba∇dblBh/ba∇dblL2\nT(Hs−2)≤C(ε/ba∇dblAh/ba∇dblL2\nT(Hs−1)+/ba∇dblBh/ba∇dblL2\nT(Hs−2)).\nTherefore in view of (5.7) we are led to\nC−1/ba∇dbl(c,d)h/ba∇dblL2\nT(Hs)≤(εν−1\nε)1/2M0+ν−1\nε/parenleftbig\n/ba∇dblAh/ba∇dblL2\nT(Hs−1)+ε−1/ba∇dblBh/ba∇dblL2\nT(Hs−2)/parenrightbig\n.(5.8)\nThird step .\nThe last step consists in choosing suitable AandB. We recall that\n(F,G) = ((1−2−1ε2∆)1/2f,(−∆)1/2divg),DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 1\nand\nf(b,z) =νεf0(b,z)+f1(b,z), g(b,z) =g1(b,z)+εg2(b,z).\nNow, for the intermediate frequencies we define\n/braceleftigg\nAm=/parenleftbig\n(1−2−1ε2∆)1/2fm,(−∆)−1/2div(g1)m/parenrightbig\nBm=/parenleftbig\n0,ε(−∆)−1/2div(g2)m/parenrightbig\n,\nand for the high frequencies\n/braceleftigg\nAh=/parenleftbig\nνε(1−2−1ε2∆)1/2(f0)h,(−∆)−1/2div(g1)h/parenrightbig\nBh=/parenleftbig\n(1−2−1ε2∆)1/2(f1)h,ε(−∆)−1/2div(g2)h/parenrightbig\n.\nClearlyAm+Bm= (F,G)mandAh+Bh= (F,G)h. Moreover, we readily check that\n/ba∇dblAm/ba∇dblHs−1≈ /ba∇dbl(f,g1)m/ba∇dblHs−1and/ba∇dblAh/ba∇dblHs−1≈ /ba∇dbl(νεε∇f0,g1)h/ba∇dblHs−1(5.9)\nand\n/ba∇dblBm/ba∇dblHs−2≈ε/ba∇dbl(g2)m/ba∇dblHs−2and/ba∇dblBh/ba∇dblHs−2≈ /ba∇dbl(ε∇f1,εg2)h/ba∇dblHs−2.(5.10)\nOn the one hand we have\n/ba∇dblg1/ba∇dblHs−1+/ba∇dblg2/ba∇dblHs−2≤C(/ba∇dblz·z/ba∇dblHs+/ba∇dblbImz/ba∇dblHs)≤CL(b,z). (5.11)\nOn the other hand, the support properties of/hatwidest(f0)mimply that\n/ba∇dbl(f0)m/ba∇dblHs−1≤Cmin(1,ν−1\nε)/ba∇dbl(f0)m/ba∇dblHs,\nso that\n/ba∇dblfm/ba∇dblHs−1≤νε/ba∇dbl(f0)m/ba∇dblHs−1+/ba∇dbl(f1)m/ba∇dblHs−1≤C(/ba∇dbl(f0)m/ba∇dblHs+/ba∇dbl(f1)m/ba∇dblHs−1),\nand finally\n/ba∇dblfm/ba∇dblHs−1≤CL(b,z). (5.12)\nArguing similarly we obtain\nνε/ba∇dbl(ε∇f0)h/ba∇dblHs−1≤Cνεε/ba∇dblf0/ba∇dblHs≤CL(b,z) (5.13)\nand\n/ba∇dbl(ε∇f1)h/ba∇dblHs−2≤ε/ba∇dblf1/ba∇dblHs−1≤CεL(b,z). (5.14)\nWe infer from (5.9), (5.11), (5.12) and (5.13) that\n/ba∇dblAm/ba∇dblHs−1+/ba∇dblAh/ba∇dblHs−1≤CL(b,z). (5.15)\nMoreover (5.10), (5.11) and (5.14) yield\n/ba∇dblBm/ba∇dblHs−2+/ba∇dblBh/ba∇dblHs−2≤CεL(b,z), (5.16)\nso that the conclusion of Lemma 5.4 finally follows from (5.6) , (5.8), (5.15) and (5.16). /square\nNext, in order to establish the second part of Proposition 2. 4 involving the norm /ba∇dblb/ba∇dblL2(L∞),\nwe show the following analogs of Lemmas 5.3 and 5.4 involving /ba∇dblc/ba∇dblL2(Hs−1).\nLemma 5.5. Under the assumptions of Proposition 2.4 we have for T∈[0,T0]\nC−1/ba∇dblc/ba∇dblL2\nT(Hs−1)≤(εν−1\nε)1/2M0+εmax(1,ν−1\nε)/ba∇dblL(b,z)/ba∇dblL2\nT.22 EVELYNE MIOT\nProof.We closely follow the proofs of Lemmas 5.3 and 5.4, handling a gain the regions R1,\nR2andR3separately.\nFirst step : low frequencies |ξ| ≤rνε.\nFor low frequencies one may even improve the estimates given by Lemma 5.2 for the semi-\ngroup acting on c. Indeed, according to identity (7.1) stated in the proof of L emma 5.2, we\nget the bound\n|/hatwidecs(t,ξ)| ≤C(I(t,ξ)+J(t,ξ)),\nwhere\nI(t,ξ) =e−νε\n2εt/vextendsingle/vextendsingle/hatwide(c,d)s(0,ξ)/vextendsingle/vextendsingle+/integraldisplayt\n0e−νε\n2ε(t−τ)|/hatwider(F,G)(τ,ξ)|dτ\nand\nJ(t,ξ) =e−c|ξ|2\nνεεt/vextendsingle/vextendsingle(|ξ|2ν−2\nε/hatwidecs,|ξ|ν−1\nε/hatwideds)(0)/vextendsingle/vextendsingle+/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|(|ξ|2ν−2\nε/hatwiderFs,|ξ|ν−1\nε/hatwiderGs)|dτ\n=JL(t,ξ)+JNL(t,ξ).\nHere again we set ˇI=F−1IandˇJ=F−1J. In view of the first step in the proof of Lemma\n5.3 (see (5.5)) we already know that\n/ba∇dblˇI/ba∇dblL2\nT(Hs)≤C/parenleftbig\n(εν−1\nε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2\nT/parenrightbig\n.\nNext, since |ξ|ν−1\nε≤rwe have\n/ba∇dblˇJL/ba∇dblL2\nT(Hs−1)≤/vextenddouble/vextenddoublee−c|ξ|2\nνεεt(1+|ξ|s−1)/parenleftbig\n|ξ|2ν−2\nε|/hatwidecs(0)|+|ξ|ν−1\nε|/hatwideds(0)|/parenrightbig/vextenddouble/vextenddouble\nL2\nT(L2)\n≤Cν−1\nε/vextenddouble/vextenddoublee−c|ξ|2\nνεεt|ξ|(1+|ξ|s−1)/parenleftbig\n|/hatwidecs(0)|+|/hatwideds(0)|/parenrightbig/vextenddouble/vextenddouble\nL2\nT(L2)\n≤Cν−1\nε(ενε)1/2M0,\nwhere the last inequality is a consequence of Lemma 7.1.\nOn the other side we have\n/ba∇dblˇJNL/ba∇dblL2\nT(Hs−1)≤C/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)(1+|ξ|s−1)/parenleftbig\n|ξ|2ν−2\nε|/hatwiderFs|+|ξ|ν−1\nε|/hatwiderGs|/parenrightbig\ndτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n≤Cν−2\nε/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|2(1+|ξ|s−1)|/hatwiderFs|dτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)\n+Cν−1\nε/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−c|ξ|2\nνεε(t−τ)|ξ|2(1+|ξ|s−1)|ξ|−1|/hatwiderGs|dτ/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2).\nApplying Lemma 7.1 to each term we obtain\n/ba∇dblˇJNL/ba∇dblL2\nT(Hs−1)≤C(νεεν−2\nε/ba∇dblFs/ba∇dblL2\nT(Hs−1)+νεεν−1\nε/ba∇dblD−1Gs/ba∇dblL2\nT(Hs−1))\n≤C(εν−1\nε/ba∇dblfs/ba∇dblL2\nT(Hs−1)+ε/ba∇dblD−1gs/ba∇dblL2\nT(Hs−1))\n≤Cε/ba∇dblL(b,z)/ba∇dblL2\nT.\nWe have used the support properties of fsin the last inequality above.\nFinally, we gather the previous inequalities to find\nC−1/ba∇dblcs/ba∇dblL2\nT(Hs−1)≤(εν−1\nε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2\nT. (5.17)DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 3\nSecond step : intermediate frequencies rνε≤ |ξ| ≤Rε−1.\nIn contrast with the previous step, we may here imitate the fir st step of the proof of Lemma\n5.4, estimating the Hs−1norm instead :\n/ba∇dblcm/ba∇dblL2\nT(Hs−1)≤ /ba∇dbl(c,d)m/ba∇dblL2\nT(Hs−1)≤C/parenleftbig\n(εν−1\nε)1/2/ba∇dbl(c,d)(0)/ba∇dblHs−1+εν−1\nε/ba∇dbl(F,G)m/ba∇dblL2\nT(Hs−1)/parenrightbig\n.\nRecalling that ( F,G)m=Am+Bm, where/hatwidestAmand/hatwiderBmare compactly supported in the region\n{|ξ| ≤Rε−1}, we obtain\n/ba∇dbl(F,G)m/ba∇dblL2\nT(Hs−1)≤ /ba∇dblAm/ba∇dblL2\nT(Hs−1)+/ba∇dblBm/ba∇dblL2\nT(Hs−1)≤ /ba∇dblAm/ba∇dblL2\nT(Hs−1)+Cε−1/ba∇dblBm/ba∇dblL2\nT(Hs−2).\nIn view of the third step of the proof of Lemma 5.4 (see (5.15) a nd (5.16)) we get\n/ba∇dbl(F,G)m/ba∇dblHs−1≤CL(b,z)\nand we conclude that\nC−1/ba∇dblcm/ba∇dblL2\nT(Hs−1)≤(εν−1\nε)1/2M0+εν−1\nε/ba∇dblL(b,z)/ba∇dblL2\nT. (5.18)\nThird step : high frequencies |ξ| ≥Rε−1.\nWith (F,G)h=Ah+Bhwe obtain, arguing exactly as in the second step of the proof o f\nLemma 5.4, the analog of (5.7):\n/ba∇dbl(c,d)h/ba∇dblL2\nT(Hs−1)≤C/parenleftbig\n(εν−1\nε)1/2M0+(ενε)−1/ba∇dbl(F,G)h/ba∇dblL2\nT(Hs−3)/parenrightbig\n≤C/parenleftbig\n(εν−1\nε)1/2M0+ν−1\nε/ba∇dbl(F,G)h/ba∇dblL2\nT(Hs−2)/parenrightbig\n≤C/parenleftbig\n(εν−1\nε)1/2M0+ν−1\nεε(/ba∇dblAh/ba∇dblL2\nT(Hs−1)+ε−1/ba∇dblBh/ba∇dblL2\nT(Hs−2))/parenrightbig\n.\nHence we infer from estimates (5.15) and (5.16) for AhandBhthat\nC−1/ba∇dblch/ba∇dblL2\nT(Hs−1)≤(εν−1\nε)1/2M0+εν−1\nε/ba∇dblL(b,z)/ba∇dblL2\nT. (5.19)\nThe conclusion finally follows from estimates (5.17), (5.18 ) and (5.19). /square\nInvoking the previous results we may now complete the\nProof of Proposition 2.4 .\nFirst, Cagliardo-Nirenberg inequality yields\n/ba∇dbl|z|2/ba∇dblHs+/ba∇dblb2/ba∇dblHs+/ba∇dblbz/ba∇dblHs+/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblHs≤C/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHs\nand\n/ba∇dblεb|z|2/ba∇dblHs≤Cε/ba∇dbl(b,z)/ba∇dbl2\n∞/ba∇dbl(b,z)/ba∇dblHs,\nso that\nL(b,z)≤C(1+ε/ba∇dbl(b,z)/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHs.\nBy Sobolev embedding and Cauchy-Schwarz inequality we obta in\n/ba∇dblL(b,z)/ba∇dblL2\nT≤C/parenleftbig\n1+ε/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)/parenrightbig\n/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)/ba∇dbl(b,z)/ba∇dblL2\nT(Hs)\nand\n/ba∇dblL(b,z)/ba∇dblL1\nT≤C/parenleftbig\n1+ε/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nL2\nT(Hs).\nProposition 2.4 finally follows from both estimates above to gether with Lemmas 5.3, 5.4 and\n5.5. /square\nWe conclude this section with a result that will be needed in t he course of the next section.\nWe omit the proof, which is a straightforward adaptation of t he proof of Lemma 5.5.24 EVELYNE MIOT\nProposition 5.1. Under the assumptions of Proposition 2.4 we have for all T∈[0,T0]\nC−1/ba∇dblc/ba∇dblL2\nT(Hs)≤(εν−1\nε)1/2M0+ν−1\nε/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)/ba∇dbl(b,z)/ba∇dblL2\nT(Hs)(1+ε/ba∇dbl(b,z)/ba∇dblL∞\nT(Hs)).\n6.Proofs of Theorems 1.1 and 1.3.\n6.1.Proof of Theorem 1.1. This paragraph is devoted to the proof of Theorem 1.1. Let\nΨ0∈ W+Hs+1such that\nΨ0=ρ0exp(iϕ0) =/parenleftbig\n1+ε√\n2a0/parenrightbig1/2exp(iϕ0),\nwhere (a0,ϕ0) satisfies the assumptions of Theorem 1.1. Let Ψ ∈ W+C([0,T∗),Hs+1) denote\nthe corresponding solution to (C ε) provided by Theorem 3.1.\nWithc(s,N) denoting a constant corresponding to the Sobolev embeddin gHs(RN)⊂\nL∞(RN), we first assume that the constant K1(s,N) in Theorem 1.1 satisfies\nK1(s,N)>√\n2c(s,N). (6.1)\nHence\n/ba∇dbl|Ψ0|2−1/ba∇dbl∞=ε√\n2/ba∇dbla0/ba∇dbl∞<1\n2,\nso that the assumptions of Corollary 3.1 are satisfied. Let ( b,v) be the solution given by\nCorollary 3.1 on [0 ,T0), withT0≤T∗maximal.\nWe introduce the following control function\n\n\nH(t) =/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)+/ba∇dbl(b,z)/ba∇dblL2\nt(Hs)\nκ1/2max(1,ν−1ε)+/ba∇dblb/ba∇dblL2\nt(L∞)\n(εν−1ε)1/2,\nH0=H(0).(6.2)\nNote that, according to (5.4) we have\nH0≤C1(s,N)M0and/ba∇dbl(b,v)(t)/ba∇dblHs+ε/ba∇dblb(t)/ba∇dblHs+1≤C1(s,N)H(t),\nwhere the constant C1(s,N) depends only on sandN. We recall that M0is defined in\nTheorem 1.1. Increasing possibly the number K1(s,N) introduced in Theorem 1.1, we may\nassume that C1(s,N)< K1(s,N).\nWe define the stopping time\nTε= sup{t∈[0,T0) such that H(t)< C2(s,N)M0},\nwhereC2(s,N) denotes a constant (to be specified later) satisfying\nC1(s,N)< C2(s,N)< K1(s,N). (6.3)\nWe remark that Tε>0 by continuity of t/ma√sto→H(t).\nWe next choose κ0(s,N) in such a way that\nκ0(s,N)C2(s,N)<K1(s,N)√\n2c(s,N). (6.4)\nBy assumption on M0, this implies that for κ≤κ0(s,N)\nC2(s,N)M0<C2(s,N)νε\nK1(s,N)≤1√\n2c(s,N)ε.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 5\nIn particular, since /ba∇dbl(b,z)(t)/ba∇dblHs≤H(t), it follows that\n/ba∇dbl|Ψ|2(t)−1/ba∇dbl∞<1\n2,∀t∈[0,Tε). (6.5)\nOur next purpose is to show that Tε=T0=T∗= +∞.\nFirst, mollifying possibly ( a0,u0) we may asume that ( b,z)∈C1([0,T0),Hs+1). By (6.5),\nPropositions 2.3 and 2.4 hold on [0 ,Tε), so that\nC(s,N)−1H≤H0+H2/parenleftig/parenleftbig\n1+κH/parenrightbig\nκmax(1,ν−1\nε)2+/parenleftbig\n1+εH/parenrightbig\n(κmax(1,ν−1\nε)2+ε+ν−1\nε)/parenrightig\n≤H0+H2/parenleftbig\n1+max( ε,κ)H/parenrightbig\n(κmax(1,ν−1\nε)2+ε+ν−1\nε).\nObserving that\nκmax(1,ν−1\nε)2+ε+ν−1\nε≤3(κ+ν−1\nε),\nwe find\nC3(s,N)−1H≤H0+max(κ,ν−1\nε)H2/parenleftbig\n1+max( ε,κ)H/parenrightbig\n.\nHereC3(s,N) is a constant depending only on sandN, which can be assumed to be larger\nthan max( C1(s,N),1). On the other side, for t∈[0,Tε] we have according to (6.3) and by\nassumption on M0\nmax(ε,κ)H≤max(ε,κ)C2(s,N)M0≤1,\nso that\nH≤2C3(s,N)/parenleftig\nM0+max(κ,ν−1\nε)H2/parenrightig\n. (6.6)\nAt this stage we may choose the constants C2(s,N) andK1(s,N) as follows:\nC2(s,N) = 4C3(s,N) and K1(s,N)>16C3(s,N)2max(√\n2c(s,N),1),\nso that all conditions (6.1), (6.3) and (6.4) are met.\nWe now show that Tε=T0: otherwise Tεis finite. Hence, considering (6.6) at time Tεwe\nobtain\n4C3(s,N)M0≤2C3(s,N)(M0+16max( κ,ν−1\nε)C3(s,N)2M2\n0),\nwhence\n1≤16C3(s,N)2max(κ,ν−1\nε)M0≤16C3(s,N)2\nK1(s,N).\nBy definition of K1(s,N), this leads to a contradiction, therefore Tε=T0.\nNow, since (6.5) holds on [0 ,T0), Corollary 3.1 and a standard continuation argument imply\nthatT0=T∗. Invoking again (6.5) we easily show that\n/ba∇dbl∇Ψ(t)/ba∇dblHs≤C/parenleftbig\n1+/ba∇dbl(b,v)(t)/ba∇dbl2\nHs+1×Hs/parenrightbig\n,∀t∈[0,T∗)\nfor a constant C. In view of the previous estimates we obtain\nlimsup\nt→T∗/ba∇dbl∇Ψ(t)/ba∇dblHs≤limsup\nt→T∗C(1+H(t)2)<∞.\nWe finally conclude that T∗= +∞thanks to Theorem 3.1. /square26 EVELYNE MIOT\n6.2.Proof of Theorem 1.3. We present here the proof of Theorem 1.3. Here again,\nCalways stands for a constant depending only on sandN. We define ( bℓ,vℓ)(t,x) =\n(aℓ,uℓ)(ε−1t,x), where ( aℓ,uℓ) is the solution to the linear equation (1.6) with initial da tum\n(b0,v0) = (a0,u0). Introducing ( b,v) = (b−bℓ,v−vℓ), we have\n\n\n∂tb+√\n2\nεdivv+2νε\nεb−κ∆b=f(b,z)\n∂tv+√\n2\nε∇b−κ∆v=g(b,z) +ε√\n2∇∆b.\nThe proof of Theorem 1.3 relies on energy estimates, since th e method used in Section 5 is not\nconvenient to establish uniform in time estimates. For 0 ≤k≤swe compute by integration\nby parts\n1\n2d\ndt/ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2\nL2=/integraldisplay\nRNDkbDk∂tb+Dkv·Dk∂tv\n=−2νε\nε/integraldisplay\nRN|Dkb|2−κ/integraldisplay\nRN|∇Dkb|2−κ/integraldisplay\nRN|∇Dkv|2\n+/integraldisplay\nRNDkbDkf(b,z)+/integraldisplay\nRNDkv·Dkg(b,z) +ε√\n2/integraldisplay\nRNDkv·Dk∇∆b.\nWe recall the decompositions f=νεf0+f1andg=g1+εg2=∇h0+ε∇h1, where the\nfi,gi,hi,i= 0,1,2, which have been defined in Paragraph 5.2, are i-order derivatives of\nquadratic functions in ( b,z). We obtain\n1\n2d\ndt/ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2\nL2≤I+J+K,\nwhere\nI=−2νε\nε/integraldisplay\nRN|Dkb|2+νε/integraldisplay\nRNDkbDkf0(b,z)\nJ=/integraldisplay\nRNDkbDkf1(b,z)+/integraldisplay\nRNDkv·Dkg1(b,z),\nK=−κ/integraldisplay\nRN|∇Dkv|2+ε/integraldisplay\nRNDkv·Dkg2(b,z)+ε√\n2/integraldisplay\nRNDkv·Dk∇∆b.\nEstimates for IandJ.\nBy virtue of Lemma 7.4 and by Sobolev embedding we find\nI≤ −νε\nε/integraldisplay\nRN|Dkb|2+Cενε/integraldisplay\nRN|Dkf0|2≤Cκ/ba∇dblf0/ba∇dbl2\nHk≤Cκ/ba∇dbl(b,z)/ba∇dbl4\nHs.\nNext, Cauchy-Schwarz inequality yields\nJ≤ /ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(f1,g1)/ba∇dblHk≤C/ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(b,z)/ba∇dbl2\nHk+1.\nEstimate for K.\nWe perform an integration by parts in the last two integrals a nd insert the fact that g2=\n∇h1to obtain\nK=−κ/integraldisplay\nRN|∇Dkv|2−ε/integraldisplay\nRNdivDkvDkh1−ε√\n2/integraldisplay\nRNdivDkvDk∆b\n≤ −κ\n4/integraldisplay\nRN|∇Dkv|2+Cε2\nκ/integraldisplay\nRN|Dkh1|2+C\nκ/integraldisplay\nRN|ε∆Dkb|2.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 7\nFirst, by virtue of Cagliardo and Sobolev inequalities we ha ve\n/ba∇dblDkh1/ba∇dblL2≤C/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHk+1≤C/ba∇dbl(b,z)/ba∇dblHs/ba∇dbl(b,z)/ba∇dblHk+1.\nTherefore:\n•If 0≤k≤s−2 we find\nK≤Cκ−1ε2/parenleftbig\n/ba∇dbl(b,z)/ba∇dbl4\nHs+/ba∇dblb/ba∇dbl2\nHs/parenrightbig\n.\n•Ifk=s−1 we observe that /ba∇dblε∆Dkb/ba∇dblL2≤C/ba∇dblc/ba∇dblHs,wherec= (1−ε2∆/2)1/2bis defined\nin the beginning of Section 5. So we find\nK≤Cκ−1/parenleftbig\nε2/ba∇dbl(b,z)/ba∇dbl4\nHs+/ba∇dblc/ba∇dbl2\nHs/parenrightbig\n.\n•Ifk=s, similar arguments using that /ba∇dblε∆Dkb/ba∇dblL2≤C/ba∇dblc/ba∇dblHs+1≤C/ba∇dbl(b,z)/ba∇dblHs+1(see\n(5.4)) yield\nK≤Cκ−1/ba∇dbl(b,z)/ba∇dbl2\nHs+1/parenleftbig\n1+ε2/ba∇dbl(b,z)/ba∇dbl2\nHs/parenrightbig\n.\nIntegrating the previous estimates for I,JandKon [0,t] we find:\n•If 0≤k≤s−2,\n/ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2\nL2≤C/integraldisplayt\n0/ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(b,z)/ba∇dbl2\nHsdτ\n+C/integraldisplayt\n0/parenleftig\n(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4\nHs+κ−1ε2/ba∇dbl(b,z)/ba∇dbl2\nHs/parenrightig\ndτ.\nAppyling Young inequality to the first term in the right-hand side we infer that\nC−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2\nL∞\nt(L2)≤ /ba∇dbl(b,z)/ba∇dbl4\nL2\nt(Hs)\n+(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)+κ−1ε2/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs).(6.7)\n•Similarly, if k=s−1 we have\nC−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2\nL∞\nt(L2)≤ /ba∇dbl(b,z)/ba∇dbl4\nL2\nt(Hs)\n+(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)+κ−1/ba∇dblc/ba∇dbl2\nL2\nt(Hs).(6.8)\n•Ifk=sthen\nC−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2\nL∞\nt(L2)≤ /ba∇dbl(b,z)/ba∇dbl4\nL2\nt(Hs+1)\n+κ/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)+κ−1/parenleftbig\n1+ε2/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/parenrightbig\n/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs+1).(6.9)\nProof of the uniform in time comparison estimates in Theorem 1.3.\nWe control each term in the right-hand sides in (6.7), (6.8) a nd (6.9) by means of the various\nestimates established in the previous sections. We recall t hat the control function H(t),\nwhich is defined in (6.2), satisfies H(t)≤CM0. This controls the quantities /ba∇dbl(b,z)/ba∇dblL2\nt(Hs)\nand/ba∇dbl(b,z)/ba∇dblL∞\nt(Hs)in terms of M0. We use Proposition 5.1 to estimate /ba∇dblc/ba∇dblL2\nt(Hs). Finally, to\ncontrol/ba∇dbl(b,z)/ba∇dblL2\nt(Hs+1)we rely on the second inequality in Proposition 2.3. Straigh tforward\ncomputations then lead to the uniform comparison estimates in Theorem 1.3.\nProof of the time dependent comparison estimates in Theorem 1.3.\nWe go back to the previous energy estimates.\n•If 0≤k≤s−2 we apply Cauchy-Schwarz inequality in (6.7) to obtain\nC−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2\nL∞\nt(L2)≤t/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)\n+t(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4\nL∞\nt(Hs)+tκ−1ε2/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs).28 EVELYNE MIOT\n•Ifk=s−1 we similarly infer from (6.8)\nC−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2\nL∞\nt(L2)≤t/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs)/ba∇dbl(b,z)/ba∇dbl2\nL2\nt(Hs)\n+t(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4\nL∞\nt(Hs)+tκ−1/ba∇dbl(b,z)/ba∇dbl2\nL∞\nt(Hs).\nUsing that H(t)≤CM0, the assumptions on M0as well as the fact that ( aε,uε)(t) =\n(bε,vε)(εt) we are led to the desired estimates. We omit the details. /square\n7.Appendix.\nIn this appendix we gather some helpful results.\n7.1.Some parabolic estimates and useful tools. The following result is an immediate\nconsequenceof maximal regularity fortheheat operator et∆. We referto[8] forfurtherdetails.\nLemma 7.1. There exists C >0such that for all λ >0,a0∈L2(RN),a=a(s)∈L2(R+×\nRN)andT >0\n/ba∇dbleλt∆a0/ba∇dblL2\nT(˙H1)≤C√\nλ/ba∇dbla0/ba∇dblL2\nand /vextenddouble/vextenddouble/vextenddouble/vextenddouble∆/integraldisplayt\n0eλ(t−s)∆a(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)≤C\nλ/ba∇dbla/ba∇dblL2\nT(L2).\nWe also have the following\nLemma 7.2. There exists C >0such that for all λ >0andH∈L2(R+×RN)\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0eλ(t−s)∆H(s)ds/vextenddouble/vextenddouble/vextenddouble\nL2\nT(˙H1)≤C√\nλ/integraldisplayT\n0/ba∇dblH(t)/ba∇dblL2dt.\nProof.We may assume that His smooth, compactly supported, and that the function u(t) =/integraltextt\n0eλ(t−s)∆H(s)dsis the smooth solution to\n∂tu−λ∆u=Handu(0) = 0.\nWe infer that\n1\n2d\ndt/ba∇dblu(t)/ba∇dbl2\nL2=/integraldisplay\nRNuH−λ/integraldisplay\nRN|∇u|2,\nso that\nλ/ba∇dbl∇u/ba∇dbl2\nL2\nT(L2)≤C/integraldisplayT\n0/integraldisplay\nRN|u||H|dtdx≤Csup\nt∈[0,T]/ba∇dblu(t)/ba∇dblL2/ba∇dblH/ba∇dblL1\nT(L2).\nButu(0) = 0, therefore we also have /ba∇dblu(t)/ba∇dbl2\nL2≤C/integraltextt\n0/integraltext\n|uH|. This yields\nsup\nt∈[0,T]/ba∇dblu(t)/ba∇dblL2≤C/ba∇dblH/ba∇dblL1\nT(L2)\nand the conclusion follows. /square\nLemma 7.3. There exists C >0such that for all λ >0,a0∈L2,a∈L2(R+×RN)and\nT >0\n/ba∇dble−λta0/ba∇dblL2\nT≤C√\nλ/ba∇dbla0/ba∇dblL2\nand/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−λ(t−s)a(s,·)ds/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)≤C\nλ/ba∇dbla/ba∇dblL2\nT(L2).DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 9\nProof.We only establish the second estimate. We set ˜ a(s) =a(s) fors∈[0,T] and ˜a= 0 for\ns /∈[0,T], so that\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−λ(t−s)a(s)ds/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayT\n0e−λ(t−s)/ba∇dbl˜a(s)/ba∇dblL2(RN)ds/vextenddouble/vextenddouble/vextenddouble\nL2\nT=/ba∇dble−λ·∗/ba∇dbl˜a(·)/ba∇dblL2(RN)/ba∇dblL2.\nBy Young inequality for the convolution, we then have\n/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0e−λ(t−s)a(s)ds/vextenddouble/vextenddouble/vextenddouble\nL2\nT(L2)≤C/ba∇dble−λ·/ba∇dblL1/ba∇dbl˜a/ba∇dblL2(R+,L2).\nWe conclude by definition of ˜ a. /square\nWe conclude this paragraph with the following result, which is a consequence of Gagliardo-\nNirenberg inequality.\nLemma 7.4 (see [2], Lemma 3) .Letk∈Nandj∈ {0,...,k}. There exists a constant\nC(k,N)such that\n/ba∇dblDjuDk−jv/ba∇dblL2≤C(k,N)/parenleftig\n/ba∇dblu/ba∇dbl∞/ba∇dblDkv/ba∇dblL2+/ba∇dblv/ba∇dbl∞/ba∇dblDku/ba∇dblL2/parenrightig\nand\n/ba∇dbluv/ba∇dblHk≤C(k,N)(/ba∇dblu/ba∇dbl∞/ba∇dblv/ba∇dblHk+/ba∇dblv/ba∇dbl∞/ba∇dblu/ba∇dblHk).\n7.2.Proof of Lemma 5.2. In all the following Cdenotes a numerical constant. In order to\nsimplify the notations we introduce the quantities\nω=ε2|ξ|2andµ=1\nνε|ξ|√\n2+ω,\nand we express Mas follows\nM=νε\nε/parenleftbigg\n2+ω µ\n−µ ω/parenrightbigg\n.\nFirst we compute the eigenvalues λ1andλ2ofM. Setting\n∆ = 1−µ2,\nwe have\nλ1=νε\nε(ω+1−C√\n∆) and λ2=νε\nε(ω+1+C√\n∆),\nwhereC√\n∆is√\n∆if∆≥0andisi√\n−∆if∆<0. Hence M=P−1DP, whereD= diag(λ1,λ2)\nand\nP−1=1\nµ2−α2/parenleftbigg\n−µ α\nα−µ/parenrightbigg\n, P=/parenleftbigg\n−µ−α\n−α−µ/parenrightbigg\n,withα= 1+C√\n∆.\nFinally for all ( a,b)∈C2we have\ne−tM/parenleftbigg\na\nb/parenrightbigg\n=P−1e−tDP/parenleftbigg\na\nb/parenrightbigg\n=1\nµ2−α2/parenleftbigg\n(µ2a+αµb)e−λ1t−(α2a+αµb)e−λ2t\n(αµa+µ2b)e−λ2t−(αµa+α2b)e−λ1t/parenrightbigg\n=e−νε\nε(1+ω)t\nµ2−α2/parenleftigg\n(µ2a+αµb)etνε\nεC√\n∆−(α2a+αµb)e−tνε\nεC√\n∆\n(αµa+µ2b)e−tνε\nεC√\n∆−(αµa+α2b)etνε\nεC√\n∆/parenrightigg\n,\nor equivalently\ne−tM/parenleftbigg\na\nb/parenrightbigg\n=e−νε\nε(1+ω)t/bracketleftigg\ne−tνε\nεC√\n∆/parenleftbigg\na\nb/parenrightbigg\n+etνε\nεC√\n∆−e−tνε\nεC√\n∆\nµ2−α2/parenleftbigg\nαµb+µ2a\n−αµa−α2b/parenrightbigg/bracketrightigg\n.(7.1)30 EVELYNE MIOT\nFirst case |ξ|2≥3ν2\nε/8.\nThenµ2≥3/4, hence ∆ ≤1/4. We need to examine the following subcases.\n•0≤∆≤1/4.\nIt follows thatC√\n∆ =√\n∆ andµ2−α2=−2(∆+√\n∆), so that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleexp(tνε\nε√\n∆)−exp(−tνε\nε√\n∆)\nµ2−α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sinh/parenleftig\ntνε\nε√\n∆/parenrightig\n√\n∆≤Csinh/parenleftbiggνεt\n2ε/parenrightbigg\n,\nwhere the second inequality is due to the fact that x/ma√sto→sinh(x)/xis an increasing function\nonR+. We infer that\n/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νε\n2εt/parenrightig\nexp/parenleftig\n−νεω\nεt/parenrightig/parenleftbig\n|a|+|b|/parenrightbig\n. (7.2)\n•−1≤∆<0.\nThenC√\n∆ =i√\n−∆ andµ2−α2=−2(∆+i√\n−∆), therefore\n|µ2−α2|= 2/radicalbig\n∆2−∆≥2√\n−∆.\nIt follows that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleexp(itνε\nε√\n−∆)−exp(−itνε\nε√\n−∆)\nµ2−α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/vextendsingle/vextendsinglesin/parenleftbig\ntνε\nε√\n−∆/parenrightbig/vextendsingle/vextendsingle\n√\n−∆≤Cνεt\nε,\nwhere in the last inequality we have inserted that |sinx| ≤xfor allx≥0. Since |µ| ≤Cand\n|α| ≤Cthis yields\n/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νε\nε(1+ω)t/parenrightig/parenleftig\n1+νε\nεt/parenrightig/parenleftbig\n|a|+|b|/parenrightbig\n,\nso finally/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νε\n2ε(1+ω)t/parenrightig/parenleftbig\n|a|+|b|/parenrightbig\n. (7.3)\n•∆≤ −1.\nWe have\n|µ2−α2|= 2/radicalbig\n∆2−∆≥2|∆| ≥Cµ2,\nwhile|α|=√\n1−∆ =µ. Hence we find\n/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νε\nε(1+ω)t/parenrightig/parenleftbig\n|a|+|b|/parenrightbig\n. (7.4)\nSecond case |ξ|2≤3ν2\nε/8.\nWe check that µ2≤3(2 + 3κ2/8)/8, therefore 1 /8≤∆≤1 whenever κ < κ 0=/radicalbig\n8/9.\nMoreover\nC−1≤ |µ2−α2| ≤C, α≤C, µ≤Candµ≤C|ξ|\nνε.\nIn addition,\nνε\nε(−1+√\n∆) =−νε\nε1−∆\n1+√\n∆=−νε\nεµ2\n1+√\n∆≤ −Cνε\nεµ2.\nTherefore in view of (7.1)\n/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νε\nε(1+ω)t/parenrightig/parenleftbig\n|a|+|b|/parenrightbig\n+Cexp/parenleftig\n−νεω\nεt/parenrightig\nexp/parenleftbigg\n−Cνεµ2\nεt/parenrightbigg/parenleftbigg|ξ|\nνε|a|+|b|/parenrightbigg\n.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 3 1\nNow, since\nC|ξ|2\nν2ε≥µ2=|ξ|2\nν2ε(2+ω)≥|ξ|2\nν2ε\nwe obtain\n/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig\n−νεω\nεt/parenrightig/parenleftbigg\nexp/parenleftig\n−νε\nεt/parenrightig\n+exp/parenleftbigg\n−C|ξ|2\nνεεt/parenrightbigg/parenrightbigg/parenleftbigg|ξ|\nνε|a|+|b|/parenrightbigg\n.(7.5)\nGathering estimates (7.2) to (7.5) and setting r=/radicalbig\n3/8 we are led to the conclusion of\nthe Lemma.\n/square\nAcknowlegments. I warmly thank Didier Smets for his help. This work was partly sup-\nported by the grant JC05-51279 of the Agence Nationale de la R echerche.\nReferences\n[1] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation , Rev. Mod. Phys. 74\n(2002), 99-143.\n[2] F. Bethuel, R. Danchin and D. Smets, On the linear wave regime for the Gross-Pitaevskii equation , J.\nAnal. Math. , to appear.\n[3] A. Capella, C. Melcher and F. Otto, Wave-type dynamics in ferromagnetic thin films and the motio n of\nN´ eel walls , Nonlinearity 20(2007), 2519-2537.\n[4] C. Gallo, The Cauchy problem for defocusing nonlinear Schr¨ odinger e quations with non-vanishing initial\ndata at infinity , Comm. Partial Differential Equations 33(2008), no. 4-6, 729-771.\n[5] P. G´ erard, The Cauchy problem for the Gross-Pitaevskii equation , Ann. Inst. H. Poincar´ e Anal. Non\nLin´ eaire 23(2006), no. 5, 765-779.\n[6] M. Kurzke, C. Melcher, R. Moser and D. Spirn, Dynamics of Ginzburg-Landau vortices in a mixed flow ,\nIndiana Univ. Math. Jour. , to appear.\n[7] M. Kurzke, C. Melcher, R. Moser and D. Spirn, Ginzburg-Landau vortices driven by the Landau-Lifschitz-\nGilbert equations , preprint.\n[8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and quasilinear equations of parabolic\ntype, American Mathematical Society, Providence, R.I. (1967), vol. 23.\n[9] E. Miot, Dynamics of vortices for the complex Ginzburg-Landau equat ion, Analysis and PDE 2(2009),\nno. 2, 159-186.\n(E. Miot) Dipartimento di Matematica G. Castelnuovo, Universit `a di Roma “La Sapienza”, Italy\nE-mail address :miot@ann.jussieu.fr"    },    {        "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 field 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": "1004.1127v6.Concatenated_quantum_codes_can_attain_the_quantum_Gilbert_Varshamov_bound.pdf",        "content": "Concatenated quantum codes can attain the quantum\nGilbert-Varshamov bound\nYingkai Ouyang\nDepartment of Combinatorics and Optimization,\nInstitute of Quantum Computing,\nUniversity of Waterloo,\n200 University Avenue West,\nWaterloo, Ontario N2L 3G1, Canada.\ny3ouyang@math.uwaterloo.ca\nAbstract\nA family of quantum codes of increasing block length with positive rate is asymptotically good if\nthe ratio of its distance to its block length approaches a positive constant. The asymptotic quantum\nGilbert-Varshamov (GV) bound states that there exist q-ary quantum codes of su\u000eciently long\nblock length Nhaving \fxed rate Rwith distance at least NH\u00001\nq2((1\u0000R)=2), where Hq2is the\nq2-ary entropy function. For q < 7, only random quantum codes are known to asymptotically\nattain the quantum GV bound. However, random codes have little structure. In this paper, we\ngeneralize the classical result of Thommesen [1] to the quantum case, thereby demonstrating the\nexistence of concatenated quantum codes that can asymptotically attain the quantum GV bound.\nThe outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random\nindependently chosen stabilizer codes, where the rates of the inner and outer codes lie in a speci\fed\nfeasible region.\n1arXiv:1004.1127v6  [quant-ph]  16 Jan 2014I. INTRODUCTION\nA family of q-ary quantum codes [2] of increasing block length with positive rate is de\fned\nto be asymptotically good if the ratio of its distance to its block length approaches a positive\nconstant. Designing good quantum codes is highly nontrivial, just as it is in the classical\ncase. The quantum Gilbert-Varshamov (GV) bound [3{8] is a lower bound on an achievable\nrelative distance of a quantum code of a \fxed rate, and is attainable for various families of\nrandom quantum codes [3, 5, 7]. Explicit families of quantum codes, both unconcatenated\n[9, 10] and concatenated [11{14], have been studied, but do not attain the quantum GV\nbound forq <7 [15]. We show that concatenated quantum codes can attain the quantum\nGV bound.\nWe are motivated by the historical development of the idea of concatenating a sequence\nof increasingly long classical Reed-Solomon (RS) outer codes with various types of classical\ninner codes. In both cases where the inner codes are all identical [16] or all distinct [17],\nthe resultant sequence of concatenated codes while asymptotically good nonetheless fail to\nattain the GV bound. A special case of Thommesen's result [1] shows that even if the inner\ncodes all have a rate of one, if they are chosen uniformly at random, the resultant sequence\nof concatenated codes almost surely attains the GV bound. Our work extends this classical\nobservation to the quantum case.\nWe show the quantum analog of Thommesen's result { the sequence of concatenated\nquantum codes with the outer code being a quantum generalized RS code [14, 18{20] and\nrandom inner stabilizer codes almost surely attains the quantum GV bound when the rates\nof the inner and outer codes lie in feasible region (III.1) with an example depicted in Figure 2.\nThe property of the outer code that we need is that the normalizer of its stabilizer is classical\nmaximal distance separable (MDS) code [20]. Our work is closest in spirit to that of Fujita\n[12], where quantum equivalents of the Zyablov and the Blokh-Zyablov bounds are obtained\n(not attaining the quantum GV bound) by choosing a quantum RS code with essentially\nrandom inner codes.\nIn the proof of the classical result, Thommesen uses a random coding argument to\ncompute the probability that any codeword of weight less than the target minimum distance\nbelongs to the random code. Subsequently, he uses the union bound, the spectral property\n2of the Reed-Solomon outer code, and properties of the q-ary entropy function (de\fned in\nII.1), to prove that the proposed random code almost surely does not contain any codeword\nof weight less than the prescribed minimum distance.\nThe proof of our quantum result follows a similar strategy, with codewords replaced by\nelements of the normalizer not in the stabilizer. However the feasible region for the rates\nof the inner and outer codes for the classical and the quantum result are not analogous,\nbecause the monotonicity of the q-ary entropy function applies in a di\u000berent feasible region\nfrom that of the classical case.\nThe organization of this paper is as follows: Section II introduces the notation and\npreliminary material used in this paper. This section lays out the formalism of concatenating\nstabilizer codes, which is crucial to the proof of the main result. We state our main result\nin Theorem III.1 of Section III, and the remainder of the paper is dedicated to its proof.\nII. PRELIMINARIES\nLetL(Cq) denote the set of complex qbyqmatrices. De\fne 1qto be a size qidentity\nmatrix and !q:=e2\u0019i=qto be a primitive q-th root of unity, where q\u00152 is an prime power.\nDe\fne 0 logq0:= 0. De\fne the q-ary entropy function and its inverse to be Hq: [0;1]![0;1]\nandH\u00001\nq: [0;1]![0;q\u00001\nq] respectively where\nHq(x):=xlogq(q\u00001)\u0000xlogqx\u0000(1\u0000x) logq(1\u0000x): (II.1)\nTheq-ary entropy function is important here because it helps us to count the size of sets\nwithqsymbols. The base- qlogarithm of the number of vectors from Fn\nqthat di\u000ber in at\nmostxncomponents from the zero-vector is dominated by nHq(x) asnbecomes large.\nFor a ground set \n and n-tuples x2\nn, de\fnexjto bej-th element of the n-tuple\nx. Given tuples x2\nnandy2\nm, de\fne the pasting of the tuples xandyto be\n(xjy):= (x1;:::;xn;y1;:::;ym). WhenM1andM2are matrices with the same number\nof columns, de\fne ( M1;M2):=0\n@M1\nM21\nA:For positive integer `, de\fne [`]:=f1;:::;`g.\nDe\fne the Hamming distance dH(x;y) between x2\nnandy2\nnas the number of\nindices on which xandydi\u000ber. De\fne the minimum distance of any subset C\u001a\nnto be\nmindist(C):= min x;y2CfdH(x;y) :x6=yg:\n3A code over a vector \feld Fn\nqisq-ary linear code of length nif it is a subspace of Fn\nq.\nAn additive code is a subgroup of the \feld under the \feld addition operation. A classical\nq-ary linear code [16] of block length nandkgenerators with minimum distance of dis said\nto be an [n;k]qcode or an [ n;k;d ]qcode. A classical [ n;k;d ]qcode is maximally distance\nseparated (MDS) if d=n\u0000k+ 1. A quantum q-ary stabilizer code [2] of block length n\nencodingkqudits is said to be an Jn;k Kqcode. The rates of an Jn;k Kqcode and an [ n;k]q\ncode are both de\fned to bek\nn.\nA. Finite Fields and q-ary Error Bases\nWe brie\ry review q-ary error bases [5]. Given a prime number p, letq=pkwherekis a\npositive integer. Let generalizations of the qubit Pauli matrices be\nX:=p\u00001X\nj=0j(j+ 1) modpihjj\nZ:=p\u00001X\nj=0(!p)jjjihjj (II.2)\nwhich satisfy the commutation property XaZb= (!p)abZbXafor non-negative integers a\nandb. We de\fne the matrix\nXaZb:=Xa1Zb1\n:::\nXakZbk (II.3)\nas a single qudit q-ary error basis element. We de\fne a q-ary error basis on a single qudit\nas the setEq:=fXaZb:a;b2Zk\npg:Aq-ary error basis on nqudits is de\fned as E\nn\nqand\nits basis elements have the form\nXa(1)Zb(1)\n:::\nXa(n)Zb(n)=X(a(1)j:::ja(n))Z(b(1)j:::jb(n)):\nNow lettbe any positive integer. Observe that for a;b;c;d2Zt\np, the matrices XaZband\nXcZdsatisfy the commutation relation\n(XaZb)(XcZd) = (XcZd)(XaZb)(!p)Pt\ni=1aidi\u0000bici:\nHence the symplectic scalar product\nh(ajb);(cjd)is:=tX\ni=1aidi\u0000bici=adT\u0000bcT\n4quanti\fes the commutation relation between the matrices XaZbandXcZd. When this scalar\nproduct is zero, we say that the vectors ( ajb) and ( cjd) ares-orthogonal, and the matrices\nXaZbandXcZdcommute under matrix mutiplication.\nWe now elucidate the connection between q-ary error bases and \fnite \felds. De\fne\nthe trace function from the \feld FqtoFpto be Tr : x7!Pk\u00001\ni=0xpi. Also letf\r;\rqgbe\na basis of Fq2over Fq, where\rand\rqare the distinct roots of an irreducible degree-2\npolynomial over Fq. Now let a:= (\u000b1;:::;\u000bk) and b:= (\f1;:::;\fk) be dual bases of Fq\nso that aTbis a sizekidentity matrix. Also let a;b;c, and dbe vectors from Zk\np. Then\nTr((aaT)(bbT)) = Tr( aaTbbT)) =abT;which implies that\nTr((aaT)(dbT)\u0000(baT)(cbT)) =adT\u0000bcT: (II.4)\nGiven the vectors xandyinFn\nq2, the Hermitian scalar product (see (28) of [5]) between x\nandyis\nhx;yih:=nX\ni=1(xi)qyi:\nWhen this Hermitian scalar product is zero, we say that xandyareh-orthogonal. This\nscalar product is called Hermitian because taking an element of Fq2to theq-th power is\nanalogous to conjugation over the complex \feld. For any subset C\u001aFn\nq2, we also de\fne its\nHermitian dual to be C?h:=fy2Fn\nq2:hx;yih= 0;x2Cg.\nThe following proposition shows that if two error basis elements are to commute, it su\u000eces\nfor theirq2-ary \fnite \feld counterparts to be h-orthogonal.\nProposition II.1 ([5]).Letx;y2Fn\nq2, and suppose that hx;yih= 0. For alli2[n], letxi\nandyihave the decompositions\nxi=xi;1\r+xi;2\rq=a(i)aT\r+b(i)bT\rq;\nyi=yi;1\r+yi;2\rq=c(i)aT\r+d(i)bT\rq;\nwherexi;1;xi;2;yi;1;yi;22Fqand a(i);b(i);c(i);d(i)2Zk\np. Then the matrices\nX(a(1)j:::ja(n))Z(b(1)j:::jb(n))andX(c(1)j:::jc(n))Z(d(1)j:::jd(n))from the setE\nn\nqcommute under matrix\nmultiplication.\nProof. Sincehx;yih= (hy;xih)qand 0q= 0, we havehx;yih= 0 implying that hy;xih= 0.\n5W2E\nn\nq\nw2F2n\nq~w2Fn\nq2~''\n \nFIG. 1: Equivalent representations of an n-qudit q-ary error basis element.\nThushx;yih\u0000hy;xih= 0, which implies thatPn\ni=1xq\niyi\u0000yq\nixi= 0. Hence\n0 =nX\ni=1((xi;1\rq+xi;2\r)(yi;1\r+yi;2\rq)\u0000(yi;1\rq+yi;2\r)(xi;1\r+xi;2\rq))\n= (\r\u0000\r2)nX\ni=1(xi;1yi;2\u0000xi;2yi;1): (II.5)\nIf\r=\r2, then\r=\rqwhich is a contradiction. Hence \r6=\r2which implies that\nnX\ni=1(xi;1yi;2\u0000xi;2yi;1) = 0:\nLeta= (a(1)j:::ja(n)),b= (b(1)j:::jb(n)),c= (c(1)j:::jc(n)), and d= (d(1)j:::jd(n)).\nTracing both sides of the above equation gives h(ajb);(cjd)is= 0, which implies that the\nmatricesXaZbandXcZdcommute.\nIn view of Proposition II.1 and (II.4), each element of a q-ary error basis over nqubits\nW=X(a(1)j:::ja(n))Z(b(1)j:::jb(n))can be represented by the codewords '(W):=w2F2n\nqand\n~'(W):=~w2Fn\nq2, where for i2[n],\nwi=a(i)aT;\nwi+n=b(i)bT;\n~wi=wi\r+wi+n\rq:\nWe de\fne the map  to take wto~w. Let the maps  ;' and ~'act component-wise on\nsets and matrices. Consequently, elements of an error basis can be studied in their di\u000berent\n\fnite \feld representations, with the bijective maps ';~'and depicted in Figure 1.\n6B. Stabilizer Codes\nGiven a prime number p, letq=pkwherekis a positive integer. Given a subset\nS\u001a E\nn\nqwhere'(S) is an additive group with sindependent additive generators, the\nmaximal subspace of ( Cq)\nnleft invariant under the action of all elements of Sis called an\nJn;n\u0000s\nkKqstabilizer code. The sets S,'(S) and ~'(S) are the stabilizers of our stabilizer code\nin the matrix representation, the F2n\nq-representation and the Fn\nq2-representation respectively.\nWe study stabilizer codes in the language of \fnite \felds [3, 5].\nConsider the full rank generator matrix G= (Gstb;Gx;Gz) over Fqwith (2kn\u0000s) rows\nand 2ncolumns where the stabilizer generator Gstb= (s(1);:::;s(s)), the logical-X generator\nGx= (x(1);:::;x(kn\u0000s)), and the logical-Z generator Gz= (z(1);:::;z(kn\u0000s)) are submatrices\nof G. We also require G= (Gstb;Gx;Gz) to have the properties:\n1. Each row of Gstbiss-orthogonal to every row of G.\n2. For alli;j2[kn\u0000s],hx(i);z(i)is=\u000ei;j, where\u000ei;jis the Kronecker delta.\nThe error basis elements corresponding to the rows of GxandGzare generators for logical\noperations that can be applied on the stabilizer code.\nWe denote the additive (not necessarily linear) classical codes generated by GstbandG\nunder \feld addition by CstbandCnrmrespectively. The set of all elements in F2n\nqthat are\ns-orthogonal to all elements in CstbisCnrm. The minimum distance of our stabilizer code\nis the minimum distance of the punctured code ~Cpnc:=fx2 (Cnrm) :x =2 (Cstb)g[5].\nWe denote an Jn;n\u0000s\nkKqstabilizer code with distance dasJn;n\u0000s\nk;dKq. The rate of the\nstabilizer code is 1 \u0000s\nknand its relative distance isd\nn.\nWe de\fne a random Jn;n\u0000s\nkKqstabilizer code to be a stabilizer code corresponding to a\ngenerator matrix G= (Gstb;Gx;Gz) chosen uniformly at random from all possible generator\nmatrices with (2 kn\u0000s) rows and 2 ncolumns over the vector \feld F2n\nq.\nLet the rates and relative distances of an in\fnite code sequence of fJn;nrn;n\u000enKqgn\nconverge to the positive numbers rand\u000erespectively. If\n\u000e\u0015H\u00001\nq2\u00121\u0000r\n2\u0013\n; (II.6)\nwe say that the code sequence attains the asymptotic quantum q-ary GV bound.\n7C. Concatenation of Stabilizer Codes\nConcatenation makes a longer code from an appropriately chosen set of shorter codes.\nWe consider only the concatenation of stabilizer codes. Let q=pkwherepis prime.\nThe quantum message that we wish to encode into a concatenated quantum code is\naqK-dimension quantum state which is \frst encoded into an JN;K Kqouter code . Let\nour JN;K Kqouter code be generated by G(out)= (G(out)\nstb;G(out)\nx;G(out)\nz). The outer code\ncomprises of Nblocks of dimension qcomplex Euclidean spaces, with each of these N\nblocks further encoded as an Jn;k Kpinner code . Let thej-th Jn;k Kpinner code be generated\nbyG(j)= (G(j)\nstb;G(j)\nx;G(j)\nz), withG(j)\nx= (x(j);1;:::;x(j);k) andG(j)\nz= (z(j);1;:::;z(j);k) for\nj2[N]. The resultant code is a concatenated code with parameters JnN;kK Kpgenerated\nbyG(concat)= (G(concat)\nstb ;G(concat)\nx ;G(concat)\nz ).\nWe now elucidate the construction of the generator of the concatenated code G(concat)\nusing the generator of the outer code G(out)and the generators of the inner codes G(j)for\nj2[N].\nUsing the notation de\fned in Section II A, let the letter w2Fq2have the decomposition\nw=aaT\r+bbT\rqwhere a;b2Zk\np. We de\fne the image of wover the smaller \feld Fp2\nwith respect to the j-th inner code to be the  (C(j)\nstb)-coset representative given by\n\u0019(j)(w):=kX\n`=1\u0000\na`x(j);`+b`z(j);`\u0001\n: (II.7)\nGiven vectors s2[N]mandw2Fm\nq2, we de\fne \u0019s(w):= (\u0019(s1)(w1)j:::j\u0019(sm)(wm)):As a\nshorthand we de\fne \u0019:=\u0019(1;:::;N ). Let\u0019also act component-wise on both matrices and\nsets. Then the Fp2-representations of the stabilizer generator, the X-logical generator and\nthe Z-logical generator of our concatenated code are given by\n (G(concat)\nstb ) =0\nBBBBBB@\u0019( (G(out)\nstb));0\nBBBBBB@ (G(1)\nstb)0 0 0\n0 (G(2)\nstb)0 0\n0 0...0\n0 0 0  (G(N)\nstb)1\nCCCCCCA1\nCCCCCCA\n (G(concat)\nx ) =\u0019( (G(out)\nx));  (G(concat)\nz ) =\u0019( (G(out)\nz)) (II.8)\n8respectively. The Fp2-representations of the stabilizer and the normalizer of the\nconcatenated code are  (C(concat)\nstb ):=\u0019( (C(out)\nstb)) + (C(1)\nstb\u0002::::\u0002C(N)\nstb) and\n (C(concat)\nnrm ):=\u0019( (C(out)\nnrm)) + (C(concat)\nstb ) respectively.\nIn this paper, we use some of the q-ary quantum codes of Li, Xing and Wang [20] as the\nouter codes of our concatenated codes. The stabilizers and normalizers of these codes are\nclassical MDS codes in the Fq2-representation, which is not necessarily the case for other\nquantum codes [19].\nTheorem II.2 (Li, Xing, Wang [20] ) .LetNbe a prime power and Kbe an even integer\nin[0;N]such thatN\u0000K\n2is also an integer. Then there exists a quantum generalized Reed-\nSolomon code with parameters JN;K;N\u0000K\n2+ 1 KN. Moreover, the stabilizer  (Cstb)and\nnormalizer  (Cnrm)of this code in the FN2-representation are classical generalized Reed-\nSolomon codes (are hence classical MDS codes), with  (Cnrm) = (Cstb)?h.\nIII. THE MAIN RESULT\nOur main result is that our sequence of concatenated p-ary quantum codes asymptotically\nattains the quantum GV bound. The outer code is a quantum generalized RS code with\n (Cnrm) = (Cstb)?hgiven by [20], and the inner codes are independently chosen random\nstabilizer codes. Theorem III.1 is our main result.\nTheorem III.1. Letr;R2Q\\[0;1]be the rates of the inner and outer code respectively.\nLetpbe a prime number and nbe a positive integer such that rn,N=prn, and1\u0000R\n2N2Z\nare also integers. Also suppose that\nR< min\b\n1\u00002Hp2(1\u0000pr\u00001);1\t\n: (III.1)\nLet JnN;rRnN;d Kpbe a concatenated quantum code with a JN;RN KNouter code of given by\nTheorem II.2 concatenated with Nindependent and identically distributed random Jn;rn Kp\ninner quantum codes. Then with probability at least 1\u00001\np2\u00001p\u00002N(1\u0000R\n2),\nd\nnN>H\u00001\np2\u00121\u0000rR\n2\u0013\n\u00003c(p2;1+r\n2)\n2n\nwherec(p2;1+r\n2)is a continuity constant as de\fned in the Appendix in equation (IV.4).\n9Corollary III.2. Letpbe a prime and r;R2[0;1]such that the inequality (III.1) holds.\nFor all positive integers n, letkn=dnre,Nn=pknandKn=Nn\u00002d1\u0000R\n2e. LetCnbe a\ncode formed by concatenating an JNn;KnKNnouter code given by Theorem II.2 with Nn\nindependent and identically distributed random Jn;knKpstabilizer codes. Then the code\nsequencefCngn2Z+asymptotically attains the quantum Gilbert-Varshamov bound.\nrFeasible Region: 0 ≤R < 1−2H4(1−2r−1)R\n0.70.750.80.850.90.95 100.20.40.60.81\nFIG. 2: When p= 2, the shaded region depicts the rates randRfor which Theorem III.1 applies.\nWe proceed to introduce Proposition III.3 and Lemma III.4, which are used in the random\ncoding aspects of the proof of Theorem III.1.\nProposition III.3. Letwbe any nonzero element of Fn\np2. Let (Cnrm)and (Cstb)\nbe the normalizer and stabilizer over Fp2of a random Jn;k Kpstabilizer code, and\n10let the corresponding punctured code be ~Cpnc:=fw2 (Cnrm) :w=2 (Cstb)g. Then\nPr[w2~Cpnc]<p\u0000(n+k).\nProof. LetU\u001aF2n\npbe a set of independent mutually s-orthogonal vectors. Then the\nnumber of vectors in F2n\npthat ares-orthogonal to all elements of Uisp2n\u0000jUj. Hence\nPr[w2 (Cnrm)] =Qn\u0000k\u00001\ni=0(p2n\u00001\u0000i\u0000pi)Qn\u0000k\u00001\ni=0(p2n\u0000i\u0000pi)<p\u0000(n\u0000k):The number of cosets of CstbinCnrmdistinct\nfromCstbisp2k\u00001. Hence Pr[ w2~Cpnc]<p\u0000(n\u0000k)1\np2k\u00001<pn+k.\nLemma III.4. LetWbe any nonzero vector in FN\nq2of weightw, andhbe a positive integer\nno greater thanp2\u00001\np2nw. LetS=\u0019(W) + (C(1)\nstb\u0002:::\u0002C(N)\nstb)be a random coset. Then\nPr [minwt(S)\u0014h]<(p2)nwHp2(h\nnw)p\u0000(n+k)w:\nProof. The minimum weight of Sis equal to the minimum weight of the random coset\nS0=\u0019((W1;:::;Ww))+ (C(1)\nstb\u0002:::\u0002C(w)\nstb), whereW1;:::;Wware the nonzero letters of W.\nWhenh\u0014p2\u00001\np2nw, there are at most ( p2)nwHp2(h\nnw)members of Fn\np2of weight no more than\nh(see [16]). Let w= (v1j:::jvw) be any such member of Fnw\np2, where v1;:::;vw2Fn\np2. If\nwis also an element of S0, each viis necessarily an element of the non-trivial random coset\n\u0019(i)(Wi) + (C(i)\nstb), the probability of which is less than p\u0000(n+k)by the Proposition III.3.\nHence the probability that wis an element of the random set S0is less than p\u0000(n+k)w.\nSubsequently, applying the union bound on the number of wwith a weight no more than h\ngives the result.\nNow we proceed to prove our main result, Theorem III.1.\nProof of Theorem III.1. To prove our main result, we have to \fnd a designed distance h>0\nsuch that:\n1. The probability that the distance of our concatenated quantum code is less that his\nnegligible.\n2. The designed relative distanceh\nnNasymptotically attains the quantum GV bound.\nWe \frst determine a su\u000ecient condition for Pr[ d\u0014h] to vanish as nbecomes large.\nNow our outer code's normalizer  (C(out)\nnrm) is a classical MDS code [20] with parameters\n[N;NR nrm;D]q2whereD=N(1\u0000Rnrm) + 1 andRnrm:=1+R\n2. The MDS property of\n11our outer code's normalizer implies that the spectrum of the normalizer Aw, de\fned as the\nnumber of codewords in  (C(out)\nnrm) with weight w2[D;N ], is at most\u0012N\nw\u0013\n(p2k)w\u0000D+1(see\nthe references [1, 16]). Let ~C(concat)\npnc :=fW2 (C(concat)\nnrm ) :W=2 (C(concat)\nstb )g. Our upper\nbound on the spectrum Aw, the union bound and Lemma III.4 imply that\nPr[d\u0014h] = Pr[minwt( ~C(concat)\npnc )\u0014h]\n\u0014X\nW2 (C(out)\nnrm )\nW6=0Prh\nminwt(\u0019(W) + (C(1)\nstb)\u0002:::\u0002C(N)\nstb))\u0014hi\n<NX\nw=D2N(p2k)w\u0000D0+1(p2)nwHp2(h\nnw)\u0000n+k\n2w\u00141X\nw=D(p2)\u0000nw\u0011;\nwhere\n\u0011:=\u0000N\n2nw\u0000r\u0012\n1\u0000D\nw+1\nw\u0013\n\u0000Hp2\u0012h\nnw\u0013\n+1 +r\n2: (III.2)\nNow let\u0012= 1\u0000D\nw+1\nwand observe that 0 \u0014\u0012<R nrmfor our feasible values of w. If\u0011\u00151\nn\nfor allw2[D;N ], then Pr[d\u0014h]\u0014(p2)\u0000D 1\n1\u0000p\u00002:We will determine feasible values of the\ndesigned distance hfor which the inequality \u0011\u00151\nnholds.\nSince the inverse entropy function is monotone increasing on the open unit interval, it\nsu\u000eces to require that our choice of hsatis\fes the inequality\nh\nnN\u0014w\nNH\u00001\np2\u00121 +r\n2\u0000r\u0012\u0000N\n2nw\u00001\nn\u0013\n: (III.3)\nIt su\u000eces to haveh\nnNequal to some lower bound on the right hand side of the inequality\n(III.3). Continuity of the inverse entropy (Lemma IV.1) and the substitutionw\nN=1\u0000Rnrm\n1\u0000\u0012\ngives\n1\u0000Rnrm\n1\u0000\u0012H\u00001\np2\u00121 +r\n2\u0000r\u0012\u00001\nn\u0012N\n2w+ 1\u0013\u0013\n\u00151\u0000Rnrm\n1\u0000\u0012H\u00001\np2\u00121 +r\n2\u0000r\u0012\u0013\n\u0000\u00121\n2+w\nN\u0013c(p2;1+r\n2\u0000r\u0012)\nn: (III.4)\nThe inequality (III.1) together with our restriction that r;R2[0;1] imply that r\nandRsatisfy the requirements of Lemma IV.1. Hence Lemma IV.1 implies that\n1\u0000Rnrm\n1\u0000\u0012H\u00001\np2\u00001+r\n2\u0000r\u0012\u0001\nis a monotonic non-increasing function of \u0012. Sincec(p2;1+r\n2\u0000r\u0012)\nnis also\na monotonic non-increasing function of \u0012for feasible values of randR, the right hand side\nof (III.4) is at least H\u00001\np2\u00001\u0000rR\n2\u0001\n\u00003c(p2;1+r\n2)\n2nby setting\u0012to beRnrm. We seth\nnNto be this\nlower bound so that the inequality (III.3) holds, from which the result follows.\n12IV. APPENDIX : THE Q-ARY ENTROPY AND ITS INVERSE\nIn this section, we derive properties of the q-ary entropy function and its inverse. Since\nHqis a strictly increasing concave function on (0 ;q\u00001\nq),H\u00001\nqis a strictly increasing convex\nfunction on the open interval (0 ;1). Observe that for x2(0;1),\nH0\nq(x):=d\ndxHq(x) = logq(q\u00001)\u0000logqx+ logq(1\u0000x); (IV.1)\n(1\u0000x)H0\nq(1\u0000x) =Hq(1\u0000x) + logqx: (IV.2)\nSinceHq(y) is a continuously di\u000berentiable function for y2(0;1\u00001\nq), by the inverse function\ntheorem, we have that\n(H\u00001\nq)0(y) =1\nH0\nq(H\u00001\nq(y))(IV.3)\nfory2(0;1), where ( H\u00001\nq)0(y):=d\ndyH\u00001\nq(y). These technical properties of the q-ary\nentropy function are used to obtain Lemma IV.1 which pertains to the monotonicity of\n1\n1\u0000\u0012H\u00001\nq\u00001+r\n2\u0000r\u0012\u0001\nwith respect to \u0012, and Lemma IV.2 which is about continuity.\nNow de\fne f:= 1\u0000H\u00001\nq(1+r\n2\u0000r\u0012). Observe thatdf\nd\u0012=r(H\u00001\nq)0(1+r\n2\u0000r\u0012) =\nr\nH0q(H\u00001\nq(1+r\n2\u0000r\u0012))=r\nH0q(1\u0000f). We now introduce Lemma IV.1 which makes an assertion on\nthe monotonicity of the function1\u0000f\n1\u0000\u0012.\nLemma IV.1. [Monotonicity] Let pbe prime,q=p2, andr;R2[0;1]such that (III.1)\nholds. Then1\u0000f\n1\u0000\u0012is a non-increasing function with respect to \u00122[0;1+R\n2].\nProof. Nowd\nd\u00121\u0000f\n1\u0000\u0012=1\u0000f\n(1\u0000\u0012)2\u00001\n1\u0000\u0012df\nd\u0012anddf\nd\u0012=r\nH0q(1\u0000f). Henced\nd\u00121\u0000f\n1\u0000\u0012\u00140 if and only if\n(1\u0000f)H0\nq(1\u0000f)\u0014r(1\u0000\u0012). From (IV.2), we get\n(1\u0000f)H0\nq(1\u0000f) =Hq(1\u0000f) + logqf=\u00121 +r\n2\u0000r\u0012\u0013\n+ logqf:\nThus (1\u0000f)H0\nq(1\u0000f)\u0014r(1\u0000\u0012) holds if and only if r\u00151 + 2 logqf, the latter inequality\nof which holds because of (III.1).\nLemma IV.2. [Continuity] Let x;y2(0;q\u00001\nq)where the integer qis greater than 2 and\nx>y . ThenH\u00001\nq(y)\u0015H\u00001\nq(y)\u0000(x\u0000y)c(q;x);where our continuity constant is\nc(q;x):=\u0012\nlogq(q\u00001) + logq\u00121\nH\u00001\nq(x)\u00001\u0013\u0013\u00001\n: (IV.4)\n13Proof. The convexity and continuous di\u000berentiability of H\u00001\nqon the unit open interval imply\nthatH\u00001\nq(y)\u0015H\u00001\nq(x)\u0000(x\u0000y)(H\u00001\nq)0(x):Use of (IV.1) with (IV.3) then gives the result.\n[1] C. Thommesen, \\The existence of binary linear concatenated codes with Reed-Solomon outer\ncodes which asymptotically meet the Gilbert-Varshamov bound,\" IEEE Transactions on\nInformation Theory , vol. 29, no. 6, pp. 850{853, 1983.\n[2] E. Rains, \\Nonbinary quantum codes,\" IEEE Transactions on Information Theory , vol. 45,\npp. 1827 {1832, Sep 1999.\n[3] D. Gottesman, Stabilizer Codes and Quantum Error Correction . PhD thesis, California\nInstitute of Technology, 1997. quant-ph/9705052.\n[4] A. Ashikhmin, A. Barg, E. Knill, and S. Litsyn, \\Quantum error detection .II. bounds,\" IEEE\nTransactions on Information Theory , vol. 46, pp. 789 {800, May 2000.\n[5] A. Ashikhmin and E. Knill, \\Nonbinary quantum stabilizer codes,\" IEEE Transactions on\nInformation Theory , vol. 47, pp. 3065 {3072, Nov 2001.\n[6] K. Feng and Z. Ma, \\A \fnite Gilbert-Varshamov bound for pure stabilizer quantum codes,\"\nIEEE Transactions on Information Theory , vol. 50, no. 12, pp. 3323{3325, 2004.\n[7] Y. Ma, \\The asymptotic probability distribution of the relative distance of additive quantum\ncodes,\" Journal of Mathematical Analysis and Applications , vol. 340, pp. 550{557, 2008.\n[8] L. Jin and C. Xing, \\Quantum Gilbert-Varshamov bound through symplectic self-orthogonal\ncodes,\" in IEEE International Symposium on Information Theory Proceedings (ISIT) , pp. 455\n{458, Aug 2011.\n[9] A. Ashikhmin, S. Litsyn, and M. A. Tsfasman, \\Asymptotically good quantum codes,\" Phys.\nRev. A , vol. 63, p. 032311, Feb 2001.\n[10] R. Matsumoto, \\Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes,\"\nIEEE Transactions on Information Theory , vol. 48, pp. 2122 {2124, Jul 2002.\n[11] H. Chen, S. Ling, and C. Xing, \\Asymptotically good quantum codes exceeding the\nAshikhmin-Litsyn-Tsfasman bound,\" IEEE Transactions on Information Theory , vol. 47,\npp. 2055 {2058, Jul 2001.\n14[12] H. Fujita, \\Several classes of concatenated quantum codes: Constructions and bounds,\"\nIEIC Technical Report (Institute of Electronics, Information and Communication Engineers) ,\nvol. 105, no. 662, pp. 195{200, 2006.\n[13] M. Hamada, \\Concatenated quantum codes constructible in polynomial time: E\u000ecient\ndecoding and error correction,\" IEEE Transactions on Information Theory , vol. 54, pp. 5689\n{5704, Dec 2008.\n[14] Z. Li, L. Xing, and X. Wang, \\A family of asymptotically good quantum codes based on code\nconcatenation,\" IEEE Transactions on Information Theory , vol. 55, pp. 3821 {3824, Aug\n2009.\n[15] A. Niehage, \\Nonbinary quantum Goppa codes exceeding the quantum Gilbert-Varshamov\nbound,\" Quantum Information Processing , vol. 6, no. 3, pp. 143{158, 2007.\n[16] F. J. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes . North-Holland\npublishing company, \frst ed., 1977.\n[17] J. Justesen, \\Class of constructive asymptotically good algebraic codes,\" IEEE Transactions\non Information Theory , vol. 18, pp. 652{656, Sep 1972.\n[18] M. Grassl, W. Geiselmann, and T. Beth, \\Quantum Reed-Solomon codes,\" Proceedings\nApplied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-13), Springer\nLecture Notes in Computer Science , p. 1719, 1999.\n[19] M. Grassl, T. Beth, and M. Roetteler, \\On optimal quantum codes,\" International Journal\nof Quantum Information , vol. 2, no. 1, pp. 55{64, 2004.\n[20] Z. Li, L.-J. Xing, and X.-M. Wang, \\Quantum generalized Reed-Solomon codes: Uni\fed\nframework for quantum maximum-distance-separable codes,\" Phys. Rev. A , vol. 77, p. 012308,\nJan 2008.\n15"    },    {        "title": "1005.0023v1.Limit_theory_for_planar_Gilbert_tessellations.pdf",        "content": "Limit theory for planar Gilbert tessellations\nTomasz Schreiber\u0003and Natalia Soja,\nFaculty of Mathematics & Computer Science,\nNicolaus Copernicus University,\nToru\u0013 n, Poland,\ne-mail: tomeks,natas at mat.umk.pl\nAbstract A Gilbert tessellation arises by letting linear segments (cracks) in R2unfold\nin time with constant speed, starting from a homogeneous Poisson point process of germs\nin randomly chosen directions. Whenever a growing edge hits an already existing one, it\nstops growing in this direction. The resulting process tessellates the plane. The purpose of\nthe present paper is to establish law of large numbers, variance asymptotics and a central\nlimit theorem for geometric functionals of such tessellations. The main tool applied is the\nstabilization theory for geometric functionals.\nkeywords Gilbert crack tessellation, stabilizing geometric functionals, central limit the-\norem, law of large numbers.\nMSC classi\fcation Primary: 60F05; Secondary: 60D05.\n1 Introduction and main results\nLetX\u0012R2be a \fnite point set. Each x2X is independently marked with a unit length\nrandom vector ^ \u000bxmaking a uniformly distributed angle \u000bx2[0;\u0019) with thex-axis, which is\nreferred to as the usual marking in the sequel. The collection \u0016X=f(x;\u000bx)gx2Xdetermines\na crack growth process (tessellation) according to the following rules. Initially, at the time\nt= 0;the growth process consists of the points (seeds) in X. Subsequently, each point\nx2X gives rise to two segments growing linearly at constant unit rate in the directions of\n^\u000bxand\u0000^\u000bxfromx:Thus, prior to any collisions, by the time t>0 the seed has developed\ninto the edge with endpoints x\u0000t^\u000bxandx+t^\u000bx;consisting of two segments, say the upper\none [x;x+t^\u000bx] and the lower one [x;x\u0000t^\u000bx]:Whenever a growing segment is blocked by\n\u0003Research supported by the Polish Minister of Science and Higher Education grant N N201 385234\n(2008-2010)\n1arXiv:1005.0023v1  [math.PR]  30 Apr 2010an existing edge, it stops growing in that direction, without a\u000becting the behaviour of the\nsecond constituent segment though. Since the possible number of collisions is bounded,\neventually we obtain a tessellation of the plane. The resulting random tessellation process\nis variously called the Gilbert model/tessellation, the crack growth process, the crack\ntessellation, and the random crack network, see e.g. [8, 12] and the references therein.\nLetG(\u0016X) denote the tessellation determined by \u0016X:We shall write \u0018+(\u0016x;\u0016X); x2X;\nfor the total length covered by the upper segment emanating from xinG(\u0016X);and likewise\nwe let\u0018\u0000(\u0016x;\u0016X) stand for the total length of the lower segment from x:Note that we\nuse \u0016xfor marked version of x;according to our general convention of putting bars over\nmarked objects. For future use we adopt the convention that if \u0016 xdoes not belong to \u0016X;\nwe extend the de\fnition of \u0018+=\u0000(\u0016x;\u0016X) by adding \u0016 xto\u0016Xand endowing it with a mark\ndrawn according to the usual rules. Observe that for some xthe values of \u0018+=\u0000may be\nin\fnite. However, in most cases in the sequel Xwill be a realization of the homogeneous\nPoisson point process P=P\u001cof intensity \u001c > 0 in growing windows of the plane. We\nshall use the so-called stabilization property of the functionals \u0018+and\u0018\u0000;as discussed in\ndetail below, to show that the construction of G(\u0016X) above can be extended to the whole\nplane yielding a well de\fned process G(\u0016P);where, as usual, \u0016Pstands for a version of P\nmarked as described above. This yields well de\fned and a.s. \fnite whole-plane functionals\n\u0018+(\u0001;\u0016P) and\u0018\u0000(\u0001;\u0016P):\nThe conceptually somewhat similar growth process whereby seeds are the realization\nof a time marked Poisson point process in an expanding window of R2and which subse-\nquently grow radially in all directions until meeting another such growing seed, has received\nconsiderable attention [1, 3, 4, 5, 6, 11, 16], where it has been shown that the number of\nseeds satis\fes a law of large numbers and central limit theorem as the window size in-\ncreases. In this paper we wish to prove analogous limit results for natural functionals\n(total edge length, sum of power-weighted edge lengths, number of cracks with lengths ex-\nceeding a given threshold etc.) of the crack tessellation process de\fned by Poisson points\nin expanding windows of R2:We will formulate this theory in terms of random measures\nkeeping track not only the cumulative values of the afore-mentioned functionals but also\ntheir spatial pro\fles.\nAnother interesting class of model bearing conceptual resemblance to Gilbert tessella-\ntions are the so-called lilypond models which have recently attracted considerable attention\n[2, 7, 9, 10] and where the entire (rather than just directional) growth is blocked upon a\ncollision of a growing object (a ball, a segment etc.) with another one.\nTo proceed, consider a function \u001e: [R+[f+1g]2!Rwith at most polynomial growth,\ni.e. for some 0 <q< +1\n\u001e(r1;r2) =O((r1+r2)q): (1)\nWithQ\u0015:= [0;p\n\u0015]2standing for the square of area \u0015inR2;we consider the empirical\nmeasure\n\u0016\u001e\n\u0015:=X\nx2P\\Q\u0015\u001e\u0000\n\u0018+(\u0016x;\u0016P);\u0018\u0000(\u0016x;\u0016P)\u0001\n\u000ex=p\n\u0015: (2)\n2Thus,\u0016\u001e\n\u0015is a random (signed) measure on [0 ;1]2for all\u0015>0:The large\u0015asymptotics of\nthese measures is the principal object of study in this paper. Recalling that \u001cstands for\nthe intensity ofP=P\u001c;we de\fne\nE(\u001c) :=E\u001e\u0000\n\u0018+(\u00160;\u0016P);\u0018\u0000(\u00160;\u0016P)\u0001\n: (3)\nThe \frst main result of this paper is the following law of large numbers\nTheorem 1 For any continuous function f: [0;1]2!Rwe have\nlim\n\u0015!11\n\u0015Z\n[0;1]2fd\u0016\u001e\n\u0015=\u001cE(\u001c)Z\n[0;1]2f(x)dx\ninLp; p>1:\nNote that this theorem can be interpreted as stating that E(\u001c) is the asymptotic mass per\npoint in\u0016\u001e\n\u0015;since the expected cardinality of P\\Q\u0015is\u001c\u0015:To characterize the second order\nasymptotics of random measures \u0016\u001e\n\u0015we consider the pair-correlation functions\nc\u001e[x] :=E\u001e2\u0000\n\u0018+(x;\u0016P);\u0018\u0000(x;\u0016P)\u0001\n; x2R2(4)\nand\nc\u001e[x;y] :=E\u001e\u0000\n\u0018+(x;\u0016P[fyg);\u0018\u0000(x;\u0016P[fyg)\u0001\n\u0001\u001e\u0000\n\u0018+(y;\u0016P[fxg);\u0018\u0000(y;\u0016P[fxg)\u0001\n\u0000[E(\u001c)]2: (5)\nIn fact, it easily follows by translation invariance that c\u001e[x] above does not depend on x\nwhereasc\u001e[x;y] only depends on y\u0000x:In terms of these functions we de\fne the asymptotic\nvariance per point\nV(\u001c) =c\u001e[\u00160] +\u001cZ\nR2c\u001e[\u00160;x]dx: (6)\nNotice that in a special case when function \u001e(\u0001;\u0001) is homogeneous of degree k(i.e. forc2R\nwe have\u001e(cr1;cr2) =ck\u001e(r1;r2)) one can simplify (3) and (6). Then the following remark\nis a direct consequence of standard scaling properties of Gilbert's tessellation construction\nand those of homogeneous Poisson point processes, whereby upon multiplying the intensity\nparameter\u001cby some factor \u001awe get all lengths in G(\u0016P) re-scaled by factor \u001a\u00001=2:\nRemark 1 For\u001e: [R+[f+1g]2!Rhomogeneous of degree kwe have\nE(\u001c) =\u001c\u0000k=2E(1)\nV(\u001c) =\u001c\u0000kV(1): (7)\nIn other words, E(\u0001)andV(\u0001)are homogeneous of degree \u0000k=2and\u0000k, respectively.\nOur second theorem gives the variance asymptotics for \u0016\u001e\n\u0015:\n3Theorem 2 The integral in (6) converges and V(\u001c)>0for all\u001c >0:Moreover, for each\ncontinuous f: [0;1]2!R\nlim\n\u0015!11\n\u0015Var\u0014Z\n[0;1]2fd\u0016\u001e\n\u0015\u0015\n=\u001cV(\u001c)Z\n[0;1]2f2(x)dx:\nOur \fnal result is the central limit theorem\nTheorem 3 For each continuous f: [0;1]2!Rthe family of random variables\n\u001a1p\n\u0015Z\n[0;1]2fd\u0016\u001e\n\u0015\u001b\n\u0015>0\nconverges in law to N\u0010\n0;\u001cV(\u001c)R\n[0;1]2f2(x)dx\u0011\nas\u0015!1:Even more, we have\nsup\nt2R\f\f\f\f\f\f\f\fP8\n>><\n>>:R\n[0;1]2fd\u0016\u001e\n\u0015r\nVarhR\n[0;1]2fd\u0016\u001e\n\u0015i6t9\n>>=\n>>;\u0000\b(t)\f\f\f\f\f\f\f\f6C(log\u0015)6\np\n\u0015(8)\nfor all\u0015>1, whereCis a \fnite constant.\nPrincipal examples of functional \u001ewhere the above theory applies are\n1.\u001e(l1;l2) =l1+l2:Then the total mass of \u0016\u001e\n\u0015coincides with the total length of edges\nemitted in G(\u0016P) by points inP\\Q\u0015:Clearly, the so-de\fned \u001eis homogeneous of\norder 1 and thus Remark 1 applies.\n2. More generally, \u001e(l1;l2) = (l1+l2)\u000b; \u000b\u00150:Again, the total mass of \u0016\u001e\n\u0015is seen\nhere to be the sum of power-weighted lengths of edges emitted in G(\u0016P) by points in\nP\\Q\u0015:The so-de\fned \u001eis homogeneous of order \u000b:\n3.\u001e(l1;l2) =1fl1+l2\u0015\u0012g;where\u0012is some \fxed threshold parameter. In this set-up, the\ntotal mass of \u0016\u001e\n\u0015is the number of edges in G(\u0016P) emitted from points in P\\Q\u0015and\nof lengths exceeding threshold \u0012:This is not a homogeneous functional.\nThe main tool used in our argument below is the concept of stabilization expressing in\ngeometric terms the property of rapid decay of dependencies enjoyed by the functionals\nconsidered. The formal de\fnition of this notion and the proof that it holds for Gilbert\ntessellations are given in Section 2 below. Next, in Section 3 the proofs of our Theorems\n1, 2 and 3 are given.\n42 Stabilization property for Gilbert tessellations\n2.1 Concept of stabilization\nConsider a generic real-valued translation-invariant geometric functional \u0018de\fned on pairs\n(x;X) for \fnite point con\fgurations X\u001aR2and withx2X:For notational convenience\nwe extend this de\fnition for x62X as well, by putting \u0018(x;X) :=\u0018(x;X[fxg) then.\nMore generally, \u0018can also depend on i.i.d. marks attached to points of X;in which case\nthe marked version of Xis denoted by \u0016X:\nFor an input i.i.d. marked point process \u0016PonR2;in this paper always taken to be\nhomogeneous Poisson of intensity \u001c;we say that the functional \u0018stabilizes atx2R2on\ninput \u0016Pi\u000b there exists an a.s. \fnite random variable R[x;\u0016P] with the property that\n\u0018(\u0016x;\u0016P\\B(x;R[x;\u0016P])) =\u0018(\u0016x;(\u0016P\\B(x;R[x;\u0016P]))[\u0016A) (9)\nfor each \fnite A\u001aB(x;R[x;\u0016P])c;with \u0016Astanding for its marked version and with B(x;R)\ndenoting ball of radius Rcentered at x:Note that here and henceforth we abuse the notation\nand refer to intersections of marked point sets with domains in the plane { these are to\nbe understood as consisting of those marked points whose spatial locations fall into the\ndomain considered. When (9) holds, we say that R[x;\u0016P] is a stabilization radius for \u0016Pat\nx:By translation invariance we see that if \u0018stabilizes at one point, it stabilizes at all points\nofR2;in which case we say that \u0018stabilizes on (marked) point process \u0016P:In addition, we\nsay that\u0018stabilizes exponentially on input \u0016Pwith rateC > 0 i\u000b there exists a constant\nM > 0 such that\nPfR[x;\u0016P]>rg6Me\u0000Cr(10)\nfor allx2R2andr >0:Stabilizing functionals are ubiquitous in geometric probability,\nwe refer the reader to [1, 13, 14, 15, 16, 17, 18, 19] for further details, where prominent ex-\namples are discussed including random geometric graphs (nearest neighbor graphs, sphere\nof in\ruence graphs, Delaunay graphs), random sequential packing and variants thereof,\nBoolean models and functionals thereof, as well as many others.\n2.2 Finite input Gilbert tessellations\nLetX \u001a R2be a \fnite point set in the plane. As already mentioned in the intro-\nduction, each x2X is independently marked with a unit length random vector ^ \u000bx=\n[cos(\u000bx);sin(\u000bx)] making a uniformly distributed angle \u000bx2[0;\u0019) with thex-axis and the\nso marked con\fguration is denoted by \u0016X:In order to formally de\fne the Gilbert tessellation\nG(\u0016X) as already informally presented above, we consider an auxiliary partial tessellation\nmappingG(\u0016X) :R+!F (R2) whereF(R2) is the space of closed sets in R2and where,\nroughly speaking, G(\u0016X)(t) is to be interpreted as the portion of tessellation G(\u0016X);identi-\n\fed with the set of its edges, constructed by the time tin the course of the construction\nsketched above.\n5Figure 1\nFinite input Gilbert tessellation.\nWe proceed as follows. For each \u0016 x= (x;\u000bx)2\u0016Xat the time moment 0 the point\nxemits in directions ^ \u000bxand\u0000^\u000bxtwo segments, referred to as the \u0016 x+- and \u0016x\u0000-branches\nrespectively. Each branch keeps growing with constant rate 1 in its \fxed direction until\nit meets on its way another branch already present, in which case we say it gets blocked ,\nand it stops growing thereupon. The moment when this happen is called the collision\ntime. Fort>0 byG(\u0016X)(t) we denote the union of all branches as grown by the time t:\nNote that, withX=fx1;::: ;xmg;the overall number of collisions admits a trivial bound\ngiven by the number of all intersection points of the family of straight lines ffxj+s^\u000bjs2\nRg;j= 1;2;::: ;mgwhich ism(m\u00001)=2. Thus, eventually there are no more collisions\nand all growth unfolds linearly. It is clear from the de\fnition that G(\u0016X)(s)\u001aG(\u0016X)(t) for\ns < t: The limit set G(\u0016X)(+1) =S\nt2R+G(\u0016X)(t) is denoted by G(\u0016X) and referred to as\ntheGilbert tessellation . Obviously, since the number of collisions is \fnite, the so-de\fned\nG(\u0016X) is a closed set arising as a \fnite union of (possibly in\fnite) linear segments. For\n\u0016x2\u0016Xby\u0018+(\u0016x;\u0016X) we denote the length of the upper branch \u0016 x+emanating from xand,\nlikewise, we write \u0018\u0000(\u0016x;\u0016X) for the length of the corresponding lower branch.\nFor future reference it is convenient to consider for each x2X thebranch history\nfunctions \u0016x+(\u0001);\u0016x\u0000(\u0001) de\fned by requiring that \u0016 x+=\u0000(t) be the growth tip of the respective\nbranch \u0016x+=\u0000at the time t2R+:Thus, prior to any collision in the system, we have just\n\u0016x+=\u0000(t) =x+=\u0000^\u000bxt;that is to say all branches grow linearly with their respective speeds\n+=\u0000^\u000bx:Next, when some \u0016 y+=\u0000; y2X gets blocked by some other \u0016 x+=\u0000; x2X at time\nt;i.e. \u0016y+=\u0000(t) = \u0016x+=\u0000(s) for somes6t;the blocked branch stops growing and its growth\ntip remains immobile ever since. Eventually, after all collisions have occured, the branches\nnot yet blocked continue growing linearly to 1:\n62.3 Stabilization for Gilbert tessellations\nWe are now in a position to argue that the functionals \u0018+and\u0018\u0000arising in Gilbert\ntessellation are exponentially stabilizing on Poisson input P=P\u001cwith i.i.d. marking\naccording to the usual rules. The following is the main theorem of this subsection.\nTheorem 4 The functionals \u0018+and\u0018\u0000stabilize exponentially on input \u0016P:\nBefore proceeding to the proof of Theorem 4 we formulate some auxiliary lemmas.\nLemma 1 LetXbe a \fnite point set in R2and \u0016Xthe marked version thereof, according\nto the usual rules. Further, let y62X:Then for any t>0we have\nG(\u0016X)(t)4G(\u0016X[f \u0016yg)(t)\u001aB(y;t)\nwith4standing for the symmetric di\u000berence.\nProof For a point setY\u001aR2andx2Y we will use the notation (\u0016 x;\u0016Y)+and (\u0016x;\u0016Y)\u0000\nto denote, respectively, the upper and lower branch outgrowing from \u0016 xinG(\u0016Y):Also, we\nuse the standard extension of this notation for branch-history functions. Note \frst that,\nby the construction of G(\u0016Y) and by the triangle inequality\n(\u0016x;\u0016Y)\"(s0)2B(y;s0))8s>s0(\u0016x;\u0016Y)\"(s)2B(y;s); s0>0;\"2f\u0000 1;+1g: (11)\nThis is a formal version of the obvious statement that, regardless of the collisions, each\nbranch grows with speed at most one throughout its entire history.\nNext, writeX0=X[fygand \u0001(t) =G(\u0016X)(t)4G(\u0016X0)(t) fort>0. Further, let\nt1< t2< t3< ::: < t nbe the joint collection of collision times for con\fgurations \u0016Xand\n\u0016X0:\nChoose arbitrary p2\u0001(t):Then there exist unique Y=Y(p)2fX;X0gandx2Yas\nwell as\"2f+;\u0000gwith the property that p= (\u0016x;\u0016Y)\"(u) for some u6t:We also write\nY0for the second element of fX;X0g;i.e.fY;Y0g=fX;X0g:With this notation, there is\na uniquei=i(p) withtimarking the collision time in Y0where the branch (\u0016 x;Y0)\"gets\nblocked inG(\u0016Y0);clearlyu>tithen and for s<tiwe have (\u0016x;\u0016Y)\"(s)=2\u0001(t).\nWe should show that p2B(y;t). We proceed inductively with respect to i:Fori= 0\nwe havex=yandY=X0:Since (\u0016y;\u0016X0)\"(0) =y2B(y;0), the observation (11) implies\nthatp= (\u0016y;\u0016X)\"(u)2B(y;u)\u001aB(y;t). Further, consider the case i >0 and assume\nwith no loss of generality that Y(p) =X, the argument in the converse case being fully\nsymmetric. The fact that p2G(\u0016X)(t)4G(\u0016X0)(t) and that p= (\u0016x;\u0016X)\"(u) implies the\nexistence of a point z2X0such that a branch emitted from zdoes block \u0016 x\"inG(\u0016X0)\n(by de\fnition necessarily at the time ti) but does not block it in G(\u0016X). In particular,\nwe see that (\u0016 z;\u0016X0)\u000e(s) = (\u0016x;\u0016X)\"(ti) and (\u0016z;\u0016X0)\u000e(s0)2\u0001(s0) for some \u000e;s;s0such that\n\u000e2f+;\u0000gands0< s6ti. By the inductive hypothesis we get (\u0016 z;\u0016X0)\u000e(s0)2B(y;s0).\nUsing again observation (11) we conclude thus that (\u0016 x;\u0016X)\"(ti) = (\u0016z;\u0016X0)\u000e(s)2B(y;s) and\nhencep= (\u0016x;\u0016X)\"(u)2B(y;u)\u001aB(y;t). This shows that p2B(y;t) as required. Since\npwas chosen arbitrary, this completes the proof of the lemma. 2\nOur second auxiliary lemma is\n7Lemma 2 For arbitrary \fnite point con\fguration X\u001aR2and\u0016x2\u0016Xwe have\n\u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016X\\B(x;2\u0018+(\u0016x;\u0016X)))\n\u0018\u0000(\u0016x;\u0016X) =\u0018\u0000(\u0016x;\u0016X\\B(x;2\u0018\u0000(\u0016x;\u0016X))): (12)\nProof We only show the \frst equality in (12), the proof of the second one being fully\nanalogous. De\fne A(\u0016X;\u0016x) = \u0016XnB(x;2\u0018+(\u0016x;\u0016X)). Clearly, A(\u0016X;\u0016x) is \fnite and we will\nproceed by induction in its cardinality.\nIfjA(\u0016X;\u0016x)j= 0, our claim is trivial. Assume now that jA(\u0016X;\u0016x)j=nfor somen>1\nand let \u0016y= (y;\u000by)2A(\u0016X;\u0016x). Putt=\u0018+(\u0016x;\u0016X) and \u0016X0=\u0016Xnf\u0016yg. Applying Lemma 1\nwe see that G(\u0016X)(t)4G(\u0016X0)(t)\u001aB(y;t). We claim that \u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016X0). Assume\nby contradiction that \u0018+(\u0016x;\u0016X)6=\u0018+(\u0016x;\u0016X0). Then for arbitrarily small \u000f > 0 we have\n(G(\u0016X)(t)4G(\u0016X0)(t))\\B(x;t+\u000f)6=;. On the other hand, since kx\u0000yk>2tasy =2B(x;2t);\nfor\"0>0 small enough we get B(x;t+\u000f0)\\B(y;t) =;. Thus, we are led to\n;6= (G(\u0016X)(t)4G(\u0016X0)(t))\\B(x;t+\"0)\u001aB(y;t)\\B(x;t+\"0) =;\nwhich is a contradiction. Consequently, we conclude that t=\u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016X0) as\nrequired. SincejA(\u0016X0;\u0016x)j=n\u00001, the inductive hypothesis yields \u0018+(\u0016x;\u0016X0) =\u0018+(\u0016x;\u0016X0\\\nB(x;2\u0018+(\u0016x;\u0016X0)) =\u0018+(\u0016x;\u0016X0\\B(\u0016x;2t)). Moreover, \u0016X0\\B(x;2t) = \u0016X\\B(x;2t). Putting\nthese together we obtain\n\u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016X0) =\u0018+(\u0016x;\u0016X0\\B(x;2t)) =\u0018+(\u0016x;\u0016X\\B(x;2t));\nwhich completes the proof. 2\nIn full analogy to Lemma 2 we obtain\nLemma 3 For a \fnite point con\fguration X\u001aR2andx2X we have\n\u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016X[ \u0016A1)and\u0018\u0000(\u0016x;\u0016X) =\u0018\u0000(\u0016x;\u0016X[ \u0016A2)\nfor arbitrary A1\u001aB(x;2\u0018+(\u0016x;\u0016X))c,A2\u001aB(x;2\u0018\u0000(\u0016x;\u0016X))c.\nCombining Lemmas 2 and 3 we conclude\nCorollary 1 Assume that \fnite marked con\fgurations \u0016Xand\u0016Ycoincide on B(x;2\u0018+(\u0016x;\u0016X)):\nThen\n\u0018+(\u0016x;\u0016X\\B(x;2\u0018+(\u0016x;\u0016X))) =\u0018+(\u0016x;\u0016X) =\u0018+(\u0016x;\u0016Y):\nAnalogous relations hold for \u0018\u0000:\nWe are now ready to proceed with the proof of Theorem 4.\n8Proof of Theorem 4 We are going to show that the functional \u0018+stabilizes expo-\nnentially on input process \u0016P:The corresponding statement for \u0018\u0000follows in full analogy.\nConsider auxiliary random variables \u0018+\n%; %> 0 given by\n\u0018+\n%=\u0018+(\u0016x;\u0016P\\B(x;%))\nwhich is clearly well de\fned in view of the a.s. \fniteness of \u0016P\\B(x;%):We claim that\nthere exist constants M;C > 0 such that for %>t>0\nP(\u0018+\n%>t)6Me\u0000Ct: (13)\nFigure 2\nIndeed, let %>0. Consider the branch \u0016 x+:= (\u0016x;\u0016P\\B(x;%))+and planar regions Biand\nDi; i>1 along the branch as represented in \fgure 2. Say that the event Eioccurs i\u000b\n\u000fthe regionBicontains exactly one point yofPand the angular mark \u000bylies within\n(\u000bx+\u0019=2\u0000\u000f;\u000bx+\u0019=2 +\u000f);\n\u000fand there are no further points of Pfalling into Di;\nwhere\u000fis chosen small enough so as to ensure that with probability one on Eithe branch\n\u0016x+does not extend past Bi;either getting blocked in Bior in an earlier stage of its growth,\nfor instance \u000f= 0:01 will do. Let pstand for the common positive value of P(Ei); i>0:\nBy standard properties of Poisson point process the events Eiare collectively independent.\nWe conclude that, for N3n6%=3\nP(\u0018+\n%>3n)6P n\\\ni=1Ec\ni!\n= (1\u0000p)n\n9which decays exponentially whence the desired relation (13) follows.\nOur next step is to de\fne a random variable R+=R+[\u0016x;P;\u0016] and to show it is a\nstabilization radius for \u0018+atxfor input process \u0016P:We shall also establish exponential\ndecay of tails of R+:For%>0 we putR+\n%= 2\u0018+\n%:Further, we set ^ %= inffm2NjR+\nm6mg:\nSince P(T\nm2NfR+\nm>mg)6infm2NP(R+\nm>m) which is 0 by (13), we readily conclude\nthat so de\fned ^ %is a.s. \fnite. Take\nR+:=R+\n^%: (14)\nThen, using that by de\fnition R+6^%;for any \fnite A\u001aB(x;R+)cwe get a.s. by Lemma\n3 and Corollary 1\n\u0018+(\u0016x;(\u0016P\\B(x;R+))[A) =\u0018+(\u0016x;\u0016P\\B(x;^%)\\B(x;2\u0018+(\u0016x;\u0016P\\B(x;^%)))[A) =\n=\u0018+(\u0016x;\u0016P\\B(x;^%)\\B(x;2\u0018+(\u0016x;\u0016P\\B(x;^%)))) =\u0018+(\u0016x;(\u0016P\\B(x;R+))):\nThus,R+is a stabilization radius for \u0018+on\u0016Pas required. Further, taking into account\nthatR+\nk=R+for allk>^%by Corollary 1, we have for m2N\nP(R+>m) =P( lim\nk!1R+\nk>m) = lim\nk!1P(R+\nk>m) =\n= lim\nk!1P(\u0018+\nk>m=2)6Me\u0000Cm=2(15)\nwhence the desired exponential stabilization follows. 2\nUsing the just proved stabilization property of \u0018+and\u0018\u0000we can now de\fne\n\u0018+(\u0016x;\u0016P) =\u0018+(\u0016x;\u0016P\\B(x;R+)) = lim\n%!1\u0018+(\u0016x;\u0016P\\B(x;%)) =R+=2 (16)\nand likewise for \u0018\u0000:Clearly, the knowledge of these in\fnite volume functionals allows us\nto de\fne the whole-plane Gilbert tessellation G(\u0016P):\n3 Completing proofs\nTheorems 1,2 and 3 are now an easy consequence of the exponential stabilization Theorem\n4. Indeed, observe \frst that, by (1), (16) and (15) the geometric functional\n\u0018(\u0016x;\u0016X) :=\u001e(\u0018+(\u0016x;\u0016X);\u0018\u0000(\u0016x;\u0016X))\nsatis\fes the p-th bounded moment condition [19, (4.6)] for all p >0:Hence, Theorem 1\nfollows by Theorem 4.1 in [19]. Further, Theorem 2 follows by Theorem 4.2 in [19]. Finally,\nTheorem 3 follows by Theorem 4.3 in [19] and Theorem 2.2 and Lemma 4.4 in [13].\nAcknowledgements Tomasz Schreiber acknowledges support from the Polish Minister\nof Science and Higher Education grant N N201 385234 (2008-2010). He also wishes to\nexpress his gratitude to J.E. Yukich for helpful and inspiring discussions.\n10References\n[1] Yu. Baryshnikov and J. E. Yukich, Gaussian limits for random measures in geometric\nprobability. Annals Appl. Prob. 15, 1A (2005), pp. 213-253.\n[2] C. Cotar and S. Volkov, A note on the Lilypond model Adv. Appl. Prob. 36(2004),\n325-339\n[3] S. N. Chiu, A central limit theorem for linear Kolmogorov's birth-growth models.\nStochastic Proc. and Applic. 66(1997), pp. 97-106.\n[4] S. N. Chiu and M. P. Quine, Central limit theory for the number of seeds in a growth\nmodel in Rdwith inhomogeneous Poisson arrivals. Annals of Appl. Prob. 7(1997),\npp. 802-814.\n[5] S. N. Chiu and M. P. Quine , Central limit theorem for germination-growth models in\nRdwith non-Poisson locations. Advances Appl. Prob. 33no. 4 (2001).\n[6] S. N. Chiu and H. Y. Lee, A regularity condition and strong limit theorems for linear\nbirth growth processes. Math. Nachr. 241(2002), pp. 21 - 27.\n[7] D.J. Daley and G. Last, Descsnding chains, the lilypond model and mutual-nearest-\nneighbour matching. Adv. Appl. Probab. 37(2005), 604-628.\n[8] N.H. Gray, J.B. Anderson, J.D. Devine and J.M. Kwasnik, Topological Properties of\nRandom Crack Networks , Mathematical Geology 8(1976), 617-626.\n[9] M. Heveling and G. Last, Existence, Uniqueness, and Algorithmics Computation of\nGeneral Lilyponcd Systems Random Structures and Algorithms 29(2006), 338-350.\n[10] O. Haeggstroem and R. Meester, Nearest neighbour and hard sphere models in con-\ntinuum percolation , Random Structures and Algorithms 9(1996), 295-315.\n[11] L. Holst, M. P. Quine and J. Robinson, A general stochastic model for nucleation and\nlinear growth. Annals Appl. Prob. 6(1996), pp. 903-921.\n[12] M.S. Makisack and R.E. Miles, Homogeneous rectangular tessellations. Adv. Appl.\nProbab. 28(1996), 993-1013.\n[13] M. D. Penrose, Gaussian limits for random geometric measures. European Journal of\nProbability 12(2007), pp. 989-1035.\n[14] M. D. Penrose, Laws of large numbers in stochastic geometry with statistical applica-\ntions. Bernoulli 13(2007), pp. 1124-1150.\n[15] M. D. Penrose and J. E. Yukich, Central limit theorems for some graphs in computa-\ntional geometry. Ann. Appl. Probab. 11(2001), pp. 1005-1041.\n11[16] M. D. Penrose and J. E. Yukich, Limit theory for random sequential packing and\ndeposition. Ann. Appl. Probab. 12(2002), pp. 272-301.\n[17] M. D. Penrose and J. E. Yukich, Weak laws of large numbers in geometric probability.\nAnn. Appl. Probab. 13(2004), pp. 277-303.\n[18] M. D. Penrose and J. E. Yukich, Normal approximation in geometric probability. In:\nStein's Method and Applications, Lecture Note Series, Institute for Mathematical\nSciences, National University of Singapore, 5, A. D. Barbour and Louis H. Y. Chen,\nEds. (2005), pp. 37-58.\n[19] T. Schreiber, Limit theorems in stochastic geometry. In: New Perspectives in Stochas-\ntic Goemetry, W.S. Kendall and I. Molchanov, Eds., Oxford University Press, 2009,\npp. 111-144.\n12"    },    {        "title": "1005.4595v3.Structural__static_and_dynamic_magnetic_properties_of_CoMnGe_thin_films_on_a_sapphire_a_plane_substrate.pdf",        "content": "Structural, static and dynamic magnetic properties of Co 2MnGe thin films on a\nsapphire a-plane substrate\nMohamed Belmeguenai1, Fatih Zighem2, Thierry Chauveau1, Damien Faurie1, Yves\nRoussigné1, Salim Mourad Chérif1, Philippe Moch1, Kurt Westerholt3and Philippe Monod4\n1LPMTM, Institut Galilée, UPR 9001 CNRS, Université Paris 13,\n99 Avenue Jean-Baptiste Clément F-93430 Villetaneuse, France\n2LLB (CEA CNRS UMR 12), Centre d’études de Saclay, 91191 Gif-Sur-Yvette, France\n3Institut für Experimentalphysik/Festkörperphysik,\nRuhr-Universität Bochum, 44780 Bochum, Germany and\n4LPEM, UPR A0005 CNRS, ESPCI, 10 Rue Vauquelin, F-75231 Paris cedex 5, France\nMagnetic properties of Co 2MnGe thin films of different thicknesses (13, 34, 55, 83, 100 and 200\nnm), grown by RF sputtering at 400\u000eC on single crystal sapphire substrates, were studied using\nvibrating sample magnetometry (VSM) and conventional or micro-strip line (MS) ferromagnetic\nresonance (FMR). Their behavior is described assuming a magnetic energy density showing twofold\nand fourfold in-plane anisotropies with some misalignment between their principal directions. For\nall the samples, the easy axis of the fourfold anisotropy is parallel to the c-axis of the substrate\nwhile the direction of the twofold anisotropy easy axis varies from sample to sample and seems to\nbe strongly influenced by the growth conditions. Its direction is most probably monitored by the\nslight unavoidable miscut angle of the Al 2O3substrate. The twofold in-plane anisotropy field Hu\nis almost temperature independent, in contrast with the fourfold field H4which is a decreasing\nfunction of the temperature. Finally, we study the frequency dependence of the observed line-width\nof the resonant mode and we conclude to a typical Gilbert damping constant á value of 0.0065 for\nthe 55-nm-thick film.\nPACS numbers:\nKeywords:\nI. Introduction\nFerromagnetic Heusler half metals with full spin po-\nlarization at the Fermi level are considered as potential\ncandidates for injecting a spin-polarized current from a\nferromagnetintoasemiconductorandfordevelopingsen-\nsitive spintronic devices [1]. Some Heusler alloys, like\nCo2MnGe, are especially promising for these applica-\ntions, due to their high Curie temperature (905 K) [2]\nand to their good lattice matching with some techno-\nlogically important semiconductors [3]. Therefore, great\nattention was recently paid to this class of Heusler alloys\n[4-10].\nIn a previous work [11], we used conventional and\nmicro-strip line (MS) ferromagnetic resonance (FMR),\nas well as Brillouin light scattering (BLS) to study\nmagnetic properties of 34-nm-, 55-nm- and 83-nm-thick\nCo2MnGe films at room temperature. We showed that\nthe in-plane anisotropy is described by the superposition\nof a twofold and of a fourfold term. The easy axes of\nthe fourfold anisotropy were found parallel to the c-axis\nof the Al 2O3substrate (and, consequently, the hard\naxes lie at\u000645\u000eofc). The easy axes of the twofold\nanisotropy were found at \u000645\u000eofcfor the 34-nm- and\n55-nm-thick films and slightly misaligned with this ori-\nentation in the case of the 83-nm-thick sample. However,\na detailed study of the in-plane anisotropy, involving\ntemperature and thickness dependence, allowing for\ntheir physical interpretation is still missing. Therefore,\nit forms the aim of the present paper. Rather completex-rays diffraction (XRD) measurements over a large\nthickness range of Co 2MnGe films are reported below\nin an attempt to find correlations between in-plane\nanisotropies, thickness and crystallographic textures.\nThe thickness- and the temperature-dependence of these\nanisotropies are investigated using vibrating sample\nmagnetometry (VSM) and the above mentioned FMR\ntechniques. In addition, we present intrinsic damping\nparametersdeducedfrombroadbandFMRdataobtained\nwiththehelpofavectornetworkanalyzer(VNA)[12-14].\nI. Sample properties and preparation\nCo2MnGe films with 13, 34, 55, 83, 100 and 200\nnm thickness were grown on sapphire a-plane substrates\n(showing an in-plane c-axis) by RF-sputtering with a fi-\nnal 4 nm thick gold over layer. A more detailed descrip-\ntion of the sample preparation procedure can be found\nelsewhere [11, 15].\nThe static magnetic measurements were carried out\nat room temperature using a vibrating sample mag-\nnetometer (VSM). The dynamic magnetic properties\nwere investigated with the help of 9.5 GHz conven-\ntional FMR and of MS-FMR [11]. The conventional\nFMR set-up consists in a bipolar X-band Bruker ESR\nspectrometer equipped with a TE 102resonant cavity\nimmersed is an Oxford cryostat, allowing for exploring\nthe 4-300 K temperature interval. The MS-FMR set-up\nis home-made designed and, up to now, only works at\nroom temperature. The resonance fields (conventionalarXiv:1005.4595v3  [cond-mat.mtrl-sci]  26 Aug 20102\nFMR) and frequencies (MS-FMR) are obtained from\na fit assuming a Lorentzian derivative shape of the\nrecorded data. The experimental results are analyzed in\nthe frame of the model presented in [11].\nXRD experiments were performed using four circles\ndiffractometers in Bragg-Brentano geometry in order\nto determine \u0012\u00002\u0012patterns and pole figures. The\ndiffractometer devoted to the \u0012\u00002\u0012patterns was\nequipped with a point detector (providing a precision of\n0:015\u000ein2\u0012scale). The instrument used for recording\npole figures was equipped with an InelTMcurved linear\ndetector ( 120\u000eaperture with a precision of 0:015\u000ein2\u0012\nscale). The X-rays beams (Cobalt line focussource at\n\u0015= 1:78897) were emitted by a BrukerTMrotating an-\node. define a direct macroscopic ortho-normal reference\n(1,2,3), where the 3axis stands for the direction normal\nto the film. 'and are the so-called diffraction angles\nused for pole figure measurements.  is the declination\nangle between the scattering vector and the 3-axis, '\nis the rotational angle around the 3-axis. The \u0012\u00002\u0012\npatterns (not shown here) indicate that, for all the\nCo2MnGe thin films, the <110>axis can be taken along\nthe3-axis. The Co 2MnGe deduced lattice constant\n(a= 5:755is in good agreement with the previously\npublished ones [6, 16]. Due to the [111] preferred orien-\ntation of the gold over layer along the 3-axis, only partial\n{110} pole figures could be efficiently exploited. They\nbehave as {110} fiber textures containing well defined\nzones showing significantly higher intensities (Figure 1\n(a) and (b)). These regions correspond to orientation\nvariants which can be grouped into two families (see\nFigure 1). The first one, where the threefold [111]or\nthe[111]axis is oriented along the crhombohedral\ndirection, consists of two kinds of distinct domains with\nthe [001] axis at \u000654:5\u000efrom the c-axis. The second\nfamily, which is rotated around the 3-axis by 90\u000efrom\nthe first one, also contains two variants. This peculiar\nin-plane domain structure is presumably induced by the\nunderlying vanadium seed layer. As illustrated in Figure\n1b, which represents '-scans at = 60\u000e, we do not\nobserve major differences between the crystallographic\ntextures of the 55-nm and of the 100-nm-thick samples :\nthe first family shows a concentration twice larger than\nthe second one ; at least for the first family, which allows\nfor quantitative evaluations, the concentrations of the\ntwo variants do not appreciably differ from each other;\nfinally, about 50% of the total scattered intensity arises\nfrom domains belonging to these oriented parts of the\nscans. In the 200-nm-thick sample the anisotropy of the\nfiber is less marked but the two families remain present.\nIII. Results and discussion\n1- Static magnetic measurements\nIn order to study the magnetic anisotropy at room\ntemperature, the hysteresis loops were measured for all\n0 25 50 75 100 125 150 175 0.0 0.2 0.4 0.6 0.8 1.0 (b)Normalized Intensity \nφ (degrees) 55 nm \n 100 nm a)\n13 nm \n100 nm 55 nm \n200 nm Figure 1: (Color online) (a) Partial {110} X-rays pole figures\n(around 60\u000e) of 13, 55, 100 and 200-nm-thick films. (b) Dis-\nplay of the angular variations of the intensity derived from\nthe above figures for the 55 and 100-nm-thick samples (the\nblue and pink vertical dashed lines respectively refer to the\ntwo expected positions of the diffraction peak relative to the\ntwo variants belonging to family 1).\nthe studied films with an in-plane applied magnetic field\nalong various orientations as shown in Figure2 ( 'His the\nin-plane angle between the magnetic applied field Hand\nthec-axis of the substrate). The variations of the coer-\ncive field ( Hc) and of the reduced remanent magnetiza-\ntion ( Mr/Ms) were then investigated as function of 'H.\nThetypicalbehaviorisillustratedbelowthroughtworep-\nresentative films which present different anisotropies.\nFigure 2a shows the loops along four orientations for\nthe 100-nm-thick sample. One observes differences in\nshape of the normalized hysteresis loops depending upon\nthe field orientation. For Halong c-axis ('H=0\u000e) we\nobserve a typical easy axis square-shaped loop with a\nnearly full normalized remanence ( Mr/Ms=0.9), a co-\nercive field of about 20 Oe and a saturation field of 100\nOe. As'Hincreases away from the c-axis, the coerciv-3\n-100 -50 0 50 100 -1.0 -0.5 0.0 0.5 1.0 Normalized magnetization (M/M s)\nApplied magnetic field (Oe)  ϕΗ=0° \n ϕΗ=45° \n ϕΗ=90° \n ϕΗ=135° 100 nm \n(a) \n-100 -50 0 50 100 -1.0 -0.5 0.0 0.5 1.0 \n(b) Normalized magnetization (M/M s)\nApplied magnetic field (Oe)  ϕH=0° \n ϕH=45° \n ϕH=90° \n ϕH=135° 55 nm \nFigure 2: (Color online) VSM magnetization loops of the (a)\n100-nm-thick and the (b) 55-nm-thick samples. The magnetic\nfield is applied parallel to the film surface, at various angles\n('H)with the c-axis of the sapphire substrate.\nity increases and the hysteresis loop tends to transform\ninto a hard axis loop. When 'Hslightly overpasses 90\u000e\n(90\u000e< 'H<100\u000e) the loop evolves into a more com-\nplicated shape: it becomes composed of three (or two)\nopen smaller loops. Further increasing the in-plane rota-\ntion angle, it changes from such a split-open curve up to\nan almost rectangular shape. The results for 'H= 45\u000e\nand'H= 135\u000eare different: they show a rounded loop\nwithMr=Msequal to 0.75 and 0.63 and with saturation\nfields of about 170 Oe and 200 Oe, respectively. This\nresult qualitatively agrees with a description of the in-\nplane anisotropy in terms of four-fold and two-fold con-\ntributions with slightly misaligned easy axes.\nThe variations of HcandMr=Msversus'Hare il-\nlustrated in Figures 3a and 3b for the 100-nm-thick film.\nThe presence of a fourfold anisotropy contribution is sup-\nported by the behavior of Hc(Figure 3a), since two min-ima appear within each period ( 180\u000e, as expected), as\nshown in Figure 3a. The minimum minimorum is mainly\nrelated to the uniaxial anisotropy term. In the same way,\nasdisplayedinFigure3b,thebehaviorof Mr/Msisdom-\ninated by the uniaxial anisotropy. It is worth to notice\nthat the minimum minimorum position slightly differs\nfrom 90\u000e(lying around 96\u000e), thus arguing for a misalign-\nment between the twofold and the fourfold anisotropy\naxes.\nFigure 2b shows a series of hysteresis loops, recorded\nwithanin-planeappliedfield, forthe55-nm-thickfilm. A\ncareful examination suggests that the fourfold anisotropy\ncontribution is the dominant one and that the related\neasy axis lies along c-axis. The Mr/Msvariation versus\n'H, reported in Figure 3c, is consistent with an easy\nuniaxial axis oriented at 45\u000eof this last direction. Both\nfourfold and uniaxial terms are smaller than for the 100-\nnm-thick sample.\n2- Dynamic magnetic properties\nAs previously published [11], the dynamic properties\nare tentatively interpreted assuming a magnetic energy\ndensity which, in addition to Zeeman, demagnetizing\nand exchange terms, is characterized by the following\nanisotropy contribution:\nEanis: =K?sin2\u0012M\u00001\n2(1 +cos(2('M\u0000\n'u))Kusin2\u0012M\u00001\n8(3 + cos 4( 'M\u0000'4))K4sin4\u0012M\n(1)\nIn the above expression, \u0012Mand'Mrespectively\nrepresent the out-of-plane and the in-plane (referring\nto the c-axis of the substrate) angles defining the\ndirection of the magnetization Ms;'uand'4stand\nfor the angles of the uniaxial axis and of the easy\nfourfold axis, respectively, with this c-axis. With these\ndefinitions KuandK4are necessarily positive. As\ndone in ref. [11], it is often convenient to introduce\nthe effective magnetization 4\u0019Meff= 4\u0019Ms\u00002K=Ms,\nthe uniaxial in-plane anisotropy field Hu= 2Ku=Ms\nand the fourfold in-plane anisotropy field H4= 4K4=Ms.\nFor an in-plane applied magnetic field H, the studied\nmodel provides the following expression for the frequen-\ncies of the experimentally observable magnetic modes:\nFn:=\r\n2\u0019(Hcos('H\u0000'M) +2K4\nMscos 4('M\u0000'4) +\n2Ku\nMscos 2('M\u0000'u) +2Aex:\nMs\u0000n\u0019\nd\u00012)\u0002\n(Hcos('H\u0000'M)+4\u0019Meff+K4\n2Ms(3+cos 4('M\u0000'4))+\nKu\nMs(1 + cos 2('M\u0000'u)) +2Aex:\nMs(n\u0019\nd)2)(2)\nIn the above expression \ris the gyromagnetic factor:\n(\r=2\u0019) =g\u00021:397\u0002106Hz/Oe. The uniform mode\ncorresponds to n=0. The other modes to be considered\n(perpendicular standing modes) are connected to integer\nvalues of n: their frequencies depend upon the exchange4\n0 50 100 150 200 250 300 3501520253035 VSM measurements\n(a)Coercive field (Oe) \nϕΗ (degrees)100 nm\n0 50 100 150 200 250 300 350 0.2 0.4 0.6 0.8 \n(b) Reduced remanent magnetization (M r/M s)\nϕΗ(degrees) VSM measurements 100 nm \n0 50 100 150 200 250 300 350 0.75 0.80 0.85 0.90 0.95 Reduced remanent magnetization (M r/M s)\nϕΗ(degrees)  VSM measurements 55 nm \n(c) \nFigure 3: (a) Coercive field and (b) reduced remanent mag-\nnetization of the 100-nm-thick sample as a function of the\nin-plane field orientation ( 'H). (c) Reduced remanent mag-\nnetization of the 55-nm-thick-film\nstiffness constant Aexand upon the film thickness d.\nFor all the films the magnetic parameters at room\ntemperature were derived from MS-FMR measurements.\nThe deduced gfactor is equal to 2.17, as previously\npublished [11].\nThe in-plane MS-FMR spectrum of the 100 nm-thick\n6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 -400 -300 -200 -100 0100 200 300 Amplitude (arb. units) \nFrequency (GHz) (a) H=520 Oe ϕΗ=0° Mode 2 \nMode 1 \n0 300 600 900 1200 1500 246810 12 14 \n(b) Frequency (GHz) \nApplied magnetic field (Oe)  Mode 2 \n Mode 1 \n Fit mode 2 \n Fit mode 1 \nϕΗ=0° Figure 4: (Color online) (a) MS-FMR spectrum under a mag-\nneticfieldapplied(H=520Oe)parallelthec-axisand(b)field-\ndependence of the resonance frequency of the uniform excited\nmodes, in the 100-nm-thick thin film. The fits are obtained\nusing equation (2) with the parameters indicated in Table I.\nsample (Figure 4a) submitted to a field of 520 Oe shows\ntwo distinct modes: a main one (mode 2), with a wide\nline-width (about 0.6 GHz) and a second weaker one\n(mode 1) at lower frequency with a narrower line-width\n(0.2 GHz)). Their field-dependences are presented in\nFigure 4b. In contrast with mode 2, which presents\nsignificant in-plane anisotropy, the measured resonance\nfrequency of mode 1 does not vary versus the in-plane\nangular orientation of the applied magnetic field: such\na different behavior prevents from attributing mode 1\nto a perpendicular standing excitation. Consequently,\nmode 1 is presumably a uniform mode arising from the\npresence of an additional magnetic phase in the film,\npossessing a lower effective demagnetizing field. In the\nfollowing, we focus on mode 2 which is assumed to be\nthe uniform mode arising from the main phase. As\npreviously published, only one uniform mode is observed\nwith the 55-nm-thick sample.\nFigures 5b and 5d illustrate the experimental in-plane\nangular-dependencies of the resonance frequency of the5\nuniform mode for the 100- and for the 55-nm-thick\nsamples, compared to the obtained fits using equation\n(2). As expected from the VSM measurements, in\nthe 100-nm sample the fourfold and uniaxial axes of\nanisotropy are misaligned: it results an absence of\nsymmetry of the representative graphs around 'H=\n90\u000e. The best fit is obtained for the following values of\nthe magnetic parameters: 4\u0019M e\u000b= 9800Oe,Hu= 55\nOe,H4= 110Oe,'4=0\u000e,'u\u000e= 12\u000e. As previously\npublished, in the case of the 55-nm sample the direction\nof the easy uniaxial axis does not coincide with the\nobserved one for the fourfold axis. The best fit for this\nfilm corresponds to: 4\u0019M e\u000b= 9800Oe,Hu= 10Oe,\nH4= 54Oe,'4= 0\u000e,'u=45\u000e. In both samples, the\nfourfold anisotropy easy direction is parallel to the c\naxis of the substrate: this presumably results from an\naveraging effect of the above described distribution of\nthe crystallographic orientations, in spite of the facts\nthat such a conclusion requires equal concentrations\nof the two main variants, a condition which, strictly\nspeaking, is not fully realized, and that the observed\nvalue of'4does not derive from probably oversimplified\naveraging model that we attempted to use, based on\nindividual domain contributions showing their principal\naxis of anisotropy along their cubic direction.\nAs usual, attempts to interpret the in-plane hysteresis\nloops using the coherent rotation model do not provide a\nquantitative evaluation of the anisotropy terms involved\nin the expression of magnetic energy density. However,\nthe experimentally measured Mr/Msangular variation,\nwhich, with this model, is given by cos('M-'H)) in\nzero-applied field and is easily calculated knowing '\n'u,'4andHu/H4, is in agreement with the values of\nthese coefficients fitted from resonance data, as shown\nin Figures 5a and 5c.\nt (nm) 4\u0019M e\u000b\n(kG)Hu\n(Oe)H4\n(Oe)'u\n(deg.)'4\n(deg.)\n13 8000 45 40 12 0\n34 9000 6 20 45 0\n55 9800 10 54 45 0\n89 9200 15 22 -5 0\n100 9800 60 110 12 0\n200 9900 24 0\nTable I : Magnetic parameters obtained from the best fits\nto our experimental results. 'uand'4are the angles of\nin-plane uniaxial and of fourfold anisotropy easy axes,\nrespectively\nThe magnetic parameters deduced from our resonance\nmeasurements are given in Table I for the complete set\nof the studied films. In contrast with the direction of\nthe fourfold axis which does not vary, the orientation\nof the uniaxial axis is sample dependent: for some of\nthem (34 and 55nm) the easy uniaxial direction lies at\n0 50 100 150 200 250 300 350 6.8 7.2 7.6 8.0 0.0 0.3 0.6 0.9 \n100 nm \n(b) Frequency (GHz) \nϕH (degrees)  MS-FMR measurements \n Fit 100 nm \n(a) \n Mr/M s\n Fit \nVSM measurements \n0 50 100 150 200 250 300 350 2.7 3.0 3.3 3.6 0.6 0.8 1.0 \n(d) 55 nm \nH=130 Oe Frequency (GHz) \nϕH (degrees)  MS-FMR \n Fit (c) 55 nm \n Mr/M s\n VSM measurements \n Fit Figure5: Reducedremanentmagnetizationofthe(a)100-nm-\nandofthe(c)55-nm-thickfilms. Thesimulationsareobtained\nfrom the energy minimization using the parameters reported\nin Table I. (b) and (d) show the compared in-plane angular-\ndependencesoftheresonancefrequencyoftheuniformmodes.\nThe fit is obtained using equation (2) with the parameters\nindicated in Table I.\n45\u000efrom the c-axis of the substrate (thus coinciding\nwith the hard fourfold direction); for other ones (13,\n83, 100 nm) it shows a variable misalignment; finally,\nthe uniaxial anisotropy field vanishes for the thickest\nsample (200 nm). We tentatively attribute at least a\nfraction of the uniaxial contribution as originating from\na slight misorientation of the surface of the substrate.\nThe amplitudes of both in-plane anisotropies are sample\ndependent and cannot be simply related to the film\nthickness. It should be mentioned that some authors [17]\nhave reported on strain-dependent uniaxial and fourfold\nanisotropies in Co 2MnGa. This suggests a forthcoming\nexperimental X-rays study of the strains present in our\nfilms.\nIn addition, it is useful to get information about the\ndamping terms involved in the dynamics of magnetic ex-\ncitations in the above samples. Notice that in order to6\n4 5 6 7 8 9 10 25 30 35 40 45 50 Field linewidth ∆H (Oe)\nFrequency (GHz) VNA-FMR measurements \n Fit\nFigure 6: Line-width \u0001Has a function of the resonance fre-\nquency for 55-nm-thick film. \u0001His derived from the experi-\nmental VNA-FMR frequency-swept line-width.\nintegrate these films in application devices like, for in-\nstance, MRAM, it is important to make sure that their\ndamping constant is small enough. The damping of the\n55-nm-thick film was studied by VNA-FMR [12-14]: it is\nanalyzedintermsofaGilbertcoefficient \u000bintheLandau-\nLifschitz-Gilbert equation of motion. The frequency line-\nwidth \u0001fof the resonant signal around frobserved us-\ning this technique is related to the field line-width \u0001H\nmeasured with conventional FMR excited with a radio-\nfrequency equal to frthrough the equation [18]:\n\u0001H= \u0001@H(f)\n@fjf=fr(3)\n\u0001His given by:\n\u0001H= \u0001H0+4\u0019fr\nj\rj\u000b(4)\n(where \u0001H0stands for a small contribution arising\nfrom inhomogeneous broadening). The measured linear\ndependence of \u0001His shown versus frin Figure 6. We\nthen obtain the damping coefficient: \u000b=0.0065. This\nvalue lies in the range observed in the Co 2MnSi thin\nfilms [19-21].\nFinally, the temperature dependence was studied for\nthe 55-nm-thick sample using conventional FMR. The\nfits of the magnetic parameters were performed assum-\ning that gpractically does not vary versus the tempera-\nture T, as generally expected. We then take: g= 2:17.\nThe results for the uniaxial and for the fourfold in-plane\nanisotropy fields are reported in Figure 7. Huis temper-\nature independent while H4is a significantly decreasing\nfunction of T. This behavior of H4is presumably related\nto the magneto-crystalline origin of this anisotropy term.\n0 50 100 150 200 250 300 10 20 30 40 50 60 70 80 90 100 Anisotropy fields (Oe)\nTemperature (K) H u\n H 4Figure7: (Coloronline)Temperature-dependenceofthefour-\nfoldanisotropyfield( H4)andtheunixialanisotropyfield( Hu)\nof the 55-nm-thick film, measured by FMR at 9.5 GHz.\nV Summary\nThe static and dynamic magnetic properties of\nCo2MnGe films of various thicknesses sputtered on\na-plane sapphire substrates have been studied. The\npresent work focused on the dependence of the pa-\nrameters describing the magnetic anisotropy upon the\ncrystallographic texture and upon the thickness of\nthe films. The crystallographic characteristics were\nobtained through X-ray diffraction which reveals the\npresence of a majority of two distinct (110) domains.\nMagnetometric measurements were performed by VSM\nand magnetization dynamics was analyzed using conven-\ntional and micro-strip resonances (FMR and MS-FMR).\nThe main results concern the in-plane anisotropy which\ncontributes to the magnetic energy density through two\nterms: a uniaxial one and a fourfold one. The easy axis\nrelated to the fourfold term is always parallel to the\nc-axis of the substrate while the easy twofold axis shows\na variable misalignment with the c-axis. The fourfold\nanisotropy is a decreasing function of the temperature:\nit is presumably of magneto-crystalline nature and its\norientation is related to the above noticed domains. The\nobserved misalignment of the two-fold axis is tentatively\ninterpreted as induced by random slight miscuts affect-\ning the orientation of the surface of the substrate. The\ntwo-fold anisotropy does not significantly depend on\nthe temperature. There is no evidence of a well-defined\ndependence of the anisotropy versus the thickness of the\nfilms. Finally, we show that the damping of the magne-\ntization dynamics can be interpreted as arising from a\nGilbert term in the equation of motion, that we evaluate.\nReferences\n[1] S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, Y.\nAndo, Appl. Phys. Lett. 93, 112506 (2008)7\n[2] S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev.\nB 66, 094421 (2002).\n[3] S. Picozzi, A. Continenza, and A. J. Freeman, J. Phys.\nChem. Solids 64, 1697 (2003).\n[4] T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys.\n89, 7522 (2001).\n[5] T. Ishikawa, T. Marukame, K. Matsuda, T. Uemura, M.\nArita, and M. Yamamoto, J. Appl. Phys. 99, 08J110 (2006)\n[6] F. Y. Yang, C. H. Shang, C. L. Chien, T. Ambrose, J. J.\nKrebs, G. A. Prinz, V. I. Nikitenko, V. S. Gornakov, A. J.\nShapiro, and R. D. Shull, Phys. Rev. B 65, 174410 (2002).\n[7] H. Wang, A. Sato, K. Saito, S. Mitani, K. Takanashi, and\nK. Yakushiji, Appl. Phys. Lett. 90, 142510 (2007)\n[8] Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato,\nA. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett.\n88, 192508 (2006).\n[9] T. Marukame, T. Ishikawa, K. Matsuda, T. Uemura, and\nM. Yamamoto, Appl. Phys. Lett. 88, 262503 (2006).\n[10] D. Ebke, J. Schmalhorst, N.-N. Liu, A. Thomas, G. Reiss,\nand A. Hütten, Appl. Phys. Lett. 89, 162506 (2006).\n[11] M. Belmeguenai, F. Zighem, Y. Roussigné, S-M. Chérif,\nP. Moch, K. Westerholt, G. Woltersdorf, and G. Bayreuther\nPhys. Rev. B 79, 024419 (2009).[12] M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier,\nand G. Bayreuther, Phys. Rev. B 76, 104414 (2007).\n[13] T. Martin, M. Belmeguenai, M. Maier, K. Perzlmaier,\nand G. Bayreuther, J Appl. Phys. 101, 09C101 (2007)\n[14] M. Belmeguenai, T. Martin, G. Woltersdorf, G.\nBayreuther, V. Baltz, A. K; Suszka and B. J. Hickey, J. Phys.:\nCondens. Matter 20, 345206 (2008).\n[15] U. Geiersbach, K.Westerholt and H. Back J. Magn.\nMagn. Mater. 240, 546 (2002).\n[16] T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys.\n87, 5463 (2000)\n[17] M. J. Pechana, C. Yua, D. Carrb, C. J. Palmstrøm, J.\nMag. Mag. Mat. 286, 340 (2005)\n[18] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl.\nPhys. 99, 093909 (2006)\n[19] R. Yilgin, M. Oogane, Y. Ando, T. Miyazaki, J. Mag.\nMag. Mat. 310, 2322 (2007)\n[20] R. Yilgin, Y. Sakuraba, M. Oogane, S. Mizumaki, Y.\nAndo, and T. Miyazaki, Japan. J. Appl. Phys. 46, L205\n(2007)\n[21] S. Trudel, O. Gaier, J. Hamrle, and B.Hillebrands, J.\nPhys. D: Appl. Phys. 43, 193001 (2010)"    },    {        "title": "1007.3508v1.Precessing_vortices_and_antivortices_in_ferromagnetic_elements.pdf",        "content": "Precessing vortices and antivortices in ferromagnetic elements\nA. Lyberatos,1S. Komineas,2and N. Papanicolaou3, 4\n1Department of Materials Science,University of Crete,PO BOX 2208,71003 Heraklion,Greece\n2Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece\n3Department of Physics, University of Crete, 71003 Heraklion, Crete, Greece\n4Institute for Theoretical and Computational Physics, University of Crete, Heraklion, Greece\nA micromagnetic numerical study of the precessional motion of the vortex and antivortex states\nin soft ferromagnetic circular nanodots is presented using Landau-Lifshitz-Gilbert dynamics. For\nsu\u000eciently small dot thickness and diameter, the vortex state is metastable and spirals toward the\ncenter of the dot when its initial displacement is smaller than a critical value. Otherwise, the vor-\ntex spirals away from the center and eventually exits the dot which remains in a state of in-plane\nmagnetization (ground state). In contrast, the antivortex is always unstable and performs damped\nprecession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies\nof precession are compared with the response expected on the basis of Thiele's theory of collective\ncoordinates. We also calculate the vortex restoring force with an explicit account of the magne-\ntostatic and exchange interaction on the basis of the 'rigid' vortex and 'two-vortices side charges\nfree' models and show that neither model explains the vortex translation mode eigenfrequency for\nnanodots of su\u000eciently small size.\nPACS numbers: 75.70.Kw,75.75.Fk,75.75.Jn,75.78.Cd,75.78.Fg\nI. INTRODUCTION\nThe vortex state is one of the equilibrium states of thin\nsoft ferromagnetic elements of micrometer size and below\n(magnetic dots). The interplay between the magneto-\nstatic and exchange energy favours an in-plane, closed\n\rux domain structure with a 10 \u000020 nm central core,\nwhere the magnetization turns out of plane to avoid the\nhigh energetic cost of anti-aligned moments. Core rever-\nsal can be triggered by application of an in-plane pulsed\n\feld or pulsed current allowing the possibility of applica-\ntion of patterned thin \flm elements in data storage and\nmagnetic and magneto-electronic random access mem-\nory [1]. Core reversal is usually assumed to arise from\nthe spontaneous creation of a vortex-antivortex (VA) pair\n(vortex dipole) of opposite polarity with respect to the\noriginal vortex, followed by collision of the pair with the\noriginal vortex. A fundamental understanding of the dy-\nnamics of vortices and antivortices is therefore necessary\nto control the switching of the magnetization.\nThe basic excitation mode of the vortex or antivortex\nstate from its equilibrium position is in-plane gyrotropic\nmotion. It is a low frequency (GHz) mode corresponding\nto the displacement of the whole structure. The gener-\nalized dynamic force can be determined using Thiele's\ncollective-variable approach [2]-[3]. Theoretical [4]-[5]\nand experimental [6] studies of the dynamics of magnetic\nvortices in 2D \flms have shown a connection with the\ntopology of the magnetization structure . Magnetic vor-\ntices con\fned in circular dots can be described by ana-\nlytical models based on di\u000berent methods for accounting\nfor the magnetostatic interaction [7]-[9]. The vortex may\nbe 'rigid' or deform so that no magnetic charges appear\nat the side of the cylinder. The latter (two-vortices side\ncharge free model) provides a good description of the dy-\nnamic behavior of vortices in submicron-sized permalloydots, in particular the increase of vortex eigenfrequency\nwith dot aspect ratio L=R, whereLis the dot thickness\nandRis the dot radius. The basic assumption in these\ncalculations is that the vortex displacement lfrom equi-\nlibrium, at the dot center, is small l<<R .\nThe main objective here is to consider the response of\nthe vortex state to large perturbations, induced for in-\nstance by thermal activation. We focus our attention to\ndots of su\u000eciently small radius and thickness so that the\nvortex state is metastable . Using micromagnetic calcu-\nlations it is shown that vortex stability can be de\fned\nin terms of a critical displacement lcleading to an irre-\nversible transition to the ground state characterized by\nin-plane magnetization. We test the accuracy of the col-\nlective coordinate representation for vortex and antivor-\ntex dynamics and discuss the limitations of the approxi-\nmate analytical models [8],[9].\nII. VORTEX AND ANTIVORTEX PRECESSION\nThe dynamical behavior of a single vortex or antivor-\ntex trapped in a ferromagnetic nanodot was studied\nusing a \fnite-di\u000berence Landau-Lifshitz-Gilbert micro-\nmagnetic model. The material parameters for permal-\nloy (Ni 80Fe20) were used in the calculations: saturation\nmagnetization Ms= 800 emu/cc and exchange sti\u000b-\nness coe\u000ecient A= 1:3\u000210\u00006erg/cm. The mag-\nnetic anisotropy was neglected and the exchange length\nislex=p\nA=(2\u0019M2s) = 5:7 nm. The disk thickness is in\nthe rangeL= 5\u000010 nm , comparable to the exchange\nlength, so the magnetization dependence along the dot\nnormal can be neglected to a \frst approximation. The\ndisk radius is not large compared to the vortex core and\nis initially taken as R= 30 nm. Inspection of the phase\ndiagram of magnetic ground states for this material [10]arXiv:1007.3508v1  [cond-mat.mtrl-sci]  20 Jul 20102\nFIG. 1: a) Vortex snd b) antivortex structure in permalloy\ndot of thickness L= 5 nm and radius R= 30 nm.\nindicates that for the particular choice of disk dimensions,\nin-plane magnetization is the ground state of the system.\nThe integration of the LLG equation is carried out using\na damping parameter \u000b= 0:01, which is appropriate for\npermalloy [11].\nThe initial vortex or antivortex structure is de\fned as\nfollows: Using the coordinate system with the z axis par-\nallel to the dot cylindrical axis, the magnetization compo-\nnents aremz=\u0015cos \u0002(\u001a);mx+imy= sin \u0002 exp( i\u0014(\u001e\u0000\n\u001eo)), where m(r) =M(r)=Msis the reduced magneti-\nzation,\u001a;\u001eare cylindrical coordinates, \u0015=\u00061 is the\nvortex polarity, and \u0002( \u001a) is the magnetization angle to\nthe normal (z) direction. The vortex number is \u0014= 1\nfor a vortex and \u0014=\u00001 for an antivortex structure.\nThe constant \u001eo=\u0006\u0019=2 de\fnes the chirality i.e. the\ndirection of the curl of the vortex. The speci\fc choice\nof chirality is largely insigni\fcant for vortex dynamics,\nin contrast to the vortex number \u0014and polarity \u0015which\nplay an important role.\nThe initial pro\fle \u0002( \u001a) is assumed to have the modelform cos \u0002 = (cosh( c\u001a=lex))\u00001which is appropriate for\neasy-plane ferromagnets [12]. The parameter cde\fnes\nthe core radius b. The core radius is determined by the\ndot thickness Land the exchange length lexbut is inde-\npendent of the dot radius [13]. In the following, dot thick-\nness isL= 5 nm, unless speci\fed otherwise. Numerical\ncalculations of relaxed vortex and antivortex structures\nat the dot center were found to be in good agreement\nwith the model pro\fle for c'1. Typical examples of a\nvortex and antivortex structure are shown in Fig. 1. It\nshould be noted that other choices for the vortex pro-\n\fle are possible [14], for instance tan(\u0002 =2) =\u001a=bwas\nobtained by Usov et al. [15] using a variational proce-\ndure to minimize the exchange energy whereas the vor-\ntex core radius bwas determined by minimization also\nof the magnetostatic energy. This form is employed in\nanalytical calculations of vortex states in ferromagnetic\ndisks [7]-[9], however, micromagnetic modelling by Scholz\net al. [10], con\frmed by our calculations, has shown that\nit underestimates the core radius b, de\fned in Ref.[10] as\nthe \frstmz= 0 crossover from the vortex center.\nThe dynamical behavior of the magnetic vortex ( \u0014= 1)\ndepends on the initial displacement l(t= 0) =lofrom\nthe disk center. If the displacement is equal or smaller\nthan some critical value lc= 0:52R, the damped preces-\nsion of the vortex leads to relaxation to the disk center.\nFor positive vortex polarity \u0015= 1, an anticlockwise pre-\ncession is observed as shown in Fig. 2a. At the start of\nthe simulation the precession is not smooth as a result of\nthe internal relaxation (deformation) of the model vortex\nto minimize the energy during precession. If lo>lc, the\ndamped precession is clockwise and the distance from the\ndot center increases, as shown in Fig. 2b, until the vor-\ntex is annihilated and the magnetization is aligned along\nthe in-plane direction with quasi-uniform magnetization,\nthe so-called 'leaf' state [16] which is the ground state\nof the system. Irregularities in the precessional motion\narise from the uncertainty on the position of the vortex.\nThe antivortex instead is always unstable. For any\nchoice of initial displacement lo, the antivortex performs\ndamped precession to the edge of the disk and is an-\nnililated. For positive polarity, anticlockwise precession\nis observed. The sense of gyrotropic motion of a vortex\nor antivortex is switched on reversal of the polarity.\nIII. THE COLLECTIVE COORDINATE\nAPPROACH\nThe damped precession of the vortex or antivortex can\nbe described using Thiele's equation [2],[3] augmented by\na dissipative term.\nG\u0002dl\ndt+ 2QG\u0011dl\ndt\u0000@E(l)\n@l= 0 (1)\nwhere l= (lx;ly) is the position of the vortex center\nandE(lx;ly) is the potential energy of the shifted vor-3\n-0.6-0.4-0.20.00.20.40.6-0.6-0.4-0.20.00.20.40.6(a)\n   y/Rx/R\n-0.8-0.6-0.4-0.20.00.20.40.60.8-0.8-0.6-0.4-0.20.00.20.40.60.8\n   y/Rx/R(b)\nFIG. 2: Trajectory of a vortex of positive polarity \u0015= 1 in\nzero \feld for a time interval 0 < t < 2\u0002104. The initial\nposition vector of the vortex is a) lo= (0:52R;0) and b)\n(0:53R;0). The anisotropy is neglected and the damping is\n\u000b= 0:01.\ntex. The gyroforce G\u0002dl=dtdepends on the topolog-\nical structure of the magnetization and is proportional\nto the gyrovector G=\u0000G^z, where the gyroconstant is\nG= 2\u0019\u0014\u0015LMs=\rand\r= 1:76\u0002107rad Oe\u00001s\u00001is the\ngyromagnetic ratio. Q=\u00001\n2\u0014\u0015is the skyrmion number\nand\u0011is the dissipation constant.\nFor axially symmetric energy potential E=E(l) wherel=q\nl2x+l2y,@E=@lx=E0lx=land it is straightforward\nto show that\n_lx\u00002Q\u0011_ly=\u0000!ly (2)\n2Q\u0011_lx+_ly=!lx (3)\nwhere the angular frequency is\n!=1\nGl@E\n@l(4)\nThe vortex motion in complex form is given by\n(1 + 2Q\u0011i)\u0010\n_lx+i_ly\u0011\n=i!(lx+ily) (5)\nIntroducing polar coordinates lx+ily=lei\u001e\n_l=2Q\u0011\n1 +\u00112!(l)l (6)\n_\u001e=!(l)\n1 +\u00112(7)\nDividing and integrating over the time interval of the\ndamped precession, the time dependence of the preces-\nsion angle is\n\u001e(t) =2Q\n\u0011ln\u0012l\nlo\u0013\n(8)\nwherelois the initial displacement of the vortex center.\nThe clockwise or anticlockwise sense of gyration is there-\nfore dependent on the skyrmion number Q.\nMicromagnetic simulations of vortex motion were per-\nformed and the position of the vortex center ( lx;ly) was\ndetermined by a method of interpolation for the position\nof maximum mz. The precession angle \u001e= arctan(ly=lx)\nwas found to vary linearly with the logarithm of the vor-\ntex shift, in agreement with Eq.(8). Fig. 3 shows numeri-\ncal data for damped precession of a vortex of positive po-\nlarity (Q=\u00001=2) with initial displacement lo= 0:52R.\nThe relaxation to the disk center involves many revolu-\ntions (Fig. 2a) and the gradient \u00001=\u0011provides an accu-\nrate estimate of the dissipation constant \u0011= 0:013. It is\nevident that for permalloy nanodots, the damping in the\nvortex motion is weak and the angular frequency of pre-\ncession in Eq.(7) can be approximated using !'d\u001e=dt .\nFitting the numerical \u001e(t) curve to a 4th order poly-\nnomial, the time variation of the angular frequency !(t)\ncan be determined. The vortex shift l(t) exhibits oscilla-\ntions that are neglected by \ftting to a 4th order polyno-\nmial and the !(l) dependence is obtained using the !(t)\ncurve. The same procedure is employed for damped pre-\ncession leading to vortex annihilation ( lo= 0:53R). The4\nFIG. 3: Azimuthal angle \u001eof the vortex position as a function\nof the logarithm of the reduced vortex displacement s=l=R.\nThe initial radial position is lo= 0:52R.\nFIG. 4: Precession frequency of a vortex !=2\u0019as a function\nof the reduced diplacement s=l=Rfrom the center of the\ndot.\ncombined results for the dependence of the frequency of\nprecession!=2\u0019on reduced vortex shift s=l=Rare il-\nlustrated in Fig. 4.\nFor small displacement of the vortex center from its\nequilibrium position ( l= 0), the potential energy is\nE(l) =E(0) + (1=2)\u0014l2, where\u0014is the sti\u000bness coe\u000e-\ncient and the eigenfrequency is !o=\u0014=G [9]. At the\ncritical displacement lc, corresponding to a maximum in\nthe potential energy E(l) the precession frequency van-\nishes.\n0.00.10.20.30.40.50.60.70.80.1000.1020.1040.1060.1080.1100.1120.1140.116\n   Reduced energy density  e\nReduced vortex displacement  seMAX\nsc(a)\n0.00.10.20.30.40.50.60.70.80.90.000.020.040.060.080.100.12eeexReduced energy density  \nReduced vortex shift  sed(b)FIG. 5: (a) Reduced energy density of a permalloy dot\n\u000f=E=4\u0019M2\nsVas a function of normalized vortex displace-\nments. Results are shown for two sets of micromagnetic simu-\nlations (markers) and analytical (solid line) calculations using\nThiele's collective variable theory (Eq.10). The maximum en-\nergy density \u000fmaxoccurs at vortex displacement sc. (b) The\ncontribution of the magnetostatic and exchange terms to the\ntotal energy, obtained from micromagnetic calculations.\nThe motion of vortices and antivortices is driven by\nthe restoring force @E=@ l(Eq.1). The potential energy\nof the shifted vortex is axially symmetric E=E(l) and\ncan be written\nE(l) =E(0) +GZl\n0!(\u001a)\u001ad\u001a (9)\nA simpler form in terms of the reduced energy density\n\u000f=E=4\u0019M2\nsVover the dot volume Vis\n\u000f(l) =\u000f(0)\u00004QZs\n0!(s0)s0ds0(10)5\nwheres0=\u001a=R,s=l=Ris the reduced displacement\nof the vortex and the time associated with !is scaled\nby\u001co= 1=4\u0019\rMs. The function !(s0) was determined\nfrom micromagnetic simulations of the vortex precession\nand using Eq(10) the curve \u000f(l) expected from appli-\ncation of the theory of collective coordinates, was ob-\ntained, as shown in Fig. 5a. Superimposed is the energy\nevaluated directly from simulations of vortex relaxation.\nThiele's theory appears to provide a good description of\nthe vortex precession despite its limitations, for instance,\na) Thiele's approach is known to be an approximate de-\nscription of vortex dynamics in in\fnite \flms b) the vortex\nis here con\fned in a nanodot ( R= 5:3lex) and c) the vor-\ntex structure does not remain rigid during the relaxation\nprocess but is modi\fed as a result of the change in the\ndistribution of the demagnetizing \felds. The magneto-\nstatic and exchange contribution to the energy variation\n\u000f(l) was obtained from micromagnetic calculations and\nis shown in Fig. 5b. It should be noted that incorporat-\ning the demagnetizing energy to the total anisotropy, is\nstrictly valid for in\fnite thin \flms and results in mono-\ntonically decreasing energy \u000f(l), as reported in Ref. [17].\nThe oscillations in the energy variation during precession\nare related to the \fnite micromagnetic grid.\nThe potential energy of the vortex attains a maximum\nvalue at some critical value of the displacement sc= 0:52\ncorresponding to a zero crossover of the precession fre-\nquency!(Fig. 4). The stability of the vortex at the dot\ncenter arises from the magnetostatic energy, in particular\nthe volume magnetic charges resulting from vortex defor-\nmation and the surface charges at the side of the cylinder\n[9]. The face charges do not depend on ssince the charge\ndistribution on the top and bottom surfaces of the disk is\nunchanged with the vortex displacement. The exchange\nenergy decreases with increasing vortex shift s[9]. The\nmagnetostatic and exchange contributions to the restor-\ning force are in exact balance at the point of maximum\nenergy.\nA similar analysis was carried out for an antivortex\nstructure in a dot of identical dimensions. The potential\nenergy decreases monotonically with increasing displace-\nment s, as shown in Fig. 6. Application of the collec-\ntive coordinates treatment results in the solid curve in\nFig. 6 of slightly smaller curvature. The vortex insta-\nbility arises from the uncompensated magnetic charge\ndistribution within the antivortex core (Fig. 1b), so the\nmagnetostatic energy is reduced upon motion away from\nthe dot center. The precession frequency increases dur-\ning the relaxation process as shown in Fig. 7, as a result\nof the steeper energy gradient for large displacement s.\nAssuming identical position, it is evident that the an-\ntivortex precesses faster than the vortex as a result of\nthe larger magnetostatic energy gradient.\nFIG. 6: Energy density \u000f=E=4\u0019M2\nsVvs antivortex displace-\nments. The notation is similar to Fig. 5a.\nFIG. 7: Antivortex precession frequency !=2\u0019as a function of\nthe reduced diplacement s=l=Rfrom unstable equilibrium\nposition at the dot center.\nIV. DEPENDENCE OF VORTEX PRECESSION\nON DISK SIZE\nThe maximum potential energy Emax of the shifted\nmagnetic vortex, evaluated from plots such as Fig. 5a,\ndepends on the radius of the dot, where it is con\fned.\nIn Fig. 8, micromagnetic calculations of the reduced dot\nenergy density \u000fare shown as function of dot radius R,\nscaled by the exchange length. The curves correspond\nto the maximum vortex energy Emax and the minima\nassociated with the three equilibrium states of the mag-\nnetization (in-plane,perpendicular,vortex). The vortex6\nFIG. 8: Micromagnetic calculations of scaled dot energy den-\nsity\u000fvs dot radius R(in units of lex) for the three equilibrium\nstates of the magnetization and the vortex state of maximum\nenergy.\nFIG. 9: Barrier to vortex escape ( \u000fB), de\fned by the relation\n\u000fB=\u000fmax\u0000\u000fvortex as a function of dot radius (in units of\nlex).\nstate is unstable for small dots R < Rs, metastable for\nRs< R < R eqand a ground state for R > Reqwhere\nthe values Rs= 2:5lexandReq= 16lexare obtained\nfor the absolute and equilibrium single domain radius re-\nspectively. The corresponding variation of vortex barrier\nenergy\u000fB=Emax\u0000Evortex and displacement lc=scR\nfor vortex escape are shown in Figs. 9 and 10 respectively.\nFor small dots R= 3:5lex, the vortex is unstable and any\nshift from equilibrium at the dot center results in relax-\nation to the ground state (in-plane magnetization). The\nFIG. 10: Reduced displacement for maximum energy of the\nvortex state scas a function of dot radius (in units of lex).\ncritical size for vortex instability is larger than Rssince\nthe latter is de\fned assuming a random perturbation dif-\nferent than shifting the whole vortex. The vortex barrier\nenergy increases for larger dots and attains a maximum\nvalue, forR > 10lex, related to the vortex annihilation\n\feld [7]. The displacement lcfor vortex escape increases\nwith disk size to the maximum value imposed by the\ndisk perimeter ( lc=R!1), attained for sub-micron dots\nR>>lex. For su\u000eciently large dots, the vortex is within\nthe domain of attraction of the dot center, irrespective\nof the initial position.\nMicromagnetic calculations of the dependence of the\nfundamental vortex eigenfrequency !o=2\u0019, obtained for\nsmall perturbation s<< 1 as in Fig. 4, on the dot aspect\nratio\f=L=R are shown in Fig. 11. The dot thickness\nwas \fxed (L= 10 nm) and the radius Rwas allowed to\nvary. The eigenfrequency attains a maximum value at\n\f= 0:35 and vanishes for smaller radius \f'0:5 when\nthe vortex becomes unstable. The maximum value arises\nfrom the change in the relative contribution of the magne-\ntostatic and exchange energy to the sti\u000bness coe\u000ecient\n\u0016. For instance, the eigenfrequency assuming a 'rigid'\nvortex is [9]\n!o=\u0019\rMs\nQ\u0014\nF1(\f)\u00001\n(R=lex)2\u0015\n(11)\nwhereF1(x) =R\ndtt\u00001f(xt)J2\n1(t) corresponds to the av-\neraged in-plane dot demagnetizing factor, f(x) = 1\u0000\n[1\u0000exp(\u0000x)]=xandJ1is the Bessel function. Using\nthe approximation F1(\f)'(\f=2\u0019)[ln(8=\f)]\u00001=2], valid\nfor\f << 1, it can be shown that the eigenfrequency is\nmaximum at radius R= 4\u0019=L[ln(8=\f)\u00003=2]\u00001where\nall lengths are in units of lex. Previous studies were re-\nstricted to sub-micron sized dots where the second term7\nFIG. 11: Micromagnetic (markers) and analytical (solid lines)\ncalculations of vortex eigenfrequency !o=2\u0019as a function of\ndot aspect ratio \f=L=R a) the rigid vortex model and b)\nthe two-vortices model. The dot thickness is here L= 10 nm.\nof Eq.(11) could be neglected [9], so a monotonically in-\ncreasing eigenfrequency !o(\f) arising from the magneto-\nstatic energy only was reported.\nThe micromagnetic calculations are compared in\nFig. 11 with the curves obtained using the 'rigid' vor-\ntex and 'two-vortices' approximations for the magneto-\nstatic energy. For sub-micron sized disks with aspect\nratio\f < 0:05 corresponding to a radius R > 200 nm,\nthe micromagnetic calculations are in good quantitative\nagreement with the 'two vortices' model, as reported in\nRef.[9]. In this regime, the eigenfrequency is determined\nprimarily by the magnetostatic energy. The rigid vor-\ntex approximation fails to describe the dynamic behav-\nior since the magnetostatic energy can be decreased by\nelimination of the surface charges at the disk perime-\nter at the expense of some contribution from volume\nmagnetic charges arising from vortex deformation. Forsmaller disks R < 200 nm, the vortex eigenfrequency is\nbetween the predictions of the two models. A reduction\nin side charges occurs but is not complete as a result of\nthe large expense in exchange energy arising from vortex\ndeformation. Similar results can be obtained in principle\nfor thinner disks, however, the 'two vortices' approxima-\ntion is then valid for larger cylinders which are not easily\namenable to micromagnetic simulations.\nV. CONCLUSIONS\nMicromagnetic calculations were carried out of the pre-\ncessional behaviour of a single magnetic vortex or an-\ntivortex con\fned in a permalloy circular nanodot. The\nexistence of two domains of attraction for the vortex state\nare identi\fed arising from a maximum in the potential\nenergy of the shifted vortex. This e\u000bect is atrributed\nto the competition between the magnetostatic attractive\nand exchange repulsive forces on the shifted vortex. An-\ntivortices instead are always unstable and trace a spi-\nral trajectory of increasing distance from the dot center\nfollowed by annihilation at the dot envelope. The pre-\ncessional behaviour of vortices and antivortices is satis-\nfactorily described by Thiele's theory of collective coor-\ndinates, relating the angular frequency of precession to\nthe gradient of the potential energy. For small nano-\nsized dots, however, the 'rigid' vortex and 'two-vortices'\napproximation for the magnetostatic energy is not sat-\nisfactory and the exchange forces have a signi\fcant ef-\nfect on the translation mode vortex eigenfrequency. For\nantivortices, the development of an analytical model to\naccount for the magnetostatic interaction in circular dots\nis clearly needed to provide further insight on the results\nof our micromagnetic calculations.\nVortex stability is necessary in applications of nanos-\ntructured patterned media for data storage. Microfabri-\ncation downscaling implies that the displacement lcand\nassociated energy barrier may become useful characteri-\nzation parameters of the thermal stability of the recorded\ninformation.\n[1] B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T.\nTyliszczak, R. Hertel, M. F ahnle, H. Br uckl, K. Rott, G.\nWeiss, I. Neudecher, D. Weiss, C.H. Back and G. Sch utz,\nNature (London) 444, 461 (2006).\n[2] A.A. Thiele, PHys. Rev. Lett. 30, 230 (1973).\n[3] D.L. Huber, Phys. Rev. B 26, 3758 (1982).\n[4] N. Papanicolaou and T.N. Tomaras, Nucl. Phys. B 360,\n425 (1991).\n[5] S. Komineas and N. Papanicolaou, Physica (Amsterdam)\n99D , 81 (1996).\n[6] S.-B Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran,\nJ. St ohr and H.A. Padmore, Nature 304, 420 (2004).\n[7] K. Yu Guslienko, V. Novosad, Y. Otani, H. Shima andK. Fukamichi, Appl. Phys. Lett. 78, 3848 (2001).\n[8] K.L. Metlov and K. Yu Guslienko, J. Magn. Magn.\nMater. 242-245 , 1015 (2002).\n[9] K. Yu Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H.\nShima and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002).\n[10] W. Scholz, K. Yu Guslienko, V. Novosad, D. Suess,\nT.Schre\r, R.W. Chantrell and J. Fidler, J. Magn. Magn.\nMater. 266, 155 (2003).\n[11] D.V. Berkov and J. Miltat, J. Magn. Magn. Mater. 320,\n1238 (2008).\n[12] S. Komineas and N. Papanicolaou, in Electromagnetic,\nmagnetostatic and exchange-interaction vortices in con-\n\fned magnetic structures , edited by E.O. Kamenetskii,8\n(Transworld Research Network, Kerala,2008).\n[13] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T.\nOno, Science 289, 034318 (2005).\n[14] P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d' Albu-\nquerque e Castro and P. Vargas, Phys. Rev. B 71, 094435\n(2005)\n[15] N.A. Usov and S.E. Peschany, J. Magn. Magn. Mater.118, L290 (1993).\n[16] K.L. Metlov and K. Yu Guslienko, Phys. Rev. B 70,\n052406 (2004)\n[17] D.D. Sheka, J.P. Zagorodny, J. Caputo, Y. Gaididei and\nF.G. Mertens, Phys. Rev. B 71, 134420 (2005)"    },    {        "title": "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": "1008.2177v1.Magnetization_dynamics_in_the_inertial_regime__nutation_predicted_at_short_time_scales.pdf",        "content": "arXiv:1008.2177v1  [cond-mat.mes-hall]  12 Aug 2010Magnetization dynamics in the inertial regime: nutation pr edicted at short time scales\nM.-C. Ciornei and J.-E. Wegrowe\nEcole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palais eau F-91128, France.\nJ. M. Rub´ ı\nDepartement de Fisica Fonamental, Universitat de Barcelon a, Diagonal 647, Barcelona 08028, Spain.\n(Dated: April 23, 2022)\nThe dynamical equation of the magnetization has been recons idered with enlarging the phase\nspace of the ferromagnetic degrees of freedom to the angular momentum. The generalized Landau-\nLifshitz-Gilbert equation that includes inertial terms, a nd the corresponding Fokker-Planck equa-\ntion, are then derived in the framework of mesoscopic non-eq uilibrium thermodynamics theory. A\ntypical relaxation time τis introduced describing the relaxation of the magnetizati on acceleration\nfrom the inertial regime towards the precession regime defin ed by a constant Larmor frequency.\nFor time scales larger than τ, the usual Gilbert equation is recovered. For time scales be lowτ,\nnutation and related inertial effects are predicted. The ine rtial regime offers new opportunities for\nthe implementation of ultrafast magnetization switching i n magnetic devices.\nPACS numbers: 75.78.jp, 05.70.Ln\nIn 1935 Landau and Lifshitz proposed an equation for\nthe dynamicsofthe magnetization M(ofconstantmodu-\nlus), composedofaprecessionterm M×Handa longitu-\ndinalrelaxationterm M×(M×H)thatdrivesthemagne-\ntization towards equilibrium along the magnetic field H\n[1]. Two decades later T. L. Gilbert derived the equation\nthat bearshisname in whichthe relaxationtowardsequi-\nlibrium is described by a damping term η[2] through the\ndynamic equation dM/dt=γM×(H−ηdM/dt), withγ\nas the gyromagnetic ratio. The two equations (Landau-\nLifshitz and Gilbert) are mathematically equivalent.\nThe range of validity of the Landau-Lifshitz-Gilbert\n(LLG) equation was established one decade later by W.\nF. Brown, with a description of a magnetic moment cou-\npled to a heat bath (”thermal fluctuations of a single-\ndomain particle”, 1963 [3]). The magnetic moment is\ntreated as a Brownian particle described by the slow de-\ngrees of freedom (10−9s), the angles {θ,φ}. The remain-\ning degrees of freedom of the system relax in a much\nshorter time scale ( <10−12s). The time scale separa-\ntion between the rapidly relaxing environmental degrees\nof freedom and the slow magnetic degrees of freedom al-\nlows the coupling between the magnetization and the en-\nvironment to be reduced to a single phenomenological\ndamping parameter η, whatever the complexity of the\nmicroscopic relaxation involved [4, 5].\nHowever, important experimental advances towards\nvery short time-resolved response of the magnetization\n(sub-picosecondsresolution,i.e. belowthelimitproposed\nby Brown) have been reported in the last decade [6]. In\nparallel, industrial needs for very fast memory storage\ntechnologies are approaching the limits imposed by the\nprecessional switching [7]. In these experiments and in\nthe corresponding applications, time scale separation be-\ntween the conserved degrees of freedom {θ,φ}and the\nother degrees of freedom, assumed by Brown [3], finds its\nlimit.\nThe purpose of this Letter is to investigate the dynam-ics of the magnetization beyond this limit by extending\nthephasespacetoadditionaldegreesoffreedomexpected\nto be also out-of-equilibrium at short time scales [5, 9].\nAccording to the well-known gyromagnetic relation [10],\nthe next relevant degree of freedom of the ferromagnetic\nsystem (beyond the coordinates of position; i.e. the an-\ngles{θ,φ}) is the angular momentum L. As will be\nshown below, the consequence of considering also the\nconservation of the angular momentum is that inertial\nterms, i.e. acceleration terms proportional to d2M/dt2,\nappear in the equation of motion. The existence of in-\nertial terms in the dynamics of the magnetization opens\nthe way to deterministic ultrafast magnetization switch-\ning strategies, beyond the limitations of the precessional\nregime [11]. We assume however that the microscopic re-\nlaxationchannelsareinactiveat the time consideredhere\n(e.g. by choosing the adequate materials and excitations\nof the magnetization). Otherwise, a non-deterministic\nregime would take place [7, 12, 13].\nWe derive below the generalized Gilbert equation and\nthe corresponding Fokker-Planck equation that includes\nthe inertial effects for a uniform magnetic moment. The\nderivation is performed in the framework of mesoscopic\nnonequilibrium thermodynamics (MNET) [14–16], and is\nbased on the expression of the conservation laws, ther-\nmodynamic laws, and symmetry properties.\nIt is convenient to model the dynamics of a magnetic\nmoment M=Mse(submitted to an applied magnetic\nfieldH=−1\nMs∂VF\n∂eand coupled to a heat bath) with a\nstatistical ensemble composed by non-interacting identi-\ncal uniform magnetic moments found in the same given\nconditions (ergodic property). Here, e,MsandVFare\nrespectively the radial unit vector of angles {θ,φ}, the\nmagnetization at saturation, and the ferromagnetic po-\ntential energy. The ensemble of magnetic moments of\nconstant modulus Msdefines a sphere surface Σ and the\nnumber of magnetic moments oriented within ( e,e+de)\ndefines the density n(e) of magnetic moments over Σ.2\nWe have shown in previous works that associating two\ndegrees of freedom {θ,φ}to a magnetic moment is suf-\nficient to derive both the Gilbert equation and the cor-\nresponding rotational Fokker-Planck equation from non-\nequilibrium thermodynamics principles only [13, 17].\nExtending the configuration space to the magnetic an-\ngular momentum L, the space Σ is extended from a\ntwo dimensions space, to a priori five dimensions space\n{θ,φ,L}[18]. A distribution function f(e,L) of mag-\nnetic momentswith the magnetizationorientationwithin\n(e,e+de) and the angularmomentum within ( L,L+dL)\nshould then be defined, where fis assumed to vanish for\ninfinite values of Las: lim L→±∞f(e,L) = 0. The an-\ngular momentum Lassociated to a magnetic moment is\neither changed by an applied torque N=M×Has/parenleftbigdL\ndt/parenrightbig\ns=N, either by the interaction with the heat bath.\nWhen considering the statistical ensemble, the interac-\ntion with the bath is modeled through a phase space flux\nJL(defined below) which vanishes for large values of L:\nlimL→±∞JL= 0.\nThe kinetic energy expression that Gilbert associated\nto the magnetization [2] is written as: K=LL:¯¯I−1/2,\nwhere the magnetic inertial tensor¯¯I, is related to the\nmagnetic moment (and not to the inertia of matter). It\nis assumed that¯¯Ikeeps the symmetry of the magnetic\nmoment, i.e. is axial symmetric of symmetry axis e:¯¯I=\nI1/parenleftBig¯¯U−ee/parenrightBig\n+I3ee, with¯¯Uthedyadicunit(where I1=I2\nandI3arethe diagonalcoefficients ofthe inertial tensor).\nIn the space-fixed reference frame denoted by the sub-\nscripts, the conservation law for the number of axial\nsymmetric moments f(e,L) writes [19]:\n∂\n∂tf(e,Ls) =−/braceleftbigg∂(f˙ e)\n∂e/bracerightbigg\nLs−Ns·∂f\n∂Ls−∂JL\n∂Ls(1)\nwhere the derivatives with respect to the angles are made\nwhile holding the Cartesian components of Lsconstant:\n/braceleftbigg∂(f˙ e)\n∂e/bracerightbigg\nLs=1\nsinθ/braceleftBigg\n∂(fsinθ˙θ)\n∂θ/bracerightBigg\nLs+/braceleftBigg\n∂(f˙φ)\n∂φ/bracerightBigg\nLs(2)\nThe density n(e) of magnetic moments in the space Σ\nis recovered by integrating over the angular momentum\ndegree of freedom n(e)=/integraltextf(e,L)d3L. The conservation\nlaw for the magnetic moments in the Σ space is hence\ndeduced from (1):\n∂n\n∂t=/integraldisplay∂f\n∂td3Ls=−∂\n∂e·/integraldisplay\nf˙ ed3Ls=−∂(n˙e)\n∂e(3)\nBeyond, the conservationlaw for the mean value of themagneticangularmomentum /an}bracketle{tLs/an}bracketri}htisalsoderived[19,20]:\n∂n/an}bracketle{tLs/an}bracketri}ht\n∂t=/integraldisplay∂f\n∂tLsd3Ls\n(1)=−∂\n∂e·/integraldisplay\nf˙ eLsd3Ls+nNs(e)+/integraldisplay\nJLd3Ls\nnd/an}bracketle{tLs/an}bracketri}ht\ndt=−∂\n∂e·/parenleftBig\ne×Ps/parenrightBig\n+nNs(e)+/integraldisplay\nJLd3Ls(4)\nwhere the magnetic pressure tensor is defined as Ps=\nI−1/integraltext\n(L− /an}bracketle{tLs/an}bracketri}ht)(L− /an}bracketle{tLs/an}bracketri}ht)f d3Ls.\nThe conservation equation (4) states that the rate of\nthe average angular momentum /an}bracketle{tLs/an}bracketri}htis due to three con-\ntributions: an applied torque Ns, an average interaction\nwith the bath/integraltext\nJLd3Ls(i.e. damping), and a torque due\nto pressure (i.e. rotational diffusion).\nThe expressionfor JLis deduced fromthe entropypro-\nduction expression σ(e) [14, 15, 20]. Defining the fer-\nromagnetic chemical potential µ(e,L), the power Tσ(e)\ndissipated by the magnetic system is the product of the\ngeneralized flux by the generalized force:\nTσ(e) =−/integraldisplay\nJL·∂µ\n∂Lsd3Ls (5)\nwhere the chemical potential takes the canonical form\n[14, 16]:\nµ(e,L) =kT ln[f(L,e)]+K(e,L)+VF(e) (6)\nThe application of the second law of thermodynam-\nics, together with the local equilibrium hypothesis in the\n(e,L) space, lead us to the introduce the Onsager matrix\nLsuch that: JL=−L ·∂µ\n∂Ls. As the Onsager coeffi-\ncients are a reflection of the system’s symmetry [15], the\nrelaxation tensor defined as τ−1=1\nfLI−1\nis also axial\nsymmetric: τ−1=τ−1\n1(U−ee)+τ−1\n3ee(whereτ1=τ2\nandτ3are the diagonal coefficients), and is related to\ndamping. Moreover, as eis an axis of symmetry for the\nferromagnetic potential VF(θ,φ), the relaxation tensor\nτ−1is not expected to have any components in the e\ndirection [19], leading to τ−1\n3= 0.\nThe dynamic equation (4) can be rewritten:\nd/an}bracketle{tLs/an}bracketri}ht\ndt=Ns−τ−1\ns·/an}bracketle{tLs/an}bracketri}ht−1\nn∂\n∂e·/parenleftBig\ne×Ps/parenrightBig\n(7)\nAs the inertial tensor Iand the relaxation tensor τ−1\nare time independent in the rotating frame (or magne-\ntization frame), a simpler expression of Eq. (7) can be\nobtained in this frame. After introducing the averagean-\ngular velocity Ωsuch that /an}bracketle{tL/an}bracketri}ht=I·Ω, Eq. (7) rewrites\nas:\ndΩr\ndt=I−1\nr·/bracketleftbigg\nNr−1\nn∂\n∂e·/parenleftBig\ne×P/parenrightBig/bracketrightbigg\n−τ−1\nrot·Ωr(8)3\nThe rotating frame is denoted by the subscript rand\nτ−1\nrot=τ−1\nr−/parenleftBig\nI3\nI1−1/parenrightBig\nΩ3e×U, or\nτ−1\nrot= (τ1α∗)−1\nα∗1 0\n−1α∗0\n0 0 0\n (9)\nwhereα∗=α(I3/I1−1)−1withα= (Ω3τ1)−1.\nThe three components of Eq. (8) read:\n\n\n˙Ω1=−Ω1\nτ1−/parenleftbiggI3\nI1−1/parenrightbigg\nΩ3Ω2−MsH2\nI1−/bracketleftBigg\n1\nI1n∂(e×P)\n∂e/bracketrightBigg\n1\n˙Ω2=−Ω2\nτ1+/parenleftbiggI3\nI1−1/parenrightbigg\nΩ3Ω1+MsH1\nI1−/bracketleftBigg\n1\nI1n∂(e×P)\n∂e/bracketrightBigg\n2\n˙Ω3=−τ−1\n3Ω3= 0\n(10)\nSince the quantity L3=I3Ω3is a constant of motion,\nthe well-known gyromagnetic ratio γcan be defined as\nthe ratio of the magnetization at saturation Msby the\naxial angular momentum γ=Ms\n/angbracketleftL3/angbracketright[2, 10].\nAlso,theaverageddynamicEq. (10)introducesachar-\nacteristic time scale τ1, which separates the behavior of\nthe magnetic system of particles in two regimes: the dif-\nfusion regime or the long time scale limit t≫τ1, and the\ninertial regime or the short time scale limit t≪τ1. Since\nthe modulus of the magnetization Mis conserved, the\nrelationdM\ndt=Ω×Mholds. Cross-multiplying by M\nand using the above definition of γleads to the identity\nΩ=M\nM2s×dM\ndt+M\nI3γ.\nIn a diffusive regime, i.e. for t≫τ1, the inertial terms\ndΩ1\ndtanddΩ2\ndtare negligible with respect toΩ1\nτ1andΩ2\nτ1.\nEq. (8) then rewrites as the Gilbert equation with an\ninertial correction performed on the previously defined\ngyromagnetic coefficient γ∗=γ\n1−I1/I3:\ndM\ndt=γ∗M×/parenleftbigg\nHeff−ηdM\ndt/parenrightbigg\n(11)\nThe Gilbert damping coefficient ηis now defined as:\nη=I1\nτ1M2s(so that α∗=γ∗ηMsis the corresponding\ndimensionless coefficient), and Heffis an effective field\nthat includes the diffusion term.\nAt the diffusive limit, the magnetic moments follow a\ndistribution function f(e,L) close to a Maxwellian cen-\ntered on the average angular momentum /an}bracketle{tL/an}bracketri}ht[9]. This\nleads to a diagonal form for the pressure tensor: P=\nnkT/UandHeff=H−kT\nn1\nMs∂n\n∂e[13, 17].\nEq. (11) containsthe density n(e) so that the equation\nis not closed. However, inserting Eq. (11) into the con-\nservation law (3) leads to the rotational Fokker-Planck\nequation of n(e), derived by Brown [3]:∂n\n∂t=∂(ne×Ω)\n∂e\nFor short enough time scales t≈τ1, the inertial terms\ncannot be neglected and the Gilbert approximation is\nno longer valid. The dynamic equation (10) takes the\nfollowing generalized form:dM\ndt=γM×/bracketleftbigg\nH−η/parenleftbiggdM\ndt+τ1d2M\ndt2/parenrightbigg/bracketrightbigg\n−γ\nn∂(M×P)\n∂M\n(12)\nThe corresponding generalized rotational Fokker-\nPlanck equation for the statistical distribution fis ob-\ntained with replacing JLby the Onsager relation derived\nearlier into the conservation law (1) and rewriting the\nlaw in the rotating frame [20]:\n∂f(e,Lr)\n∂t=\n\n∂(fe×I−1\nrLr)\n∂e\n\n\nLr+\n∂\n∂Lr·/bracketleftbigg\nfτ−1\nrot·Lr−fNr+kTτ−1\nrIr·∂f\n∂Lr/bracketrightbigg\n(13)\nAt short time scales t≈τ1and due to the inertial\neffect, the usual precessional behavior is enriched by a\nnutation effect. The simplest way to understand nuta-\ntion is to imagine that the effective field is switched off\nsuddenly with zero damping: the precession stops sud-\ndenly because the Larmore frequency ωL=γ∗Hdrops\nto zero at the same time. However, in the absence of\ninertial terms, the magnetic moment also stops at this\nposition within an arbitrarily short time scale. But if the\nkinetic energy is not zero (and this is the case for magne-\ntomechanical measurements of the magnetization [10]),\nthe movement cannot be stopped suddenly: the preces-\nsion (around the magnetic field) stops but the magnetic\nmoment starts to rotate around the angular momentum\nvector in order to conserve the energy: the precession is\ntransformed into nutation.\nFig. 1 shows the numerical resolution of Eq. (12) (ne-\nglecting thermal fluctuations) with a field along z axis\nand for a parameters τ1fixed to 2 pswith|α|= 0.05.\nThe trajectories are plotted on the sphere Σ. The usual\ntrajectory deduced from the LLG equation ( τ1≪ps)\nis also plotted for comparison. The motion of the mag-\nnetic moment displays the familiar curve due to Larmor\nprecession,withsuperimposedloopsgeneratedbythenu-\ntation effect. Fig.1(b) shows a trajectory starting with-\nout initial velocity under an effective field of 1 MA/m,\nchanged suddenly to 3 MA/mand once again down to\nzero. Four curves are represented, two for the Eq. (12)\nwith|α|= 0.05 (red continuous line) and α= 0 (red\ndashed line), and two for the usual LLG equation with\nand without damping (blue). At the end of the motion\n(left), the field is set to zero and the precession is de-\nstroyed, with the nutation effect shaping a circle (with-\nout damping) or a spiral (with damping). Note that the\nprofile of the nutation loops depends on the initial condi-\ntions (the cusp presented in Fig.1 instead of loops is due\nto zero initial velocity). Fig. 1 (c) shows the time deriva-\ntive of the angle φas a function of time for the trajectory\ndisplayedinFig. 1(b). Thehorizontallinesrepresentthe4\n01\n-110 20dφ/dt(1012 s-1)\nH = 1 MA/mH = 3 MA/m\nt(ps)H = 0\n(c)(b)\nLLG2 psτ\n1 = (a)\nFIG. 1: Numerical resolution of Eq. (12) with τ1= 2ps. (a)\nTrajectories at two different fields with |α|= 0.05 (red) and\ncurves deduced from the LLG equation (blue). (b) Trajec-\ntoryofthemagnetization withchangingsuddenlytheeffecti ve\nfields from H=1 MA/m to H=3 MA/m and H=0, with damp-\ning (continuous line) and without (dotted line). (c) Time\nderivative of the azimuth angle φplotted as a function of\ntime for the trajectory of Fig. 1(b).\nconstant Larmorfrequencies, and the oscillations are dueto nutation (for |α|= 0.05 andα= 0).\nInthe caseoftwostablestatesseparatedbyapotential\nbarrier(e.g. for magnetic memory units), efficient strate-\ngiesbased on the inertial mechanismcan be perormed for\nultrafast magnetization reversal. Such a strategy has al-\nready been implemented in the case of antiferromagnets\n[11], in which inertial terms are present due to the energy\nstoredbydeformationinthemagneticdomainwalls. The\ninertia has been used to overcomean energy barrier after\nhaving push the magnetization with a very short optical\nimpulsion. The novelty of our results is that any kind\nof ferromagnets could in principle be used for ultra-short\ninertial magnetization switching.\nIn conclusion, we have shown that extending the phase\nspace of the magnetization to the degrees of freedom of\nthe magnetic angular momentum leads to considere a\ngeneralized Landau-Lifshitz-Gilbert equation that con-\ntains inertial terms. This extension is justified by the\nwell-known gyromagnetic relation that relates the mag-\nnetization to the angularmomentum. It is predicted that\ninertial effects should be observed at short enough time\nscales (typicaly below the picosecond), e.g. by measur-\ning nutation loops superimposed to the usual precession\nmotion of a magnetic moment. The inertial regime at\nshort time scales would also offer possibilities for new ex-\nperiments and devices based on ultrafast magnetization\nswitching.\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153\n(1935).\n[2] T. L. Gilbert, Phys. Rev. 100, 1243 (1955) (Abstract\nonly), reprint in IEEE Trans. Mag. 40, 3443 (2004).\n[3] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[4] W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, The\nLangevin equation , World Scientific Series in contempo-\nrary Chemical Physics Vol. 11, 1996.\n[5] E. Fick and S. Sauermann, the quantum statistics of Dy-\nnamic Processes , Springer Series in Solid-States Sciences\n86, 1990.\n[6] E. Beaurepaire etal., Phys.Rev.Lett. 76, 4250 (1996), J.\nHohlfeld etal. Phys.Rev.Lett. 78, 4861 (1997), A.Scholl\net al. Phys. Rev. Lett. 79, 5146 (1997), M. Aeschlimann\net al. Phys. Rev. Lett. 79, 5158 (1997), C. H. Back et\nal. Phys. Rev. Lett. 81, 3251 (1998 ), H. S. Rhie et al.\nPhys. Rev. Lett. 90, 247201 (2003), A. V. Kimel et al.\nNature435, 655 (2005). C. D. Stanciu et al. Phys. Rev.\nLett.99, 047601 (2007),\n[7] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C.\nSiegmann, J. Steohr, G. Ju, B. Lu and D. Weller, Nature\n428, 831 (2004).\n[8] R. Kikuchi, J. of Appl. Phys. 27, 1352 (1956).\n[9] J. M. Rub´ ı, A. P´ erez-Madrid, Physica A, 264(1999) 492.\n[10] The gyromagnetic relation M=γLis observed through\nmagnetomechanical measurements (in isolated systems),as shown by S. J. Barnett (see Rev. Mod. Phys. 7, 129\n(1935)), and A. Einstein and W. J. de Haas (Verh. d. D.\nPhys. Ges. 17, 152 (1915)). Typically |γ| ≈2 105m/As.\n[11] A. V. Kimel et al., Nature Physics, 5, 727 (2009).\n[12] J.-Y. Bigot et al., Nature Physics 5, 515 (2009), B. Koop-\nmans et al., Nature Materials, 9, 259 (2009) and K. H.\nBennemann, Ann. Phys. 18, 480 (2009).\n[13] J.-E. Wegrowe et al. Phys. Rev. B 77, 174408 (2008), and\nJ.-E. Wegrowe, Solid State Com. 150519 (2010).\n[14] D. Reguera, J. M. G. Vilar and J. M. Rub´ ı, J. Phys.\nChem. B, 109, 21502 (2005).\n[15] P. Mazur, Physica A 261, 451 (1998).\n[16] S. R. De Groot and P. Mazur, non-equilibrium thermo-\ndynamics Amsterdam : North-Holland, 1962.\n[17] J.-E. Wegrowe, M. C. Ciornei, H.-J. Drouhin, J. Phys.:\nCondens. Matter 19, 165213 (2007).\n[18] The symmetry of the ferromagnetic potential VF(θ) im-\nposes a constraint on the axial component of the angular\nmomentum, fixing one degree of freedom.\n[19] D. W. Condiff and J. S. Dahler, J. Chem. Phys. 10, 3988\n(1966).\n[20] M.-C. Ciornei, Role of magnetic inertia in damped\nmacrospin dynamics , PhD Thesis, Ecole Polytechnique,\nFrance, January 2010."    },    {        "title": "1010.0268v2.Ferromagnetic_resonance_study_of_Co_Pd_Co_Ni_multilayers_with_perpendicular_anisotropy_irradiated_with_Helium_ions.pdf",        "content": "Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular\nanisotropy irradiated with Helium ions\nJ-M. L. Beaujour, A. D. Kent,1D. Ravelosona,2and I. Tudosa and E. E. Fullerton3\n1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\n2Institut d'Electronique Fondamentale, UMR CNRS 8622,\nUniversite Paris Sud, 91405 Orsay Cedex, France\n3University of California, San Diego, Center for Magnetic Recording Research, La Jolla, CA 92093-0401, USA\n(Dated: April 24, 2022)\nWe present a ferromagnetic resonance (FMR) study of the e\u000bect of Helium ion irradiation on the\nmagnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd\nbilayers. The perpendicular magnetic anisotropy decreases linearly with He ion \ruence, leading to\na transition to in-plane magnetization at a critical \ruence of 5 \u00021014ions/cm2. We \fnd that the\ndamping is nearly independent of \ruence but the FMR linewidth at \fxed frequency has a maximum\nnear the critical \ruence, indicating that the inhomogeneous broadening of the FMR line is a non-\nmonotonic function of the He ion \ruence. Based on an analysis of the angular dependence of the\nFMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude\nof the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be\nused to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy\nparameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies.\nPACS numbers:\nCo/Ni multilayers are of great interest in informa-\ntion technology and in spin-transfer devices because they\ncombine high spin polarization with large perpendicu-\nlar magnetic anisotropy (PMA) [1{3]. Perpendicular\nanisotropy was predicted in Co/Ni multilayers and has\nbeen shown experimentally to be a function of layer com-\nposition and thin \flm growth conditions [4]. Recently\nit has been shown that the coercivity and perpendicu-\nlar anisotropy of a multilayer can be tailored by Helium\nion irradiation [5], making it possible to modify \flms af-\nter growth to tune their magnetic properties. This is\nof great interest for applications and basic physics stud-\nies, as in many cases the perpendicular anisotropy of a\nstructure sets importance device metrics, and ion irradi-\nation o\u000bers the possibility of changing these properties,\nboth locally and globally, after device fabrication. For\ninstance, in spin-transfer magnetic random access mem-\nories (STT-MRAM) the current threshold for switching\nis proportional to the PMA [2, 6].\nLight ion irradiation has been used to vary the mag-\nnetic properties of multilayer \flms in many earlier studies\n[7{11]. For instance, the coercivity of Co/Pt multilayers\nwas found to decrease with ion dose [8]. This behavior\nwas attributed to interface mixing and strain relaxation\nreducing the PMA. Very recently, it was reported that\nthe coercive \feld of Co/Ni multilayers decreases linearly\nwith increasing He+irradiation \ruence up to F= 1015\nions/cm2, suggesting changes in the magnetic anisotropy\nof the \flm [10]. The e\u000bect of ion irradiation on the FMR\nlinewidth has also been studied in Au/Fe multilayer \flms\nwith PMA [11]. The PMA is reduced by He+irradi-\nation and the authors explained this by a reduction of\nthe inhomogeneous contribution to the FMR linewidth.\nIn a recent paper, we presented a FMR study of the\nanisotropy and the linewidth of a Co/Ni multilayer \flmexposed to a relatively high He+irradiation \ruence ( F\n= 1015ions/cm2) [12]. In addition to a strong decrease\nof the PMA, the contribution to the linewidth from spa-\ntial variation of the anisotropy, was reduced compared\nto that of a non irradiated Co/Ni multilayer. Further-\nmore, a correlation between the anisotropy distribution\nand the linewidth broadening from two-magnon scatter-\ning (TMS) mechanism was observed. However, a system-\natic study of the e\u000bect of the He+irradiation on the FMR\nspectra as a function of \ruence has yet to be reported.\nIn this paper, we present a FMR study of a Co/Ni\nmultilayer coupled to Co/Pd bilayers exposed to Helium\nion irradiation of \ruence up to 1015ions/cm2. The PMA\nand the contributions to the FMR linewidth, including\nthose from Gilbert damping ( \u000b), are studied as a function\nof \ruence.\nThe samples had the following layer structure: jj3 Taj1\nPdj0.3 Coj1 Pdj0.14 Coj[0.8 Nij0.14 Co]\u00023j1 Pdj0.3 Coj1\nPdj3 Tajj(layer thickness in nm) and was fabricated by dc\nmagnetron sputtering. The Co/Ni multilayer is embed-\nded between Co/Pd bilayers to enhance the overall PMA\nof the \flm and to have resonance frequencies in which the\nfull angular dependence of the FMR response could be\ninvestigated in a 1 T electromagnet. The substrate was\ncleaved into several pieces that were then exposed to dif-\nferent doses of Helium ion irradiation of energy 20 keV\nwith \ruence in the range 1014\u0014F\u00141015ions/cm2.\nFMR measurements were conducted at room tempera-\nture using a coplanar waveguide (CPW). Details of the\nexperimental setup can be found in [13]. The \feld swept\nCPW transmission signal ( S21) was recorded as a func-\ntion of frequency for dc magnetic \felds normal to the \flm\nplane and as a function of the out-of-plane \feld angle at\n20 GHz. The magnetization density of the \flm at room\ntemperature was measured with a SQUID magnetome-arXiv:1010.0268v2  [cond-mat.mes-hall]  5 Oct 20102\n03 06 09 00.40.60.8K\nK0\n5 1 001020H\n /s615490Hres  (T) 0 \n1 \n2.5 \n F\nield angle /s61542H  (deg)20 GHz 5 \n7.5 \n10HH\n-0.40.00.40.80\n102030f  (GHz)(a) \n(b) /s615490Hres    (T) \n/s61549 0Meff (\nc)  \nF (1014 ions/cm2)K (105 J/m3)21\nFIG. 1: a) Angular dependence of the resonance \feld for dif-\nferent irradiation \ruence ( \u00021014ions/cm2). The solid lines\nare guides to the eye. b) Frequency dependence of the reso-\nnance \feld when the applied \feld is normal to the \flm plane\n(\u001eH= 90o) at selected \ruences. The solid lines are the \fts\nto Eq. 1. The zero frequency intercept gives the e\u000bective de-\nmagnetization \feld, \u00160Me\u000b. c) The second and fourth order\nperpendicular anisotropy constants, K1andK2, versus \ru-\nence.\nter:Ms'4:75\u0002105A/m. Within the measurement\nuncertainty, Msremains unchanged after irradiation.\nFig. 1a shows the out of plane angular dependence\nof the resonance \feld at 20 GHz for di\u000berent \ruences.\nFor the non-irradiated \flm, the resonance \feld when H\nis normal to the \flm plane ( \u001eH= 90o) is smaller than\nthat when the \feld is in the \flm plane ( \u001eH= 0o). This\nshows that the magnetic easy axis is normal to the \flm\nplane. As the \ruence increases, Hresat\u001eH= 90oin-\ncreases whereas that at \u001eH= 0odecreases. In the high\n\ruence range, F>5\u00021014ions/cm2,Hresis the larger\nwhen the \feld is normal to the \flm plane, i.e. the mag-\nnetic easy axis is in the \flm plane. Fig. 1b shows the\nfrequency dependence of the resonance \feld for di\u000berent\n\ruences when the dc \feld is normal to the \flm plane.\nThis data is \ftted to the resonance condition [14]:\nf=1\n2\u0019\r\u00160(Hres\u0000Me\u000b); (1)\nwhere\ris the gyromagnetic ratio. \u00160Me\u000b, the e\u000bec-\ntive easy plane anisotropy, is given by: Me\u000b=Ms\u0000\n2K1=(\u00160Ms), whereK1is the second order anisotropy\nconstant. We \fnd that \u00160Me\u000bis negative at low \ru-\nence which implies that the PMA is su\u000ecient to over-\ncome the demagnetizing energy and hence the easy axis\nis normal to the \flm plane. As the \ruence is further\nincreased,\u00160Me\u000bbecomes positive. These results con-\n\frm that there is a re-orientation of the easy axis, as was\ninferred indirectly through magnetic hysteresis loop mea-\nsurements in Ref. [10]. \u00160Me\u000bchanges sign for \ruence\nbetween 5 and 7.5 \u00021014ions/cm2. Therefore, by expos-\ning the \flm to a speci\fc \ruence, it is posible to engineer\n01000\n1000\n3 06 09 001000\n102030020406080 \n \nF=0Δ\nHα       ΔHinh         ΔHtot  \nF=5/s615490ΔH  (mT)f\n/s61472 ( GHz )  F=10F\nield angle /s61542H  ( deg )HH /s615490ΔH  (mT) \n \nF=7.5H\nFIG. 2: On the left, the linewidth as a function of frequency\nfor the \flm irradiated at 7.5 \u00021014ions/cm2. The solid line\nis a linear \ft to the experimental data. On the right, the\nangular dependence of the linewidth at 20 GHz for a selec-\ntion of \ruences. The solid lines represent the \fts to the total\nlinewidth \u0001 Htot= \u0001H\u000b+\u0001Hinh, , where the intrinsic damp-\ning and the inhomogeneous contribution are represented by\nthe dashed line and the dotted line respectively.\nthe anisotropy so that the PMA \feld just compensates\nthe demagnetization \feld.\nThe second order perpendicular anisotropy constant\nK1decreases linearly with \ruence (Fig. 1c). The \flm\nirradiated at 1015ions/cm2has an anisotropy constant\n40% smaller than that of the non-irradiated \flm. The\n4thorder anisotropy constant K2is determined from\nthe angular dependence of Hresfor magnetization angles\n45\u0014\u001e\u001490o[13].K2is smaller than K1by a factor 10,\nand is nearly independent of \ruence.\nThe FMR linewidth \u00160\u0001Hwhen the dc \feld is ap-\nplied normal the \flm plane was measured as a function\nof frequency. Fig. 2 shows \u00160\u0001Hversusffor the \flm\nirradiated at F= 7:5\u00021014ions/cm2. The linewidth in-\ncreases linearly with frequency, characteristic of Gilbert\ndamping, an intrinsic contribution to the linewidth \u0001 H\u000b\n[15]:\n\u0001H\u000b=4\u0019\u000b\n\u00160\rf: (2)\nFrom a linear \ft to the experimental data, the magnetic\ndamping constant is estimated from the slope of the line:\n\u000b= 0:037\u00060:004. The \flms irradiated at F= 0;1 and\n10\u00021014ions/cm2shows a similar frequency dependence\nof the linewidth and have about the same damping con-\nstant,\u000b\u00190:04. At intermediate \ruence F= 2:5, 5\u00021014\nions/cm2, the linewidth is enhanced and is frequency in-\ndependent, i.e. the linewidth is dominated by an inhomo-\ngeneous contribution, \u0001 Hinh. The angular dependence\nof the linewidth measured at 20 GHz is shown in Fig.\n2 for \flms irradiated at selected \ruences. For the non-\nirradiated \flm and the \flm irradiated at 1015ions/cm2,\nthe linewidth is practically independent of the \feld an-3\n05 1 01 53060901200\n5100.030.04µ0 Δ/s61512/s61472 \n ⊥ \n(mT)F\n  (1014 ions/cm2)ΔK1  ( 105 J/m3 )H\n0.00.10.20.30.4 \nα-dampingF\n (1014 ions/cm2)\nFIG. 3: The \ruence dependence of the linewidth at 20 GHz\nwhen the dc \feld is normal to the \flm plane (squares). The\nsolid circles represent the \ruence dependence of the distribu-\ntion in the PMA constant K1determined from \ftting \u0001 Hvs.\n\u001eH. The inset shows the Gilbert damping constant \u000bas a\nfunction of \ruence.\ngle from about 30oup to 90o. For the \flm irradiated at\n5\u00021014ions/cm2, \u0001His clearly angular dependent and\nshows a minimum at an intermediate \feld angle.\nThe angular dependence of the linewidth was \ft to\na sum of the intrinsic linewidth \u0001 H\u000band an inhomo-\ngeneous contribution \u0001 Hinhfor magnetization angles\n45o\u0014\u001e\u001490o, an angular range in which TMS does\nnot contribute to the linewidth [13]. The inhomogeneous\nlinewidth is given by:\n\u0001Hinh:(\u001eH) =j@Hres=@K 1j\u0001K1+j@Hres=@\u001ej\u0001\u001e;(3)\nwhere \u0001K1is the width of the distribution of anisotropies\nand \u0001\u001eis the distribution of the angles of the magnetic\neasy axis relative to the \flm normal. The computed\nlinewidth contributions are shown for the \flm irradiated\natF= 5\u00021014ions/cm2in Fig. 2. Note that the intrin-\nsic contribution \u0001 H\u000bis practically independent of \feld\nangles, as expected when the angle between the magne-\ntization and the applied \feld is small. For this sample,\nthe maximum angle is about 5oand it is due to the fact\nthat the resonance \feld ( Hres'0:6 T) is much larger\nthan the e\u000bective demagnetization \feld ( Me\u000b'0). Theinhomogeneous contribution from the distribution in the\nanisotropy \feld directions does not signi\fcantly a\u000bect the\n\ft. For the \flm irradiated at the lower and upper \ruence\nrange, the angular dependence of the intrinsic linewidth\nis computed \fxing the value of \u000bto that obtained from\nthe \ft of the frequency dependence of the linewidth. For\nthe other \flms ( F=2.5 and 5\u00021014ions/cm2),\u000bwas\na \ftting parameter.\nThe \ruence dependence of \u0001 K1and the linewidth at\n20 GHz are shown in Fig. 3. The inset shows the Gilbert\ndamping constant as a function of \ruence. The linewidth\nat 20 GHz when the \feld is normal to the \flm plane is\na non monotonic function of \ruence. \u0001 Hincreases as\nthe \ruence increases, reaching a maximum value at F\n\u00195\u00021014ions/cm2. Then, as the \ruence is further in-\ncreased, \u0001Hdecreases and falls slightly below the range\nof values at the lower \ruence range. Interestingly, the\nlarger linewidth is observed just at the \ruence for which\n\u00160Me\u000b= 0. The magnetic damping is practically not\na\u000bected by irradiation within the error bars: \u000b\u00190:04.\nThe distribution of PMA constants, \u0001 K1, shows a similar\n\ruence dependence as the total linewidth, with a maxi-\nmum at F\u00195\u00021014ions/cm2, clearly indicating that\nthis is at the origin of the \ruence dependence of the mea-\nsured linewidth. The distribution in PMA anisotropy is\nalmost zero when the \ruence is above 7 \u00021014ions/cm2.\nThe largest value of \u0001 K1corresponds to variation of K1\nofabout 8%, which is much larger than that of non irradi-\nated \flm and the highly irradiated \flm, \u0001 K1=K1\u00192%\nand 0.3% respectively.\nIn summary, irradiation of Co/Pd/Co/Ni \flms with\nHelium ions leads to clear changes in its magnetic char-\nacteristics, a signi\fcant decrease in magnetic anisotropy\nand a change in the distribution of magnetic anisotropies.\nImportantly, this is achieved without a\u000becting the \flm\nmagnetization density and magnetic damping, which re-\nmain virtually unchanged. It would be of interest to have\na better understanding of the origin of the maximum in\nthe distribution of magnetic anisotropy at the critical\n\ruence, the \ruence needed to produce a reorientation of\nthe magnetic easy axis. Nonetheless, these results clearly\ndemonstrate that ion irradiation may be used to system-\natically tailor the magnetic properties of Co/Pd/Co/Ni\nmultilayers for applications and basic physics studies.\n[1] S. Mangin et al: , Nat. Mater. 5, 210 (2006).\n[2] S. Mangin et al: , Appl. Phys. Lett. 94, 012502 (2009).\n[3] D. Bedau et al: , Appl. Phys. Lett. 96, 022514 (2010).\n[4] G. H. O. Daalderop et al: , Phys. Rev. Lett. 68, 682\n(1992); S. Girod et al: , Appl. Phys. Lett. 94, 262504\n(2009).\n[5] C. Chappert et al: , Science 280, 1919 (1998).\n[6] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[7] A. Traverse et al: , Europhysics Letters 8, 633 (1989).\n[8] T. Devolder, Phys. Rev. B 62,5794 (2000).\n[9] J. Fassbender etal: , J. Phys. D: J. Appl. 37, R179 (2004).[10] D. Stanescu et al: , J. Appl. Phys. 103, 07B529 (2008).\n[11] C. Bilzer et al: , J. Appl. Phys. 103, 07B518 (2008).\n[12] J.-M. L. Beaujour et al: , Phys. Rev. B 80, 180415(R)\n(2009).\n[13] J.-M. L. Beaujour etal: , Eur. Phys. J. B 59, 475 (2007).\n[14] S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon,\nOxford, 1966).\n[15] Spin Dynamics in Con\fned Magnetic Structures II\n(edited by B. Hillebrands and K. Ounadjela (Springer,\nHeidelberg, 2002))."    },    {        "title": "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": "1010.1537v1.Power_optimization_for_domain_wall_motion_in_ferromagnetic_nanowires.pdf",        "content": "arXiv:1010.1537v1  [cond-mat.mes-hall]  7 Oct 2010Power optimization for domain wall motion in ferromagnetic nanowires\nO. A. Tretiakov,1Y. Liu,1and Ar. Abanov1\nDepartment of Physics & Astronomy, Texas A&M University, Co llege Station, Texas 77843-4242,\nUSA\n(Dated: October 7, 2010)\nThe current mediated domain-wall dynamics in a thin ferromagnetic w ire is investigated. We derive the\neffective equations of motion of the domain wall. They are used to stu dy the possibility to optimize the power\nsupplied by electric current for the motion of domain walls in a nanowire . We show that a certain resonant\ntime-dependent current moving a domain wall can significantly reduc e the Joule heating in the wire, and\nthus it can lead to a novel proposal for the most energy efficient me mory devices. We discuss how Gilbert\ndamping, non-adiabatic spin transfer torque, and the presence o f Dzyaloshinskii-Moriya interaction can effect\nthis power optimization.\nIntroduction. Due to its direct relevance to future\nmemory and logic devices, the dynamics of domain walls\n(DW) in magnetic nanowires has become recently a very\npopulartopic.1–3Therearemainlytwogoalswhichscien-\ntists try to achieve in this field. One goal is to move the\ndomain walls with higher velocity in order to make faster\nmemory or computer logic. The other one is inspired by\nthe modern trend of energy conservation and concerns a\npower optimization of the domain-wall devices.\nGenerally, the domain walls can be manipulated\nwhether by a magnetic field3,4or electric current.1,5Al-\nthough the latter method is preferred for industrial ap-\nplications due to the difficulty with the application of\nmagnetic fields locally to small wires. For this reason, we\nconsider in this paper the current induced domain-wall\ndynamics. We make a proposal on how to optimize the\npowerfor the DWmotionby meansofreducingthe losses\non Joule heating in ferromagnetic nanowires.6Moreover,\nbecause the averaged over time (often called drift) veloc-\nity of a DW generally increases with applied current, we\nalso address the first goal. Namely, our proposal allows\nto move the DWs with higher current densities without\nburning the wire by the excessive heat and thus archive\nhigher drift velocities of DWs. The central idea of this\nproposal is to employ resonant time-dependent current\nto move DWs, where the period of the current pulses is\nrelated to the periodic motion of DW internal degrees of\nfreedom.\nThe schematic view of a domain wall in a narrow fer-\nromagnetic wire is shown in Fig. 1. These DWs are char-\nFIG. 1. (color online) A schematic view of a current-driven\ndomain wall in a ferromagnetic wire. The DW width is ∆.acterized by their width ∆ which is mainly determined\nby exchange interaction and anisotropy along the wire λ.\nAnother important quantity is the transverse anisotropy\nacrossthewire K, whichgovernsthepinningofthetrans-\nverse component of the DW magnetization. When no\ncurrent is applied to the wire it leads to two degenerate\npositions of the transverse magnetization component of\nthe wall: as shown in Fig. 1 and anti-parallel to it.\nTo describe the dynamics of DW in a thin wire we\nderived the effective equations of motion from general-\nized Landau-Lifshitz-Gilbert7,8(LLG) equation with the\ncurrentJ,\n˙S=S×Heff−J∂S\n∂z+βJS×∂S\n∂z+αS×˙S,(1)\nwhereSis magnetization unit vector, Heff=δH/δSis\nthe effective magnetic field given by the Hamiltonian H\nof the system, βis non-adiabatic spin torque constant,\nandαis Gilbert damping constant. The derivation of\nthe effective equations of motion is based on the fact\nthat in thin ferromagnetic wires the static DWs are rigid\ntopologically constrained spin-textures. Therefore, for\nnot too strong drive, their dynamics can be described\nin terms of only a few collective coordinates associated\nwith the DW degrees of freedom.9In very thin wires,\nthere are two collective coordinates corresponding to two\nsoftest modes of the DW motion: the DW position along\nthe wire z0and the magnetization angle φin the DW\naround the wire axis. All other degrees of freedom are\ngapped by strong anisotropic energy along the wire.\nBy applying the orthogonality condition to LLG, one\ncan obtain the equations of motion for the two DW soft-\nest modes, z0(t) andφ(t),10\n˙z0=AJ+B[J−jcsin(2φ)], (2)\n˙φ=C[J−jcsin(2φ)], (3)\nwhereJ(t) is a time-dependent current. The co-\nefficients A,B,C, and critical current jccan be\nevaluated for a particular model in terms of α,β\nand other microscopic parameters. Following Ref. 10,\nfor the model with Dzyaloshinskii-Moriya interaction\n(DMI) one can find A=β/α,B= (α−β)(1 +\nαΓ∆)/[α(1 +α2)],C= (α−β)∆/[(1 +α2)∆2\n0], and2\nFIG. 2. (color online) DW motion characteristics for dc cur-\nrents. (a) Drift velocity Vdof DW as a function of current J\nforB >0 andB <0, see Eq. (2). The slope at J < jcis given\nbyA, whereas at J≫jcit isA+B. (b) Power of Ohmic\nlossespdc(Vd/Vc) =J2/j2\ncas a function of drift velocity Vd.\nForB <0 the power has a discontinuity at Vd/Vc= 1.\njc= (αK∆/|α−β|)[πΓ∆/sinh(πΓ∆)], where Jexis ex-\nchange constant, Dis DMI constant, and Γ = D/Jex.\nAlso, ∆ = ∆ 0//radicalbig\n1−Γ2∆2\n0where ∆ 0is the DW width in\nthe absence of DMI.\nAlternatively, Eqs. (2) and (3) can be obtained in a\nmore general framework by means of symmetry argu-\nments. We note that because of the translational invari-\nance ˙z0and˙φcannot depend on z0. Furthermore, to the\nfirst order in small transverse anisotropy K,˙φand ˙z0are\nproportional to the first harmonic sin(2 φ). Then the ex-\npansion in small current Jup to a linear in Jorder gives\nEqs. (2) and (3). In this case the coefficients A,B,C,\nandjchave to be determined directly from experimental\nmeasurements.11,12\nFor the dc current applied to the wire the DW dy-\nnamics governed by Eqs. (2) and (3) can be obtained\nexplicitly.10ForJ < j candA/negationslash= 0 the DW only\nmoves along the wire and is tilted on angle φ0from\nthe transverse-anisotropy easy axis given by condition\nsin(2φ0) =J/jc. Thedriftvelocityis Vd=/angbracketleft˙z0(J)/angbracketright=AJ,\nsee Eq. (2). Therefore, the linear slope of Vd(J) belowjc\ngives constant A, see Fig. 2 (a). The value of jcis deter-\nmined as the endpoint of this linear regime. At J=jc\nthe magnetization angle becomes perpendicular to the\neasy axis, φ0=π/2. ForJ > jcthe DW both moves and\nrotates, and Eqs. (2) and (3) give Vd=AJ+B/radicalbig\nJ2−j2c,\nso that the slope of Vd(J) at large JgivesA+B.\nPower optimization. The largestlossesin the nanowire\nwith a DW are the Ohmic losses of the current. In gen-\neral, the influence of the DW on the resistance is negli-\ngible and therefore we can assume that the resistance of\nthe wire is constant with time. Then the time-averaged\npower of Ohmic losses is proportional to /angbracketleftJ2(t)/angbracketright. Since\nthe resistance is almost constant, in this paper we will\ncalculate P=/angbracketleftJ2(t)/angbracketrightand loosely call it the power of\nOhmic losses. Our goal is to minimize the Ohmic losses\nwhile keeping the DW moving with a given constant drift\nvelocity.\nFor the following it will be convenient to introduce the\ndimensionless variables for time, drift velocity, current,\npower, and the ratio of slopes of Vd(J) at large and smallcurrents,\nτ=Cjct, vd=Vd\nVc, j=J\njc, p=P\nj2c, a=A+B\nA.\n(4)\nAlthough we note that in the special case of α=β,\nit can be shown that C=B= 0 and one cannot use\ndimensionless variables (4). However, in this case the\nDW dynamics is trivial:13the DW does not rotate φ=\n0,πand moves with the velocity ˙ z0=J.\nFirst, we consider the case of dc current and the power\nas a function of drift velocity. For vd<1 we find pdc=\nv2\nd. For currents above jcthe power pdc(vd) =j2is given\nintermsofdriftvelocity vd=j+(B/A)/radicalbig\nj2−1asshown\nin Fig. 2 (b). The poweris quadraticin vd, and for B <0\nit has a discontinuity at vd= 1.\nIn general, the DW motion has some period Tand\ncurrentj(τ) must be a periodic function with the same\nTto minimize the Ohmic losses. Measuring the angle\nfrom the hard axis instead of easy axis and scaling it\nby 2, i.e, 2 φ=θ−π/2, we can write the dimensionless\ncurrent drift velocity as6\nj(τ) =˙θ/2−cosθ, vd=a\n2/angbracketleft˙θ/angbracketright−/angbracketleftcosθ/angbracketright,(5)\nwhere˙θ=∂θ/∂τ.\nTo minimize the power of Ohmic losses we need to find\nthe minimum of /angbracketleftj2(τ)/angbracketrightat fixedvd,\np=/angbracketleftBig\n(˙θ/2−cosθ)2−2ρ(a˙θ/2−cosθ−vd)/angbracketrightBig\n,(6)\nwhere we use a Lagrange multiplier 2 ρto account for the\nconstraint given by vdfrom Eq. (5). Power (6) can be\nconsidered as an effective action for a particle in a peri-\nodic potential U, and its minimization gives the equation\nof motion ¨θ/2 =−∂U/∂θwhich in turn can be reduced\nto\n˙θ=±2/radicalbig\nd−U(θ,ρ), U(θ,ρ) =−cos2θ−2ρcosθ.\n(7)\nwheredis an arbitrary constant. Since changing ρ→ −ρ\ninUof Eq. (7) is equivalent to changing θ→π+θ, below\nwe can consider only positive ρ.\nEq. (7) shows that there are two different regimes: 1)\nthe bounded regime where d <max[U(θ,ρ)] in which\ncaseθis bounded, and the particle oscillates in potential\nwellU(θ), see inset of Fig. 3 (a); and 2) the rotational\nregime where d >max[U(θ,ρ)] with freely rotating mag-\nnetization in the DW.\nIn the bounded regime the particle moves between the\ntwo turning points −θ0andθ0given by d=U(±θ0,ρ).\nSinceθis a bounded function /angbracketleft˙θ/angbracketright= 0 and vd=−/angbracketleftcosθ/angbracketright.\nOne can show6that in this regime the power of Ohmic\nlosses is minimal for dc current, i.e., p=v2\nd.\nIn the rotational regime the term in Eq. (5) with /angbracketleft˙θ/angbracketright\nshould be kept because θis not bounded. The equation\nof motion is the same as for a nonlinear oscillator.6Using3\n0 00 0.2 0.4 0.6 0.8 1 1.2 1.402468\n1-2-\n--\n01\n10\nFIG. 3. (color online) (a) Minimal power of Ohmic losses\np=/angbracketleftJ2/angbracketright/j2\ncas a function of drift velocity Vdshown by solid\nline fora= 0.5. The dashed line depicts pfor dc current. The\ninset shows the potential U(θ) in which a “particle” is moving\nin the bounded (pendulum-like) and unbounded (rotational)\nregimes. A sketch of /angbracketleftJ2/angbracketright(Vd) shown by solid line in (b) for\nβ≫α(a≪1) and (c) for β≪α(a≫1).\nthe minimization condition ∂p/∂ρ|vd= 0 one finds\n/integraldisplayπ\n−π/radicalbig\nd−U(θ,ρ)dθ= 2πaρ. (8)\nThis equation defines the relationship between dandρ.\nThe results for the minimal power of Ohmic losses\np(vd) are presented in Fig. 3. For a >1 there is a crit-\nical velocity vrc<1, such that at vd< vrcthe power\nof Ohmic losses is p=v2\nd=pdc. Above vrcone can\nminimize the Ohmic losses by moving DW with resonant\ncurrent pulses. Right above vrcthere is a certain rangeof\nvdwherep= 2ρ0vd−ρ2\n0withρ0(a)<1 given by Eq. (8)\nwithd=ρ2. The critical velocity is found as vrc=ρ0(a).\nFora <1, see e.g. Fig. 3 (a), we find that vrc= 1,\nwhereas at vd>1 minimal power pis significantly lower\nthanpdc. Immediately above vd= 1 we find that there\nis a range of vdwherepis linear in vd. At large vdthe\nminimal power is always smaller than pdc, the difference\nbetween them then approaches pdc−p= (1−1/a)2/2.\nWe note that even in the limiting cases of the systems\nwith weak ( β≪α) or strong ( β≫α) non-adiabatic spin\ntransfer torque, see Fig. 3 (b) and (c), where the power\nof Ohmic losses is high for dc currents, the optimized ac\ncurrent gives dramatic reduction in heating power thus\ngreatly expanding the range of materials which can be\nused for spintronic devices.1,3We also note that DMI\nsuppresses critical current jcand affects parameter a.\nForvd< vrcthe optimal current coincides with the dc\ncurrent, above vrcthe resonant current j(t) is plotted in\nFig. 4 for a= 2 and two different velocities vd. Atvd>\nvrcthe current’s maximum jmaxincreases from 2 −vrc\nat small enough vd<∼1 up to jmax≈vd/aatvd≫\n1. The current’s minimum increases monotonically from0 10 20 30 40 50 600123\nFIG. 4. (color online) Resonant time-dependent current J(τ)\nwithτ=Cjctfor drift velocities vd= 0.5 (dashed line) and\nvd= 4.5 (solid line) for a= 2.\nsmall positive values jmin=vrcatvd∼1 up tojmin=\njmax−2|1−a|/aatvd≫1. Atvd<∼1 (fora >1) the\ntime between the current picks decreases with increasing\nvelocity as T≃(πa−2arcsinvrc)/(vd−vrc), whereas the\npick’s width is given by ≈1.3//radicalbig\n(1−vrc). Therefore, at\nsmallvd−vrcthe picks are widely separated, then as vd\nincreases the time between the picks decreases. At vd≫\n1 the optimal current has a large constant component\nand small-amplitude ac modulations on top of it.\nConclusions. We have studied the current driven DW\ndynamicsinthinferromagneticwires. Theultimatelower\nbound for the Ohmic losses in the wire has been found\nfor any DW drift velocity Vd. We have obtained the ex-\nplicit time-dependence of the current which minimizes\nthe Ohmic losses. We believe that the use of these res-\nonant current pulses instead of dc current can help to\ndramatically reduce heating of the wire for any Vd.\nWe thank Jairo Sinova for valuable discussions. This\nwork was supported by the NSF Grant No. 0757992 and\nWelch Foundation (A-1678).\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190\n(2008); M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and\nS. S. P. Parkin, ibid.320, 209 (2008).\n2D.A. Allwood, G. Xiong, M. D.Cooke, C. C. Faulkner, D.Atkin-\nson, N. Vernier, and R. P. Cowburn, Science 296, 2003 (2002).\n3D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit ,\nand R. P. Cowburn, Science 309, 1688 (2005).\n4T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and\nT. Shinjo, Science 284, 468 (1999).\n5A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and\nT. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n6O. A. Tretiakov, Y. Liu, and Ar. Abanov, Phys. Rev. Lett., in\npress; arXiv:1006.0725.\n7Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004).\n8A. Thiaville et al., Europhys. Lett. 69, 990 (2005).\n9O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008); D. J.\nClarke, O. A. Tretiakov, G.-W. Chern, Y. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. B 78, 134412 (2008).\n10O. A. Tretiakov and Ar. Abanov, Phys. Rev. Lett. 105, 157201\n(2010).\n11S. A. Yang, G. S. D. Beach, C. Knutson,\nD. Xiao, Q. Niu, M. Tsoi, and J. L. Erskine,\nPhys. Rev. Lett. 102, 067201 (2009).\n12Y. Liu, O. Tretiakov, and Ar. Abanov, (unpublished).\n13S. E. Barnes and S. Maekawa,\nPhys. Rev. Lett. 95, 107204 (2005)."    },    {        "title": "1010.1626v3.A_unified_first_principles_study_of_Gilbert_damping__spin_flip_diffusion_and_resistivity_in_transition_metal_alloys.pdf",        "content": "A uni\fed \frst-principles study of Gilbert damping, spin-\rip di\u000busion and resistivity\nin transition metal alloys\nAnton A. Starikov,1Paul J. Kelly,1Arne Brataas,2Yaroslav Tserkovnyak,3and Gerrit E. W. Bauer4\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands\n2Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n4Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: October 25, 2018)\nUsing a formulation of \frst-principles scattering theory that includes disorder and spin-orbit\ncoupling on an equal footing, we calculate the resistivity \u001a, spin \rip di\u000busion length lsfand the\nGilbert damping parameter \u000bfor Ni 1\u0000xFexsubstitutional alloys as a function of x. For the tech-\nnologically important Ni 80Fe20alloy, permalloy, we calculate values of \u001a= 3:5\u00060:15\u0016Ohm-cm,\nlsf= 5:5\u00060:3 nm, and \u000b= 0:0046\u00060:0001 compared to experimental low-temperature values in\nthe range 4 :2\u00004:8\u0016Ohm-cm for \u001a, 5:0\u00006:0 nm forlsf, and 0:004\u00000:013 for\u000bindicating that the\ntheoretical formalism captures the most important contributions to these parameters.\nPACS numbers: 72.25.Rb, 71.70.Ej, 72.25.Ba, 75.40.Gb, 76.60.Es\nIntroduction. The drive to increase the density and\nspeed of magnetic forms of data storage has focussed at-\ntention on how magnetization changes in response to ex-\nternal \felds and currents, on shorter length- and time-\nscales [1]. The dynamics of a magnetization Min an ef-\nfective magnetic \feld He\u000bis commonly described using\nthe phenomenological Landau-Lifshitz-Gilbert equation\ndM\ndt=\u0000\rM\u0002He\u000b+M\u0002\"~G(M)\n\rM2sdM\ndt#\n;(1)\nwhereMs=jMjis the saturation magnetization, ~G(M)\nis the Gilbert damping parameter (that is in general a\ntensor) and the gyromagnetic ratio \r=g\u0016B=~is ex-\npressed in terms of the Bohr magneton \u0016Band the Land\u0013 e\ngfactor, which is approximately 2 for itinerant ferromag-\nnets. The time decay of a magnetization precession is\nfrequently expressed in terms of the dimensionless pa-\nrameter\u000bgiven by the diagonal element of ~G=\rMsfor\nan isotropic medium. If a non-equilibrium magnetization\nis generated in a disordered metal (for example by inject-\ning a current through an interface), its spatial decay is\ndescribed by the di\u000busion equation\n@2\u0001\u0016\n@z2=\u0001\u0016\nl2\nsf(2)\nin terms of the spin accumulation \u0001 \u0016, the di\u000berence be-\ntween the spin-dependent electrochemical potentials \u0016s\nfor up and down spins, and the spin-\rip di\u000busion length\nlsf[2, 3]. In spite of the great importance of \u000bandlsf,\nour understanding of the factors that contribute to their\nnumerical values is at best sketchy. For clean ferromag-\nnetic metals [4] and ordered alloys [5] however, recent\nprogress has been made in calculating the Gilbert damp-\ning using the Torque Correlation Model (TCM) [6] and\nthe relaxation time approximation in the framework ofthe Boltzmann equation. Estimating the relaxation time\nfor particular materials and scattering mechanisms is in\ngeneral a non-trivial task and application of the TCM to\nnon-periodic systems entails many additional complica-\ntions and has not yet been demonstrated. Thus, the the-\noretical study of Gilbert damping or spin-\rip scattering\nin disordered alloys and their calculation for particular\nmaterials with intrinsic disorder remain open questions.\nMethod. In this paper we calculate the resistivity \u001a,\nspin-\rip di\u000busion length lsfand Gilbert damping param-\neter\u000bfor substitutional Ni 1\u0000xFexalloys within a single\n\frst-principles framework. To do so, we have extended a\nscattering formalism [7] based upon the local spin den-\nsity approximation (LSDA) of density functional theory\n(DFT) so that spin-orbit coupling (SOC) and chemical\ndisorder are included on an equal footing. Relativistic\ne\u000bects are included by using the Pauli Hamiltonian.\nFor a disordered region of ferromagnetic (F) alloy sand-\nwiched between leads of non-magnetic (N) material, the\nscattering matrix Srelates incoming and outgoing states\nin terms of re\rection ( r) and transmission matrices ( t)\nat the Fermi energy. To calculate the scattering ma-\ntrix, we use a \\wave-function matching\" (WFM) scheme\n[7] implemented with a minimal basis of tight-binding\nlinearized mu\u000en-tin orbitals (TB-LMTOs) [8]. Atomic-\nsphere-approximation (ASA) potentials [8] are calculated\nself-consistently using a surface Green's function (SGF)\nmethod also implemented [9] with TB-LMTOs. Charge\nand spin densities for binary alloy AandBsites are calcu-\nlated using the coherent potential approximation (CPA)\n[10] generalized to layer structures [9]. For the transmis-\nsion matrix calculation, the resulting spherical potentials\nare assigned randomly to sites in large lateral supercells\n(SC) subject to maintenance of the appropriate concen-\ntration of the alloy [7]. Solving the transport problem\nusing lateral supercells makes it possible to go beyondarXiv:1010.1626v3  [cond-mat.mtrl-sci]  19 May 20112\ne\u000bective medium approximations such as the CPA. Be-\ncause we are interested in the properties of bulk alloys,\nthe leads can be chosen for convenience and we use Cu\nleads with a single scattering state for each value of crys-\ntal momentum, kk. The alloy lattice constants are de-\ntermined using Vegard's law and the lattice constants of\nthe leads are made to match. Though NiFe is fcc only\nfor the concentration range 0 \u0014x\u00140:6, we use the fcc\nstructure for all values of x.\nFor the self-consistent SGF calculations (without\nSOC), the two-dimensional (2D) Brillouin zone (BZ) cor-\nresponding to the 1 \u00021 interface unit cell was sampled\nwith a 120\u0002120 grid. Transport calculations including\nspin-orbit coupling were performed with a 32 \u000232 2D BZ\ngrid for a 5\u00025 lateral supercell, which is equivalent to\na 160\u0002160 grid in the 1 \u00021 2D BZ. The thickness of\nthe ferromagnetic layer ranged from 3 to 200 monolay-\ners of fcc alloy; for the largest thicknesses, the scattering\nregion contained more than 5000 atoms. For every thick-\nness of ferromagnetic alloy, we averaged over a number\nof random disorder con\fgurations; the sample to sample\nspread was small and typically only \fve con\fgurations\nwere necessary.\nResistivity. We calculate the electrical resistivity to\nillustrate our methodology. In the Landauer-B uttiker\nformalism, the conductance can be expressed in terms of\nthe transmission matrix tasG= (e2=h)Tr\b\ntty\t\n[11, 12].\nThe resistance of the complete system consisting of ideal\nleads sandwiching a layer of ferromagnetic alloy of thick-\nnessLisR(L) = 1=G(L) = 1=GSh+ 2Rif+Rb(L) where\nGSh=\u0000\n2e2=h\u0001\nNis the Sharvin conductance of each lead\nwithNconductance channels per spin, Rifis the interface\nresistance of a single N jF interface, and Rb(L) is the bulk\nresistance of a ferromagnetic layer of thickness L[7, 13].\nWhen the ferromagnetic slab is su\u000eciently thick, Ohmic\nbehaviour is recovered whereby Rb(L)\u0019\u001aLas shown in\nthe inset to Fig. 1 for permalloy (Py = Ni 80Fe20) and\nthe bulk resistivity \u001acan be extracted from the slope\nofR(L) [14]. For currents parallel and perpendicular to\nthe magnetization direction, the resistivities are di\u000berent\nand have to be calculated separately. The average resis-\ntivity is given by \u0016 \u001a= (\u001ak+ 2\u001a?)=3, and the anisotropic\nmagnetoresistance ratio (AMR) is ( \u001ak\u0000\u001a?)=\u0016\u001a.\nFor permalloy we \fnd values of \u0016 \u001a= 3:5\u00060:15\u0016Ohm-\ncm and AMR = 19 \u00061%, compared to experimental low-\ntemperature values in the range 4 :2\u00004:8\u0016Ohm-cm for\n\u0016\u001aand 18% for AMR [15]. The resistivity calculated as a\nfunction of xis compared to low temperature literature\nvalues [15] in Fig. 1. The plateau in the calculated values\naround the Py composition appears to be seen in the\nexperiments by Smit and Jaoul et al. [15]. The overall\nagreement with previous calculations is good [16]. In\nspite of the smallness of the SOC, the resistivity of Py\nis underestimated by more than a factor of four when it\nis omitted, underlining its importance for understanding\ntransport properties.\n0 20 40 60 80 1000123456ρ [µΩ ⋅ cm]\nFe concentration [%]With SOC\nWithout SOCCadeville\nMcGuire\nJaoul\nSmit\n0 10 20 30\n123R|| [fΩ ⋅ m2]L [nm]FIG. 1. Calculated resistivity as a function of the concen-\ntrationxfor fcc Ni 1\u0000xFexbinary alloys with (solid line) and\nwithout (dashed-dotted line) SOC. Low temperature experi-\nmental results are shown as symbols [15]. The composition\nNi80Fe20is indicated by a vertical dashed line. Inset: resis-\ntance of CujNi80Fe20jCu as a function of the thickness of the\nalloy layer. Dots indicate the calculated values averaged over\n\fve con\fgurations while the solid line is a linear \ft.\nThree sources of disorder which have not been taken\ninto account here will increase the calculated values of\n\u001a; short range potential \ructuations that go beyond the\nsingle site CPA, short range strain \ructuations re\recting\nthe di\u000bering volumes of Fe and Ni and spin disorder.\nThese will be the subject of a later study.\nGilbert Damping. Recently, Brataas et al. showed\nthat the energy loss due to Gilbert damping in an N jFjN\nscattering con\fguration can be expressed in terms of the\nscattering matrix S[17]. Using the Landau-Lifshitz-\nGilbert equation (1), the energy lost by the ferromagnetic\nslab is,\ndE\ndt=d\ndt(He\u000b\u0001M) =He\u000b\u0001dM\ndt=1\n\r2dm\ndt~G(M)dm\ndt\n(3)\nwhere m=M=Msis the unit vector of the magnetization\ndirection for the macrospin mode. By equating this en-\nergy loss to the energy \row into the leads [18] associated\nwith \\spin-pumping\" [19],\nIPump\nE =~\n4\u0019Tr\u001adS\ndtdSy\ndt\u001b\n=~\n4\u0019Tr\u001adS\ndmdm\ndtdSy\ndmdm\ndt\u001b\n;\n(4)\nthe elements of the tensor ~Gcan be expressed as\n~Gi;j(m) =\r2~\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (5)\nPhysically, energy is transferred slowly from the spin de-\ngrees of freedom to the electronic orbital degrees of free-\ndom from where it is rapidly lost to the phonon degrees\nof freedom. Our calculations focus on the role of elastic\nscattering in the rate-limiting \frst step.\nAssuming that the Gilbert damping is isotropic for cu-\nbic substitutional alloys and allowing for the enhance-\nment of the damping due to the F jN interfaces [19{21],3\n0 20 40 60 80 10002468101214α [x 10−3]\nFe concentration [%]Rantschler\nIngvarsson\nMizukami\nNakamuraPatton\nBailey\nBonin\nNibargerInaba\nLagae\nOogane0 5 10 15 20 2500.050.10.15G/(γ ⋅ µs A) [nm]\nL [nm]\nFIG. 2. Calculated zero temperature (solid line) and exper-\nimental room temperature (symbols) values of the Gilbert\ndamping parameter as a function of the concentration xfor\nfcc Ni 1\u0000xFexbinary alloys [21{23]. Inset: total damping of\nCujNi80Fe20jCu as a function of the thickness of the alloy\nlayer. Dots indicate the calculated values averaged over \fve\ncon\fgurations while the solid line is a linear \ft.\nthe total damping in the system with a ferromagnetic slab\nof thickness Lcan be written ~G(L) =~Gif+~Gb(L) where\nwe express the bulk damping in terms of the dimension-\nless Gilbert damping parameter ~Gb(L) =\u000b\rMs(L) =\n\u000b\r\u0016sAL, where\u0016sis the magnetization density and Ais\nthe cross section. The results of calculations for Ni 80Fe20\nare shown in the inset to Fig. 2, where the derivatives of\nthe scattering matrix in (5) were evaluated numerically\nby taking \fnite di\u000berences. The intercept at L= 0, ~Gif,\nallows us to extract the damping enhancement [20] but\nhere we focus on the bulk properties and leave consid-\neration of the material dependence of the interface en-\nhancement for later study. The value of \u000bdetermined\nfrom the slope of ~G(L)=(\r\u0016sA) is 0:0046\u00060:0001 that\nis at the lower end of the range of values 0 :004\u00000:013\nmeasured at room temperature for Py [21{23].\nFig. 2 shows the Gilbert damping parameter as a func-\ntion ofxfor Ni 1\u0000xFexbinary alloys in the fcc structure.\nFrom a large value for clean Ni, it decreases rapidly to a\nminimum at x\u00180:65 and then grows again as the limit\nof clean fccFe is approached. Part of the decrease in\n\u000bwith increasing xcan be explained by the increase in\nthe magnetic moment per atom as we progress from Ni\nto Fe. The large values of \u000bcalculated in the dilute al-\nloy limits can be understood in terms of conductivity-like\nenhancement at low temperatures [24] that has been ex-\nplained in terms of intraband scattering [4, 6]. The trend\nexhibited by the theoretical \u000b(x) is seen to be re\rected\nby experimental room temperature results. In spite of\na large spread in measured values, these seem to be sys-\ntematically larger than the calculated values. Part of this\ndiscrepancy can be attributed to an increase in \u000bwith\ntemperature [22, 25].\nSpin di\u000busion. When an unpolarized current is in-\njected from a normal metal into a ferromagnet, the polar-\nization will return to the value characteristic of the bulk\n0 5 10 15 20 25 3000.20.40.60.81\nz [nm]  \n1+β\n2\n1−β\n2p↑\np↓FIG. 3. Fractional spin-current densities for electrons injected\natz= 0 from Cu into Ni 80Fe20alloy. Calculated values (sym-\nbols) and \fts to Eq. (6) (solid lines).\nferromagnet su\u000eciently far from the injection point, pro-\nvided there are processes which allow spins to \rip. Fol-\nlowing Valet-Fert [3] and assuming there is no spin-\rip\nscattering in the N leads, we can express the fractional\nspin current densities p\"(#)=J\"(#)=Jas a function of\ndistancezfrom the interface as\np\"(#)(z) =1\n2\u0006\f\n2\u0014\n1\u0000exp(\u0000z=lsf)r\u0003\nif(\f\u0000\r+\rsech\u000e)\n\f(r\u0003\nif+lsf\u000e\u001a\u0003\nFtanh\u000e)\u0015\n;\n(6)\nwhereJis the total current through the device, J\"\nandJ#are the currents of majority and minority elec-\ntrons, respectively, lsfis the spin-di\u000busion length, \u001a\u0003\nF=\n(\u001a#+\u001a\")=4 is the bulk resistivity and \fis the bulk spin\nasymmetry ( \u001a#\u0000\u001a\")=(\u001a#+\u001a\"). The interface resistance\nr\u0003\nif= (r#\nif+r\"\nif)=4, the interface resistance asymmetry\n\r= (rif#\u0000r\"\nif)=(r#\nif+r\"\nif) and the interface spin-relaxation\nexpressed through the spin-\rip coe\u000ecient \u000e[26] must be\ntaken into consideration resulting in a \fnite polarization\nof current injected into the ferromagnet. The correspond-\ning expressions are plotted as solid lines in Fig. 3.\nTo calculate the spin-di\u000busion length we inject non-\npolarized states from one N lead and probe the transmis-\nsion probability into di\u000berent spin-channels in the other\nN lead for di\u000berent thicknesses of the ferromagnet. Fig. 3\nshows that the calculated values can be \ftted using ex-\npressions (6) if we assume that J\u001b=J=G\u001b=G, yielding\nvalues of the spin-\rip di\u000busion length lsf= 5:5\u00060:3 nm\nand bulk asymmetry parameter \f= 0:678\u00060:003 for\nNi80Fe20alloy compared to experimentally estimated val-\nues of 0:7\u00060:1 for\fand in the range 5 :0\u00006:0 nm for\nlsf[27].\nlsfand\fare shown as a function of concentration x\nin Fig. 4. The convex behaviour of \fis dominated by\nand tracks the large minority spin resistivity \u001a#whose\norigin is the large mismatch of the Ni and Fe minority\nspin band structures that leads to a \u0018x(1\u0000x) concen-\ntration dependence of \u001a#(x) [16]. The majority spin band\nstructures match well so \u001a\"is much smaller and changes\nrelatively weakly as a function of x. The increase of lsf\nin the clean metal limits corresponds to the increase of4\n0 20 40 60 80 100510152025l sf [nm]\nFe concentration [%]← lsfβ →\n0 20 40 60 80 1000.50.60.70.80.9\nβ\nFIG. 4. Spin-di\u000busion length (solid line) and polarization \fas\na function of the concentration xfor Ni 1\u0000xFexbinary alloys.\nthe electron momentum and spin-\rip scattering times in\nthe limit of weak disorder.\nIn summary, we have developed a uni\fed DFT-based\nscattering theoretical approach for calculating transport\nparameters of concentrated alloys that depend strongly\non spin-orbit coupling and disorder and have illustrated\nit with an application to NiFe alloys. Where comparison\nwith experiment can be made, the agreement is remark-\nably good o\u000bering the prospect of gaining insight into\nthe properties of a host of complex but technologically\nimportant magnetic materials.\nThis work is part of the research programs of \\Sticht-\ning voor Fundamenteel Onderzoek der Materie\" (FOM)\nand the use of supercomputer facilities was sponsored by\nthe \\Stichting Nationale Computer Faciliteiten\" (NCF),\nboth \fnancially supported by the \\Nederlandse Organ-\nisatie voor Wetenschappelijk Onderzoek\" (NWO). It was\nalso supported by \\NanoNed\", a nanotechnology pro-\ngramme of the Dutch Ministry of Economic A\u000bairs and\nby EC Contract No. IST-033749 DynaMax.\n[1] See the collection of articles in Ultrathin Magnetic Struc-\ntures I-IV , edited by J. A. C. Bland and B. Heinrich\n(Springer-Verlag, Berlin, 1994-2005).\n[2] P. C. van Son, H. van Kempen, and P. Wyder, Phys.\nRev. Lett., 58, 2271 (1987); 60, 378 (1988).\n[3] T. Valet and A. 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Phys., 3, L115 (1973).\n[16] J. Banhart and H. Ebert, Europhys. Lett., 32, 517 (1995);\nJ. Banhart, H. Ebert, and A. Vernes, Phys. Rev. B, 56,\n10165 (1997).\n[17] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett., 101, 037207 (2008).\n[18] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun,\nPhys. Rev. Lett., 87, 236601 (2001); M. Moskalets and\nM. B uttiker, Phys. Rev. B, 66, 035306 (2002); 66,\n205320 (2002).\n[19] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett., 88, 117601 (2002); Phys. Rev. B, 66, 224403\n(2002).\n[20] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, Phys. Rev. B, 71, 064420 (2005).\n[21] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. &\nMagn. Mater., 226{230 , 1640 (2001); Jpn. J. Appl.\nPhys., 40, 580 (2001).\n[22] W. Bailey, P. Kabos, F. Manco\u000b, and S. Russek, IEEE\nTrans. Mag., 37, 1749 (2001).\n[23] C. E. Patton, Z. Frait, and C. H. Wilts, J. Appl. Phys.,\n46, 5002 (1975); S. Ingvarsson, G. Xiao, S. Parkin, and\nR. Koch, Appl. Phys. Lett., 85, 4995 (2004); H. Naka-\nmura, Y. Ando, S. Mizukami, and H. Kubota, Jpn. J.\nAppl. Phys., 43, L787 (2004); J. O. Rantschler, B. B.\nMaranville, J. J. Mallett, P. Chen, R. D. McMichael,\nand W. F. Egelho\u000b, IEEE Trans. Mag., 41, 3523 (2005);\nR. Bonin, M. L. Schneider, T. J. Silva, and J. P.\nNibarger, J. Appl. Phys., 98, 123904 (2005); L. La-\ngae, R. Wirix-Speetjens, W. Eyckmans, S. Borghs, and\nJ. de Boeck, J. Magn. & Magn. Mater., 286, 291 (2005);\nJ. P. Nibarger, R. Lopusnik, Z. Celinski, and T. J. Silva,\nAppl. Phys. Lett., 83, 93 (2003); N. Inaba, H. Asanuma,\nS. Igarashi, S. Mori, F. Kirino, K. Koike, and H. Morita,\nIEEE Trans. Mag., 42, 2372 (2006); M. Oogane, T. Wak-\nitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and\nT. Miyazaki, Jpn. J. Appl. Phys., 45, 3889 (2006).\n[24] S. M. Bhagat and P. Lubitz, Phys. Rev. B, 10, 179 (1974);\nB. Heinrich, D. J. Meredith, and J. F. Cochran, J. Appl.5\nPhys., 50, 7726 (1979).\n[25] D. Bastian and E. Biller, Phys. Stat. Sol. A, 35, 113\n(1976).\n[26] W. Park, D. V. Baxter, S. Steenwyk, I. Moraru, W. P.Pratt, Jr., and J. Bass, Phys. Rev. B, 62, 1178 (2000).\n[27] J. Bass and W. P. Pratt Jr., J. Magn. & Magn. Mater.,\n200, 274 (1999); J. Phys.: Condens. Matter, 19, 183201\n(2007)."    },    {        "title": "1011.5054v1.Ultra_fast_magnetisation_rates_within_the_Landau_Lifshitz_Bloch_model.pdf",        "content": "arXiv:1011.5054v1  [cond-mat.mtrl-sci]  23 Nov 2010Ultra-fastmagnetisationrates withintheLandau-Lifshit z-Bloch model.\nU. Atxitia and O. Chubykalo-Fesenko\nInstituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco, 28049 Madrid, Spain\nThe ultra-fast magnetisation relaxation rates during the l aser-induced magnetisation process are analyzed\nin terms of the Landau-Lifshitz-Bloch (LLB) equation for di fferent values of spin S. The LLB equation is\nequivalentinthelimit S→∞totheatomisticLandau-Lifshitz-Gilbert(LLG)Langevind ynamicsandfor S=1/2\nto the M3TM model [B. Koopmans, et al.Nature Mat. 9(2010) 259]. Within the LLB model the ultra-fast\ndemagnetisation time( τM)and the transverse damping ( α⊥) are parameterizedby theintrinsic coupling-to-the-\nbath parameter λ, defined by microscopic spin-flip rate. We show that for the ph onon-mediated Elliott-Yafet\nmechanism, λis proportional to the ratio between the non-equilibrium ph onon and electron temperatures. We\ninvestigate the influence of the finite spin number and the sca ttering rate parameter λon the magnetisation\nrelaxation rates. The relation between the fs demagnetisat ion rate and the LLG damping, provided by the LLB\ntheory, is checked basing on the available experimental dat a. A good agreement is obtained for Ni, Co and\nGd favoring the idea that the same intrinsic scattering proc ess is acting on the femtosecond and nanosecond\ntimescale.\nPACS numbers: 75.40Gb,78.47.+p, 75.70.-i\nI. INTRODUCTION\nMagnetisation precession and the spin-phonon relaxation\nratesat picosecondtimescale wereconsideredto bethe limi t-\ningfactorforthespeedofthemagnetisationswitching1,2, un-\ntil using optical excitation with fs pulsed lasers the possi bil-\nity to influence the magnetisation on femtosecond timescale\nwasdemonstrated3–6. Theultra-fastlaser-induceddemagneti-\nsation immediately became a hot topic of solid state physics\ndue to an appealing possibility to push further the limits of\noperation of magnetic devices. This ultra-fast process has\nnow been shown to proceed with several important charac-\nteristictimescales6: (i)thefemtoseconddemagnetisationwith\ntimescale τM(ii) the picosecond recovery with timescale τE\nand (iii) the hundredpicoseconds-nanosecondmagnetisati on\nprecession, traditionally characterized by the ferromagn etic\nresonance frequency ωFMRand the Landau-Lifshitz-Gilbert\ndampingparameter αLLG(see Fig.1).\nThephysicsofthemagnetisationchangesonfemto-second\ntimescales is obviously not-trivial and will require novel the-\norieswithintherelativisticquantumelectrodynamicsofm any\nelectronsystems. From theoretical pointof view, the exist ing\nmodels try to answer an open question of the role of differ-\nentsubsystems(photons,phonons,electronsandspins)int he\nultra-fast angular momentumtransfer7. This common goal is\nstimulated by experimental findings provided by the XMCD\nmeasurements showing the important role of the spin-orbit\ninteractions8,9. For the present state of art quantummechani-\ncal descriptions8,10–13of ultra-fast demagnetisationprocesses\ninvolveunavoidablesimplificationsandsometimesevensom e\nad-hoc assumptions necessary to explain experimental find-\nings, such as reduced exchange interactions, enhanced spin -\norbit coupling or a Gaussian distribution of occupied state s\naroundthe Fermi level. While some degreeof agreementhas\nbeen achieved in modelling of the ultra-fast demagnetisati on\n(τM)scale14,themodellingofallthreeultra-fastdemagnetisa-\ntionrateswithinthesameapproachisoutsidethepossibili ties\nofthequantummechanicalapproaches.\nThe three-temperature (3T) phenomenological model in-volves the rate equations for the electron, phonon and spin\ntemperatures (energies)10,15–17. Recently, it has been shown\nthattheintroductionofthespintemperatureisnotadequat e18\nsince the spin system is not in the equilibrium on the fem-\ntosecond timescale. It has been suggested to couple the spin\ndynamics to the two-temperature (2T) model for phonon and\nelectrontemperatures18–22. Thesemodelsarebasedontheen-\nergy flow picture and leave unidentified the angular momen-\ntum transfer mechanism and the underlying quantum mech-\nanism responsible for the spin flip22. They essentially inter-\nprettheultra-fastdemagnetisationas\"thermal\"processe s,un-\nderstanding the temperature as energy input from photon to\nelectron and then to the spin system. By using these mod-\nels the important role of the linear reversal path in the femt o-\nseconddemagnetisationhasbeenidentified23,24. Thecompar-\nison with experiment seems to indicate that in order to have\nmagnetisation switching in the ultra-fast timescale, a com -\nbinedactionof\"heat\"andlargefieldcomingfromtheinverse\nFaradayeffectisnecessary24.\nThe most successful recent phenomenological models de-\nscribing the ultra-fast magnetisation dynamics are (i) the\nLangevin dynamics based on the Landau-Lifshitz-Gilbert\n(LLG) equationand classical HeisenbergHamiltonian for lo -\ncalized atomic spin moments18,19, (ii) the Landau-Lifshitz-\nBloch (LLB) micromagnetics21,22and (iii) the Koopmans’s\nmagnetisation dynamics model (M3TM)25. The spin dy-\nnamicscouldbe coupledtotheelectrontemperaturefromthe\n2T model, underlying the electronic origin of the spin-flip\nprocess18,19,21,22,24or to both electron and phonon temper-\natures, underlying the Elliott-Yafet mechanism mediated b y\nphonons25. When the 2T model was carefully parameterised\nfromthemeasuredreflectivity,it gaveanexcellentagreeme nt\nwith the experiment in Ni22using the former approach or in\nNi,Co andGdusingthe latterapproach25.\nIn the classical derivation of the LLB equation the ther-\nmal averaging has been performed analytically within the\nmean field (MFA) approximation26. Thus, the LLB equa-\ntion for classical spins ( S→∞) is equivalent to an ensem-\nbleofexchange-coupledatomisticspinsmodelledbystocha s-2\ntic LLG equations20,27. At the same time, in some cases the\nLLBequationmaybepreferablewith respectto theatomistic\nHeisenbergmodel,sincebeingmicromagneticit canincorpo -\nrate quantum nature of magnetism and the quantum deriva-\ntion of LLB also exists28. In particular the limits of validity\nforthestatisticalmechanicsbasedontheclassicalHeisen berg\nmodel for the description of materials with delocalized mag -\nnetism of d-electrons in transition metals or magnetism of f-\nelectronsin rare earthsare not clear. An alternativestati stical\nsimplified descriptionof d-metalsconsists of a two level sys-\ntem with spin-up and spin-down bands (i.e. S=±1/2), as\nhas been done by B. Koopmans et al.25. Their model, as we\nshow in the present article, is also equivalent to the quantu m\nLLB equation with spin S=1/2. An additional advantage\nin the use of the LLB equation is the possibility to model\nlarger spatial scales20,21. Therefore the LLB micromagnet-\nicsisanimportantparadigmwithinthemultiscalemagnetis a-\ntiondynamicsdescription. TheLLBequationhasbeenshown\nto describe correctly the three stages of the ultra-fast dem ag-\nnetisationprocesses: thesub-picoseconddemagnetisatio n,the\npicosecondmagnetisationrecoveryand the nanosecondmag-\nnetisationprecession20–22,see Fig.1.\nThe intrinsic quantum mechanical mechanisms responsi-\nble for the ultra-fast demagnetisation in the LLB model\nare included in the intrinsic coupling-to-the-bath parame -\nterλ22,28. The coupling process is defined by the rate of\nthe spin flip. Several possible underlying quantum mech-\nanisms are currently under debate: the Elliott-Yafet (EY)\nelectron scattering mediated by phonons or impurities13,25,\nor other electrons14and electron-electron inelastic exchange\nscattering29,30. By combining the macroscopic demagnetisa-\ntion equation (M3TM model) with the rate of spin flip calcu-\nlatedonthebasisoffullHamiltonian,Koopmans etal.25have\nbeenabletorelatetheultra-fastdemagnetisationtime τMwith\nthespinfliprateofthephonon-mediatedElliott-Yafetscat ter-\ning. The authors fitted experimentaldemagnetisation rates in\nNi,Co,GdtothephenomenologicalM3TMmodelandfound\nthemtobeconsistentwiththevaluesestimatedonthebasiso f\nab-initiotheory13. Thecoupling-to-the-bathparameter λ(mi-\ncroscopicdampingparameterinatomisticLLGmodel)should\nbe distinguished from that of the macroscopic damping αLLG\n(α⊥in the LLB model), a more complicated quantity which\nincludesthe magnon-magnonprocesses.\nThefirst attempttorelatethesub-picoseconddemagnetisa-\ntion time with the macroscopicdampingprocesses was given\nbyKoopmans etal.6whosuggestedtherelation τM∼1/αLLG.\nSubsequently and with the aim to check this relation several\nexperiments in doped permalloy were performed32–34. The\npermalloythinfilmsweredopedwithrareearthimpurities,a l-\nlowingtoincreaseinacontrolledwaythedampingparameter\nαLLG. The effect on the demagnetisation time τMwas shown\nto be opposite34or null32, in contrast to the above relation.\nHowever, it should be noted that the analysis leading to this\nexpression was performed in terms of the Landau-Lifshitz-\nGilbert equation, relating the ultra-fast demagnetisatio n time\nτMtothetransversedampingwithouttakingintoaccounttheir\ntemperature dependence. Moreover, one should take into ac-\ncount that the rare-earth impurities may introduce a differ ent\nFigure 1. Characteristic time scales in ultrafast laser-in duced mag-\nnetisation dynamics experiments. The curve is obtained by t he in-\ntegration of the Landau-Lifshitz-Bloch equation coupled t o the two-\ntemperature model with the parameters from Ref.21. For the m od-\nelling of precession the applied field Hap=1T at 30 degrees was\nused.\nscatteringmechanismwith aslowertimescale33.\nPartially basing on the above mentioned experimental re-\nsults andfroma generalpointof view,the longitudinalrela x-\nation (the ultra-fast demagnetisation rate τM) and the trans-\nverse relaxation (the LLG damping αLLG) may be thought\nto be independent quantities. Indeed, different intrinsic and\nextrinsic mechanisms can contribute to the demagnetisatio n\nrates at different timescales. One can, for example, men-\ntionthatduringthefemtoseconddemagnetisationtheelect ron\ntemperature is often raised up to the Curie temperature22,24.\nAt this moment, the high frequency THz spinwaves35,36in-\ncludingtheStonerexcitations30contribute. At thesame time,\nthe transverse relaxation is related to the homogeneous pre -\ncessional mode. The LLBequationtakescare of thedifferent\nnaturesoflongitudinalandtransverserelaxation,arisin gfrom\nthe spin disordering. The LLB model calculates them inde-\npendently but basing on the same intrinsic scattering mecha -\nnismparameterizedbythe parameter λ. Theincrementof the\nnumber of scattering events is mimicked by the increases of\nthe electron temperature. Consequently, the relation betw een\nthe ultra-fast demagnetisation and precession remains val id\nbutwithatemperature-dependentcorrection. Ifthisrelat ionis\nconfirmedexperimentally,a uniqueintrinsic couplingpara m-\netermeansthatthesamemainmicroscopicmechanismisact-\ningonbothtimescales. Inthepresentarticlewewillshowth at\nthe analysis of the available experimental data seems to ind i-\ncate towards this possibility, at least in pure transition m etals\nsuch as Ni or Co and in rare earth metal Gd. We did not find\nvalidityofthecorrespondingrelationin Fe.\nUp to now only classical version ( S→∞) of the LLB\nequation was used to model the ultra-fast demagnetisation\nprocesses20,21,24. In the present article we show the impor-\ntant role of the choice of the quantum spin value, resulting\ninthedifferencesinthecorrespondinglongitudinalrelax ation\ntimes. The article is organized as follows. In section II we\npresentdifferentformulationsofthequantumLLBmodeland\nits main features for different spin values S. In section III3\nwe present results on the modelling of the demagnetisation\nprocesses within LLB models with different choices of the\nquantumspinsnumber Sandofthe intrinsic scatteringmech-\nanisms. In section IV we present our attempts to link the\nultra-fast demagnetisation rates in transition metals and Gd\nand comparison with available experimental data. Section\nV concludes the article. In the Appendix to the article we\ndemonstrate the equivalence of the LLB model with S=1/2\nandtheM3TMmodelbyB.Koopmans et al.25.\nII. THE LANDAU-LIFSHITZ-BLOCHMODELWITH\nQUANTUMSPINNUMBER S.\nTheLLBequationforaquantumspinwasderivedfromthe\ndensity matrix approach28. Although the model Hamiltonian\nwas rather the simplest form of the spin-phonon interaction ,\nthegeneralizationoftheapproachshouldbepossibleto mor e\ncomplex situations. The macroscopic equation for the mag-\nnetisationdynamics,validatalltemperatures,iswritten inthe\nfollowingform:\n˙n=γ[n×H]+γα/bardbl\nn²[n·Heff]n−γα⊥\nn2[n×[n×Heff]](1)\nwheren=M/Me(T) =m/meis the reduced magnetisation,\nnormalizedtotheequilibriumvalue Meatgiventemperature T\nandm=M/Me(T=0K). Theeffectivefield Heff,containsall\nusualmicromagneticcontributions,denotedby Hint(Zeeman,\nanisotropy,exchangeandmagnetostatic)andisaugmentedb y\nthecontributioncomingfromthetemperature\nHeff=Hint+me\n2/tildewideχ/bardbl/parenleftbig\n1−n2/parenrightbig\nn, (2)\nwhere/tildewideχ/bardbl(T) = (∂m/∂H)H→0is the longitudinal susceptibil-\nity . The LLB equation contains two relaxational parame-\nters: transverse α⊥andlongitudinal α/bardbl,relatedtotheintrinsic\ncoupling-to-the-bathparameter λ. Inthe quantumdescription\nthe couplingparameter λcontains the matrix elements repre-\nsenting the scattering events and, thus, is proportional to the\nspin-fliprateduetotheinteractionwiththeenvironment. T his\nparameter,inturn,couldbetemperaturedependentand,ino ur\nopinion, it is this microscopic parameter which should be re -\nlatedtotheGilbertparametercalculatedthroughab-initi ocal-\nculations as in Refs.38,39, since the contribution coming from\nthespindisorderingisnotproperlytakenintoaccountinth ese\nmodels. In the quantum case the temperature dependence of\nthe LLB damping parameters is given by the following ex-\npressions:\nα/bardbl=λ\nme2T\n3TC2qS\nsinh(2qS)=⇒\nS→∞λ\nme2T\n3TC, (3)\nα⊥=λ\nme/bracketleftbiggtanh(qS)\nqS−T\n3TC/bracketrightbigg\n=⇒\nS→∞λ\nme/bracketleftbigg\n1−T\n3TC/bracketrightbigg\n,(4)withqS=3TCme/[2(S+1)T], whereSis the quantum spin\nnumber and TCis the Curie temperature. In the case S→∞\nthedampingcoefficientshavetheformsusedinseveralprevi -\nouslypublishedworks40,suitableforthecomparisonwiththe\nLangevindynamicssimulationsbasedontheclassicalHeise n-\nbergHamiltonianandinagreementwiththem20,27.\nEq.(1) is singular for T>TC, in this case it is more con-\nvenient to use the LLB equation in terms of the variable\nm=M/Me(T=0K)27. The corresponding LLB equation is\nindistinguishable from Eq.(1) but with different relaxati onal\nparameters/tildewideα/bardbl=meα/bardbl,/tildewideα⊥=meα⊥and/tildewideα⊥=/tildewideα/bardblforT>TC,\nin this case the contribution of temperature to Heff[the sec-\nondterminEq.(2)]is (−1//tildewideχ/bardbl)[1−3Tcm2/5(T−Tc)m]m. Al-\nthough this formulation is more suitable for the modelling o f\nthe laser-induced demagnetisation process, during which t he\nelectronic temperature is usually raised higher than TC, it is\nthe expression (4) which should be compared with the trans-\nverse relaxation parameter αLLGdue to the similarity of the\nformulationoftheEq.(1)withthemacromagneticLLGequa-\ntion. In the classical case and far from the Curie temperatur e\nT≪TC,λ=α⊥=/tildewideα⊥(αLLG).\nS→∞S=7/2S=3/2S=1/2\nT/T CαLLG\n10.90.80.70.60.50.40.09\n0.06\n0.03\nS→∞S=7/2S=3/2S=1/2\nT/T Cτ/bardbl[ps]\n1.2 1 0.8 0.6 0.46\n4\n2\n0.1\nFigure 2. (Up) The transverse damping parameter α⊥(αLLG) as\na function of temperature within the LLB model for different spin\nvaluesS. The intrinsic coupling parameter was set to λ=0.03.\n(Down) The longitudinal relaxation time τ/bardblas a function of tem-\nperature within the LLB model for different spin values S. The\ntemperature-dependent magnetisation and the longitudina l suscepti-\nbility/tildewideχ/bardblwereevaluatedinbothcasesintheMFAapproachusingthe\nBrillouinfunction.\nIn the \"thermal\" model the nature of the longitudinal and4\nthe transverse relaxation differs from the point of view of\ncharacteristicspinwavefrequencies. Thetransverserela xation\n(knownastheLLGdamping)isbasicallytherelaxationofthe\nFMR mode. The contributionof other spinwave modes is re-\nducedtothethermalaveragingofthemicromagneticparame-\nters and the main effect comes fromthe decrease of the mag-\nnetisation at high temperature. Consequently, the transve rse\ndamping parameter increases with temperature (see Fig.2),\nconsistentwithatomisticmodellingresults27andwell-known\nFMRexperiments37,41.\nOn the contrary, the main contribution to the longitudinal\nrelaxation comes from the high-frequency spin waves. This\nprocessoccursinastrongexchangefield. Asaresult,thelon -\ngitudinal relaxation time (the inverse longitudinal relax ation)\nis much faster and increases with temperature,knownas crit -\nical slowing down, see Fig.2. This slowing down has been\nshown to be responsible for the slowing down of the femto-\nsecond demagnetisation time τMas a function of laser pump\nfluency18,22. The characteristic longitudinal timescale is not\nonly defined by the longitudinal damping parameter (3) but\nalso by the temperature-dependentlongitudinal susceptib ility\n/tildewideχ/bardbl(T)27, accordingtothefollowingequation:\nτ/bardbl(T)=/tildewideχ/bardbl(T)\nγ/tildewideα/bardbl(T). (5)\nAs it can be observed in Fig. 2 the transverse relaxation\nparameter α⊥(αLLG) and the longitudinal relaxation time τ/bardbl\nhaveastrongdependenceonthequantumspinnumber Scho-\nsen to describe system’s statistics. We conclude here about\nthe occurrence of quite different relaxation rates for the t wo\nextremecases S=1/2andS=∞.\nB. Koopmans et al.recently used a different equation\nto describe the ultrafast demagnetisation dynamics25, called\nM3TMmodel:\ndm\ndt=RmTp\nTC/parenleftbigg\n1−mcoth/parenleftbiggmTC\nTe/parenrightbigg/parenrightbigg\n. (6)\nEq.(6)hasbeenobtainedthroughthegeneralMasterequatio n\napproach for the dynamics of the populations of a two level\nsystem (spin S=1/2 was used) with the switching probabil-\nityevaluatedquantum-mechanicallyforthephonon-mediat ed\nEY spin-flips. Here TpandTeare phonon and electron tem-\nperatures,respectively,and Risa materialspecificparameter,\nrelatedtothespin-flipprobabilityinthephonon-mediated EY\nscatteringevents asf, as\nR=8asfGepµBkBVaT2\nC\nµatE2\nD, (7)\nwhereVaandµataretheatomicvolumeandmagneticmoment,\nrespectively, Gepistheelectron-phononcouplingconstant, kB\nistheBoltzmannconstant, µBistheBohrmagnetonand EDis\nthe Debye energy. This equation has allowed to fit the ultra-\nfast demagnetisation time ( τM) obtaining the values of Rin\nNi, Co and Gd25and relating them to the phonon-mediated\nEYscatteringrates asf.\nAs we show in the Appendix, the M3TM equation (6) cor-\nresponds to the longitudinal part of the LLB equation withthermal field only ( Hint=0) and with spin S=1/2, i.e. it is\nequivalentto\ndm\ndt=γ/tildewideα/bardblHeff. (8)\nThisgivesarelationbetweentheintrinsiccouplingparame ter\nλand the material specific parameter Rand finally with the\nphonon-mediatedEYspin-flipprobability asfviatheformula:\nλ=3R\n2γµat\nkBTCTp\nTe=λ0Tp\nTe. (9)\nThus the two approaches are reconciled, provided that the\ntemperature-dependent coupling rate (9) is used in the LLB\nequation,incontrasttootherworks18,21,22wherethecoupling\nλis considered to be temperature-independent. Combining\nexpressions (5) (7) and (9), one can immediately see that in\nthecaseofthephonon-mediatedEYprocess,thelongitudina l\nrelaxationtimeisdeterminedby\nτ/bardbl∝/tildewideχ/bardbl\nasfE2\nD\nGepVaTp. (10)\nInRef.25andbasingonthephonon-mediatedEYpicture,the\nclassification of materials on the basis of the \"magnetic in-\nteraction strength\" parameter µat/Jwas proposed, where Jis\nthematerialexchangeparameter. Accordingtotheexpressi on\nabove,thedemagnetisationratedependsonmoreparameters ,\namong which the important one is also the electron-phonon\ncoupling Gepdefining how fast the electron system can pass\ntheenergytothephononone. Anotherimportantparameteris\nthe microscopicspin-fliprate asf. Comparingto theB. Koop-\nmanset al.25materialsclassification,thelongitudinalsuscep-\ntibilityinEq.(10) isindeeddefinedbythevalueoftheatomi c\nmomentµatandbythefactthatthisfunctionrapidlyincreases\nwithtemperatureanddivergescloseto TC∝J. AtT≈TCone\nobtainsasimplelinearrelation27/tildewideχ/bardbl∝µat/J,thusshowingthe\ndependenceof the demagnetisation rate on this parameter, a s\nsuggestedinRef.25.\nIn the case of the phonon-mediated EY process the tem-\nperature dependence of the longitudinal relaxation is comi ng\nfrom the longitudinalsusceptibiliy only (cf. Eq. (10)), as op-\nposed to the case λ=const (cf. Eq.(5)). (We do not discuss\nherethepossibilitythatthephonon-mediatedEYspin-flipr ate\nasfmay be also temperature dependent.) However, the tem-\nperature dependence of the susceptibility is characterize d by\nitsexponentialdivergencecloseto TC. Inthesecircumstances\nan additional linear temperature dependence provided by th e\nlongitudinal damping is difficult to distinguish in the fitti ng\nprocedureofexperimentaldata.\nIII. MODELLINGOF THELASER-INDUCED\nULTRA-FASTDEMAGNETISATIONWITHINTHE LLB\nMODELS.\nInthespiritofRefs.18,20–22,25forthemodellingofultra-fast\ndemagnetisationdynamics,theLLBequationmaybecoupled5\nto the electron temperature Teonly, understanding the elec-\ntrons as the main source for the spin-flip mechanism18,20–22\nor to both phonon and electron temperatures in the spirit of\nthe phonon-mediatedElliott-Yafet process25. In both cases it\nis the electrontemperature T=Tewhich couplesto the mag-\nnetisation in the LLB formalism, since the phonon tempera-\nture could only enter into the temperature dependence of the\ncoupling-to-the bath parameter λvia Eq.(9) . Note that the\ntemperature Tis not the spin temperature, since the resulting\ndynamicsistakingplaceout-of-equilibrium.\nTheelectron Teandphonon Tptemperaturesaretakenfrom\nthe two-temperature (2T) model15,45,46. Within this model\ntheirdynamicsisdescribedbytwodifferentialequations:\nCedTe\ndt=−Gep(Te−Tp)+P(t),\nCpdTp\ndt=Gep(Te−Tp). (11)\nHereCe=γeTe(γe=const) and Cpare the specific heats of\ntheelectronsandthelattice. TheGaussiansourceterm P(t)is\na function which describes the laser power density absorbed\nin the material. The function P(t)is assumed to be propor-\ntional to the laser fluence Fwith the proportionality coeffi-\ncient which could be obtained from the long time scale de-\nmagnetization data (for which Te=Tp)22. The dynamics of\nthe electron temperature can be also measured directly in th e\ntime-resolvedphotoemissionexperiment47.\nThe first of Eqs.(11) may also include a diffusion term\n∇z(κ∇zTe)taking into account a final penetration depth of\nthe deposited energy into the film thickness25and a term,\nCe(Te−300K)/τthdescribing the heat diffusion to the exter-\nnal space22. In the present article, the parameters for the 2T-\nmodel were taken either from Koopmans et al.25or from U.\nAtxitiaet al.22(for Ni only), where they were carefully pa-\nrameterized through the reflectivity measurements. The Ni\n(Co, Gd etc) parameters, such as magnetisation as a function\nof temperature were taken assuming the Brilloiun (Langevin\nforS→∞)function.\nThe coupling of the 2T model to the LLB equation ade-\nquately describes all three stages of the ultra-fast demagn eti-\nsation rates: sub-ps demagnetisation, ps recovery and sub- ns\nprecession21,22, see Fig.1. As a consequence of the temper-\nature dependenceof both longitudinal dampingand suscepti -\nbility, and since the temperature is dynamically changed ac -\ncording to Eqs.(11), the longitudinal relaxation time is ti me-\ndependent via Eq.(5). It is also strongly dependent on the\nparameters of the 2T model and its dynamics is not simple.\nConsequently,thesub-psultra-fastdemagnetisationgene rally\nspeaking is not exponentialand cannot be described in terms\nof one relaxation time τM. Simple analytical expression is\npossible to obtain with the supposition of a square-shaped\ntemperaturepulse23. The two-exponentialfitting is also often\nused22,36. In our approachthe fs demagnetisation is fitted di-\nrectlytothesolutionoftheLLBequationwithoutassumptio n\nof the one- or two-exponential decay. However, to comply\nwith the existing approaches, we still discuss the demagnet i-\nsationratein termsofauniqueparameter τM.\nIn the experiment performedin the same material the onlyremainingfittingparameterfortheLLBmodelisthecoupling\nparameter λ. The choice of λtogether with the parameters\nof the 2T model defines all magnetisation rates. In Fig.3 we\npresent modelling of the ultra-fast demagnetisation and re -\nmagnetisationforvariousvaluesofthecouplingparameter λ,\nchosen to be independent on temperature, as in Ref. 22. If\nfor some reason the scattering channel was suppressed, this\nwouldleadtoasmallscatteringrateandconsequentlyasmal l\ndemagnetisation and a slow recovery. Indeed, the value of λ\nforGdwasfoundtobe60timessmallerthanforNi(seeTable\nI).Thissmallvalueof λassuresalargedelayinthemagnetis-\narionrelaxationtowardstheequilibriumelectrontempera ture.\nThus this parameter defines the diversity of the demagnetisa -\ntion rates in larger extend than the ratio µat/J, suggested in\nRef.25anddiscussedintheprevioussubsection.\nλ=0.001\nλ=0.01\nλ=0.1\nt[ps]∆m/m 0\n20 10 00\n-0.1\n-0.2\n-0.3\nFigure 3. The result of integration of the LLB model ( S→∞) with\ndifferent parameters λ(increasing from top to the bottom). In this\ncase the the 2T model parameters were taken from Ref.22withl aser\nfluenceF=30 mJ/cm2\nAnother parameter strongly influencing the demagnetisa-\ntion rates is the phonon-electron coupling Gepdefining the\nrate of the electron temperature equilibration time. This i s\nthe main parameter governing the magnetisation recovering\ntimeτE. Indeed, in Ref.25the phonon-electroncoupling Gep\nwas chosen to be 20 times smaller for Gd than for Ni. By\nadjustingthis parameter,the ultra-slowdemagnetisation rates\nobserved in TbFe alloy48, Gd49and in half-metals50as well\nas the two time-scales demagnetisation25,49are also well-\nreproduced (see, as an example, Fig.4). Within this model\nthetwo-timescaleprocessconsistsofarelativelyfastdem ag-\nnetisation (however much slower than in Ni), defined by the\nelectrontemperatureandsmallvalueof λ,followedbyamuch\nslowerprocessduetoaslowenergytransferfromtheelectro n\ntothelattice system.\nAs it was mentioned in the previous subsection, the\nphonon-mediatedEY mechanismpredictsthe couplingto the\nbath parameter λto be dependent on the ratio between the\nphonon and electron temperature through the relation (9). A\ndecrease of λup to two times at high fluencies is observed\nfor Ni and Co. The analysis of the data presented in Ref. 25\nand47forGdhasshownthatduringthedemagnetisationpro-\ncess the ratio Te/Tphas increased almost 6 times. In Fig.5\nwe present the magnetisation dynamics for Ni evaluated for6\nt[ps]400300200100m/m0\n32101\n0.8\n0.6\n0.4\nFigure 4. The result of integration of the LLB model ( S→∞) with\nconstantλ0=0.0015 (seeTableI).Inthiscase the2Tmodel param-\neters were takenfrom Ref.25corresponding toGd.\ntwo laser pulse fluencies, assumingvariousvaluesof the spi n\nSandtemperature-dependentandindependent λvalues. Note\nquite different demagnetisation rates at high fluency for tw o\nlimiting cases S=1/2, used in Ref.25 and S=∞, used in\nRef.22. The differences in the choice of λare pronounced\nat high pump fluency but are not seen at low fluency. One\ncan also hope that in the fitting procedure of experimental\ndata it would be possible to distinguish the two situations.\nUnfortunately, the fitting to experimental data procedure i s\ncomplicated and the changes coming from the two cases de-\nscribed above are competing with several different possibi l-\nities such as an additional temperature dependency in elec-\ntron or phonon specific heats51. Additionally, we would like\nto mention different electron-phonon coupling constants Gep\nused in Refs. 22 and 25. Fitting to experimental data from\nRef.25 for Ni for high fluence, we have found that the case\nof the temperature-dependent λ=λ0(Tp/Te)can be equally\nfitted with the temperature-independent λ≈λ0/2. To answer\ndefinitely which fitting is better, more experimental data pr o-\nmoting one or another intrinsic mechanism and varying laser\nfluencyisnecessary.\nIV. LINKING DIFFERENTTIMESCALES\nSince the longitudinal relaxation occurs under strong ex-\nchange field and the transverse relaxation - under external\napplied field, their characteristic timescales are quite di ffer-\nent. However, the LLB equation provides a relation be-\ntween the ultra-fast demagnetisation(longitudinalrelax ation)\nand the transverse relaxation (ordinaryLLG dampingparam-\neter) via the parameter λ0(λ=λ0orλ=λ0(Tp/Te)for\nTp=Te). The two demagnetisation rates could be measured\nindependently by means of the ultra-fast laser pump-probe\ntechnique52. It has been recently demonstrated53that the\ndamping of the laser-induced precession coincides with the\nmeasuredby FMR intransitionmetals. By separatemeasure-\nmentsofthetwomagnetisationrates,therelations(4)and( 5)\ngiven by the LLB theory could be checked. This can pro-\nvidethevalidationoftheLLBmodel,aswellastheanswerto\nthe question if the same microscopic mechanism is acting onM3TMLLB\nS→∞S→1/2S→∞S→1/2\nt[ps]∆m/m 0\n3 2 1 00.1\n0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\nFigure 5. Magnetisation dynamics during laser-induced dem agneti-\nsation process calculated within the LLB model with differe nt spin\nnumbers and for two laser-fluencies F=10 mJ/cm2(upper curves)\nandF=40 mJ/cm2(bottom curves). Ni parameters from Ref.22\nwere used. The symbols are calculated with the LLB equation w ith\nthe intrinsic damping parameter using a constant λ0=0.003 value,\nand the solid lines with the LLB equation and the intrinsic co upling\nwiththe temperature dependent λ=λ0/parenleftbig\nTp/Te/parenrightbig\n.\nfemtosecond and picosecond timescales. Unfortunately, th e\ndamping problem in ferromagnetic materials is very compli-\ncated and the literature reveals the diversity of measured v al-\nuesinthesamematerial,dependingonthepreparationcondi -\ntions.\nThus, to have a definite answer the measurement on\nthe same sample is highly desired. The measurements of\nbothα⊥andτMare available for Ni22where an excel-\nlent agreement between ultra-fast magnetisation rates via a\nuniquetemperature-independentparameter λ=0.04hasbeen\nreported22. The resultsof thesystematic measurementsof τM\nare also available for Ni, Co, Gd in Ref. 25, as well as for\nFe55. The next problem which we encounter here is that the\ndemagnetisationratesstronglydependon the spin value S, as\nis indicatedin Figs. 2 and 5. The fitting of experimentaldata\nusing LLB model with different Svalues results in different\nvalues of the coupling parameter λ0. The use of S=1/2\nvalue25orS=∞value22is quite arbitrary and these values\ndo not coincide with the atomic spin numbers of Ni,Co, Gd.\nGenerally speaking, for metals the spin value is not a good\nquantum number. The measured temperature dependence of\nmagnetisation, however, is well fitted by the Brillouin func -\ntionwith S=1/2forNi andCoand S=7/2forGd54. These\narethevaluesof Swhichwe usein TableI.\nConsequently in Table I we present data for the coupling\nparameter λ0extractedfromRef.25. Differentlytothisarticle,\nfor Gd we corrected the value of the parameter Rto account\nforadifferentspinvaluebytheratioof thefactors,i.e. RS1=\n(fS2/fS1)RS2with\nfS=2qS\nsinh(2qS)1\nm2\ne,SχS\n/bardbl, (12)\nwhere the parameters are evaluated at 120 Kusing the MFA\nexpressions for each spin value S. The data are evaluated7\nMaterial S R25λ0α⊥ αLLG\nNi 1/2 17.2 0.0974 0 .032 0 .01942-0.02841\nCo 1/2 25.3 0.179 0.025 0.003641-0.00643-0.01144\nGd 7/2 0.009 0.0015 0.00036 0 .000533\nTable I. The data for ultra-fast demagnetisation rate param eters for\nthree different metals from ultrafast demagnetization rat es and from\nFMR mesurements. The third column presents the demagnetisa tion\nparameter Rfrom Ref. 25, corrected in the case of Gd for spin\nS=7/2. The fourth column presents the value of the λ0parame-\nter, as estimated from the M3TM model25and the formula Eq.(9).\nThe fifth column presents the data for α⊥estimated via the LLB\nmodel Eq.(4) and the λ0value from the third column, at room tem-\nperatureT=300Kfor Co and Ni and at T=120Kfor Gd . The\nlast column presents the experimentally measured Gilbert d amping\ncollectedfrom different references.\nfor the phonon-mediated EY process with the temperature-\ndependent parameter λvia the expression (9). The value of\ntheGilbertdampingparameter α⊥wasthenestimatedthrough\nformula(4)at300 K(forNiandCo)andat120 KforGd. Note\nthat for temperature-independent λ=λ0the resulting λ0and\nα⊥valuesareapproximatelytwotimessmallerforNiandCo.\nThelastcolumnpresentsexperimentalvaluesforthesamepa -\nrameterfoundinliteratureforcomparisonwiththeonesint he\nfifth column, estimated through measurements of the ultra-\nfast demagnetisation times τMand the relation provided by\ntheLLBequation.\nGiven the complexityof the problem, the results presented\ninTableIdemonstratequiteasatisfactoryagreementbetwe en\nthe values, extractedfrom the ultra-fast demagnetisation time\nτMand the Gilbert damping parameter α⊥via one unique\ncoupling-to-the-bath parameter λ. The agreement is particu-\nlarlygoodforNi,indicatingthatthesamespinflipmechanis m\nisactingonbothtimescales. Thisistrueforbothexperimen ts\nin Refs.22 and 25. For Co the value is some larger. For the\ntemperature-independent λ, the resulting value is two times\nsmaller and the agreement is again satisfactory. We would\nlike to note that no goodagreementwas obtainedforFe. The\nreporteddampingvalues41are5-10timessmallerasestimated\nfromthedemagnetisationratesmeasuredinRef. 55.\nV. CONCLUSIONS\nThe Landau-Lifshitz-Bloch(LLB) equation providesa mi-\ncromagnetic tool for the phenomenological modelling of the\nultra-fast demagnetisation processes. Within this model o ne\ncan describe the temperature-dependent magnetisation dy-\nnamics at arbitrary temperature, including close and above\nthe Curie temperature. The micromagnetic formulation can\ntake into account the quantum spin number. The LLB model\nincludes the dynamics governed by both the atomistic LLG\nmodel and the M3TM model by Koopmans et al.25. In\nthe future it represents a real possibility for the multisca le\nmodelling20.\nWe have shown that within this model the ultra-fast de-\nmagnetisation rates could be parameterized through a uniqu etemperature dependent or independent parameter λ, defined\nby the intrinsic spin-flip rate. The magnetisation dynamics\nis coupled to the electron temperature throughthis paramet er\nand is always delayed in time. The observed delay is higher\nforhigherelectrontemperature. Thisisinagreementwitht he\nexperimentalobservationthatdifferentmaterialsdemagn etize\natdifferentrates25,50andthattheprocessissloweddownwith\nthe increase of laser fluency. We have shown that for the\nphonon-mediatedEY mechanism the intrinsic parameter λis\ndependent on the ratio between phonon and electron temper-\natures and therefore is temperature dependent on the femto\nsecond-severalpicosecondtimescale. TheLLBequationcan\nreproduce slow demagnetizing rates observed in several ma-\nterials such as Gd, TbFe and half metals. This is in agree-\nment with both phonon-mediated EY picture since in Gd a\nlowerspin-flipprobabilitywaspredictedandalsowiththei n-\nelastic electron scattering picture, since the electron di ffusive\nprocesses are suppressed in insulators and half-metals31,50.\nHowever, we also stress the importance of other parameters\ndetermining the ultra-fast demagnetisation rates, such as the\nelectron-latticecoupling.\nThe macroscopic damping parameters (longitudinal and\ntransverse) have different natures in terms of the involved\nspinwaves and in terms of the timescales. Their temperature\ndependenceisdifferent,however,theyarerelatedbythesp in-\nflip rate. We have tried to check this relation in several tran -\nsition metals such as Ni, Co, Fe and the rare-earth metal Gd.\nA good agreement is obtained in Co and Gd and an excel-\nlent agreement in Ni. This indicates that on both timescales\nthe same main microscopic mechanism is acting. In Ni the\nagreement is good both within the assumptions λ=λ0and\nλ=λ0Tp/Te. InCotheagreementseemstobebetterwiththe\ntemperature-independentparameter λ=λ0whichdoesnotin-\ndicate towards the phonon-mediated EY mechanism. How-\never, given a small discrepancy and the complexity of the\ndamping problem, this conclusion cannot be considered defi-\nnite. Wecanneitherexcludeanadditionaltemperaturedepe n-\ndence of the intrinsic scattering probability (i.e. the par ame-\nterλ0)forbothphonon-mediatedEYandexchangescattering\nmechanismswhichwasnottakenintoaccount.\nAnopenquestionistheproblemofdopedpermalloywhere\nanattempttosystematicallychangethedampingparameterb y\ndopingwithrare-earthimpuritieswasundertaken33inorderto\nclarify the relation between the LLG damping and the ultra-\nfast demagnetisation rate32,34. The results are not in agree-\nment with the LLB model. However in this case we think\nthat the hypothesis of the slow relaxing impurities present ed\nin Ref.34 might be a plausible explanation. Indeed, if the\nrelaxation time of the rare earth impurities is high, the sta n-\ndardLLB modelis not valid since it assumes an uncorrelated\nthermal bath. The correlation time could be introduced in\ntheclassicalspindynamicsviatheLandau-Lifshitz-Miyas aki-\nSeki approach56. It has been shown that the correlation time\nof the order of 10 fs slows down the longitudinal relaxation\nindependentlyon the transverse relaxation. Thus in this ca se,\nthemodificationoftheoriginalLLBmodeltoaccountforthe\ncolorednoiseisnecessary.8\nVI. ACKNOWLEDGEMENT\nThis work was supported by the Spanish projects\nMAT2007-66719-C03-01,CS2008-023.\nAppendixA\nTo show the equivalence between the LLB model with\nS=1/2 and the M3TM model25, we compare the relaxation\nrates resulting fromboth equations. We start with the M3TM\nequation\ndm\ndt=−RTp\nTC/parenleftbigg\n1−mcoth/bracketleftbigg/parenleftbiggTC\nTe/parenrightbigg\nm/bracketrightbigg/parenrightbigg\nm(A1)\nwhereweidentifytheBrillouinfunctionforthecase S=1/2,\nB1/2=tanh(q)withq=q1/2=(TC/Te)m. Now, we use the\nidentityB1/2=2/B′\n1/2sinh(2q)towrite\ndm\ndt=−RTp\nTC/bracketleftbigg2\nsinh(2q)/bracketrightbigg/parenleftBigg\n1−B1/2\nm\nB′\n1/2/parenrightBigg\nm2(A2)\nwe multiplyanddivideby qµatβto obtain\ndm\ndt=−RTp\nTCµat\nkBTC/bracketleftbigg2q\nsinh(2q)/bracketrightbigg/parenleftBigg\n1−B1/2\nm\nµatβB′\n1/2/parenrightBigg\nm(A3)\nM3TMLLB\nt[ps]∆m/m 0\n6 4 2 00\n-0.1\n-0.2\n-0.3\nFigure 6. Longitudinal relaxation calculated with M3TM and LLB\n(S=1/2) models for Nickel parameters22andT/Tc=0.8.We expand around equilibrium me=B1/2(qe)the small\nquantity1 −B1/2/m\n1−B1/2(q)\nm∼=δm\nme/parenleftbigg\n1−/parenleftbiggTC\nTe/parenrightbigg\nB′\n1/2(qe)/parenrightbigg\n(A4)\nwhereδm=m−me. Next,weexpand maroundm2\ne\nm=me+1\n2(m2−m2\ne)\nme=⇒δm\nme=(m2−m2\ne)\n2m2e(A5)\nand,\n1−B1/2/m\nβµatB′\n1/2≈1\n2/tildewideχ/bardbl(m2−m2\ne)\nm2e(A6)\nFinally,collectingthe equations(A3)and(A6)altogether :\ndm\ndt=/parenleftbigg3R\n2µat\nkBTC/parenrightbigg2Tp\n3TC2q\nsinh(2q)/parenleftbigg1\n2/tildewideχ/bardbl(1−m2\nm2e)m/parenrightbigg\n(A7)\nComparing this to the LLB equation with longitudinal re-\nlaxation only and without anisotropy and external fields, we\ncanwriteEq. (A7)in termsof n:\ndn\ndt=γλ\nme2Te\n3TC2q\nsinh(2q)Heff=γα/bardblHeff(A8)\nwhereHeff=me\n2/tildewideχ/bardbl(1−n2)n,and\nα/bardbl=/bracketleftbigg3R\n2γµat\nkBTCTp\nTe/bracketrightbigg2Te\n3TC2q\nsinh(2q)(A9)\nThustheKoopmans’M3TMequationisequivalenttotheLLB\nequationwith S=1/2andwheretheprecessionalaspectsare\nnotconsidered. 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Common behavior is observed for three FM layers, where the addit ional\nrelaxation obeys botha strict inverse power law dependence ∆ G=Ktn,n=−1.04±\n0.06 and a similar magnitude K= 224±40 Mhz·nm. As the tested FM layers span\nan order of magnitude in spin diffusion length λSDL, the results are in support of\nspin diffusion, rather than nonlocal resistivity, as the origin of the e ffect.\n1Theprimarymaterialsparameter which describes thetemporal res ponseofmagnetization\nMto applied fields His the Gilbert damping parameter α, or relaxation rate G=|γ|Msα.\nUnderstanding of the Gilbert relaxation, particularly in structures of reduced dimension, is\nan essential question for optimizing the high speed / Ghz response o f nanoscale magnetic\ndevices.\nExperiments over the last decade have established that the Gilbert relaxation of ferro-\nmagnetic ultrathin films exhibits a size effect, some component of whic h is nonlocal. Both\nα(tFM) =α0+α′(tFM) andG(tFM) =G0+G′(tFM) increase severalfold with decreasing FM\nfilm thickness tFM, from near-bulk values α0,G0fortFM>∼20 nm. Moreover, the damp-\ning size effect can have a nonlocal contribution responsive to layers or scattering centers\nremoved, through a nonmagnetic (NM) layer, from the precessing FM. Contributed Gilbert\nrelaxation has been seen from other FM layers1as well as from heavy-element scattering\nlayers such as Pt.2\nThe nonlocal damping size effect is strongly reminiscent of the electr ical resistivity in\nferromagnetic ultrathin films. Electrical resistivity ρis size-dependent by a similar factor\noverasimilarrangeof tFM; theresistivity ρ(tFM)issimilarlynonlocal,dependentuponlayers\nnot in direct contact.3–5. It isprima facie plausible that the nonlocal damping and nonlocal\nelectrical resistivity share a common origin in momentum scattering ( with relaxation time\nτM) by overlayers. If the nonlocal damping arises from nonlocal scat teringτ−1\nM, however,\nthere should be a marked dependence upon FM layer type. Damping in materials with\nshort spin diffusion length λSDLis thought to be proportional to τ−1\nM(ref.6); the claim for\n”resistivity-like” damping hasbeenmadeexplicitly forNi 81Fe19byIngvarsson7et al. ForFM\nwith along λSDL, onthe other hand, relaxation Giseither nearly constant withtemperature\nor ”conductivity-like,” scaling as τM.\nInterpretation of the nonlocal damping size effect has centered in stead on a spin current\nmodel8advanced by Tserkovnyak et al9. An explicit prediction of this model is that the\nmagnitude of the nonlocal Gilbert relaxation rate ∆ Gis only weakly dependent upon the\nFM layer type. The effect has been calculated10as\n∆G=|γ|2¯h/4π/parenleftBig\ng↑↓\neff/S/parenrightBig\nt−1\nFM (1)\n, where the effective spin mixing conductance g↑↓\neff/Sis given in units of channels per area.\nAb-initio calculationspredictaveryweakmaterialsdependencefortheinter facialparameters\n2g↑↓/S, with±10% difference in systems as different as Fe/Au and Co/Cu, and neglig ible\ndependence on interfacial mixing.11\nIndividual measurements exist of the spin mixing conductance, thr ough the damping,\nin FM systems Ni 81Fe1912, Co13, and CoFeB14. However, these experiments do not share\na common methodology, which makes a numerical comparison of the r esults problematic,\nespecially given that Gilbert damping estimates are to some extent mo del-dependent15. In\nour experiments, we have taken care to isolate the nonlocal dampin g contribution due to Pt\noverlayers only, controlling for growth effects, interfacial interm ixing, and inhomogeneous\nlosses. The only variable in our comparison of nonlocal damping ∆ G(tFM), to the extent\npossible, has been the identity of the FM layer.\nGilbert damping αhas been measured through ferromagnetic resonance (FMR) fro m\nω/2π= 2-24 Ghz using a broadband coplanar waveguide (CPW) with broad c enter conduc-\ntor width w= 400µm, using field modulation and lock-in detection of the transmitted signa l\nto enhance sensitivity. The Gilbert damping has been separated fro minhomogeneous broad-\nening inthe filmsmeasured using the well-known relation∆ Hpp(ω) = ∆H0+/parenleftBig\n2/√\n3/parenrightBig\nαω/|γ|.\nWe have fit spectra to Lorenzian derivatives with Dysonian compone nts at each frequency,\nfor each film, to extract the linewidth ∆ Hppand resonance field Hres;αhas been extracted\nusing linear fits to ∆ H(ω).\nFor the films, six series of heterostructures were deposited of th e form Si/ SiO 2/\nX/ FM(tFM)/ Cu(3nm)[ /Pt(3nm)]/ Al(3nm), FM = {Ni81Fe19(”Py”), Co 60Fe20B20\n(”CoFeB”), pure Co }, andtFM= 2.5, 3.5, 6.0, 10.0, 17.5, 30.0 nm, for 36 heterostruc-\ntures included in the study. For each ferromagnetic layer type FM, one thickness series tFM\nwas deposited with the Pt overlayer and one thickness series tFMwas deposited without the\nPt overlayer. This makes it possible to record the additional damping ∆α(tFM) introduced\nby the Pt overlayer alone, independent of size effects present in th e FM/Cu layers deposited\nbelow. In the case of pure Co, a X=Ta(5nm)/Cu(5nm) underlayer w as necessary to sta-\nbilize low-linewidth films, otherwise, depositions were carried out direc tly upon the in-situ\nion-cleaned substrate.\nField-for-resonance data are presented in Figure 1. The main pane l showsω(H/bardbl\nB) data\nfor Ni 81Fe19(tFM). Note that there is a size effect in ω(H/bardbl\nB): the thinner films have a\nsubstantially lower resonance frequency. For tFM= 2.5 nm, the resonance frequency is\ndepressed by ∼5 Ghz from ∼20 Ghz resonance HB≃4 kOe. The behavior is fitted through\n3the Kittel relation (lines) ω(H/bardbl\nB) =|γ|/radicalbigg/parenleftBig\nH/bardbl\nB+HK/parenrightBig/parenleftBig\n4πMeff\ns+H/bardbl\nB+HK/parenrightBig\n, and the inset\nshows a summary of extracted 4 πMeff\ns(tFM) data for the three different FM layers. Samples\nwith (open symbols) and without (closed symbols) Pt overlayers sho w negligible differences.\nLinear fits according to 4 πMeff\ns(tFM) = 4πMs−(2Ks/Ms)t−1\nFMallow the extraction of bulk\nmagnetization 4 πMsand surface anisotropy Ks; we find 4 πMPy\ns= 10.7 kG, 4 πMCoFeB\ns=\n11.8 kG, 4 πMCo\ns= 18.3 kG, and KPy\ns= 0.69 erg/cm2,KCoFeB\ns= 0.69 erg/cm2,KCo\ns=\n1.04 erg/cm2. The value of gL/2 =|γ|/(e/mc),|γ|= 2π·(2.799 Mhz/Oe) ·(gL/2) is found\nfrom the Kittel fits subject to this choice, yielding gPy\nL= 2.09,gCoFeB\nL= 2.07,gCo\nL= 2.15.\nThe 4πMsandgLvalues, taken to be size-independent, are in good agreement with b ulk\nvalues.\nFMR linewidth as a function of frequency ∆ Hpp(ω) is plotted in Figure 2. The data\nfor Py show a near-proportionality, with negligble inhomogeneous co mponent ∆ H0≤4 Oe\neven for the the thinnest layers, facilitating the extraction of intr insic damping parameter\nα. The size effect in in α(tFM) accounts for an increase by a factor of ∼3, fromαPy\n0=\n0.0067 (GPy\n0= 105 Mhz) for the thickest films ( tFM= 30.0 nm) to α= 0.021 for the\nthinnest films ( tFM= 2.5 nm). The inset shows the line shapes for films with and without\nPt, illustrating the broadening without significant frequency shift o r significant change in\npeak asymmetry.\nA similar analysis has been carried through for CoFeB and Co (not pict ured). Larger\ninhomogeneous linewidths are observed for pure Co, but homogene ous linewidth still ex-\nceeds inhomogeneous linewidth by a factor of three over the frequ ency range studied, and\ninhomogeneous linewidths agree within experimental error for the t hinnest films with and\nwithout Pt overlayers. We extract for these films αCoFeB\n0= 0.0065 ( GCoFeB\n0= 111 Mhz)\nandαCo\n0= 0.0085( GCo\n0= 234 Mhz). The latter value is in very good agreement with the\naverage of easy- and hard-axis values for epitaxial FCC Co films mea sured up to 90 Ghz,\nGCo\n0= 225 Mhz.16\nWe isolate the effect of Pt overlayers on the damping size effect in Figu re 3. Values\nofαhave been fitted for each deposited heterostructure: each FM t ype, at each tFM,\nfor films with and without Pt overlayers. We take the difference ∆ α(tFM) for identical\nFM(tFM)/Cu(5nm)/Al(3nm) depositions with and without the insertion of Pt (3nm) after\nthe Cu deposition. Data, as shown on the logarithmic plot in the main pa nel, are found\n4to obey a power law ∆ α(tFM) =Ktn, withn= -1.04±0.06. This is excellent agreement\nwith an inverse thickness dependence ∆ α(tFM) =KFM/tFM, where the prefactor clearly\ndepends on the FM layer, highest for Py and lowest for Co. Note tha t efforts to extract\n∆α(tFM) =Ktnwithout the FM( tFM)/Cu baselines would meet with significant errors;\nnumerical fits to α(tFM) =KtFMnfor the FM( tFM)/Cu/Pt structures yield exponents\nn≃1.4.\nExpressing now the additional Gilbert relaxation as ∆ G(tFM) =|γ|Ms∆α(tFM) =\n|γFM|MFM\nsKFM/tFM, we plot ∆ G·tFMin Figure 4. We find ∆ G·tPy= 192±40 Mhz,\n∆G·tCoFeB= 265±40 Mhz, and ∆ G·tCo= 216±40 Mhz. The similarity of values for\n∆G·tFMis in good agreement with predictions of the spin pumping model in Equa tion 1,\ngiven that interfacial spin mixing parameters are nearly equal in diffe rent systems.\nThe similarity of the ∆ G·tFMvalues for the different FM layers is, however, at odds\nwith expectations from the ”resistivity-like” mechanism. In Figure 4 ,inset, we show the\ndependence of ∆ G·tFMupon the tabulated λSDLof these layers from Ref17. It can be seen\nthatλCo\nSDLis roughly an order of magnitude longer than it is for the other two FM layers,\nPy and CoFeB, but the contribution of Pt overlayers to damping is ve ry close to their\naverage. Since under the resistivity mechanism, only Py and CoFeB s hould be susceptible\nto a resistivity contribution in ∆ α(tFM), the results imply that the contribution of Pt to\nthe nonlocal damping size effect has a separate origin.\nFinally, we compare the magnitude of the nonlocal damping size effect with that pre-\ndicted by the spin pumping model in Ref.10. According to ∆ G·tFM=|γ|2¯h/4π=\n25.69 Mhz ·nm3(gL/2)2/parenleftBig\ng↑↓\neff/S/parenrightBig\n, our experimental ∆ G·tFMandgLdata yield effective\nspin mixing conductances g↑↓\neff/S[Py/Cu/Pt ] = 6.8 nm−2,g↑↓\neff/S[Co/Cu/Pt ] = 7.3 nm−2,\nandg↑↓\neff/S[CoFeB/Cu/Pt ] = 9.6 nm−2. The Sharvin-corrected form, in the realistic limit\nofλN\nSDL≫tN11is (g↑↓\neff/S)−1= (g↑↓\nF/N/S)−1−1\n2(g↑↓\nN,S/S)−1+ 2e2h−1ρ tN+ (˜g↑↓\nN1/N2/S)−1.\nUsing conductances 14.1nm−2(Co/Cu), 15.0nm−2(Cu), 211nm−2(bulkρCu,tN= 3nm), 35\nnm−2(Cu/Pt) would predict a theoretical g↑↓\neff,th./S[Co/Cu/Pt ] = 14.1 nm−2. Reconciling\ntheory and experiment would require an order of magnitude larger ρCu≃20µΩ·cm, likely\nnot physical.\nTo summarize, a common methodology, controlling for damping size eff ects and intermix-\ning in single films, has allowed us to compare the nonlocal damping size eff ect in different\nFM layers. We observe, for Cu/Pt overlayers, the same power law in thickness t−1.04±0.06,\n5the same materials independence, but roughly half the magnitude th at predicted by the spin\npumping theory of Tserkovnyak10. The rough independence on FM spin diffusion length,\nshown here for the first time, argues against a resistivity-based in terpretation for the effect.\nWe would like to acknowledge the US NSF-ECCS-0925829, the Bourse Accueil Pro n◦\n2715 of the Rhˆ one-Alpes Region, the French National Research A gency (ANR) Grant ANR-\n09-NANO-037, and the FP7-People-2009-IEF program no 252067 .\nREFERENCES\n1R. Urban, G. Woltersdorf, and B. Heinrich, “Gilbert damping in single a nd multilayer\nultrathin films: role of interfaces in nonlocal spin dynamics,” Physical Review Letters 87,\n217204–7 (2001).\n2S. Mizukami, Y. Ando, and T. Miyazaki, “Effect of spin diffusion onGilber t damping for a\nverythinpermalloylayer inCu/permalloy/Cu/Pt films,”Phys. Rev. B 66, 104413 (2002).\n3B. Dieny, J. Nozieres, V. Speriosu, B. Gurney, and D. Wilhoit, “Chan ge in conductance\nis the fundamental measure of spin-valve magnetoresistance,” Ap plied Physics Letters 61,\n2111–3 (1992).\n4W. H. Butler, X. G. Zhang, D. 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E. W. Bauer, “Enhanced Gilbe rt damping in thin\n6ferromagnetic films,” Phys. Rev. Lett. 88, 117601 (2002).\n10Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, “Nonloca l magnetization dy-\nnamics in ferromagnetic heterostructures,” Reviews in Modern Phy sics77, 1375 – 421\n(2005).\n11M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W . Bauer, “First-\nprinciples study of magnetization relaxation enhancement and spin t ransfer in thin mag-\nnetic films,” Phys. Rev. B 71, 064420 (2005).\n12S. Mizukami, Y. Ando, and T. Miyazaki, “Magnetic relaxation of norma l-metal\nNM/NiFe/NM films,” Journal of Magnetism and Magnetic Materials 239, 42 – 4 (2002);\n“The study on ferromagnetic resonance linewidth for NM/ 80NiFe/ N M, (NM=Cu, Ta,\nPd and Pt) films,” Japanese Journal of Applied Physics, Part 1 (Regu lar Papers, Short\nNotes and Review Papers) 40, 580 – 5 (2001).\n13J. Beaujour, W. Chen, A. Kent, and J. Sun, “Ferromagnetic reso nance study of poly-\ncrystalline cobalt ultrathin films,” Journal of Applied Physics 99, 08N503 (2006); J.-M.\nBeaujour, J. Lee, A. Kent, K. Krycka, and C.-C. Kao, “Magnetiza tion damping in ultra-\nthin polycrystalline co films: evidence for nonlocal effects,” Physical Review B (Condensed\nMatter and Materials Physics) 74, 214405 – 1 (2006).\n14H.Lee,L.Wen, M.Pathak, P.Janssen, P.LeClair,C.Alexander, C .Mewes, andT.Mewes,\n“Spin pumping in Co 56Fe24B20multilayer systems,” Journal of Physics D: Applied Physics\n41, 215001 (5 pp.) – (2008).\n15R. McMichael and P. Krivosik, “Classical model of extrinsic ferroma gnetic resonance\nlinewidth in ultrathin films,” IEEE Transactions on Magnetics 40, 2 – 11 (2004).\n16“Gilbert damping and g-factor in Fe xCo1−xalloy films,” Solid State Communications 93,\n965 – 968 (1995).\n17J. Bass and J. Pratt, W.P., “Spin-diffusion lengths in metals and alloys, and spin-flipping\nat metal/metal interfaces: an experimentalist’s critical review,” Jo urnal of Physics: Con-\ndensed Matter 19, 41 pp. –(2007); C. Ahn, K.-H. Shin, andW. Pratt, “Magnetotran sport\nproperties of CoFeB and Co/Ru interfaces in the current-perpen dicular-to-plane geome-\ntry,” Applied Physics Letters 92, 102509 – 1 (2008).\n7FIGURES\nω/ 2π (Ghz)\nH  (Oe)B     1 / t      (nm  ) FM -1 \nFIG. 1. Fields for resonance ω(HB) for in-plane FMR, FM=Ni 81Fe19, 2.5 nm ≤tFM≤30.0 nm;\nsolid lines are Kittel fits. Inset:4πMeff\nsfor all three FM/Cu, with and without Pt overlayers.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s52/s48/s48 /s49/s53/s48/s48 /s49/s54/s48/s48 /s49/s55/s48/s48/s45/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s51\n/s49/s50/s32/s71/s72/s122/s72\n/s114/s101/s115\n/s72\n/s112/s112/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41\n/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41/s45/s80/s116\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41/s45/s80/s116/s39/s39/s47 /s72/s40/s97/s46/s117/s46/s41\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41\n/s32/s50/s46/s53/s110/s109\n/s32/s51/s46/s53/s110/s109\n/s32/s54/s110/s109\n/s32/s49/s48/s110/s109\n/s32/s49/s55/s46/s53/s110/s109\n/s32/s51/s48/s110/s109/s112/s112/s40/s79/s101/s41\n/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\nFIG. 2. Frequency-dependent peak-to-peak FMR linewidth ∆ Hpp(ω) for FM=Ni 81Fe19,tFM\nas noted, films with Pt overlayers. Inset:lineshapes and fits for films with and without Pt,\nFM=Ni 81Fe19, CoFeB.\n8      t     (nm)FM       t     (nm) FM α\nFIG. 3. Inset:αno Pt(tFM) andαPtfor Py, after linear fits to data in Figure 2. Main panel:\n∆α(tFM) =αPt(tFM)−αno Pt(tFM) for Py, CoFeB, and Co. The slopes express the power law\nexponent n= -1.04±0.06.\nλPy \nCoFeB\nCo \n      t     (nm)FM ∆G·t      (Mhz·nm) FM \n∆G·t     FM \n(nm) SDL \nFIG. 4. The additional nonlocal relaxation due to Pt overlay ers, expressed as a Gilbert relaxation\nrate - thickness product ∆ G·tFMfor Py, CoFeB, and Co. Inset:dependence of ∆ G·tFMon spin\ndiffusion length λSDLas tabulated in17.\n9"    },    {        "title": "1012.5473v1.Screw_pitch_effect_and_velocity_oscillation_of_domain_wall_in_ferromagnetic_nanowire_driven_by_spin_polarized_current.pdf",        "content": "arXiv:1012.5473v1  [cond-mat.other]  25 Dec 2010Screw-pitch effect and velocity oscillation of domain-wall in ferromagnetic nanowire\ndriven by spin-polarized current\nZai-Dong Li1,2,3, Qiu-Yan Li1, X. R. Wang3, W. M. Liu4, J. Q. Liang5, and Guangsheng Fu2\n1Department of Applied Physics, Hebei University of Technol ogy, Tianjin 300401, China\n2School of Information Engineering, Hebei University of Tec hnology, Tianjin, 300401, China\n3Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China.\n4Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100080, China\n5Institute of Theoretical Physics and Department of Physics , Shanxi University, Taiyuan 030006, China\nWe investigate the dynamics of domain wall in ferromagnetic nanowire with spin-transfer torque.\nThe critical current condition is obtained analytically. B elow the critical current, we get the static\ndomain wall solution which shows that the spin-polarized cu rrent can’t drive domain wall moving\ncontinuously. In this case, the spin-transfer torque plays both the anti-precession and anti-damping\nroles, which counteracts not only the spin-precession driv en by the effective field but also Gilbert\ndamping to the moment. Above the critical value, the dynamic s of domain wall exhibits the novel\nscrew-pitch effect characterized by the temporal oscillati on of domain wall velocity and width,\nrespectively. Both the theoretical analysis and numerical simulation demonstrate that this novel\nphenomenon arise from the conjunctive action of Gilbert-da mping and spin-transfer torque. We\nalso find that the roles of spin-transfer torque are entirely contrary for the cases of below and above\nthe critical current.\nPACS numbers: 75.75.+a, 75.60.Ch, 75.40.Gb\nKeywords: Screw-Pitch effect, Velocity Oscillation of Doma in-Wall, Spin-Polarized Current.\nA magnetic domain wall (DW) is a spatially local-\nized configuration of magnetization in ferromagnet, in\nwhich the direction of magnetic moments inverses gradu-\nally. Whenaspin-polarizedelectriccurrentflowsthrough\nDW, the spin-polarization of conduction electrons can\ntransfer spin momentum to the local magnetization,\nthereby applying spin-transfer torque which can manip-\nulate the magnetic DW without the applied magnetic\nfield. This spin-transfer effect was theoretically proposed\nby Slonczewski [1] and Berger [2], and subsequently veri-\nfied experimentally [3]. As a theoretical model the mod-\nified Landau-Lifshitz-Gilbert (LLG) equation [4–6] with\nspin-transfertorquewasderivedto describesuchcurrent-\ninduced magnetization dynamics in a fully polarized fer-\nromagnet. With these novel forms of spin torque many\ninteresting phenomena have been studied, such as spin\nwave excitation [4, 7, 8] and instability [4, 9], magneti-\nzation switching and reversal [10–13], and magnetic soli-\ntons [14, 15]. For the smooth DW, this spin torque can\ndisplace DW opposite to the current direction which has\nbeenconfirmedexperimentallyinmagneticthinfilmsand\nmagnetic wires [16–20].\nWith the remarkable experimental success measur-\ning the motion of DW under the influence of current\npulse,considerableprogresshasbeenmadetounderstand\nthe current-induced DW motion in magnetic nanowire\n[5, 6, 17–20]. These studies haveimproved the pioneering\nwork of current-driven DW motion by Berger [21]. Al-\nthough both the theory and the quasi-static experiments\nhave indicated that the spin-polarized current can cause\nDW motion, the current-driven DW dynamics is not well\nunderstood. The dynamics of magnetization described\nby the LLG equation admits the static solutions for DWmotion. In the presence of spin torque and the exter-\nnal magnetic field, it is difficult to derive the dynamic\nsolutions. A circumvented approach is Walker solution\nanalysis [22] for the moving DW in response to a steady\nmagnetic field smaller than some critical value. How-\never, this approximation applying to DW motion driven\nby the electric current is unclear, and its reliability has\nto be verified theoretically and numerically.\nInthispaper,wereportanalyticallythecriticalcurrent\ncondition for anisotropic ferromagnetic nanowire driven\nonly by spin-transfer torque. Below the critical cur-\nrent, the ferromagnetic nanowire admits only the final\nstatic DW solution which implies that the spin-polarized\ncurrent can’t drive DW moving continuously. When\nthe spin-polarized current exceeds the critical value, the\ndynamics of DW exhibits the novel Screw-pitch effect\nwith the periodic temporal oscillation of DW velocity and\nwidth. A detail theoretical analysis and numerical sim-\nulation demonstrate that this novel phenomenon arises\nfrom the natural conjunction action of Gilbert-damping\nand spin-transfer torque. We also observe that the spin-\ntransfer torque plays the entirely opposite roles in the\nabove two cases. At last, our theoretical prediction can\nbe confirmed by the numerical simulation in terms of\nRKMK method [23].\nWe consider an infinite long uniaxial anisotropic fer-\nromagnetic nanowire, where the electronic current flows\nalong the long length of the wire defined as xdirection\nwhichis alsothe easyaxisofanisotropyferromagnet. For\nconvenience the magnetization is assumed to be nonuni-\nform only in the direction of current. Since the magneti-\nzation varies slowly in space, it is reasonable to take the\nadiabatic limit. Then the dynamics of the localized mag-2\nnetization can be described by the modified LLG equa-\ntion with spin-transfer torque\n∂M\n∂t=−γM×Heff+α\nMsM×∂M\n∂t+bJ∂M\n∂x,(1)\nwhereM≡M(x,t) is the localized magnetization, γ\nis the gyromagnetic ratio, αis the damping parameter,\nandHeffrepresents the effective magnetic field. The\nlast term of Eq. (1) denotes the spin-transfer torque,\nwherebJ=PjeµB/(eMs),Pis the spin polarization of\nthe current, jeis the electric current density and flows\nalong the xdirection, µBis the Bohr magneton, eis\nthe magnitude of electron charge, and Msis the sat-\nuration magnetization. For the uniaxial ferromagnetic\nnanowire the effective field can be written as Heff=/parenleftbig\n2A/M2\ns/parenrightbig\n∂2M/∂x2+HxMx/Msex−4πMzez, whereAis\nthe exchange constant, Hxis the anisotropy field, and ei,\ni=x,y,z,is the unit vector, respectively. Introducing\nthe normalized magnetization, i.e., m=M/Ms, Eq. (1)\ncan be simplified as the dimensionless form\nα1∂m\n∂t=−m×heff−αm×(m×heff)\n+αb1m×∂m\n∂x+b1∂m\n∂x, (2)\nwhereα1=/parenleftbig\n1+α2/parenrightbig\nandb1=bJt0/l0. The time\ntand space coordinate xhave been rescaled by the\ncharacteristic time t0= 1/(16πγMs) and length l0=/radicalbig\nA/(8πM2s), respectively. The dimensionless effective\nfield becomes heff=∂2m/∂x2+C1mxex−C2mzez, with\nC1=Hx/(16πMs) andC2= 0.25.\nIn the following, we seek for the exact DW solutions of\nEq. (2), and then study the dynamics of magnetization\ndriven by spin-transfer torque. To this purpose we make\nthe ansatz\nmx= tanhΘ 1,my=sinφ\ncoshΘ 1,mz=cosφ\ncoshΘ 1,(3)\nwhere Θ 1=k1x−ω1t, with the temporal and spatial\nindependent parameters φ,k1, andω1to be determined,\nrespectively. Substituting Eq. (3) into Eq. (2) we have\nk2\n1=C1+C2cos2φ, (4)\n−ω1/parenleftbig\n1+α2/parenrightbig\n=b1k1+C2sinφcosφ,(5)\nαb1k1cosφ=α/parenleftbig\nC1−k2\n1/parenrightbig\nsinφ, (6)\nαb1k1sinφ=−αC2sin2φcosφ. (7)\nFrom the above equations we can get three cases of DW\nsolutions for Eq. (2). Firstly, in the absence of damping\nEqs. (4) to (7) admit the solution\nk1=±/radicalbig\nC1+C2cos2φ,ω1=−b1k1−C2\n2sin2φ,(8)\nwith the arbitrary angle φ. This solution show that the\nspin-transfer torque contributes a dimensionless veloc-\nity−b1only without damping. The velocity of DW isformed by two parts, i.e., v=−(C2sin2φ)/(2k1)−b1,\nwhich can be affected by adjusting the angle φand the\nspin-transfer torque. Secondly, in the absence of spin\ntorque, we have the solution of Eqs. (4) to (7) as ω1= 0,\nφ=±π/2,k1=±√C1, i.e., the static DW solution. In\nterms of RKMK method [23] we perform direct numeri-\ncal simulation for Eq. (2) with various initial condition,\nand all numerical results show that the damping drives\nthe change of φwhich in turn affects the DW velocity\nand width defined by 1 /|k1|. At last φ=±π/2,ω1= 0,\ni.e., the DW loses moving, and the DW width attains\nits maximum value√C1, which confirms the Walker’s\nanalysis [22] that the damping prevents DW from mov-\ning without the external magnetic field or spin-transfer\ntorque. However, as shown later, the presence of damp-\ning is prerequisite for the novel Screw-pitch property of\nDW driven by spin-transfer torque. At last, we consider\nthe case of the presence of damping and spin-transfer\ntorque. Solving Eqs. (4) to (7) we have\nk1=±1\n2(B1−/radicalbig\nB2),ω1= 0,sin2φ=−2b1k1\nC2,(9)\nwhereB1= 2C1+C2−b2\n1,B2=/parenleftbig\nC2−b2\n1/parenrightbig2−4C1b2\n1.\nIt is clear that Eq. (9) implies the critical spin-\npolarized current condition, namely\nbJ≤(/radicalbig\nC1+C2−/radicalbig\nC1)l0/t0,\nwhich is determined by the character velocity l0/t0,\nthe anisotropic parameter C1, and the demagnetiza-\ntion parameter C2. Below the critical current, i.e.,\nb2\n1≤(√C1+C2−√C1)2, the DW width falls into the\nrange that 1 //radicalbig\nC2\n1+C1C2≤1/|k1| ≤1/C1. From\nEq. (9) we get four solutions of φ, i.e.,φ=±π/2 +\n1/2arcsin(2 b1k1/C2) fork1>0 andφ=±π/2−\n1/2arcsin(2 b1|k1|/C2) fork1<0. In fact, the signs\n“+” and “ −” in Eq. (9) denotes kink and anti-kink so-\nlution, respectively, and the corresponding solution in\nEq. (3) represents the static tail-to-tail or head-to-head\nN´ eel DW, respectively. This result shows that below the\ncritical current, the final equilibrium DW solution must\nbe realized by the condition that m×heff=b1∂m/∂x.\nIt clearly demonstrates that the spin-transfer torque has\ntwo interesting effects. One is that the term b1∂m/∂x\nin Eq. (2) plays the anti-precession role counteracting\nthe precession driven by the effective field heff. How-\never, the third term in the right hand of Eq. (2), namely\nαb1m×∂m/∂x, has the anti-damping effect counteract-\ning the damping term −αm×(m×heff). It is to say\nthat below the critical value, the spin-polarized current\ncan’t drive DW moving continuously without the applied\nexternal magnetic field.\nWhen the spin-polarized current exceed the critical\nvalue, the dynamics of DW possesses two novel prop-\nerties as shown in the following section. Above the\ncritical current, the precession term −m×heffcan’t be\ncounteracted by spin-transfer torque, and the static DW3\nsolution of Eq. (2) doesn’t exist. Because the mag-\nnetization magnitude is constant, i.e., m2= 1, so we\nhavem·∂m/∂x= 0 which shows that the direction of\n∂m/∂xis always perpendicular to the direction of m, or\n∂m/∂x= 0. It is well known that a magnetic DW sep-\narates two opposite domains by minimizing the energy.\nIn the magnetic DW the direction of magnetic moments\ngradually changes, i.e., ∂m/∂x/negationslash= 0,so the direction of\n∂m/∂xshould adopt the former case. Out of region of\nDW the normalized magnetization will site at the easy\naxis, i.e., mx= 1(or−1), in which ∂m/∂x= 0.\nWiththeaboveconsiderationwemakeadetailanalysis\nfor Eq. (2). As a characteristic view we mainly consider\nthe DW center, defined by mx= 0. The magnetic mo-\nment must be in the y-zplane, while the direction of\n∂m/∂xshould lay in x-axis (+x-axis for k1>0, and\n−x-axis for k1<0). In order to satisfy Eq. (2) the mag-\nnetic moment in DW center should include both the pre-\ncession around the effective spin-torque field αb1∂m/∂x\nand the tendency along the direction of ∂m/∂xcontinu-\nouslyfromthelasttwotermsintherighthandofEq. (2).\nThe formerprecessionmotion implies that the parameter\nφwill rotate around x-axis continuously, while the lat-\nter tendency forces the DW center moving toward to the\nopposite direction of the current, i.e., −x-axis direction,\nconfirming the experiment [16–20] in magnetic thin films\nand magnetic wires. Combining the above two effects we\nfind that this rotating and moving phenomenon is very\nsimilar to Screw-pitch effect. The continuous rotation of\nmagnetic moment in DW center, i.e., the periodic change\nofφ, can result in the periodic oscillation of DW veloc-\nity and width from Eq. (8) under the action of the first\ntwo terms in the right hand of Eq. (2). It is interesting\nto emphasize that when the current exceeds the critical\nvalue, the term αb1m×∂m/∂xplays the role to induce\nthe precession, while the term b1∂m/∂xhas the effect\nof damping, which is even entirely contrary to the case\nbelow the critical current as mentioned before. Combin-\ning the above discussion we conclude that the motion of\nmagnetic moment in the DW center will not stop, except\nit falls into the easy axis, i.e., out of the range of DW. In\nfact, all the magnetic moments in DW can be analyzed\nin detail with the above similar procedure.\nNow it is clear for the dynamics of DW driven only\nby spin-transfer torque. Coming back to Eq. (2) we can\nsee that this novel Screw-pitch effect with the periodicoscillation of DW velocity and width occurs even at the\nconjunct action of the damping and spin-transfer torque.\nTo confirm our theoretical prediction we perform direct\nnumerical simulation for Eq. (2) with an arbitrary initial\ncondition by means of RKMK method [23] with the cur-\nrent exceeding the critical value. In figure 1(a) to 1(c)\nwe plot the time-evolution of the normalized magnetiza-\ntionm, while the displacement of DW center is shown in\nfigure 1(d). The result in figure 1 confirms entirely our\ntheoretical analysis above. The evolution of cos φand\nthe DW velocity and width are shown in figure 2. From\nfigure 2 we can see that the periodic change of cos φleads\nto the periodic temporal oscillation of DW velocity and\nwidth. From Eq. (8) and the third term of Eq. (2)\nwe can infer that cos φpossesses of the uneven change\nas shown in figure 2(a), i.e., the time corresponding to\n0< φ+nπ≤π/2 is shorter than that corresponding to\nπ/2< φ+nπ≤π,n= 1,2..., in each period, and the\nDW velocity has the same character. It leads to the DW\ndisplacement firstly increases rapidly, and then slowly as\nshown in figure 1(d). This phenomenon clarifies clearly\nthe presence of Screw-pitch effect . The DW velocity os-\ncillation driven by the external magnetic field has been\nobserved experimentally [24]. Our theoretical prediction\nfor the range of DW velocity oscillation driven by the\nabove critical current could be observed experimentally.\nIn summary, the dynamics of DW in ferromagnetic\nnanowire driven only by spin-transfer torque is theoret-\nically investigated. We obtain an analytical critical cur-\nrent condition, below which the spin-polarized current\ncan’t drive DW moving continuously and the final DW\nsolution is static. An external magnetic field should be\napplied in order to drive DW motion. We also find that\nthe spin-transfer torque counteracts both the precession\ndrivenbytheeffectivefieldandtheGilbertdampingterm\ndifferent from the common understanding. When the\nspin current exceeds the critical value, the conjunctive\naction of Gilbert-damping and spin-transfer torque leads\nnaturally the novel screw-pitch effect characterized by\nthe temporal oscillation of DW velocity and width.\nThis work was supported by the Hundred Innovation\nTalents Supporting Project of Hebei Province of China,\nthe NSF of China under grants Nos 10874038, 10775091,\n90406017, and 60525417, NKBRSFC under grant No\n2006CB921400, and RGC/CERG grant No 603007.\n[1] Slonczewski J C, 1996 J. Magn. Magn. Mater. 159 L1\n[2] Berger L, 1996 Phys. Rev. B 54 9353\n[3] Katine J A, Albert F J, Buhrman R A Myers E B, and\nRalph D C 2000 Phys. Rev. Lett. 84 3149\n[4] Bazaliy Y B, Jones B A, and Zhang Shou-Cheng, 1998\nPhys. Rev. B 57 R3213\nSlonczewski J C, 1999 J. Magn. Magn. Mater. 195 L261\n[5] Tatara G, Kohno H, 2004 Phys. Rev. Lett. 92 086601\nHo J, Khanna F C, and Choi B C, 2004 Phys. Rev. Lett.92 097601\n[6] Li Z and Zhang S, 2004 Phys. Rev. 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Lett. 81\n862\nGrollier J, Boulenc, P, Cros V, Hamzic A, Vaures A, Fert\nA, and Faini G, 2003 Appl. Phys. Lett. 83 509\nTsoi M, Fontana R E, Parkin S S 2003 Appl. Phys. Lett.\n83 2617[17] Yamaguchi A, Ono T, Nasu S, Miyake K, Mibu K, Shinjo\nT, 2004 Phys. Rev. Lett. 92 077205\n[18] Saitoh E, Miyajima H, Yamaoka T and Tatara G, 2004\nNature 432 203\n[19] Lim C K, Devolder T, Chappert C, Grollier J, Cros V,\nVaures A, Fert A, and Faini G, 2004 Appl. Phys. Lett.\n84 2820\nTatara G, Saitoh E, Ichimura M and Kohno H, 2005\nAppl. Phys. Lett. 86 232504\n[20] Ohe J, Kramer B, 2006 Phys. Rev. Lett. 96 027204\nBeach G S D, Knutson C, Nistor C, Tsoi M, and Erskine\nJ L, 2006 Phys. Rev. Lett. 97 057203\nDugaev V K, Vieira V R, Sacramento P D, Barna J,\nAra´ ujo M A N, and J. Berakdar J, 2006 Phys. Rev. B\n74 054403\n[21] Berger L, 1978 J. Appl. Phys. 49 2156\nBerger L, 1984 J. Appl. Phys. 55 1954\nBerger L, 1992 J. Appl. Phys. 71 2721\nSalhi E and Berger L, 1993 J. Appl. Phys. 73 6405\n[22] Thiele A A, 1973 Phys. Rev. B 7 391\nSchryer N L, and Walker L R, 1974 J. Appl. Phys. 45\n5406\n[23] Munthe-Kaas H, 1995 BIT. 35 572\nEngo K, 2000 BIT. 40 41\n[24] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L, 2005 Nat. Mater. 4 741\nYang J, Nistor C, Beach G S D, and Erskine J L, 2008\nPhys. Rev. B 77 014413\nFigure Captions\nFig. 1. The dynamics of DW above the critical\ncurrent. (a)-(c) Evolution of the normalized magneti-\nzationm. (d) The displacement of DW driven only\nby spin-transfer torque. The parameters are α= 0.2,\nC1= 0.05,C2= 0.25,bJ= 0.6, and the initial angle\nφ= 0.01π.\nFig. 2. (a) The evolution of cos φand the periodic\noscillation of DW velocity. (b) The periodic temporal\noscillation of DW width. The parameters are same as in\nfigure 1.0 20 40 60 80 100 120 140 160 180 200-1-0.8-0.6-0.4-0.200.20.40.60.81\nTimeEvolution of cos( I) and velocity of DW center(a)Velocity of DW center\ncos(I) of DW center0 20 40 60 80 100 120 140 160 180 20012345\nTimeEvolution of DW width(b)"    },    {        "title": "1101.3144v1.Steiner_Ratio_for_Manifolds.pdf",        "content": "arXiv:1101.3144v1  [math.MG]  17 Jan 2011Steiner Ratio for Riemannian Manifolds\nD. Cieslik, A. O. Ivanov, A. A. Tuzhilin\nAbstract\nFor a metric space ( X,ρ) and any finite subset N⊂Xbyρ(SMT N)\nandρ(MST N) we denote respectively the lengths of a Steiner minimal\ntree and a minimal spanning tree with the boundary N. TheSteiner\nratiom(X,ρ) of the metric space is the value inf {N:N⊂X}ρ(SMTN)\nρ(MSTN). In\nthis paper we prove the following results describing the Ste iner ratio of\nsome manifolds:\n(1) the Steiner ratio of an arbitrary n-dimensional connected Rieman-\nnian manifold Mdoes not exceed the Steiner ratio of Rn;\n(2) the Steiner ratio of the base of a locally isometric cover ing is more\nor equal than the Steiner ratio of the total space;\n(3) the Steiner ratio of a flat two-dimensional torus, a flat Kl ein bottle,\na projective plain having constant positive curvature is eq ual to√\n3/2;\n(4) the Steiner ratio of the curvature −1 Lobachevsky space does not\nexceed 3 /4;\n(5) the Steiner ratio of an arbitrary surface of constant neg ative cur-\nvature−1 is strictly less than√\n3/2.\nKeywords: Steinerminimaltree(SMT),minimalspanningtree(MST),\nthe Steiner problem, the Steiner ratio, metric space, Riema nnian mani-\nfold.\n1 Introduction and main results\nLetVbe an arbitrary finite set. Recall that a graphGonVis the pair\n(V,E), where Eis a finite set that consists of some pairs of elements from V.\nNotice that Ecan contain several copies of some pair, and also Ecan contain\nthe pairs ofthe form {v,v}, wherev∈V. Elements from Vare called vertices of\nG, and the elements from Eare called edges of G. The edges of the form ( v,v)\nare called loops, and ifEcontains several copies of an edge e={v,v′} ∈E,\nthen the edge eis called a multiple edge . For a given graph Gwe denote the set\nof all its vertices by V(G), and the set of all its edges by E(G). For convenience,\nwe shall often denote the edge e={x,y} ∈E(G) byxy.\nSometimes it is useful to consider graphs as topological spaces glue d from\nsegments each of which corresponds to an edge of the graph. Suc h graphs are\nA. Ivanov and A. Tuzhilin were partially supported by RFBR (g rants 96–15–96142 and\n98–01–00240) and INTAS (grant 97–0808).\n1Introduction and main results. 2\ncalledtopological graphs . A continuous mapping Γ from a topological graph G\ninto a topological space is called a network; the topological graph G, and also\nthe standard graph corresponding to G, are called the typeof Γ or the topology\nof Γ. Thus, the edges of a network are continuous curves in the am bient space.\nMoreover,all the terminology of the Topological Spaces Theory is t ransferred to\nthe topological graphsand networks. If the ambient space is a smo oth manifold,\nthen a network in such space is called smooth(piecewise-smooth ), if all its edges\nare smooth (piecewise-smooth).\nAgraph Gis calledweighted ifit isgivena non-negativefunction ω:E(G)→\nRcalled the weight function . The number ω(e) is called the weight of the edge\ne∈E(G). The sum of the weights over all edges of Gis called the weight of\nthe graph Gand it is denoted by ω(G). IfGis a connected weighted graph,\nthen the set of all connected spanning subgraphs of Ghaving the least weight\ncontains a tree. Each such tree is called a minimal spanning tree and is denoted\nby MST G. Notice that if all the weights are strictly greater than zero, then any\nconnected spanning subgraph of Gof the least weight is a tree.\nLetXbe a set, ρbe a metric on X, andNbe an arbitraryfinite subset of X.\nLetGbe a complete graph on N. The metric ρgenerates the weight function\nthat assigns to each edge xy∈E(G) the number ρ(x,y). This weight function\nwill be denoted by the same letter ρ. Minimal spanning tree in the graph Gis\ndenoted by MST N. Aminimal Steiner tree on the set Nor aminimal Steiner\ntree spanning the set Nis defined to be a tree Γ, N⊂V(Γ), such that\nρ(Γ) = inf\n{¯N:¯N⊂N}ρ(MST ¯N), (1)\nwhere the least upper bound is taken over all finite subsets ¯NinXthat contain\nN. A minimal Steiner tree on the set Nis denoted by SMT N.\nNote that, generally speaking, an SMT Nexists not for any N(one of the\nreasonsof that can be the incompleteness of the metric space ( X,ρ)). Neverthe-\nless, the greatest lower bound from the definition of SMT Ndoes always exist.\nIn what follows, the greatest lower bound from (1) is always d enoted\nbyρ(SMTN), irrespective of the existence of SMTN.\nThe novelty of Steiner’s Problem is that new points, the Steiner point s, may\nbe introduced so that an interconnecting network of all these poin ts will be\nshorter. Given a set of points, it is a priori unclear how many Steiner points one\nhas to add in order to construct an SMT. Whereas Steiner’s Problem is very\nhard as well in combinatorial as in computational sense, the determ ination of a\nMinimum Spanning Tree is simple. Consequently, we are interested in\nDefinition. TheSteiner ratio m(X,ρ)of a metric space (X,ρ) is defined as\nthe following value:\nm(X,ρ) = inf\n{N:N⊂X}ρ(SMTN)\nρ(MSTN).\nIt is clear that the Steiner ratio of any metric space is always a nonne gative\nnumber with m(X,ρ)≤1. The Steiner ratio is a parameter of the consideredIntroduction and main results. 3\nspaceanddescribestheapproximationratioforSteiner’sProblem. Thequantity\nm(X,ρ)·ρ(MSTN) would be a convenient lowerbound for the length of an SMT\nforNin (X,ρ); that means, roughly speaking, m(X,ρ) says how much the total\nlength of an MST can be decreased by allowing Steiner points.\nProposition 1.1 (E.F.Moore, in [3]) For the Steiner ratio of any metric\nspace(X,ρ)the inequalities\n1\n2≤m(X,ρ)≤1\nhold.\nIt is also shown that these inequalities are the best possible ones ove r the\nclass of metric spaces.1\nAs an introductory example consider three points which form the no des of\nan equilateral triangle of unit side length in the Euclidean plane. An MST for\nthese points has length 2. An SMT uses one Steiner point, which is uniq uely\ndetermined by the condition that the three angles at this point are e qual, and\nconsequently equal 120◦. Consequently, we find the length of the SMT in 3 ·/radicalbig\n1/3 =√\n3. So we have an upper bound for the Steiner ratio of the Euclidean\nplane:\nm≤√\n3\n2= 0.86602.... (2)\nA long-standing conjecture, given by Gilbert and Pollak [3] in 1968, sa id that\nin the above inequality equality holds. This was the most important con jecture\nin the area of Steiner’s Problem in the following years. Finally, in 1990, D u and\nHwang [2] created many new methods and succeeded in proving the G ilbert-\nPollak conjecture completely: The Steiner Ratio of the Euclidean plan e equals√\n3/2 = 0.86602....2\nFor each dimension n >2, at present, exact values for the Steiner ratios of\nthe Euclidean spaces are not yet known. In particular, this is true f orn= 3.\nSMT’s have been the subject of extensive investigations during the past 30\nyears or so. Most of this research has dealt with the Euclidean metr ic, with\nmuch of the remaining work concerned with the L1-metric, or more generally,\nthe usual Lp-metric or with two-dimensional Banach spaces. An overview for\nthe Steiner ratios of these metric spaces is given in [1].\nThe first results concerning the Steiner ratios of Riemannian manifo lds dif-\nferent from Euclidean spaces were obtained by J. H. Rubinstein and J. F. Weng\nin 1997, see [7]. They have shown that the Steiner ratio for the stan dard two-\ndimensional spheres is the same as for the Euclidean plane, that is,√\n3/2.\nNow we list the main results of the present article. These results wer e\nobtainedbymeansofthetechniqueworkedoutin[1], [5],and[6]. Letus mention\nthat in [5] and [6] the authors investigateso called local minimal netwo rkswhich\nturn out to be useful in the subject.\n1And, indeed, there are metric spaces with Steiner ratios equ als 1 and equals 0 .5.\n2This mathematical fact went in The New York Times, October 30 , 1990 under the title\n”Solution to Old Puzzle: How Short a Shortcut?”Proofs of the theorems. 4\nTheorem 1.1 The Steiner ratio of an arbitrary n-dimensional connected Rie-\nmannian manifold Mdoes not exceed the Steiner ratio of Rn.\nTheorem 1.2 Letπ:W→Mbe a locally isometric covering of connected\nRiemannian manifolds. Then the Steiner ratio of the base Mof the covering is\nmore or equal than the Steiner ratio of the total space W.\nCorollary 1.1 The Steiner ratio for a flat two-dimensional torus, a flat Klei n\nbottle, a projective plain having constant positive curvat ure is equal to√\n3/2.\nThus, taking into account the results of J. H. Rubinstein and J. F. W eng [7],\nthe Steiner ratio is computed now for all closed surfaces having non -negative\ncurvature.\nTheorem 1.3 The Steiner ratio of the curvature −1Lobachevsky space does\nnot exceed 3/4.\nTheorem 1.4 The Steiner ratio of an arbitrary surface of constant negati ve\ncurvature −1is strictly less than√\n3/2.\nTheauthorswanttothankthe Ernst–Moritz–ArndtUniversityof Greifswald\nfor the opportunity to work together in Greifswald in March 2000. A . Ivanov\nand A. Tuzhilin are grateful to academic A. T. Fomenko for his kind int erest to\nour work.\n2 Proofs of the theorems\nIn the present section we give the proofs of the theorems stated above.\nWe need the following two Lemmas proved in [1] (notice that Lemma 2.1 is\nproved in [1] for the case of normalized spaces only, but the proof in the general\ncase of metric spaces is just the same.)\nLemma 2.1 LetXbe a set, and ρ1andρ2be two metrics on X. We assume\nthat for some numbers c2≥c1>0and for arbitrary points xandyfromXthe\nfollowing inequality holds: c1ρ2(x,y)≤ρ1(x,y)≤c2ρ2(x,y). Then\nc1\nc2m(X,ρ2)≤m(X,ρ1)≤c2\nc1m(X,ρ2).\nLemma 2.2 Let(X,ρ)be a metric space, and Y⊂Xbe some its subspace.\nThen\nm(Y,ρ)≥m(X,ρ).\nThe following Proposition holds.Proofs of the theorems. 5\nProposition 2.1 Letf:X→Ybe some mapping of a metric space (X,ρX)\nonto a metric space (Y,ρY). We assume that fdoes not increase the distances,\nthat is, for arbitrary points xandyfromXthe following inequality holds:\nρY/parenleftbig\nf(x),f(y)/parenrightbig\n≤ρX(x,y).\nThen for arbitrary finite set N⊂Ywe have:\nρX/parenleftbig\nMSTN/parenrightbig\n≥ρY/parenleftbig\nMSTf(N)/parenrightbig\n, ρX/parenleftbig\nSMTN/parenrightbig\n≥ρY/parenleftbig\nSMTf(N)/parenrightbig\n.\nProof.LetGbe an arbitrary connected graph constructed on N. We consider\ntwo weight functions on Gdefined on the edges xyofGas follows: ρX(xy) =\nρX(x,y), andωY(xy) =ρY/parenleftbig\nf(x),f(y)/parenrightbig\n. Sincefdoes not increasethe distances,\nthenρX(G)≥ωY(G).\nLetG′be a graph on N′=f(N), such that the number of edges joining\nthe vertices x′andy′fromN′=V(G′) is equal to the number of edges from G\njoining the vertices from f−1(x′)∩Nwith the vertices from f−1(y′)∩N. It is\nclear that G′is connected, and ρY(G′) =ωY(G).\nConversely, it is easy to see that for an arbitrary connected grap hG′con-\nstructed on f(N) there exists a connected graph GXonN, such that ρY(G′) =\nωY(GX). (To construct GXit suffices to span each set N∩f−1(x′),x′∈N′,\nby a connected graph, and then to join each pair of the construct ed graphs cor-\nresponding to some adjacent vertices G′bykedges, where kis the multiplicity\nof the corresponding edge in G′). Therefore,\nρX(MSTN) = inf\n{G:V(G)=N}ρX(G)≥inf\n{G:V(G)=N}ωY(G) =\ninf\n{G′:V(G′)=f(N)}ρY(G′) =ρY/parenleftbig\nMSTf(N)/parenrightbig\n.\nThereby, the first inequality is proved.\nNow let us prove the second inequality. We have:\nρX(SMTN) = inf\n{¯N:¯N⊃N}ρX(MST ¯N)≥inf\n{¯N:¯N⊃N}ρY(MSTf(¯N))≥\ninf\n{¯N′:¯N′⊃f(N)}ρY(MST ¯N′) =ρY(SMTf(N)).\nThe proof is complete.\nProposition 2.2 Letf:X→Ybe a mapping of a metric space (X,ρX)to a\nmetric space (Y,ρY), and let fdo not increase the distances. We assume that\nfor each finite subset N′⊂Ythere exists a finite subset N⊂X, such that\nf(N) =N′and\nρX(SMTN)≤ρY(SMTN′). (3)\nThen\nm(X,ρX)≤m(Y,ρY).Proofs of the theorems. 6\nProof.LetN⊂Xbe an arbitrary finite set. We have\nm(X,ρX) = inf\n{N:N⊂X}ρX(SMTN)\nρX(MSTN)=\ninf\n{N′:N′⊂Y}inf\n{N:f(N)=N′}ρX(SMTN)\nρX(MSTN)≤\ninf\n{N′:N′⊂Y}ρY(SMTN′)\nρY(MSTN′)=m(Y,ρY),\nwhere the inequality follows from both condition (3) and the first ineq uality of\nProposition 2.1. The proof is complete.\nProposition 2.2 can be slightly reinforced as follows.\nProposition 2.3 Letf:X→Ybe a mapping of a metric space (X,ρX)to a\nmetric space (Y,ρY), and let fdo not increase the distances. We assume that\nfor each finite subset N′⊂Ythe following inequality holds:\ninf\n{N:f(N)=N′}ρX(SMTN)≤ρY(SMTN′). (4)\nThen\nm(X,ρX)≤m(Y,ρY).\nProof.LetN⊂Xbe an arbitraryfinite set. As in the proofofProposition2.2,\nwe have:\nm(X,ρX) = inf\n{N:N⊂X}ρX(SMTN)\nρX(MSTN)= inf\n{N′:N′⊂Y}inf\n{N:f(N)=N′}ρX(SMTN)\nρX(MSTN).\nSincefdoes not increase distances, then ρX(MSTN)≥ρY(MSTf(N)) (see\nProposition 2.1); on the other hand, due to our assumption, there exists a se-\nquenceoffinitesets Ni⊂X,f(Ni) =N′, suchthat ρX(SMTNi)≤ρY(SMTN′)+\nεi, where the sequence of positive numbers εitends to 0 as i→ ∞, and the se-\nquence of positive numbers ρX(SMTNi) tends to inf {N:f(N)=N′}ρX(SMTN).\nTherefore,\nρX(SMTNi)\nρX(MSTNi)≤ρY(SMTN′)+εi\nρY(MSTN′),\nand, taking in account that {Ni} ⊂ {N:f(N) =N′}, we get:\ninf\n{N′:N′⊂Y}inf\n{N:f(N)=N′}ρX(SMTN)\nρX(MSTN)≤\ninf\n{N′:N′⊂Y}inf\n{Ni}ρX(SMTNi)\nρX(MSTNi)≤inf\n{N′:N′⊂Y}inf\niρY(SMTN′)+εi\nρY(MSTN′)=\ninf\n{N′:N′⊂Y}ρY(SMTN′)\nρY(MSTN′)=m(Y,ρY).\nThe proof is complete.Proofs of the theorems. 7\nLetMbe an arbitrary connected n-dimensional Riemannian manifold. For\neach piecewise-smooth curve γby len(γ) we denote the length of γwith respect\nto the Riemannian metric. By ρwe denote the intrinsic metric generated by\nthe Riemannian metric. We recall that\nρ(x,y) = inf\nγlen(γ),\nwhere the greatest lower bound is taken over all piecewise-smooth curvesγ\njoining the points xandy.\nLetPbe a point from M. We consider the normal coordinates ( x1,...,xn)\ncentered at P, such that the Riemannian metric gij(x) calculated at Pcoincides\nwithδij. LetU(δ) be the open convex ball centered at Pand having the radius\nδ. Any two points xandyfrom the ball are joined by a unique geodesic γ\nlying inU(δ). At that time, ρ(x,y) = len(γ). Thus, the ball U(δ) is a metric\nspace with intrinsic metric, that is, the distance between the points equals to\nthe greatest lower bound of the curves‘ lengths over all the meas urable curves\njoining the points. Notice that in terms of the coordinates ( xi) the ball U(δ) is\ndefined as follows:\nU(δ) =/braceleftbig\n(x1)2+···+(xn)2< δ2/bracerightbig\n.\nTherefore, if we define the Euclidean distance ρeinU(δ) (in terms of the normal\ncoordinates( xi)), thenthemetricspace/parenleftbig\nU(δ),ρe/parenrightbig\nalsoisthespacewithintrinsic\nmetric generated by the Euclidean metric δij.\nSince the Riemannian metric gij(x) depends on x∈U(ε) smoothly, then for\nanyε, 1/n2> ε >0, there exists a δ >0, such that\n|gij(x)−δij|< ε (5)\nfor all points x∈U(δ). The latter implies the following Proposition.\nProposition 2.4 Let/bardblv/bardblgbe the length of the tangent vector v∈TxMwith\nrespect to the Riemannian metric gij, and let /bardblv/bardblebe the length of vwith respect\nto the Euclidean metric δij. If for any iandjthe inequality (5) holds, then\n/radicalbig\n1−n2ε/bardblv/bardble≤ /bardblv/bardblg≤/radicalbig\n1+n2ε/bardblv/bardble.\nProof.Consider an orthogonal transformation (with respect to the Euc lidean\nmetricδij) reducing the matrix ( gij) to the diagonal form diag( λ1,...,λ n), and\nlet (ci\nj) be the matrix of this transformation. Then λk=/summationtext\ni,jci\nkcj\nkgij, therefore,\nusing that |ci\nj| ≤1 due to orthogonality of ( ci\nj), we get:\n|λk−1|=/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni,j(ci\nkcj\nkgij−ci\nkcj\nkδij)/vextendsingle/vextendsingle/vextendsingle≤\n/summationdisplay\ni,j|ci\nk|·|cj\nk|·|gij−δij| ≤/summationdisplay\ni,j|gij−δij| ≤n2ε.Proofs of the theorems. 8\nSo we have:\n/bardblv/bardblg=/radicalBigg/summationdisplay\nkλkvkvk≤/radicalBigg\nmax\nkλk/summationdisplay\nkvkvk≤/radicalbig\n1+n2ε/bardblv/bardble.\nSimilarly, we get\n/bardblv/bardblg≥/radicalbig\n1−n2ε/bardblv/bardble.\nThe proof is complete.\nUsing the definition of the distance between a pair of points of a conn ected\nRiemannian manifold, we obtain the following result.\nCorollary 2.1 LetMbe an arbitrary connected n-dimensional Riemannian\nmanifold, and let U(δ),ρ, andρebe as above. Then for an arbitrary ε,1/n2>\nε >0, there exists a δ >0, such that\n/radicalbig\n1−n2ερe(x,y)≤ρ(x,y)≤/radicalbig\n1+n2ερe(x,y)\nfor all points x,y∈U(δ).\nSince the Steiner ratio is evidently the same for any convex open sub sets of\nRn, Corollary 2.1 and Lemma 2.1 lead to the following result.\nCorollary 2.2 LetMbe an arbitrary n-dimensional Riemannian manifold, let\nU(ε)⊂Mbe an open convex ball of a small radius ε, and let Pbe the center of\nU(ε). Byρwe denote the metric on Mgenerated by the Riemannian metric.\nThen /radicalbigg\n1−n2ε\n1+n2εm(Rn)≤m/parenleftbig\nU(ε),ρ/parenrightbig\n≤/radicalbigg\n1+n2ε\n1−n2εm(Rn),\nwherem(Rn)stands for the Steiner ratio of the Euclidean space Rn.\nNow let us prove the main theorems stated in Introduction.\nProof of Theorem 1.1. LetMbe an arbitrary connected n-dimensional Rie-\nmannian manifold, and let ρbe the metric generated by the Riemannian metric\nofM. LetP∈Mbe an arbitrary point from M, and let U(ε) be an open\nconvex ball centered at Pand having radius ε <1/n2. As above, let ( xi) be\nnormal coordinates on U(ε), and let ρebe the metric on U(ε) generated by the\nEuclidean metric δij(with respect to ( xi)).\nFor some decreasing sequence {εi}of positive numbers with εi< εfor any\ni, whereεi→0 asi→ ∞, we consider a family of nested subsets Xi=U(εi).\nNotice that due to convexity of Euclidean balls/parenleftbig\nU(ε),ρe/parenrightbig\nwe have:\nm/parenleftbig\nU(ε),ρe/parenrightbig\n=m(Rn).\nBesides, due to convexity of the balls U(ε) with respect to the intrinsic metric ρ′\ngenerated by the Riemannian metric gij, this intrinsic metric ρ′coincides withProofs of the theorems. 9\nthe restriction of the metric ρ. Thus, the ball U(ε) with the intrinsic metric ρ′\nis a subspace in ( M,ρ).\nCorollary 2.2 implies that\nm(Xi,ρ)≤/radicalbigg\n1+n2ε\n1−n2εm(Rn).\nSince/radicalBig\n1+n2ε\n1−n2ε→1 asi→ ∞due to the choice of {εi}, we get\ninf\nim(Xi,ρ)≤m(Rn).\nBut, due to Lemma 2.2 we have:\nm(M,ρ)≤inf\nim(Xi,ρ).\nThe proof is complete.\nProof of Theorem 1.2. Letπ:W→Mbealocallyisometriccovering,where\nWandMare connected Riemannian manifolds. By ρWandρMwe denote the\nmetrics generatedby the Riemannian metrics on WandM, respectively. Notice\nthat a locally isometric covering does not increase distances, since t he image of\na measurable curve γhas the same length as γhas.\nWe consider an arbitrary finite set N′⊂M. LetG′\nibe a family of trees on\nfinite sets ¯N′\ni⊃N′such that\nρM(G′\ni)→ρM(SMTN′) asi→ ∞.\nForeach G′\nibyΓ′\niwedenoteanembeddednetworkofthetype G′\nionMsuchthat\nthe vertex set of Γ′\niisV(G′\ni) and the length of Γ′\nidiffers from ρM(G′\ni) at most\nby 1/i. Let Γ ibe a connected component of π−1(Γ′\ni), andNi=π−1(N)∩Γi.\nSince the network Γ′\niis contractible, then the restriction of the fibration π\nonto Γ′\niis trivial. Therefore the restriction of the projection πonto Γ iis a\nhomeomorphism. Since the projection πis locally isometric, then the length of\nthe network Γ iinWcoincides with the length of the network Γ′\niinM. But\nρW(SMTNi) does not exceed the length of Γ i, therefore\nρW(SMTNi)≤ρM(SMTN′)+εi,\nwhere the sequence {εi}of positive numbers tends to 0 as i→ ∞. So,\ninf\n{N:f(N)=N′}ρW(SMTN)≤ρM(SMTN′).\nIt remains to apply Proposition 2.3. The proof is complete.\nProof of Corollary 1.1. ItfollowsfromTheorems1.1, and1.2; DuandHwang\ntheorem [2] saying that the Steiner ratio of the Euclidean plane equa ls√\n3/2;\nand also from Rubinstein and Weng theorem [7] saying that the Steine r ratio\nof the standard two dimensional sphere with constant positive cur vature metric\nequals√\n3/2.Proofs of the theorems. 10\nProof of Theorem 1.3. LetusconsiderthePoincar´ emodeloftheLobachevsky\nplaneL2(−1) with constant curvature −1. We recall that this model is a ra-\ndius 1 flat disk centered at the origin of the Euclidean plane with Carte sian\ncoordinates ( x,y), and the metric ds2in the disk is defined as follows:\nds2= 4dx2+dy2\n(1−x2−y2)2.\nIt is well known that for each regular triangle in the Lobachevsky pla ne the\ncircumscribed circle exists. The radii emitted out of the center of t he circle to\nthe vertices of the triangle forms the angles of 120◦.\nLetrbe the radius of the circumscribed circle. The cosine rule implies that\nthe length aof the side of the regular triangle can be calculated as follows:\ncosha= cosh2r−sinh2rcos2π\n3= 1+3\n2sinh2r.\nIt is easy to verify that for such triangle the length of MST equals 2 a, and\nthe length of SMT equals 3 r. Therefore, the Steiner ratio m(r) for the regular\ntriangle inscribed into the circle of radius rin the Lobachevsky plane L2(−1)\nhas the form\nm(r) =3\n2·r\narccosh/parenleftbig\n1+3\n2sinh2(r)/parenrightbig.\nIt is easy to calculate that limit of the function m(r) asr→ ∞is equal to 3 /4.\nThe proof is complete.\nProof of Theorem 1.4. It is easy to see that the Taylorseriesfor the function\nm(r) atr= 0 has the following form:\n√\n3\n2−r2\n16√\n3+O(r4).\nTherefore, m(r) is strictly less than√\n3/2 in some interval (0 ,ε). The latter\nmeans that for sufficiently small regular triangles on the surfaces o f constant\ncurvature −1, the relation of the lengths of SMT and MST is strictly less than√\n3/2. The proof is complete.\nReferences\n[1]D. Cieslik , Steiner minimal trees. — Dordrecht, Boston, London, Kluwer\nAcademic Publishers, 1998.\n[2]D. Z. Du and F. K. Hwang , A proof of Gilbert–Pollak Conjecture on the\nSteiner ratio. — Algorithmica, v. 7 (1992) pp. 121–135.\n[3]E.N. Gilbert and H.O.Pollak , SteinerMinimalTrees.SIAMJ.Appl.Math.,\n16:1–29, 1968.Proofs of the theorems. 11\n[4]F. K. Hwang, D. Richards, and P. Winter , The Steiner Tree Problem. —\nElsevier Science Publishers, 1992.\n[5]A. O. Ivanov and A. A. Tuzhilin , Minimal Networks. The Steiner Problem\nand Its Generalizations. — N.W., Boca Raton, Florida, CRC Press, 199 4.\n[6]A. O. Ivanov and A. A. Tuzhilin , Branching Solutions of One-Dimensional\nVariational Problems. — World Publisher Press, 2000, to appear.\n[7]J. H. Rubinstein and J. F. Weng , Compression theorems and Steiner ratios\non spheres. — J. Combin. Optimization, v. 1 (1997) pp. 67–78."    },    {        "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. Lett. 101, 037207 (2008), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.101.037 207.\n[8] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.105.236 601.\n[9] H. Ebert, in Electronic Structure and Physical Properties\nof Solids , edited by H. Dreyss´ e (Springer, Berlin, 2000),\nvol. 535 of Lecture Notes in Physics , p. 191.\n[10] M. E. Rose, Relativistic Electron Theory (Wiley, New\nYork, 1961).\n[11] H. Ebert and S. Mankovsky, Phys.\nRev. B 79, 045209 (2009), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.79.045209 .\n[12] W. H. Butler, Phys. Rev. B 31, 3260 (1985), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.31.3260 .[13] E. M. Gololobov, E. L. Mager, Z. V. Mezhevich, and\nL. K. Pan, phys. stat. sol. (b) 119, K139 (1983).\n[14] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Jap. J. Appl. Phys. 45,\n3889 (2006).\n[15] D. K¨ odderitzsch, H. Ebert, D. A. Rowlands, and\nA. Ernst, New Journal of Physics 9, 81 (2007), URL\nhttp://dx.doi.org/10.1088/1367-2630/9/4/081 .\n[16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.\nShapiro, W. F. Egelhoff, B. B. Maranville, D. Pu-\nlugurtha, A. P. Chen, and L. M. Connors, J. Appl. Phys.\n101, 033911 (2007).\n[17] G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, a nd\nC. H. Back, Phys. Rev. Lett. 102, 257602 (2009), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.102.257 602.\n[18] B. Drittler, N. Stefanou, S. Bl¨ ugel, R. Zeller, and\nP. H. Dederichs, Phys. Rev. B 40, 8203 (1989), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.40.8203 .\n[19] N. Stefanou, A. Oswald, R. Zeller, and P. H.\nDederichs, Phys. Rev. B 35, 6911 (1987), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.35.6911 .\n[20] S. M. Bhagat andP. Lubitz, Phys. Rev.B 10, 179 (1974)."    },    {        "title": "1102.5384v2.Dynamics_of_Skyrmion_Crystals_in_Metallic_Thin_Films.pdf",        "content": "arXiv:1102.5384v2  [cond-mat.mes-hall]  16 Sep 2011Dynamics of Skyrmion Crystal in Metallic Thin Films\nJiadong Zang1,2,3,∗Maxim Mostovoy4, Jung Hoon Han5, and Naoto Nagaosa2,3†\n1Department of Physics, Fudan University, Shanghai 200433, China\n2Department of Applied Physics, University of Tokyo,\n7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Cross-Correlated Materials Research Group (CMRG),\nand Correlated Electron Research Group (CERG), RIKEN-ASI, Wako, Saitama 351-0198, Japan\n4Zernike Institute for Advanced Materials, University of Gro ningen,\nNijenborgh 4, 9747 AG Groningen, The Netherlands\n5Department of Physics, BK21 Physics Research Division,\nSungkyunkwan University, Suwon 440-746, Korea\n(Dated: June 6, 2018)\nWe study the collective dynamics of the Skyrmion crystal (Sk X) in thin films of ferromagnetic\nmetals resulting from the nontrivial Skyrmion topology. It is shown that the current-driven motion\nof the crystal reduces the topological Hall effect and the Sky rmion trajectories bend away from the\ndirection of the electric current (the Skyrmion Hall effect) . We find a new dissipation mechanism\nin non-collinear spin textures that can lead to a much faster spin relaxation than Gilbert damping,\ncalculate the dispersion of phonons in the SkX, and discuss e ffects of impurity pinning of Skyrmions.\nPACS numbers: 73.43.Cd,72.25.-b,72.80.-r\nIntroduction: Skyrmion is a topologically nontrivial\nsoliton solution of the nonlinear sigma model. It was\nnoted early on that Skyrmions in three spatial dimen-\nsions have physical properties of baryons and the peri-\nodic Skyrmion crystal (SkX) configurations were used to\nmodel nuclear matter[1, 2]. Skyrmions in two spatial\ndimensions play an important role in condensed matter\nsystems, such as quantum Hall ferromagnets[3, 4]. It\nwas suggested that SkX configurations can be stabilized\nby the Dzyaloshinskii-Moriya (DM) interaction in ferro-\nmagnetswithoutinversionsymmetry[5]. Suchastatewas\nrecently observed in a neutron scattering experiment in\nthe A-phase of the ferromagnetic metal MnSi[6].\nRecent Monte Carlo simulations indicated much\ngreater stability of the SkX when a bulk ferromagnet is\nreplaced by a thin film[7]. This result was corroborated\nby the real-space observation of SkX in Fe 0.5Co0.5Si thin\nfilm in a wide magnetic field and temperature range[8].\nLorentz force microscopy showed that Skyrmions form a\ntriangular lattice with the magnetization vector antipar-\nallel to the applied magnetic field in the Skyrmion center\nand parallel at the periphery, as was also concluded from\nthe neutron experiment[6].\nThe next important step is to explore dynamics of\nSkyrmion crystals and the ways to control them in anal-\nogy to the actively studied current- and field-driven mo-\ntion of ferromagnetic domain walls[9]. Recent observa-\ntion ofthe rotationalmotion ofthe SkX in MnSi suggests\nthat Skyrmionscanbe manipulated bymuchsmallercur-\nrents than domain walls[10].\nIn this Letter we study the coupled dynamics of spins\nand charges in the SkX, focusing on effects of the non-\n∗Electronic address: jdzang@fudan.edu.cn\n†Electronic address: nagaosa@ap.t.u-tokyo.ac.jptrivial Skyrmion topology and effective gauge fields in-\nduced by the adiabatic motion of electrons in the SkX.\nWe derive equation of motion for the collective vari-\nables describing the SkX, calculate its phonon disper-\nsion, and discuss a new form of damping, which can be\nthe dominant spin-relaxation mechanism in half-metals.\nIn addition, we consider new transport phenomena, such\nas the topological Hall effect in a sliding SkX and the\nSkyrmion Hall effect. We also discuss the Skyrmion pin-\nning by charged impurities and estimate the critical cur-\nrent above which the SkX begins to slide.\nLow-energy excitations in Skyrmion crystal: An iso-\nlated Skyrmion has two zero modes corresponding to\ntranslations along the xandydirections. Since an ap-\npliedmagneticfieldopensagapinthecontinuumofspin-\nwave excitations, the low-energy magnetic modes in SkX\nare expected to be superpositions of the Skyrmion dis-\nplacements, or the phonons. The phonon modes, as well\nasthecouplingofSkyrmiondisplacementstotheexternal\ncurrent, can be consistently described in the framework\nof elasticity theory.\nWe begin with the spin Hamiltonian HS=/integraltextd3x/bracketleftbigJ\n2a(∇n)2+D\na2n·[∇×n]−µ\na3H·n/bracketrightbig\n, whereJis\nthe exchange constant and Dis the DM coupling that\nstabilizes the SkX configuration n(x) in some interval of\nthe magnetic field H=Hˆz[5, 11]. We calculate the ‘har-\nmonic lattice energy’ by considering a deformation of the\nSkX,˜n(x,t) =n(x−u(x,t)), where the collective coor-\ndinateu(x,t) varies slowly at the scale of the SkX lattice\nconstant. The result is:\nHS=dηJ/integraldisplayd2x\nξ2[(∇ux)2+(∇uy)2],(1)\nwheredis the film thickness and ξ∼aJ\nDis the\ncharacteristic length scale of SkX[11], with abeing2\nthe lattice spacing. The dimensionless quantity η=\n1\n8π/integraltext\nucd2x(∂in·∂in) encodes the information about D\nandH, and is called shape factor in what follows.\nWhen an electroncurrentis flowingthroughthe metal-\nlic film, the conduction electrons interact with local mag-\nnetic moments through the Hund’s rule coupling HH=\n−JHSψ†σ·nψ, whereψis the electron operator. In\nthe case of small current density and the Skyrmion size\nmuch larger than the Fermi wavelength of conduction\nelectrons, one can apply the adiabatic approximation in\nwhich the electron spins align perfectly with the local\nmoment.ψis projected into the fully polarized state by\nψ=χ|n/angb∇acket∇ightwithσ·n|n/angb∇acket∇ight=|n/angb∇acket∇ight. Then the electron action\nSel=/integraltext\ndtd3x[i¯hψ†˙ψ+¯h2\n2mψ†∇2ψ+JHSψ†σ·nψ] can be\nrewritten as Sel=/integraltext\ndtd3x[i¯hχ†˙χ−ea0−1\n2mχ†(−i¯h∇−\ne\nca)2χ+JHSχ†χ], whereaµ=¯hc\n2e(1−cosθ)∂µϕwithθ\nandϕbeing the spherical angles describing the direc-\ntion of the local magnetization[12, 13]. The gauge po-\ntentialaµgives rise to internal electric and magnetic\nfields,eandh, acting on spin-polarized electrons pass-\ning through the SkX in analogy with the electromag-\nnetic gauge field. Crucially, the internal magnetic field\nb=∇×a=¯hc\n2e(n·∂xn×∂yn)ˆzis intimately re-\nlated to the topological charge Qof Skyrmions by[14]\nQ=1\n4π/integraltext\nucd2x(n·∂xn×∂yn) =±1,where the integra-\ntion goes over the unit cell of the SkX. In the language\nof internal gauge field, this topological feature is nothing\nbut thequantizationofinternalfluxΩ =/integraltexth·dSinunits\nofhc/e. The coupling of the electric current to the inter-\nnal gauge field induced by the SkX, Hint=−1\nc/integraltextd3xj·a,\nhas a simple form in terms of the collective coordinates\nintroduced above:\nHint=d¯hQ\ne/integraldisplayd2x\nξ2(uxjy−uyjx). (2)\nThe crucial difference between the SkX and a con-\nventional crystal is the form of kinetic energy. The\nspin dynamics originates from the Berry phase action,\nSBP=d\nγ/integraltext\ndtd2x(cosθ−1) ˙ϕ. Here,γ=a3\n¯h(S+x/2), where\nxis the filling of the conduction band, and S+x/2 is the\ntotal spin averagely per lattice site. In terms of uthe\nkinetic energy has the form\nSBP=dQ\nγ/integraldisplay\ndtd2x\nξ2(ux˙uy−uy˙ux). (3)\nThis form of the Berry phase shows that the collec-\ntive variables uxanduydescribing local displacements\nof Skyrmions form a pair of canonical conjugate vari-\nables, replacing cos θandϕ. This characteristic prop-\nerty of SkX leads to several unusual responses to ap-\nplied electric currents and fields. It originates from\nthe Skyrmion topology and distinguishes SkX from non-\ntopological spin textures such as spirals and domain wall\narrays.\nUsing Eqs.(1), (2) and (3), we obtain equation of mo-\ntion foru:\n˙u=−e¯hγ\n2j+QγηJ\ne¯hˆz×∇2u. (4)Two consequences follow immediately. First, the dis-\npersion of phonons in the SkX obtained from Eq.(4) is\nquadratic,\n¯hω=ηJa2\n(S+x\n2)k2, (5)\nin contrast to the linear phonon dispersion in usual crys-\ntalsandsimilartothedispersionofmagnonsin auniform\nferromagnet. Since uxanduyplay the role of the coor-\ndinate and momentum, the longitudinal and transverse\nphonon modes in the SkX merge into a single mode cor-\nresponding to the rotational motion of Skyrmions, which\nleads to the quadratic dispersion. Secondly, the SkX can\nmove as a whole driven by the charge current j, with a\nvelocityV/bardbl=˙ u=−e¯hγ\n2j. This rigid motion of SkX leads\nto several interesting results discussed below.\nHall effect due to SkX motion: In such nontrivial spin\ntextures, the external magnetic field (less than 0.2T for\nMnSi) is more than one order of magnitude smaller than\nthe internal one, so that it would be neglected in what\nfollows. As can be seen from Eq. (2), the collective\ncoordinates uxanduyplay the role of electromagnetic\ngauge potentials Ayand−Ax, respectively. It is thus\nexpected that the temporal variation of uinduced by\nthe current leads to a transverse potential drop. This\nHall-type effect can also be intuitively understood using\ntheinternalmagneticfield bintroducedabove. Amoving\nspin texture n(x−V/bardblt) induces an internal electric field\neanalogous to the electric field of a moving magnetic\nflux and related to the internal magnetic field by e=\n−1\nc/bracketleftbig\nV/bardbl×b/bracketrightbig\n.For SkX with b=bzˆz, this electric field\ngenerates an electric current in the direction transverse\ntoV/bardblresulting in the Hall conductivity:\n∆σxy\nσxx≈ −x\n2S+xe/angb∇acketleftbz/angb∇acket∇ightτ\nmc, (6)\nwheremis the electron mass and τis the relaxation\ntime. The average internal magnetic field is /angb∇acketleftbz/angb∇acket∇ight=QΦ0\n2πξ2,\nwhere Φ 0is the elementary flux and 2 πξ2is the area of\nthe unit cell of the SkX. This Hall conductivity has the\nsameorderofmagnitude asthe oneresulting fromthe so-\ncalled topologicalHall effect observedin a staticSkX[15].\nThe latter effect is nothing but the Hall effect induced\nbybviae=1\nc[v×b], andσTop\nxy/σxx≈e/angb∇acketleftbz/angb∇acket∇ightτ/mc,\nwherevis the electron velocity. Our new effect differs\nby the factor of −x\n2S+xfrom the topological Hall effect.\nIts physical origin can be easily understood by noting\nthe total force acting on a single conduction electron is\nF=−e\nc[(v−V/bardbl)×b], i.e. the Lorentz force on elec-\ntrons due to the internal magnetic field of the SkX de-\npendsontherelativevelocityofelectronsandSkyrmions.\nWhen the SkX begins to slide above the threshold elec-\ntric current jc[16], the net topological Hall voltage will\nbe suddenly reduced by the factor2S\n2S+x, which is how\nthe effect of the spin-motive force and the collective shift\nof Skyrmions can be identified experimentally.\nNew damping mechanism and Skyrmion Hall effect:\nPreviously we have systematically discussed the novel3\neffects related to the internal magnetic field. A natu-\nral question thus arises as to whether there is any new\nphenomena associated with the intrinsic internal electric\nfield, which is ei=−∂ia0−1\nc˙ai=¯h\n2e(n·∂in×˙n). Due\nto the time derivative in this expression, its effect is ab-\nsent in the static spin texture. However, in the present\ncase, the motion of SkX makes it nonvanishing, and leads\nto an additional current j′byj′=σewithσthe con-\nductivity of electrons. Substituting this current into the\nLandau-Lifshitz-Gilbert equation[12, 13]\n˙n=¯hγ\n2e[j·∇]n−γ/bracketleftbigg\nn×δHS\nδn/bracketrightbigg\n+α[˙n×n],(7)\nthe time derivative ˙nreceives a correction given by\nδ˙n=¯hγσ\n2e(e·∇)n=α′(n·∂in×˙n)∂in.(8)\nThe corresponding dimensionless damping constant is\nα′=1\n(2S+x)a3σ\nαfsξ2c, whereαfs≈1/137 is the fine struc-\nture constant. The time derivative in the r.h.s. of Eq.(8)\nshows that the current induced by internal electric field\nleads to dissipation. In contrast to Gilbert damping this\nnew mechanism does not require relativistic effects and\nonly involves the Hund’s rule coupling that conserves the\ntotal spin. The relaxation of the uniform magnetization,\ndescribed by Gilbert damping, is clearly impossible with-\nout the spin-orbit coupling, which breaks the conserva-\ntion of the total spin[17]. This argument, however, does\nnot apply to inhomogeneousmagnetic textures where the\nbreaking of the rotational symmetry by noncollinear spin\norders enables the relaxation without the spin-orbit cou-\npling (note that α′vanishes as ξ→ ∞). Despite the\nnon-relativistic origin, α′depends on the DM coupling,\nas the latter determines the Skyrmion size. Estimates of\nα′made below show that in half-metals it can greatly\nexceedα.\nThe effect of this new dissipation can be observed by\ntracing the trajectoryof Skyrmion motion. Including the\nnewdissipationterm, themodifiedequationofmotion(4)\nfor the rigid collective coordinates u(t) has the form\n˙u=−e¯hγ\n2j−Q(αη+α′η′)ˆz×˙u, (9)\nwhere the second shape factor η′is given by η′=\nQ\n4π/integraltext\nucd2x(n·∂xn×∂yn)(∂in·∂in)//integraltext\nucd2x(∂in·∂in).\nThe new dissipation term in Eq.(9) is obtained by mul-\ntiplying Eq.(8) with ∂jn, using ˙n=−(˙u·∇)n, and\nintegrating over one unit cell. The whole damping term\nleads to a transverse motion with velocity\nV⊥≈Q(αη+α′η′)/bracketleftbig\nV/bardbl׈z/bracketrightbig\n. (10)\nThis Skyrmion Hall effect can be observed by real-space\nimages of Lorentz force microscopy. The corresponding\nHall angle is θ= arctan(αη+α′η′).The estimate given\nbelow shows that main contribution to θcomes from the\nnew dissipation mechanism.Pinning of Skyrmion crystal: Next we consider the\npinning of the SkX by charged impurities. The pinning\nresults from spatial fluctuations of the impurity density\nandvariationsofthespindirectionintheSkX.Variations\nof the density of charged impurities δnigive rise to local\nvariations of the electron density neand since the double\nexchangeconstant Jisproportionaltothelatter, wehave\nδJ∼Jδni/ne. The energy per Skyrmion ES∼Jd/a.\nDenote the number of impurities in this volume by N1\nwith/angb∇acketleftN1/angb∇acket∇ight=ni2πξ2dand the variance δN1=√N1, we\nobtain the typical variation of the Skyrmion energy:\nV1=δJd\na∼J\nne2πξ2a/radicalbig\nN1=J\nneaξ/radicalbigg\nnid\n2π.(11)\nThe potential energy density is then V0=V1/(2πξ2).\nSubstituting ni∼(la2)−1, wherelis the electron mean\nfree path, and ne=x\na3, we obtain V0∼J\n(2π)3/2x/radicalBig\nd\nla\nξ3.\nThe pinning regime of the whole SkX depends on the ra-\ntioofthe pinningenergy V1andthe elasticenergy ESofa\nsingle Skyrmion[18]. Let L2be the number of Skyrmions\nin the domain where u∼ξ. The energy gain due to\nthe impurity pinning in the domain is ∼ −V1L, while\nthe elastic energy cost ∼Jd\nais independent of the do-\nmain size. Minimizing the total energy per Skyrmion,\nJd\naL2−V1\nL, we obtain L∼Jd\naV1.L≫1 corresponds to the\ncase of weak (or collective) pinning of SkX, while L∼1\ncorresponds to the strong pinning regime.\nThe pinning potential gives rise to the spin transfer\ntorque−Qγξ2\n2d/bracketleftbigˆz×δV\nδu/bracketrightbig\nin the right-hand side of Eq.(4).\nIn the steady state of moving SkX this torque has to be\ncompensated by the interaction with the electric current.\nThe critical current density is then\njc∼e\n¯hξ2\nd/angbracketleftbigg∂V\n∂u/angbracketrightbigg\nsteady state∼e\n¯hξV0\ndL,(12)\nin the weak pinning regime, while in the strong pinning\ncaseLhas to be substituted by 1. Similarly, one can\nestimate the gap in the spin wave spectrum due to the\npinning:\n¯hωpin∼¯hγξ2\nd/angbracketleftbigg∂2V\n∂u2/angbracketrightbigg\n∼¯hγξ2\ndV0L\nL2ξ2=a3\ndSV0\nL.(13)\nEstimates: For estimates we consider MnSi where Mn\nions form a (distorted) cubic sublattice with a= 2.9˚A.\nThe length of the reciprocal lattice vectors of the SkX\n∼0.035˚A−1correspondsto ξ∼77˚AandD∼0.1J. The\nkinetic energy scales as ¯ h2/mξ2<JH∼1eV, so that the\nadiabaticapproximationisjustified. Theelectrondensity\nne= 3.8·1022cm−3[15] corresponds to x=nea3∼0.9\ncharge carriers per lattice site, while the residual resis-\ntivityρ∼2µΩ·cm[15] gives an estimate of the impurity\nconcentration xi∼5·10−3. From the magnon dispersion\nin the spiral state[19], J∼3 meV.\nUsing these parameters we get α′∼0.1, which shows\nthatthedampingresultingfromtheelectriccurrentsgen-\nerated by non-collinear spin textures can be the domi-\nnant mechanism of spin relaxation in half-metals, where4\nthe Gilbert damping constant αis one-three orders of\nmagnitude smaller[20, 21]. We note, however, that since\nthe typical electron mean free path ξ∼500˚A is larger\nthan the Skyrmion size ξ, the relation between the cur-\nrent and internal electric field is nonlocal. The topo-\nlogical Hall angle, θH∼ehzτ\nmc=1\nαsfneξ2σ\ncatT= 0\nand the change in θHinduced by the sliding Skyrmion\ncrystal [see Eq.(6)] are also ∼0.1. For a 10nm thick\nfilm the parameter L∼xξ√\n2πdl\na2∼103,i.e. the pin-\nning of Skyrmions by charged impurities is exceedingly\nweak and the corresponding values of the pinning fre-\nquency ¯hωpin∼5·10−11meV and the critical current\njc∼0.2A·cm−2. This value is much lower than that\nfor domain walls[9] and smaller than the ultralow thresh-\nold current jc∼102A·cm−2observed in bulk MnSi[10],\nwhich may be attributed to other pinning mechanisms\nand the different dimensionality of the system. This low\ncurrent density also justifies the adiabatic approximation\napplied in this work.\nComparison with vortex dynamics: Finally, we com-\npare the dynamics of SkX with that of the vortex lines\n(VL) in type II superconductors. Asimilarquadraticdis-\npersion was obtained for VL [22]. However, in supercon-\nductors it results from long-ranged interactions between\nvortices, while the interactions between Skyrmions are\nshort-ranged (if one ignores the relatively weak dipole-\ndipole interactions). The absence of long-range interac-\ntions in SkX ensures the stability of the quadratic disper-\nsion. Furthermore, the kinetic terms in VL and SkX are\ncompletely different. For VL uxanduyare two indepen-\ndent variables,while forSkXtheareconjugatedvariablesas in Eq.(3). Therefore VL are massive, while Skymions\nare not. When a supercurrent flows through the type II\nsuperconductor, the charged Cooper pairs are deflected\nby the VL through the Lorentz force, which in turn gives\nrisetothetransversemotionoftheVL.TheVLdynamics\nis usually assumed to be overdamped[23], the kinetic en-\nergy of VL is neglected, and the Lorentz force is assumed\nto be counterbalanced by the friction force. In contrast,\nthe damping of Skyrmions is relatively weak. The spin\ntorque resulting from the strong Hund’s rule coupling re-\nsults in a nearly longitudinal motion Skyrmion motion.\nThe Hall motion of VL results in a longitudinal voltage\ndrop, which is not important for SkX motion due to the\nsmall Hall angle.\nDuring the completion of this paper we became aware\nof a recent paper by Kim and Onoda addressed the dy-\nnamics of Skyrmions in an itinerant double-exchange fer-\nromagnet using a Chern-Simons-Maxwell approach[24].\nTheir focus, however, seems to differ from ours. We are\ngrateful for the insightful discussions with Prof. Yoshi-\nnoriTokura. ThisworkissupportedbyGrant-in-Aidsfor\nScientific Research (No. 17105002, 19019004, 19048008,\n19048015,and21244053)fromtheMinistryofEducation,\nCulture, Sports, Science and Technology of Japan, and\nalso by Funding Program for World-Leading Innovative\nR&D on Science and Technology (FIRST Program). JZ\nis supported by Fudan Research Program on Postgrad-\nuates. MM is supported by the Stichting voor Funda-\nmental Onderzoekder Materie(FOM). HJH issupported\nby Mid-career Researcher Program through NRF grant\nfunded by the MEST Grant No. R01- 2008-000-20586-0.\n[1] T. H. R. Skyrme, Proc. Roy. Soc. London A 260, 127\n(1961); Nuc. Phys. 31, 556 (1962).\n[2] I. Klebanov, Nucl. Phys. B 262, 133 (1985).\n[3] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H.\nRezayi, Phys. Rev. B 47, 16419 (1993).\n[4] C. Timm, S. M. Girvin, and H. A. Fertig, Phys. Rev. B\n58, 10634 (1998).\n[5] A.N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101-103 (1989); U.K. Rosler, A.N. Bogdanov, and C.\nPfleiderer, Nature 4 42, 797-801 (2006).\n[6] S. M¨ uhlbauer, B. Binz, F. Joinetz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[7] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys.\nRev. B80, 054416 (2009); Ulrich K. Roeßler, Andrei A.\nLeonov, Alexei N. Bogdanov, arXiv:1009.4849.\n[8] X. Z. Yu et al., Nature (London) 465, 901 (2010).\n[9] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nand T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[10] F. Jonietz et al., Science 330, 1648 (2010).\n[11] J. H. Han, J. Zang, Z. Yang, J. H. Park, and N. Nagaosa,\nPhys. Rev. B 82, 094429 (2010).\n[12] Ya. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev.\nB57, R3213 (1998).[13] G. Tatara, H. Kohno, J. Shibata, Phys. Rep. 468, 213\n(2008).\n[14] R. Rajaraman, Solitons and Instantons , North-Holland,\n1987.\n[15] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong,\nPhys. Rev. Lett. 102, 186601 (2009); A. Neubauer et al.,\nPhys. Rev. Lett. 102, 186602 (2009).\n[16] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[17] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[18] H. FukuyamaandP. A.Lee, Phys.Rev.B 17, 535(1978);\nP. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979).\n[19] S. V. Grigoriev et al., Phys. Rev. B 74, 214414 (2006).\n[20] T. Kubota et al., Appl. Phys. Lett. 94, 122504 (2009).\n[21] C. Liu et al., Appl. Phys. Lett. 95, 022509 (2009).\n[22] P. G. de Gennes and J. Matricon, Rev. Mod. Phys. 36,\n45 (1964); A. L. Fetter, and P. C. Hohenberg, Phys. Rev.\n159, 330 (1967).\n[23] M. Tinkham, Introduction to Superconductivity ,\nMcGraw-Hill, 1996\n[24] K. S. Kim and S. Onoda, arXiv:1012.0631."    },    {        "title": "1103.1455v1.Application_of_Explicit_Symplectic_Algorithms_to_Integration_of_Damping_Oscillators.pdf",        "content": "arXiv:1103.1455v1  [math-ph]  8 Mar 2011Applicationof ExplicitSymplecticAlgorithmsto\nIntegrationofDampingOscillators\nTianshuLuoc, YimuGuod\naInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou,\nZhejiang,310027,P.R.China\nbInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou,\nZhejiang,310027,P.R.China\nURL:ltsmechanic@zju.edu.cn (YimuGuo), guoyimu@zju.edu.cn (YimuGuo)\nPreprintsubmittedtoCommunicationsinNonlinearScience andNumericalSimulationOctober23,2018Applicationof ExplicitSymplecticAlgorithmsto\nIntegrationofDampingOscillators\nTianshuLuoc, YimuGuod\ncInstituteof AppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou,\nZhejiang,310027,P.R.China\ndInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou,\nZhejiang,310027,P.R.China\nAbstract\nInthispaperanapproachisoutlined. Withthisapproachsom eexplicitalgorithms\ncanbeappliedtosolvetheinitialvalueproblemof n−dimensionaldampedoscil-\nlators. Thisapproach isbaseduponfollowingstructure: fo ranynon-conservative\nclassical mechanical system and arbitrary initial conditi ons, there exists a con-\nservative system; both systems share one and only one common phase curve;\nand, the value of the Hamiltonianof the conservativesystem is, up to an additive\nconstant, equal to the total energy of the non-conservative system on the afore-\nmentioned phase curve, the constant depending on the initia l conditions. A key\nway applying explicit symplecticalgorithms to damping osc illators is that by the\nNewton-Laplace principle the nonconservative force can be reasonably assumed\nto be equal to a function of a component of generalized coordi natesqialong a\nphase curve, such that the damping force can be represented a s a function analo-\ngous to an elastic restoring force numerically in advance. T wo numerical exam-\nplesare giventodemonstratethegoodcharacteristics ofth ealgorithms.\nKeywords: Hamiltonian,dissipation,non-conservativesystem,damp ing,explicit\nsymplecticalgorithm\n1. Introduction\nFeng[1, 2, 3, 4],Marsden[5],Neri[6] and Yoshida[7]had dev eloped a series of\nsymplectic algorithms for Hamiltonian systems. These algo rithms possess some\nURL:ltsmechanic@zju.edu.cn (YimuGuo), guoyimu@zju.edu.cn (YimuGuo)\nPreprintsubmittedtoCommunicationsinNonlinearScience andNumericalSimulationOctober23,2018advantages. But it is difficult to apply these algorithms to d amping dynamical\nsystems, because it has been stated in most classical textbo oks that the Hamilto-\nnian formalism focuses on solving conservative problems. D amping phenomena\nis very important in the modeling of dynamical systems, and c an not be avoided.\nOur aim is to apply someexplicit canonical algorithms to non linear damping dy-\nnamical systems, which is generated generally by FE-method . These canonical\nalgorithmsreportedinthispapercanbereadilyutilizedfo rcomputinglarge-scale\nnonlineardampeddynamicalsystems.\nBetch[8][9][10]attemptedtoapplydirectlysomeimplicit algorithmstodamp-\ning systems. The implicit symplectic algorithms utilized b y Betch[8] possess a\nfew good characteristic, e.g. energy-conservation, momen tum-consistence, etc...\nIn terms of energy-conservation, implicit symplectic algo rithms might be better\nthan explicit symplectic ones. But explicit symplectic sch emes might be more\nsuitablefornonlinearproblems.\nIf one needs to apply symplecticalgorithmsto a dissipative system, one must\nconvertthedissipativesystemintoaHamiltoniansystemor findsomerelationship\nbetween thedissipativesystemand aconservativeone.\nIn theliterature[11], wehavestatedapropositiondescrib ingarelationamong\nadampingdynamicalsystemand conservativeones:\nProposition1.1. For any non-conservative classical mechanical system and a r-\nbitraryinitial condition, there exists a conservative sys tem; both systems sharing\none and only one common phase curve; and the value of the Hamil tonian of the\nconservativesystemisequaltothesumofthetotalenergyof thenon-conservative\nsystemontheaforementionedphasecurveandaconstantdepe ndingontheinitial\ncondition.\nIn other words, a dissipative ordinary equation and a conser vative equation may\npossess a common particular solution. In the next section, a n analytical exam-\nples are given to explain this proposition. Readers can find t he detailed proof of\nProposition1.1 inthereference[11]\nIn the Literature [12] a basic explicit canonical integrato r is proposed. Based\non this basic scheme, Neri[6] constructed 4-order explicit canonical integrator,\nandthenYoshida[7]proposedageneralmethodtoconstructh igherorderexplicit\nsymplecticintegrator. UtilizingtheProposition1.1,wea pplythisclassofexplicit\ncanonical integrators to damping dynamical systems. This p oint will be in detail\nstatedin sec. 3.\n32. One-dimensional Analytical Example\nConsideraspecialone-dimensionalsimplemechanical syst em:\n¨x+c˙x=0, (1)\nwherecis aconstant. Theexact solutionoftheequationaboveis\nx=A1+A2e−ct, (2)\nwhereA1,A2are constants. Differentiationgivesthevelocity:\n˙x=−cA2e−ct. (3)\nFrom the initial condition x0,˙x0, we find A1=x0+˙x0/c,A2=−˙x0/c. Inverting\nEq. (2)yields\nt=−1\nclnx−A1\nA2(4)\nand bysubstitutingintoEq. (3), such wehave\n˙x=−c(x−A1) (5)\nThedissipativeforce Fin thedissipativesystem(1)is\nF=c˙x. (6)\nSubstitutingEq. (5)intoEq. (6), theconservativeforce Fisexpressed as\nF=−c2(x−A1); (7)\nClearly, the conservative force Fdepends on the initial condition of the dissi-\npative system (1), in other words, an initial condition dete rmines a conservative\nforce. Consequently,a newconservativesystemyields\n¨x+F=0→¨x−c2(x−A1)=0. (8)\nThestiffnesscoefficient in thisequationmustbenegative. Onecan readily verify\nthat the particular solution (2) of the dissipativesystem c an satisfy the conserva-\ntiveone(8). Thispointagrees withProposition(1.1).\nThepotentialoftheconservativesystem(8)is\nV=/integraldisplayx\n0/bracketleftbig\n−c2(x−A1)/bracketrightbig\ndx=−c2\n2x2+c2A1x\n4Figure1: Relationshipbetweennonconservativesystem (1) andconservativeone(8)\nThereforetheHamiltonianis\nˆH=T+V=1\n2p2−c2\n2x2+c2A1x,\nwherep=˙x.\nFurthermore, Proposition (1.1) can be depicted by Fig. 2. Th e phase flow of\nconservativesystem(2) transforms thered area in phasespa ce to the purplearea;\nthephaseflowofconservativesystem(8)transformsthereda reatothegreenarea.\nThe blue curve in Fig. 2 illustrates the common phase curve. I f one draws more\ncommon phase curves and phase flows, the picture will like a flo wer, the phase\nflow of the nonconservative system likes a pistil and phase flo ws conservative\nsystemslikepetals.\n3. Modification SymplecticNumerical Schemes\n3.1. BasicExplicitSymplecticNumericalSchemes\nIn the paper[12][6][7] a symplectic algorithm based second kind generation\nfunctionwas stated\npppi+1=pppi−τHq(pppi+1,qqqi)\nqqqi+1=qqqi+τHp(pppi+1,qqqi),(9)\nwherethesuperscript idenotesthe i-thtimenode, qqqdenotescoordinatesand pppde-\nnotescanonicalmomenta,and HdenotesHamiltonianquantity, Hq=∂H/∂qqq,Hp=\n5∂H/∂ppp. IftheHamiltonianisseperable,i.e. H=U(ppp)+V(qqq),Vq=Hq,Up=Hp,\nthenthesymplecticscheme(9)abovebecomes an explicitsym plecticscheme:\npppi+1=pppi−τVq(qqqi)\nqqqi+1=qqqi+τUp(pppi+1).(10)\nForsomenonlinearvibrationmechanicalsystem, Vq=K(qqq)qqq.\nLet usconsideran n−dimensionalnonlinearoscillator:\n¨qqq+C˙qqq+Kqqq=0, (11)\nwhereCdenotes a non-linear damping coefficient matrix which depen ds onqqq,\nandKdenotes a non-linear stiffness matrix which depends on qqqand consists of\ntwoparts K=ˇK+ˆK(ˇKis adiagonalmatrix).\nIn accordance with Proposition 1.1, a conservative mechani cal system was\nfound associated with the dissipative system (11) in additi on to its initial condi-\ntions. Subject to these initial conditions, the dissipativ e system (11) possesses a\ncommon phase curve γwith the conservativesystem. As in Eq. (7), we can con-\nsider that the components of the damping force C˙qqqdetermine the components of\naconservativeforceon thephasecurve γ\nc11˙q1=ρ11(q1)...c1n˙qn=ρ1n(q1)\n.........\ncn1˙q1=ρ21(qn)...cnn˙qn=ρnn(qn).(12)\nFor convenience, this conservative force is assumed to be an elastic restoring\nforce:\nρ11(q1)=κ11(q1)q1...ρ1n(q1)=κ1n(q1)q1\n.........\nρn1(q1)=κn1(qn)qn...ρnn(qn)=κnn(qn)qn.(13)\nIn a similar manner, the components of the non-conservative forceˆKqqqare\nequalto thecomponentsofaconservativeforceon thephasec urveγ\nˆK11q1=χ11(q1)...ˆK1nqn=χ1n(q1)\n.........\nˆKn1q1=χ21(qn)...ˆKnnqn=χnn(qn).(14)\nTheconservativeforcecan likewisebeassumedtoan elastic restoringforce:\nχ11(q1)=λ11(q1)q1...χ1n(q1)=λ1n(q1)q1\n.........\nχn1(q1)=λn1(qn)qn...χnn(qn)=λnn(qn)qn.(15)\n6Byanappropriatetransformation,anequivalentstiffness matrix˜Kthatisdiagonal\ninform can beobtained\n˜Kii=n\n∑\nl=1κil(ql)+λil(ql). (16)\nConsequently,an n-dimensionalconservativesystemisobtained\n¨qqq+(ˇK+˜K)qqq=0 (17)\nwhich shares the commonphase curve γwith then-dimensionaldamping system\ndescribed by (11). In this paper, the conservativesystem is called the ’substitute’\nconservativesystem. TheLagrangian ofEqs.(17)is\nˆL=1\n2˙qqqT˙qqq−/integraldisplayqqq\n000(ˇKqqq)Tdqqq−/integraldisplayqqq\n000(˜Kqqq)Tdqqq, (18)\nwiththeHamiltonian\nˆH=1\n2pppTppp+/integraldisplayqqq\n000(ˇKqqq)Tdqqq+/integraldisplayqqq\n000(˜Kqqq)Tdqqq, (19)\nwhere000isthezerovector,and ppp=˙qqq. HereˆHinEq. (19)isthemechanicalenergy\nof the conservativesystem (17), because/integraltextqqq\n000(˜Kqqq)Tdqqqis a potential function such\nthatˆHisindependentofthepathtaken inphasespace.\nSubjecttoacertaininitialcondition,oneneedmerelytoso lvetheconservative\nsystem(17). But one must in advance obtain the numerical app roximation of the\nmatrix˜Kfor a time step, such that one can utilize the algorithm (10) t o integrate\nthe conservative system (17) for a time step. One can repeat t his process above\nup to the end. In this way one obtains the numerical particula r solution of the\nconservativesystem(17),whichisexactlythenumericalpa rticularsolutionofthe\ndamping one. The he numerical approximation of the matrix ˇKcan be assumed\nas:\n˜K=\n˜K11...0\n.........\n0...˜Knn\n (20)\n˜Kj(qji)=cjl˙qi\nl/qji+ˆKjlqi\nl/qji\nHencetheexplicitcanonical scheme(10)can bemodified into\n˜Ki\nj(qji)=cjl˙qi\nl/qji+ˆKjlqi\nl/qji\npji+1=pji−τ[Kj+˜Ki\nj(qji)]qi)\nqji+1=qi+τpji+1(21)\n7The scheme above is a one order scheme. Furthermore one can co nstruct higher\norderexplicitcanonicalschemesutilizingthemethodrepo rtedintheliteratures[6][7].\nNowconsideramap from zzz=zzz(0)tozzz′=zzz(τ):\nzzz′′′≈(h\n∏\ni=1eritEesiτF+O(τn+1))z, (22)\nwhere\nzzz=/bracketleftbigg\nppp,\nqqq/bracketrightbigg\n,zzz′=/bracketleftbigg\nppp′,\nqqq′/bracketrightbigg\n,\nE=/bracketleftbigg\n0 0\n1 0/bracketrightbigg\nF=/bracketleftbigg\n0−(K+˜K)\n0 0/bracketrightbigg\n.\nIn fact Eq.(22)is thesuccessionofthefollowingmappings,\npj+1=pj−siτVq(qj)\nqj+1=qj+riτUp(pj+1). (23)\nInrealitythedifferencebetweentheequationsaboveand Eq .(21)isthatthecoef-\nficientssi,ribefore the time step τ. In the literature [7] a generalized method to\ndetermine si,riwere given. Therefore, thehigherorderexplicitcanonical scheme\ncan berepresented as:\n˜K(qj)=\n˜K1(qj\n1) 0\n...\n0 ˜Kn(qj\nn)\n˜Kα(qj\nα)=n\n∑\nl=1cαl˙qj\nl/qj\nα+ˆKαlqi\nl/qi\nα\nE=/bracketleftbigg0 0\n1 0/bracketrightbigg\nF=/bracketleftbigg\n0−(K+˜K)\n0 0/bracketrightbigg\nzzzj+1=(h\n∏\ni=1esiτFeriτE)zzzj(24)\n4. Numerical Examples\nTwo exampleswillbegiventoshownthisnumerical method24.\n4.1. TheFirstExample\nTo begin,weconsideraVan DerPol’soscillator\n¨x+µ˙x(x2−1)+x=0, (25)\n8-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5\n 0 10  20  30  40  50  60  70  80  90 100x\ntExplicit Symplectic\nRunge-Kutta\nFigure2: TheresonanceoftheVanDerPol’soscillator\nwhereµ=10. The initial conditions are given by x0=1,˙x=0. We employee\nthe4−order explicitsymplecticmethod(24)withcoefficients\ns1=s4=[2+(3√\n2+1/3√\n2)]/6,s2=s3=[1−(3√\n2+1/3√\n2)]/6,\nr1=r3=[2+(3√\n2+1/3√\n2)]/3,r2=−[2+(3√\n2+1/3√\n2)]/3,r4=0,\nand classical explicit 4 −order Runge-Kutta method to compute the resonance of\nthe Van Der Pol’s oscillator (26) respectively, then employ a same method to in-\ntegrate the results to the total energy, which is the sum of th e mechanical energy\nand the work done by damping forces in the system (25). The bot h methods are\nrun with the same step size τ=0.01. The resonance is shown in Fig. 2, and the\ntotalenergy isshowninFig. 3.\nIt is aparent from Fig. 3 that the explicit symplectic method (24) has qual-\nitatively different behavior to the Runge-Kutta method. Th e energy divergence\nbetween the explicit symplectic method and the exact soluti on is smaller than\nthat between Runge-Kuttamethod and the exact solution. The energy divergence\nbetween the explicit symplectic method and Runge-Kutta met hod increases with\nthe time evolution. Due to the increasement of the energy, th e phase difference\nbetween boththeresultsin Fig. 2 increases also withthetim eevolution.\n9-2.5-2-1.5-1-0.5 0 0.5 1\n 0 10  20  30  40  50  60  70  80  90 100Total Energy\ntExplicit Symplectic\nRunge-Kutta\nExact solution\nFigure3: Thetotalenergyofthe VanDer Pol’soscillator\n4.2. TheSecond Example\nInthesecondexample,weconsidera2 −dimensionaldampednonlinearDuff-\ningoscillator\n2¨q1+0.1˙q1+(2+0.1q2\n1)q1+q2=0\n3¨q2+0.2˙q2+q1+(2+0.2q2\n2)q2=0,(26)\nwith the initial conditions q1=0,q2=0,˙q1=0,˙q2=1. The program of the\nbothmethodswiththestepsize τ=0.01arecarriedouttosimulateEq. (26). The\nresonance is shown in Fig. 2, the numerical solution of the to tal energy is shown\ninFig.5.\nThere is only tiny difference between resonance results of t he two methods,\ncorrespondingly,thedifferenceamongthetotalenergy obt ainedbythenumerical\nmethods and anlytical methods is very tiny. As numerical exa mples in the other\nliteratures[13], that explicit Runge-Kutta method must ca use numerical pseudo\ndissipation which might be positive or negative. The differ ence between our nu-\nmerical examples and the examples in the literature[13] is t he total energy in our\nexamplesand themechanicalenergy intheirexamples1.\n1Fig.6.1in theliterature[13]\n10-1.5-1-0.5 0 0.5 1 1.5\n 0 20 40 60 80 100 120 140 160 180 200 220q1\ntExplicit Symplectic\nRunge-Kutta\nFigure4: The1-thdisplacementofthedampedDuffingoscilla tor\n 0.4995 0.4996 0.4997 0.4998 0.4999 0.5 0.5001\n 0 20  40  60  80 100  120  140  160  180  200Total Energy\ntExplicit Symplectic\nRunge-Kutta\nExact solution\nFigure5: Total energyofthedampeddissipativeoscillator\n115. Conclusions\nWehaveintroducedaclassofexplicitsymplecticalgorithm stodissipativeme-\nchanicalsystemssuccessfully,bychangingthesealgorith msintothescheme.(24).\nBecausethealgorithms(24)areexplicitandpossessgooden ergypreservingchar-\nacteristics, the explicit symplectic algorithms (24) is qu ite suitable for long term\nintegrationofarbitrary dimensionalnonlineardissipati vemechanical systems.\nReferences\n[1] K. Feng, On difference schemes and symplectic geometry, in: Ed. Feng\nKang Proceeding of the 1984 Beijing Symposium on differenti al geometry\nand differential equations-computationof partial differ ential equations, Sci-\nencePress, Beijing,1985,pp. 42–58.\n[2] H. Wu, M. Qin, K. Feng, Construction of canonical differe nce schemes for\nhamiltonianformalismviageneratingfunctions, JCM7 (198 9)71–96.\n[3] H.Wu,M.Qin,K.Feng, Symplecticdifferenceschemesfor thelinearhamil-\ntoniancanonicalsystems, JCM 8 (1990)371–380.\n[4] K.Feng, Thehamiltonianway forcomputinghamiltoniand ynamics, Math.\nAppl.56(1991)17–35.\n[5] J. E. Marsden, G. W. Patrick, S. Shkoller, Multisymplect ic geometry, vari-\national integrators, and nonlinear pdes, Communications i n Mathematical\nPhysics199 (1998)351–395.Cited By (since1996): 129.\n[6] F. Neri, Lie algebras and canonical integration, Techni cal Report, Depart-\nmentofPhysics,UniversityofMaryland,1988.\n[7] H. Yoshida, Construction of higher order symplectic int egrators, Physics\nLettersA 150(1990)262–268.\n[8] S. Uhlar, P. Betsch, On the derivation of energy consiste nt time stepping\nschemes for friction afflicted multibody systems, Computer s & Structures\n88(2010)737– 754.\n[9] S. Leyendecker, P. Betsch, P. Steinmann, Energy-conser ving integration of\nconstrained hamiltonian systems a comparison of approache s, Computa-\ntionalMechanics 33(2004)174–185.10.1007/s00466-003-0 516-2.\n12[10] P. Betsch, Energy-consistent numerical integration o f mechanical systems\nwith mixed holonomic and nonholonomic constraints, Comput er Methods\ninApplied Mechanics and Engineering195 (2006)7020– 7035. Multibody\nDynamicsAnalysis.\n[11] T. Luo, Y. Guo, Infinite-dimensional Hamiltonian descr iption of a class of\ndissipativemechanicalsystems, ArXive-prints (2009).\n[12] K. Feng, M. Qin, The symplectic methods for the computat ion of hamilto-\nnian equations, in: Numerical Methods for Partial Differen tial Equations,\nSpringer,Berlin, 1987,pp. 17–35.\n[13] C. Kane, J. E. Marsden, M. Ortiz, M. West, Variational in tegrators and the\nnewmarkalgorithmforconservativeanddissipativemechan icalsystems, In-\nternational Journal for Numerical Methods in Engineering 4 9 (2000) 1295–\n1325.\n13"    },    {        "title": "1103.5858v3.Spin_motive_forces_due_to_magnetic_vortices_and_domain_walls.pdf",        "content": "Spin motive forces due to magnetic vortices and domain walls\nM.E. Lucassen,1,\u0003G.C.F.L. Kruis,2R. Lavrijsen,2H.J.M. Swagten,2B. Koopmans,2and R.A. Duine1\n1Institute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Department of Applied Physics, Center for NanoMaterials and COBRA Research Institute,\nEindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 9, 2021)\nWe study spin motive forces, i.e.,spin-dependent forces, and voltages induced by time-dependent\nmagnetization textures, for moving magnetic vortices and domain walls. First, we consider the\nvoltage generated by a one-dimensional \feld-driven domain wall. Next, we perform detailed calcu-\nlations on \feld-driven vortex domain walls. We \fnd that the results for the voltage as a function of\nmagnetic \feld di\u000ber between the one-dimensional and vortex domain wall. For the experimentally\nrelevant case of a vortex domain wall, the dependence of voltage on \feld around Walker breakdown\ndepends qualitatively on the ratio of the so-called \f-parameter to the Gilbert damping constant, and\nthus provides a way to determine this ratio experimentally. We also consider vortices on a magnetic\ndisk in the presence of an AC magnetic \feld. In this case, the phase di\u000berence between \feld and\nvoltage on the edge is determined by the \fparameter, providing another experimental method to\ndetermine this quantity.\nPACS numbers: 75.60.Ch, 72.15.Gd, 72.25.Ba\nI. INTRODUCTION\nOne of the recent developments in spintronics is the\nstudy of spin motive forces1and spin pumping2. These\ne\u000bects lead to the generation of charge and spin currents\ndue to time-dependent magnetization textures. The idea\nof spin motive forces due to domain walls is easily un-\nderstood on an intuitive level: if an applied current in-\nduces domain-wall motion,3{8Onsager's reciprocity the-\norem tells us that a moving domain wall will induce a\ncurrent. This idea was already put forward in the eight-\nies by Berger9. In the case of a domain wall driven by\nlarge magnetic \felds ( i.e.,well above the so-called Walker\nbreakdown \feld), a fairly simple approach to the prob-\nlem is justi\fed where one goes to a frame of reference in\nwhich the spin quantization axis follows the magnetiza-\ntion texture.10This transformation gives rise to a vec-\ntor potential from which e\u000bective electric and magnetic\n\felds are derived. Experimentally, the domain-wall in-\nduced voltage has recently been measured above Walker\nbreakdown11, and the results are consistent with this ap-\nproach. It has also been shown that the induced voltage\nwell above Walker breakdown is determined from a topo-\nlogical argument that follows from the properties of the\nabove-mentioned vector potential.12\nThe above approach only captures the reactive con-\ntribution to the spin-motive forces. When the velocity\nof the domain wall is below or just above Walker break-\ndown, a theory is needed that includes more contribu-\ntions to the spin motive forces. Renewed interest has\nshed light on the non-adiabatic and dissipative contribu-\ntions to the spin motive forces13{15that are important in\nthis regime. In this paper, we study this regime.\nThe article is organized as follows. In Section II we\nsummarize earlier results that give a general framework\nto compute electrochemical potentials for given time-dependent magnetization textures. In Section III we con-\nsider an analytical model for a one-dimensional domain\nwall and numerically determine the form of the spin accu-\nmulation and the electrochemical potential. The results\nagree with the known results for the potential di\u000berence\ninduced by a moving one-dimensional domain wall.13In\nSection IV we turn to two-dimensional systems and study\na vortex domain wall in a permalloy strip. We use a\nmicro-magnetic simulator to obtain the magnetization\ndynamics, and numerically evaluate the reactive and dis-\nsipative contributions to the voltage below and just above\nWalker breakdown and compare with experiment.11An-\nother example of a two-dimensional system is a vortex\non a disk which we treat in Section V. For small enough\ndisks, the magnetic con\fguration is a vortex. Both ex-\nperimentally and theoretically, it has been shown that\na vortex driven by an oscillating magnetic \feld will ro-\ntate around its equilibrium position.16{20This gives rise\nto a voltage di\u000berence between the disk edge and cen-\nter as was recently discussed by Ohe et al. [21]. Here,\nwe extend this study by including both the reactive and\nthe dissipative contributions to the voltage, that turn out\nto have a relative phase di\u000berence. This gives rise to a\nphase di\u000berence between the drive \feld and voltage that\nis determined by the so-called \f-parameter.\nII. MODEL\nThe spin-motive force \feld F(~ x) induced by a time-\ndependent magnetization texture that is characterized\nat position ~ xby a unit-vector magnetization direction\nm(~ x;t) is given by13,14\nFi=~\n2[m\u0001(@tm\u0002rim) +\f(@tm\u0001rim)]:(1)arXiv:1103.5858v3  [cond-mat.other]  8 Apr 20112\nThis force \feld acts in this form on the majority spins,\nand with opposite sign on minority spins. In this ex-\npression, the \frst term is the well-known reactive term.1\nThe second term describes dissipative e\u000bects due to spin\nrelaxation13,14and is proportional to the phenomeno-\nlogical\f-parameter, which plays an important role in\ncurrent-induced domain-wall motion.3{8The spin accu-\nmulation\u0016sin the system follows from14\n1\n\u00152\nsd\u0016s\u0000r2\u0016s=\u0000r\u0001F; (2)\nwhere\u0015sd=p\n\u001cDis the spin-di\u000busion length, with \u001c\na characteristic spin-\rip time and Dthe e\u000bective spin-\ndi\u000busion constant. Here, we assume that the spin-\nrelaxation time is much smaller than the timescale for\nmagnetization dynamics. The total electrochemical po-\ntential\u0016that is generated by the spin accumulation due\nto a non-zero current polarization in the system is com-\nputed from14\n\u0000r2\u0016=P(r2\u0016s\u0000r\u0001F); (3)\nwhere the current polarization is given by P= (\u001b\"\u0000\n\u001b#)=(\u001b\"+\u001b#). Note that there is no charge accumulation\nfor\u001b\"=\u001b#, with\u001b\"(\u001b#) the conductivity of the majority\n(minority) spin electrons.\nThe magnetization dynamics is found from the\nLandau-Lifschitz-Gilbert (LLG) equation given by\n@m\n@t=m\u0002\u0012\n\u0000@Emm[m]\n~@m\u0013\n\u0000\u000bm\u0002@m\n@t: (4)\nHere,Emm[m] is the micromagnetic energy functional\nthat includes exchange interaction, anisotropy, and ex-\nternal \feld, and \u000bis the Gilbert damping constant.\nIII. ONE-DIMENSIONAL DOMAIN WALL\nFor one-dimensional problems the voltage di\u000berence\ncan be easily found. For example, an analytic expression\nfor the electric current (which is the open-circuit equiva-\nlent of the chemical-potential di\u000berence) was obtained by\none of us for an analytical model for a one-dimensional\ndriven domain wall.13In this section, we solve the po-\ntential problem for a one-dimensional domain wall and\nobtain the explicit position dependence of the spin accu-\nmulation and the chemical potential.\nA one-dimensional domain wall ( @ym=@zm= 0) is\ndescribed by23\n\u0012(x;t) = 2arctann\neQ[x\u0000X(t)]=\u0015o\n; \u001e (x) = 0;(5)\nwithm= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). Here,Q=\u00061 is\ncalled the topological charge of the domain wall since it\nindicates the way in which an external \feld a\u000bects the\ndomain-wall motion, i.e., a \feld in the direction +^ zwillmove a domain wall in the direction Q^x. Here, we choose\nQ= 1. The domain wall width is indicated by \u0015.\nTo study the time-evolution of a domain-wall, we let\n\u001e(x)!\u001e0(t) so that the wall is described by time-\ndependent collective coordinates fX(t);\u001e0(t)g, called\nposition and chirality, respectively. For external \felds\nsmaller than the Walker-breakdown \feld, there is no\ndomain-wall precession ( i.e., the chirality is constant),\nand the domain wall velocity vis constant so that\n@tm=\u0000v@xm. Since@ym=@zm= 0, we immediately\nsee that the \frst term on the right-hand side of Eq. (1)\nvanishes, and that the force is pointing along the x-axis.\nWe then \fnd that Fx= (\fv~=2)=(\u00152cosh[x=\u0015]2). Due to\nsymmetry we have that @y\u0016=@z\u0016=@y\u0016s=@z\u0016s= 0.\nIn Figs. 1 and 2 we plot the spin accumulation and the\nelectrochemical potential as a function of x.\n-10 -5 5 10x/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ\n-1-0.50.512Μs Λ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒv/HBar\nΛsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal12 Λ3\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒvs /HBar/Gradient.F\nFIG. 1: (color online) Spin accumulation as a function of\nposition for several values of the spin-di\u000busion length. The\nblack dotted line gives the value of the source term. The spin\naccumulation tends to zero for x!\u00061 .\n-10 -5 5 10xΛsd/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ20.511.522ΜΛ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΒv/HBarP\nΛsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal1\nFIG. 2: (color online) Electrochemical potential as a function\nof position. Note that the potential is proportional to the\npolarization and that on the horizontal axis the position xis\nmultiplied by the spin-di\u000busion length.\nFrom Fig. 2 we see that the total potential di\u000ber-\nence \u0001\u0016=\u0016(x!1 )\u0000\u0016(x!\u00001 ) is independent of3\nthe spin-di\u000busion length and linear in the parameter \f:\n\u0001\u0016=~P\fv=\u0015 . Note that this result is only valid below\nWalker Breakdown.\nTo \fnd the voltage for all \felds Bwe generalize the\nresults for the voltage di\u000berence in Ref. [13] for general\ndomain-wall charge Q. A general expression for the volt-\nage in one dimension is given by13\n\u0001\u0016=\u0000~P\n2jejZ\ndx[m\u0001(@tm\u0002@xm) +\f@tm\u0001@xm]:\n(6)\nWe insert the ansatz [Eq. (5) with \u001e(x) =\u001e0(t)] into\nEq. (6) and \fnd\n\u0001\u0016=\u0000~P\n2jej\"\nQ_\u001e0+\f_X\n\u0015#\n: (7)\nTo \fnd a time-averaged value for the voltage we consider\nthe equations of motion for a domain wall that is driven\nby a transverse magnetic \feld B22{24, contributing to the\nenergy\u0000gB\u0001m, withg >0. The equations of motion\nforX(t) and\u001e0(t) are ultimately derived from the LLG\nequation in Eq. (4), and given by\n(1 +\u000b2)_\u001e0=\u0000gB\n~\u0000\u000bK?\n2~sin(2\u001e0);\n(1 +\u000b2)_X\n\u0015=\u000bQgB\n~\u0000QK?\n2~sin(2\u001e0): (8)\nHere,K?is the out-of-plane anisotropy constant. These\nequations are solved by\nh_\u001e0i=\u0000Sign(B)\n1 +\u000b2Re2\n4s\u0012gB\n~\u00132\n\u0000\u0012\u000bK?\n2~\u001323\n5;\nh_Xi=\u0015Q\n1 +\u000b2 \ngB\n\u000b~+h_\u001e0i\n\u000b!\n(9)\nwhereh::idenotes a time average. It follows that the\nvoltage di\u000berence for general topological charge is\n\u0001\u0016=\u0000Sign(B)Q\n1 +\u000b2~P\n2jej(\n\f\n\u000bgjBj\n~\n\u0000\u0012\n1 +\f\n\u000b\u0013\nRe2\n4s\u0012gB\n~\u00132\n\u0000\u0012\u000bK?\n2~\u001323\n5)\n:(10)\nNote that the overall prefactor Sign( B)Qmakes sense:\ninversion of the magnetic \feld should have the same re-\nsult as inversion of the topological charge.\nIn the above, we used a domain-wall ansatz with mag-\nnetization perpendicular to the wire direction. Using\nthe topological argument by Yang et al.12one can show\nthat the result is more general and holds also for head-\nto-head and tail-to-tail domain walls. Therefore, for a\none-dimensional domain wall, the reactive and dissipa-\ntive contributions, i.e.,the contributions with and with-\nout\fin the above expression, to the voltage always have\nopposite sign.IV. VORTEX DOMAIN WALL\nFor more complicated two-dimensional structures the\nspin-motive force \feld can have rotation and the simpli-\n\fed expression in Eq. (6) is no longer valid so that we\nneed to treat the full potential problem in Eqs. (1-3).\nMotivated by recent experimental results11we consider\nin this section the voltage induced by a moving vortex\ndomain wall.\nWe study the magnetization dynamics using a micro-\nmagnetic simulator25from which we obtain the magneti-\nzation m(~ x;t). This simulator solves the LLG equation in\nEq. (4). For comparison with the experiment by Yang et\nal.,11we simulate a permalloy sample that has the same\ndimensions as this experiment, i.e. 20nm \u0002500nm\u0002\n32\u0016m, which is divided in 1 \u0002128\u00028192 lattice points.\nOn this sample, we drive a head-to-head vortex domain\nwall by means of a magnetic \feld that is pointing from\nright to left, such that the vortex moves from right to left.\nFor several \feld strengths, we obtain the magnetization\nm, and its time-derivative which allows us to compute\nthe force \feld Fat each lattice point. Next, we solve\nthe matrix problem that is the discrete equivalent to the\npotential problem in Eqs. (2) and (3). For details on this\ncalculation, see App. B.\nWe \frst investigate the velocity of the vortex domain\nwall as a function of the applied \feld. We use the value\n\u000b= 0:02 for the Gilbert-damping parameter to obtain\nthe curve in Fig. 3.\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen1050100150200250300Velocity/LParen1m/Slash1s/RParen1\nFIG. 3: Velocity of the vortex domain wall as a function of the\nmagnetic \feld strength for \u000b= 0:02. Above Walker break-\ndown, the velocity is time-averaged. The line is a guide to the\neye.\nThe decrease in velocity for B= 1:5 mT signals\nWalker breakdown. Indeed, up to \felds B= 1:4 mT, the\nvortex moves parallel to the long direction of the sample.\nForB= 1:5 mT, the vortex domain wall motion is more\ncomplicate and has a perpendicular component.26,27We\ntherefore expect that below Walker Breakdown, just like\nfor the one-dimensional domain wall, the vortex domain\nwall only has a dissipative contribution to the voltage.4\nComparison with the experimental results of Ref. [11]\nshows that our velocity is roughly a factor 2 higher.\nThis might be partly caused by a di\u000berence in damping\nand partly by the presence of defects in the experiment\nwhich causes pinning and therefore a decrease of veloc-\nity. The exact value of the Walker breakdown \feld is\nhard to compare, since this depends also on the exact\nvalue of the anisotropy. Nonetheless our value for the\nWalker breakdown \feld is of the same order as Ref. [11].\nMoreover, what is more important is the dependence of\nwall velocity and wall-induced voltage on the magnetic\n\feld normalized to the Walker-breakdown \feld, as these\nresults depend less on system details.\n500\n1000\n1500\n2000length50100\nwidth-505\nΜ/LParen1ΜV/RParen1\n500\n1000\n1500 length\nFIG. 4: Electrochemical potential as a function of position\nfor a moving vortex domain wall on the sample. The num-\nbers on the horizontal axes correspond to lattice points with\nseparation a= 3:9nm. This speci\fc \fgure is for \u000b= 0:02,\nH= 0:8mT (i.e. below Walker breakdown), P= 1 and\n\u0015sd=a. Note that the peak signals the position of the vortex\ncore.\nAn example of a speci\fc form of the electrochemical\npotential on the sample due to a \feld-driven vortex do-\nmain wall is depicted in Fig. 4. We see that there is\na clear voltage drop along the sample, like in the one-\ndimensional model. Additionally, the potential shows\nlarge gradients around the vortex core and varies along\nthe transverse direction of the sample. For each \feld\nstrength, we compute the voltage di\u000berence as a func-\ntion of time. For \feld strengths below Walker Breakdown\nwe \fnd that, as expected, only the dissipative term con-\ntributes and the voltage di\u000berence rapidly approaches a\nconstant value in time. This is understood from the fact\nthat in this regime, the wall velocity is constant after a\nshort time. The dissipative contribution to the voltage\nis closely related to the velocity along the sample, as can\nbe seen in Fig. 5.\nAbove Walker breakdown the reactive term con-\ntributes. We \fnd that for \f=\u000b, the oscillations in\nthe reactive component compensate for the oscillations\n20406080100120140\nTime-2000200400/CapDeltaΜ/Slash1P/LParen1nV/RParen1\nvelocity/CapDeltaΜreac/CapDeltaΜdissFIG. 5: (color online) reactive (blue squares) and dissipative\n(red triangles) contributions to the voltage as a function of\ntime. The numbers on the horizontal axis correspond to time\nsteps of 0.565 ns. The green line gives the velocity along the\nsample, it is scaled to show the correlation with the voltage.\nThese curves are taken for \u000b= 0:02,\f=\u000band \feld strength\nB= 1:6mT.\nin the dissipative component. If we look closely to Fig. 5\nwe see that the length of the periods is not exactly equal.\nThe periods correspond to a vortex moving to the upper\nedge of the sample, or to the lower edge. The di\u000berence\nis due to the initial conditions of our simulation. We\naverage the voltage di\u000berence over time to arrive at the\nresult in Fig. 6. We see that the dissipative contribu-\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1/CapDeltaΜdiss/Plus/CapDeltaΜreac/CapDeltaΜreac/CapDeltaΜdiss\nFIG. 6: (color online) Voltage drop along the sample for \u000b=\n0:02 and\f=\u000b.\ntion becomes smaller for \felds larger than the Walker\nbreakdown \feld, whereas the reactive contribution has\nthe same sign and increases. In fact, for \f=\u000b, the re-\nduction of the dissipative contribution is exactly compen-\nsated by the reactive contribution. The \fdependence is\nillustrated in Fig. 7. The behavior is fundamentally dif-\nferent from the one-dimensional domain-wall situation:\nfor the vortex domain wall, the dissipative contribution\nhas the same sign as the reactive contribution.\nIn order to understand the relative sign, we now dis-\ncuss general vortex domain walls. A single vortex (i.e.5\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1Β/Equal3Α/Slash12Β/EqualΑΒ/EqualΑ/Slash12Β/Equal0\nFIG. 7: (color online) Voltage drop along the sample for \u000b=\n0:02 and several values for \f.\nwith vorticity q= +1) is described by two parameters:\nthe chargep=\u00061 indicates wether the central magnetic\nmoment points in the positive or negative zdirection\nand the chirality Cindicates wether the magnetic mo-\nments align in a clockwise ( C=\u00001) or anti-clockwise\n(C= +1) fashion. We have a vortex that is oriented\nclockwiseC=\u00001. The relative sign is explained from\na naive computation of the voltage above Walker break-\ndown that does not take into account the rotation of the\nspin-motive force \feld\n\u0001\u0016/\u0000Z\ndx[\f@tm\u0001@xm+m\u0001(@tm\u0002@xm)]\n=\fvxZ\ndx(@xm)2+vyZ\ndxm\u0001(@ym\u0002@xm)\n/(\f\u000e+ 1)vx: (11)\nwhere\u000eis a positive number and we used that above\nWalker breakdown m'm(x\u0000vxt;y\u0000vyt) withvx6= 0\nandvy6= 0. Note that if vy= 0 (below Walker\nbreakdown) the reactive term, i.e., the second term in\nthe above expression, indeed vanishes. We used in the\nlast line that above Walker breakdown vx/\u0000pvyandR\ndxm\u0001(@ym\u0002@xm)/ \u0000p. The former equality is\nunderstood from a geometric consideration: consider a\nsample with a vortex characterized by C= 1,p= 1 and\nvxvy<0. By symmetry, this is equivalent to C=\u00001,\np=\u00001 andvxvy>0. It is therefore clear that the sign\nofvxvydepends on either the polarization, or the hand-\nedness of the vortex. Since we know from the vortex\ndomain wall dynamics that reversal of the polarization\nreverses the perpendicular velocity,11we conclude that\nvxvydoes not depend on the handedness of the vortex.\nThe latter equality is understood from a similar argu-\nment:RR\ndxdym\u0001(@ym\u0002@xm) changes sign under the\ntransformation m!\u0000m. During this transformation,\nbothp!\u0000pandC!\u0000C, and therefore their prod-\nuct cannot account for the total sign reversal. Therefore,\nthe integral depends on the polarization12but not on\nthe handedness of the vortex. The positive number \u000eis\nobtained from our numerical simulation, which suggeststhat the magnetic-\feld dependence of the voltage is\n\u0001\u0016=\fB\u0002constant + (1\u0000\f=\u000b)j\u0001\u0016reactivej:(12)\nNote that the sign of the relative contributions can also\nbe obtained using the topological argument by Yang et\nal.12, which gives the same result.\nWe compare our results in Fig. 7 with the experiment\nby Yang et al.11. If we assume that the voltage below\nWalker breakdown lies roughly on the same line as the\nvoltages above Walker breakdown, their results suggest\na slope of 10nV/Oe. For P\u0018 0:8, our results suggest a\nslope of (\f=\u000b)14nV/Oe. Taking into account our higher\nvelocity, we \fnd that \fin the experiment is somewhat\nlarger than \u000b. The decrease in slope of the voltage in\nRef. [11] as Walker breakdown is approached from above\nalso suggests \f >\u000b .\nIn conclusion, the behavior of the voltage around\nWalker breakdown allows us to determine the ratio \f=\u000b.\nIn experiment, the potential di\u000berence as a function of\nthe applied magnetic \feld would show an upturn or\ndownturn around Walker breakdown as in Fig. 7, which\ncorresponds to \f <\u000b and\f >\u000b , respectively.\nV. MAGNETIC VORTEX ON A DISK\nOn small disks (of size \u0016m and smaller) of ferromag-\nnetic material the lowest energy con\fguration is a vor-\ntex. It has been shown that one can let the vortex rotate\naround its equilibrium position by applying an AC mag-\nnetic \feld16{20. This motion gives rise via Eq. (1) to a\nspin motive force on the spins, which induces a voltage on\nthe edge of the disk relative to a \fxed reference voltage,\ne.g. the disk center. Ohe et al.21have shown that the\nreactive contribution to the spin motive force \feld can be\nseen as a dipole that is pointing in the radial direction,\ni.e., the divergence of the force \feld consists of a posi-\ntive and a negative peak along the radial direction (note\nthat the divergence of the force \feld can be seen as an\ne\u000bective charge). Rotation of this dipole gives rise to an\noscillating voltage on the edge of the sample. Here, we\nconsider also the dissipative contribution to the voltage.\nWe consider a vortex on a disk with radius Rthat\nmoves around its equilibrium position ( i.e., the center\nof the disk) at a distance r0from the center of the disk\nwith frequency !. We use as a boundary condition that\nthe magnetization on the edge of the disk is pointing\nperpendicular to the radial direction. In equilibrium, the\nmicro-magnetic energy density of the form \u0000Jm\u0001r2m\u0000\nK?m2\nzis minimized by\nmx(x;y) =\u0000yp\nx2+y2cosh\n2 arctan\u0010\ne\u0000Cp\nx2+y2=\u0014\u0011i\nmy(x;y) =xp\nx2+y2cosh\n2 arctan\u0010\ne\u0000Cp\nx2+y2=\u0014\u0011i\nmz(x;y) =psinh\n2 arctan\u0010\nep\nx2+y2=\u0014\u0011i\n; (13)6\nwhere the center of the vortex is chosen at x=y= 0.\nHere\u0014=p\nK?=Jis the typical width of the vortex core.\nFor permalloy this length scale is of the order \u001810nm.\nThe parameters pandCare de\fned as before, for de\f-\nniteness we choose p= 1,C=\u00001. To describe clockwise\ncircular motion of the vortex around its equilibrium po-\nsition at \fxed radius r0we substitute x!x\u0000r0sin(!t)\nandy!y\u0000r0cos(!t). Note that we assume that the\nform of the vortex is not changed by the motion, which\nis a good approximation for r0\u001cR.\nFrom the magnetization in Eq. (13), we compute the\nforce \feld using Eq. (1). The reactive and dissipative\ncontributions to the divergence of the force \feld are\nshown in Fig. 8. The direction of the dipoles follows\nΠ 0Π 0Π\nAngle00.050.1r/Slash1R\nFIG. 8: The reactive (left) and dissipative (right) contribu-\ntions to the divergence of the force \feld. White means posi-\ntive values, black is negative values. The reactive contribution\ncan be seen as a dipole in the radial direction. The dissipative\ncontribution is a dipole perpendicular to the radial direction.\ndirectly from Eq. (1) if we realize that for our system\n\u0000@tm\u0001~rm=~ v(@^vm)2is always pointing in the direc-\ntion of the velocity which shows that the dissipative con-\ntribution points along the velocity. Likewise the reactive\ncontribution is always pointing perpendicular to the ve-\nlocity.\nFrom the relative orientations of the e\u000bective dipoles,\nwe expect that the peaks in the reactive and dissipative\ncontributions to the voltage on the edge will di\u000ber by a\nphase of approximately \u0019=2 (forr0=R!0 this is exact).\nWe divide our sample in 1000 rings and 100 angles and\nuse the general method in App. B to \fnd the voltage on\nthe edge shown in Fig. 9. To compare with Ref. [21],\nwe take a frequency !=(2\u0019) = 300 MHz and P= 0:8,\nwhich yields amplitudes for the reactive contribution of\n\u0018\u0016V on the edge. However, Ohe et al. suggest that\nvoltage probes that are being placed closer to the vortex\ncore measure a higher voltage. This indeed increases the\nvoltage up to order \u001810\u0016V atr= 2r0. Placing the\nleads much closer to the vortex core does not seem to\nbe realistic because of the size of the vortex. Since the\nvoltage scales with velocity, it can also be increased by a\nlarger radius of rotation, i.e. by applying larger magnetic\n\felds. However, for disks larger than 1 \u0016m, the vortex\n0 Π 2Π 3Π 4Π\nΩt-15-10-50510152 Π /CapDeltaΜ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΩP/LParen1ΜV/Slash1GHz/RParen1\n/CapDeltaΜtotal/CapDeltaΜdiss/Slash1Β/CapDeltaΜreacFIG. 9: (color online) The reactive (red dashed curve) and\ndissipative (blue dashed curve) contributions to the voltage\ndi\u000berence between opposite points on the edge of the disk.\nThe green line gives the total voltage di\u000berence \u0001 \u0016total =\n\u0001\u0016reac+ \u0001\u0016diss, in this example for \f= 0:4. We used r0=\n10\u0015sd,R= 100\u0015sdand\u0014=\u0015sd. For realistic spin-di\u000busion\nlength\u0015sd'5 nm, these parameter values agree with the\nsystem of Ohe et al.21.\nstructure is lost.\nThe dissipative contribution becomes important for\nlarge values of \f. In principle, it is possible to determine\n\fby looking at the shift of the peak in the total volt-\nage with respect to the peak in the reactive contribution,\nwhich is in turn determined by the phase of the applied\nmagnetic \feld. The phase di\u000berence between applied\n\feld and measured voltege then behaves as tan(\u0001 \u001e)/\f.\nVI. DISCUSSION AND CONCLUSION\nWe have investigated the voltage that is induced by a\n\feld-driven vortex domain wall in detail. In contrast to\na one-dimensional model of a domain wall, the reactive\nand dissipative contribution to the voltage have the same\nsign. The qualitative di\u000berences for di\u000berent values of \f\nprovide a way to determine the ratio \f=\u000bexperimentally\nby measuring the wall-induced voltage as a function of\nmagnetic \feld. To this end the experimental results in\nRef. [11] are in the near future hopefully extended to\n\felds below Walker breakdown, which is challenging as\nthe voltages become smaller with smaller \feld.\nWe also studied a magnetic vortex on a disk. When\nthe vortex undergoes a circular motion, a voltage is in-\nduced in the sample. Earlier work computed the reactive\nvoltage on the edge of the disk,21here we include also the\ndissipative contribution to the voltage. We \fnd that the\nphase di\u000berence between voltage and AC driving \feld is\ndetermined by the \f-parameter.7\nAcknowledgments\nThis work was supported by the Netherlands Organiza-\ntion for Scienti\fc Research (NWO) and by the European\nResearch Council (ERC) under the Seventh Framework\nProgram (FP7).\nAppendix A: Boundary conditions\nAs a boundary condition for the potential problems,\nwe demand that the total spin current and charge current\nperpendicular to the upper and lower boundaries is zero:\nj?\ns=j?\n\"\u0000j?\n#= 0 andj?\ns=j?\n\"+j?\n#= 0. Therefore,\nthe majority-and minority spin currents are necessarily\nzero. They are given by j\"=\u001b\"(F\u0000r\u0016\") andj#=\n\u001b#(\u0000F\u0000r\u0016#). From this, the boundary conditions on\nthe derivatives of the potentials follow as @?\u0016s=@?(\u0016\"\u0000\n\u0016#)=2 =Fand@?\u0016=@?(\u0016\"+\u0016#)=2 = 0. We consider\na two-dimensional sample that is in\fnitely long in the\nx-direction, and of \fnite size 2\u0003 in the y-direction the\nboundary conditions are\n@y\u0016s(x;y=\u0006\u0003) =Fy(x;y=\u0006\u0003): (A1)\nTo measure the induced voltage, we also put the deriva-\ntives of the potential at in\fnity to zero so that the bound-\nary conditions for the electrochemical potential are\n@y\u0016(x;y=\u0006\u0003) = 0;\n@x\u0016(x!\u00061;y) = 0: (A2)\nAppendix B: Potential problem on a Lattice\nWe consider a two-dimensional lattice, where we have\nspin accumulation \u0016i;j\nsand an electrochemical potential\n\u0016i;jat sitei;j. Between sites ( i;j) and (i;j+ 1), there\ncan be a particle current density of majority spins\nji;j+1=2\n\";^ \u0011=\u001b\" \nFi;j+1=2\n^ \u0011+\u0016i;j\n\"\u0000\u0016i;j+1\n\"\nai;j+1=2\n^ \u0011!\n=\u001b\"\u0010\nFi;j+1=2\n^ \u0011\u0000\u000e^ \u0011\u0016i;j+1=2\n\"\u0011\n; (B1)\nwithai;j+1=2\n^ \u0011the lattice spacing in the ^ \u0011 direction between\nsites (i;j) and (i;j+ 1), and a particle current density of\nminority spins\nji;j+1=2\n#;^ \u0011=\u001b# \n\u0000Fi;j+1=2\n^ \u0011+\u0016i;j\n#\u0000\u0016i;j+1\n#\nai;j+1=2\n^ \u0011!\n=\u001b#\u0010\n\u0000Fi;j+1=2\n^ \u0011\u0000\u000e^ \u0011\u0016i;j+1=2\n\"\u0011\n; (B2)\nand equivalently for currents in the ^ \u0010 direction. The\nderivative \u000e^ \u0011is de\fned as \u000e^ \u0011Oi;j= (Oi;j+1=2\u0000\nOi;j\u00001=2)=ai;j\n^ \u0011, and likewise for \u000e^ \u0010. Note that upper in-\ndices (i;j) denote a position on the lattices and lower in-\ndices ^ \u0010 or ^ \u0011 denote a direction. We can write \u0016\"=\u0016+\u0016sand\u0016#=\u0016\u0000\u0016s. The continuity-like equations for the\ndensity of majority- and minority spins are (note that\nspins move in the direction of the current)\nAi;jni;j\n\"#\n\u001c=\u0000\u0001(`i;jji;j\n\"#)\njej; (B3)\nwith characteristic spin-\rip time \u001cand with the dimen-\nsionless operator \u0001 given by\n\u0001Oi;j=Oi+1=2;j\n^ \u0010\u0000Oi\u00001=2;j\n^ \u0010+Oi;j+1=2\n^ \u0011\u0000Oi;j\u00001=2\n^ \u0011:\n(B4)\nThese de\fnitions allows for non-square lattices with sides\nat position ( i\u00061=2;j) or (i;j\u00061=2) that have length\n`i\u00061=2;j\n^ \u0010or`i;j\u00061=2\n^ \u0011(lower index denotes the normal di-\nrection), respectively, and the area of the site itself given\nbyAi;j.\nThe equation for the electrochemical potential is ob-\ntained from the continuity equation\n0 =\u0000jejAi;jni;j\n\"+ni;j\n#\n\u001c= \u0001[`i;j(ji;j\n\"+ji;j\n#)] =\n\u001b\"\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\n\")] +\u001b#\u0001[`i;j(\u0000Fi;j\u0000\u000e\u0016i;j\n#)] =\n(\u001b\"+\u001b#)\u0001f`i;j[\u0000\u000e\u0016i;j+P(Fi;j\u0000\u000e\u0016i;j\ns)]g:\n!\u0001(`i;j\u000e\u0016i;j) =P\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\ns)]; (B5)\nwhere the current polarization is given by P= (\u001b\"\u0000\n\u001b#)=(\u001b\"+\u001b#). This result was already obtained for a\ncontinuous system in Ref. [14]. To \fnd an equation for\nthe spin accumulation, we write\n\u0000jejAi;jni;j\n\"\u0000ni;j\n#\n\u001c= \u0001[`i;j(ji;j\n\"\u0000ji;j\n#)] =\n\u001b\"\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\n\")]\u0000\u001b#\u0001[`i;j(\u0000Fi;j\u0000\u000e\u0016i;j\n#)] =\n(\u001b\"+\u001b#)\u0001f`i;j[Fi;j\u0000\u000e\u0016i;j\ns\u0000P\u000e\u0016i;j]g=\n(\u001b\"+\u001b#)(1\u0000P2)\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\ns)]: (B6)\nIf we compare this in the case of a square lattice to the\nexpression in Ref.14\n1\n\u00152\nsd\u0016s\u0000r2\u0016s=\u0000r\u0001F; (B7)\nwe \fnd that the density of spins that pile up can be\nexpressed in terms of the spin accumulation as ( ni;j\n\"\u0000\nni;j\n#)=\u001c= (\u001b\"+\u001b#)(1\u0000P2)\u0016i;j\ns=(jej\u00152\nsd). We insert this\nexpression to \fnd that the spin accumulation on a lattice\nis determined by\n\u00001\n\u00152\nsd\u0016i;j\ns=1\nAi;j\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\ns)]: (B8)8\n\u0003Electronic address: m.e.lucassen@uu.nl\n1S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n2Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n3L. Berger, J. Appl. Phys. 55, 1954 (1984).\n4P.P. Freitas and L. Berger, J. Appl. Phys. 57, 1266 (1985).\n5J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n6L. Berger, Phys. Rev. B 54, 9353 (1996).\n7S. Zhang and Z. Li, Phys Rev. Lett. 93, 127204 (2004).\n8S.E. Barnes and S. Maekawa, Phys Rev. Lett. 95, 107204\n(2005).\n9L. Berger, Phys. Rev. B 33, 1572 (1986).\n10A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n11S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Q. Niu,\nM. Tsoi, and J.L. Erskine, Phys. Rev. Lett. 102, 067201\n(2009).\n12S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Z. Zhang,\nM. Tsoi, Q. Niu, A.H. MacDonald, and J.L. Erskine, Phys.\nRev. B 82, 054410 (2010).\n13R.A. Duine, Phys. Rev. B 77, 014409 (2008); Phys. Rev.\nB79, 014407 (2009).\n14Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n15W.M. Saslow, Phys. Rev. B 76, 184434 (2007).\n16S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran,\nJ. St ohr, and A.H. Padmore, Science 304, 420 (2004).17B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T.\nTyliszczak, R. Hertel, M. F ahnle, H. Br uckl, K. Rott, G.\nReiss, I. Neudecker, D. Weiss, C.H. Back, and G. Sch utz,\nNature (London) 444, 461 (2006).\n18K.-S. Lee and S.-K. Kim, Appl. Phys. Lett. 91, 132511\n(2007).\n19M. Bolte, G. Meier, B. Kr uger, A. Drews, R. Eiselt, L.\nBocklage, S. Bohlens, T. Tyliszczak, A. Vansteenkiste, B.\nVan Waeyenberge, K.W. Chou, A. Puzic, and H. Stoll,\nPhys. Rev. Lett. 100, 176601 (2008).\n20B. Kr uger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche,\nG. Meier, Phys. Rev. B 76, 224426 (2007).\n21J.-I. Ohe, S.E. Barnes, H.-W. Lee, and S. Maekawa, Appl.\nPhys. Lett. 95, 123110 (2009).\n22N.L. Schryer and L.R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n23G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004); 96, 189702 (2006).\n24G. Tatara, H. Kohno and J. Shibata, Physics Reports 468,\n213 (2008).\n25M.R. Scheinfein, LLG Micromagnetics Simulator, http://\nllgmicro.home.mindspring.com.\n26G.S.D. Beach, c. Nistor, C. Knutson, M. Tsoi, and J.L.\nErskine, Nature Mat. 4, 741-744 (2005).\n27D.J. Clarke, O.A. Tretiakov, G.-W. Chern, Ya. B. Baziliy,\nand O. Tchernyshyov, Phys. Rev. B 78, 134412 (2008)."    },    {        "title": "1104.1625v1.Magnetization_Dissipation_in_Ferromagnets_from_Scattering_Theory.pdf",        "content": "arXiv:1104.1625v1  [cond-mat.mes-hall]  8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory\nArne Brataas∗\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nYaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\nGerrit E. W. Bauer\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\nKavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz-\nGilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza-\ntion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping\ntensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description\non the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part\ncan then be expressed in terms of the static scattering matri x. The dissipative contribution to the\nlow-frequency magnetization dynamics can be described as a n adiabatic energy pumping process\nto the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor\ndepends on the time derivative of the scattering matrix as a f unction of the magnetization direction.\nBy the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated\nin terms of the quasistatic scattering matrix. The theory is formulated for general magnetization\ntextures and worked out for monodomain precessions and doma in wall motions. We prove that the\nGilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism.\nPACS numbers: 75.40.Gb,76.60.Es,72.25.Mk\nI. INTRODUCTION\nFerromagnets develop a spontaneous magnetization\nbelow the Curie temperature. The long-wavelengthmod-\nulations of the magnetization direction consist of spin\nwaves, the low-lying elementary excitations (Goldstone\nmodes) of the ordered state. When the thermal energy is\nmuch smaller than the microscopic exchange energy, the\nmagnetization dynamics can be phenomenologically ex-\npressed in a generalized Landau-Lifshitz-Gilbert (LLG)\nform:\n˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+\nm(r,t)×/integraldisplay\ndr′[˜α[m](r,r′)˙ m(r′,t)],(1)\nwhere the magnetization texture is described by m(r,t),\nthe unit vector along the magnetization direction at po-\nsitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is\nthe gyromagnetic ratio in terms of the g-factor (≈2 for\nfree electrons) and the Bohr magneton µB. The Gilbert\ndamping ˜αis a nonlocal symmetric 3 ×3 tensor that is\na functional of m. The Gilbert damping tensor is com-\nmonly approximated to be diagonal and isotropic (i), lo-\ncal (l), and independent of the magnetization m, with\ndiagonal elements\nαil(r,r′) =αδ(r−r′). (2)\nThe linearized version of the LLG equation for small-\namplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea-\nsured response of ferromagnetic bulk materials and thin\nfilms in terms of a small number of adjustable, material-\nspecific parameters. The experiment of choice is fer-\nromagnetic resonance (FMR), which probes the small-\namplitude coherent precession of the magnet.2The\nGilbertdampingmodelinthelocalandtime-independent\napproximationhasimportantramifications, suchasalin-\near dependence of the FMR line width on resonance fre-\nquency, that have been frequently found to be correct.\nThe damping constant is technologically important since\nit governs the switching rate of ferromagnets driven by\nexternal magnetic fields or electric currents.3In spatially\ndependent magnetization textures, the nonlocal charac-\nter of the damping can be significant as well.4–6Moti-\nvated by the belief that the Gilbert damping constant is\nanimportantmaterialproperty, weset outheretounder-\nstand its physical origins from first principles. We focus\non the well studied and technologically important itiner-\nant ferromagnets, although the formalism can be used in\nprinciple for any magnetic system.\nThe reactive dynamics within the LLG Eq. (1) is de-\nscribed by the thermodynamic potential Ω[ M] as a func-\ntional of the magnetization. The effective magnetic field\nHeff[M](r)≡ −δΩ/δM(r) is the functional derivative\nwith respect to the local magnetization M(r) =Msm(r),\nincluding the external magnetic field Hext, the magnetic\ndipolar field Hd, the texture-dependent exchange energy,\nand crystal field anisotropies. Msis the saturation mag-\nnetization density. Thermal fluctuations can be included\nby a stochastic magnetic field h(r,t) with zero time av-2\nleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath (reservoirs) via metallic normal metal\nleads (N).\nerage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7\n/an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT\nγMs˜αij[m](r,r′)δ(t−t′),(3)\nwhereMsis the magnetization, iandjare the Cartesian\nindices, and Tis the temperature. This relation is a con-\nsequence ofthe fluctuation-dissipation theorem (FDT) in\nthe classical (Maxwell-Boltzmann) limit.\nThe scattering ( S-) matrix is defined in the space of\nthe transport channels that connect a scattering region\n(the sample) to real or fictitious thermodynamic (left\nand right) reservoirs by electric contacts with leads that\nare modeled as ideal wave guides. Scattering matri-\nces are known to describe transport properties, such as\nthe giant magnetoresistance, spin pumping, and current-\ninducedmagnetizationdynamicsinlayerednormal-metal\n(N)|ferromagnet (F).8–10When the ferromagnet is part\nof an open system as in Fig. 1, also Ω can be expressed\nin terms of the scattering matrix, which has been used\nto express the non-local exchange coupling between fer-\nromagnetic layers through conducting spacers.11We will\nshow here that the scattering matrix description of the\neffective magnetic fields is valid even when the system is\nclosed, provided the dominant contribution comes from\nthe electronic band structure, scattering potential disor-\nder, and spin-orbit interaction.\nScattering theory can also be used to compute the\nGilbert damping tensor ˜ αfor magnetization dynamics.15\nThe energy loss rate of the scattering region can be ex-\npressedin termsofthe time-dependent S-matrix. To this\nend, the theory of adiabatic quantum pumping has to be\ngeneralizedtodescribedissipationinametallicferromag-\nnet. The Gilbert damping tensor is found by evaluating\nthe energy pumping out of the ferromagnet and relat-\ning it to the energy loss that is dictated by the LLG\nequation. In this way, it is proven that the Gilbert phe-\nnomenology is valid beyond the linear response regime\nof small magnetization amplitudes. The key approxima-\ntion that is necessary to derive Eq. (1) including ˜ αis the\n(adiabatic) assumption that the ferromagnetic resonance\nfrequencyωFMRthat characterizesthe magnetizationdy-\nnamics is small compared to internal energy scale set by\nthe exchange splitting ∆ and spin-flip relaxation rates\nτs. The LLG phenomenology works well for ferromag-\nnets for which ωFMR≪∆//planckover2pi1, which is certainly the case\nfor transition metal ferromagnets such as Fe and Co.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from the transfer of energy\nfromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor-\nder or spin-orbit interactions in combination with impu-\nrity/phonon scattering.2Since the heat capacitance of\nthe ferromagnet is dominated by the lattice, the energy\ntransferred to the quasiparticles will be dissipated to the\nlattice as heat. Here we focus on the limit in which elas-\ntic scattering dominates, such that the details of the heat\ntransfer to the lattice does not affect our results. Our ap-\nproachformallybreaks down in sufficiently clean samples\nat high temperatures in which inelastic electron-phonon\nscattering dominates. Nevertheless, quantitative insight\ncan be gained by our method even in that limit by mod-\nelling phonons by frozen deformations.12\nIn the present formulation, the heat generated by the\nmagnetization dynamics can escape only via the contacts\nto the electronic reservoirs. By computing this heat cur-\nrent through the contacts we access the total dissipa-\ntion rate. Part of the heat and spin current that es-\ncapes the sample is due to spin pumping that causes\nenergy and momentum loss even for otherwise dissipa-\ntion less magnetization dynamics. This process is now\nwellunderstood.10For sufficiently largesamples, the spin\npumping contribution is overwhelmed by the dissipation\nin the bulk of the ferromagnet. Both contributions can\nbe separated by studying the heat generation as a func-\ntion of the length of a wire. In principle, a voltage can be\nadded to study dissipation in the presence of electric cur-\nrents as in 13,14, but we concentrate here on a common\nand constant chemical potential in both reservoirs.\nAlthough it is not a necessity, results can be simpli-\nfied by expanding the S-matrix to lowest order in the\namplitude of the magnetization dynamics. In this limit\nscattering theory and the Kubo linear response formal-\nism for the dissipation can be directly compared. We\nwill demonstrate explicitly that both approaches lead to\nidentical results, which increases our confidence in our\nmethod. The coupling to the reservoirs of large samples\nis identified to play the same role as the infinitesimals in\nthe Kubo approach that guarantee causality.\nOur formalism was introduced first in Ref. 15 lim-\nited to the macrospin model and zero temperature. An\nextension to the friction associatedwith domain wall mo-\ntion was given in Ref. 13. Here we show how to handle\ngeneral magnetization textures and finite temperatures.\nFurthermore, we offer an alternative route to derive the\nGilbert damping in terms of the scattering matrix from\nthe thermal fluctuations of the effective field. We also\nexplain in more detail the relation of the present theory\nto spin and charge pumping by magnetization textures.\nOur paper is organized in the following way. In Sec-\ntion II, we introduce our microscopic model for the fer-\nromagnet. In Section III, dissipation in the Landau-\nLifshitz-Gilbert equation is exposed. The scattering the-\nory of magnetization dynamics is developed in Sec. IV.\nWe discuss the Kubo formalism for the time-dependent\nmagnetizationsin Sec. V, before concluding our article in\nSec. VI. The Appendices provide technical derivations of\nspin, charge, and energy pumping in terms of the scat-3\ntering matrix of the system.\nII. MODEL\nOur approach rests on density-functional theory\n(DFT), which is widely and successfully used to describe\nthe electronic structure and magnetism in many fer-\nromagnets, including transition-metal ferromagnets and\nferromagnetic semiconductors.16In the Kohn-Sham im-\nplementation of DFT, noninteracting hypothetical par-\nticles experience an effective exchange-correlationpoten-\ntial that leads to the same ground-statedensity as the in-\nteractingmany-electronsystem.17Asimpleyetsuccessful\nscheme is the local-densityapproximationto the effective\npotential. DFT theory can also handle time-dependent\nphenomena. We adopt here the adiabatic local-density\napproximation (ALDA), i.e. an exchange-correlationpo-\ntential that is time-dependent, but local in time and\nspace.18,19As the name expresses, the ALDA is valid\nwhen the parametric time-dependence of the problem is\nadiabatic with respect to the electron time constants.\nHere we consider a magnetization direction that varies\nslowly in both space and time. The ALDA should be\nsuited to treat magnetization dynamics, since the typical\ntime scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com-\nparedtothethat associatedwith theFermi andexchange\nenergies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition\nmetal ferromagnets.\nIn the ALDA, the system is described by the time-\ndependent effective Schr¨ odinger equation\nˆHALDAΨ(r,t) =i/planckover2pi1∂\n∂tΨ(r,t), (4)\nwhere Ψ( r,t) is the quasiparticle wave function at posi-\ntionrand timet. We consider a generic mean-field elec-\ntronic Hamiltonian that depends on the magnetization\ndirection ˆHALDA[m] and includes the periodic Hartree,\nexchange and correlation potentials and relativistic cor-\nrectionssuchasthe spin-orbitinteraction. Impurityscat-\ntering including magnetic disorder is also represented by\nˆHALDA.The magnetization mis allowed to vary in time\nand space. The total Hamiltonian depends additionally\non the Zeeman energy of the magnetization in external\nHextand dipolar Hdmagnetic fields:\nˆH=ˆHALDA[m]−Ms/integraldisplay\ndrm·(Hext+Hd).(5)\nFor this general Hamiltonian (5), our task is to de-\nduce an expression for the Gilbert damping tensor ˜ α. To\nthis end, from the form of the Landau-Lifshitz-Gilbert\nequation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in\ntime. Such an expansion is valid provided the adiabatic\nmagnetization precession frequency is much less than the\nexchange splitting ∆ or the spin-orbit energy which gov-\nerns spin relaxation of electrons. We discuss first dissi-\npation in the LLG equation and subsequently compare\nit with the expressions from scattering theory of electron\ntransport. This leads to a recipe to describe dissipation\nby first principles. Finally, we discuss the connection to\nthe Kubo linear response formalism and prove that the\ntwo formulations are identical in linear response.\nIII. DISSIPATION AND\nLANDAU-LIFSHITZ-GILBERT EQUATION\nThe energy dissipation can be obtained from the solu-\ntion of the LLG Eq. (1) as\n˙E=−Ms/integraldisplay\ndr[˙ m(r,t)·Heff(r,t)] (6)\n=−Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7)\nThescatteringtheoryofmagnetizationdissipationcanbe\nformulated for arbitrary spatiotemporal magnetization\ntextures. Much insight can be gained for certain special\ncases. In small particles or high magnetic fields the col-\nlective magnetization motion is approximately constant\nin space and the “macrospin” model is valid in which\nall spatial dependences are disregarded. We will also\nconsider special magnetization textures with a dynamics\ncharacterized by a number of dynamic (soft) collective\ncoordinates ξa(t) counted by a:20,21\nm(r,t) =mst(r;{ξa(t)}), (8)\nwheremstis the profile at t→ −∞.This representation\nhas proven to be very effective in handling magnetiza-\ntion dynamics of domain walls in ferromagnetic wires.\nThe description is approximate, but (for few variables)\nit becomes exact in special limits, such as a transverse\ndomain wall in wires below the Walker breakdown (see\nbelow); it becomes arbitrarily accurate by increasing the\nnumber of collective variables. The energy dissipation to\nlowest (quadratic) order in the rate of change ˙ξaof the\ncollective coordinates is\n˙E=−/summationdisplay\nab˜Γab˙ξa˙ξb, (9)\nThe (symmetric) dissipation tensor ˜Γabreads4\n˜Γab=Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′∂mst(r)\n∂ξaα[m](r,r′)·∂mst(r′)\n∂ξb. (10)\nThe equation of motion of the collective coordinates un-\nder a force\nF=−∂Ω\n∂ξ(11)\nare20,21\n˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12)\nintroducing the antisymmetric and time-independent gy-\nrotropic tensor:\n˜ηab=Ms\nγ/integraldisplay\ndrmst(r)·/bracketleftbigg∂mst(r)\n∂ξa×∂mst(r)\n∂ξb/bracketrightbigg\n.(13)\nWe show below that Fand˜Γ can be expressed in terms\nof the scattering matrix. For our subsequent discussions\nit is necessary to include a fluctuating force f(t) (with\n/an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21.\nFrom Eq. (3) if follows the time correlation of fis white\nand obeys the fluctuation-dissipation theorem:\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14)\nIn the following we illustrate the collective coordinate\ndescription of magnetization textures for the macrospin\nmodel and the Walker model for a transverse domain\nwall. The treatment is easily extended to other rigid\ntextures such as magnetic vortices.\nA. Macrospin excitations\nWhen high magnetic fields are applied or when the\nsystem dimensions are small the exchange stiffness dom-\ninates. In both limits the magnetization direction and\nits low energy excitations lie on the unit sphere and its\nmagnetization dynamics is described by the polar angles\nθ(t) andϕ(t):\nm= (sinθcosϕ,sinθsinϕ,cosθ).(15)\nThe diagonal components of the gyrotropic tensor vanish\nby (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal\ncomponents are\nηθϕ=MsV\nγsinθ=−ηϕθ. (16)\nVis the particle volume and MsVthe total magnetic\nmoment. We now have two coupled equations of motion\nMsV\nγ˙ϕsinθ−∂Ω\n∂θ−/parenleftBig\n˜Γθθ˙θ+˜Γθϕ˙ϕ/parenrightBig\n= 0,(17)\n−MsV\nγ˙θsinθ−∂Ω\n∂ϕ−/parenleftBig\n˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig\n= 0.The thermodynamic potential Ω determines the ballistic\ntrajectories of the magnetization. The Gilbert damping\ntensor˜Γabwill be computed below, but when isotropic\nand local,\n˜Γ =˜1δ(r−r′)Msα/γ, (18)\nwhere˜1 is a unit matrix in the Cartesian basis and α\nis the dimensionless Gilbert constant, Γ θθ=MsVα/γ,\nΓθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ.\nB. Domain Wall Motion\nWe focus on a one-dimensional model, in which the\nmagnetization gradient, magnetic easy axis, and external\nmagnetic field point along the wire ( z) axis. The mag-\nnetic energy of such a wire with transverse cross section\nScan be written as22\nΩ =MsS/integraldisplay\ndzφ(z), (19)\nin terms of the one-dimensional energy density\nφ=A\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−Hamz+K1\n2/parenleftbig\n1−m2\nz/parenrightbig\n+K2\n2m2\nx,(20)\nwhereHais the applied field and Ais the exchange stiff-\nness. Here the easy-axis anisotropy is parametrized by\nan anisotropy constant K1. In the case of a thin film\nwire, there is also a smaller anisotropy energy associated\nwith the magnetization transverse to the wire governed\nbyK2. In a cylindrical wire from a material without\ncrystal anisotropy (such as permalloy) K2= 0.\nWhen the shape of such a domain wall is pre-\nserved in the dynamics, three collective coordinates\ncharacterize the magnetization texture: the domain\nwall position ξ1(t) =rw(t), the polar angle ξ2(t) =\nϕw(t), and the domain wall width λw(t). We con-\nsider a head-to-head transverse domain wall (a tail-\nto-tail wall can be treated analogously). m(z) =\n(sinθwcosϕw,sinθwsinϕw,cosθw), where\ncosθw= tanhrw−z\nλw(21)\nand\ncscθw= coshrw−z\nλw(22)\nminimizes the energy (20) under the constraint that the\nmagnetization to the far left and right points towardsthe5\ndomain wall. The off-diagonal elements are then ˜ ηrl=\n0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20)\nreduces to\nΩ =MsS/bracketleftbig\nA/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig\n.\n(23)\nDisregarding fluctuations, the equation of motion Eq.\n(12) can be expanded as:\n2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw,\n(24)\n−2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25)\nA/λ2\nw+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw,\n(26)\nwhereαab=γΓab/MsS.\nWhen the Gilbert dampingtensorisisotropicandlocal\nin the basis of the Cartesian coordinates, ˜Γ =˜1δ(r−\nr′)Msα/γ\nαrr=2α\nλw;αϕϕ= 2αλw;αλλ=π2α\n6λw.(27)\nwhereas all off-diagonal elements vanish.\nMost experiments are carried out on thin film ferro-\nmagnetic wires for which K2is finite. Dissipation is es-\npecially simple below the Walker threshold, the regime\nin which the wall moves with a constant drift velocity,\n˙ϕw= 0 and23\n˙rw=−2γHa/αrr. (28)\nThe Gilbert damping coefficient αrrcan be obtained di-\nrectly from the scattering matrix by the parametric de-\npendence of the scattering matrix on the center coordi-\nnate position rw. When the Gilbert damping tensor is\nisotropic and local, we find ˙ rw=λwγHa/α. The domain\nwall width λw=/radicalbig\nA/(K1+K2cos2ϕw) and the out-\nof-plane angle ϕw=1\n2arcsin2γHa/αK2. At the Walker-\nbreakdownfield ( Ha)WB=αK2/(2γ) the sliding domain\nwall becomes unstable.\nIn a cylindrical wire without anisotropy, K2= 0,ϕwis\ntime-dependent and satisfies\n˙ϕw=−(2+αϕr)\nαϕϕ˙rw (29)\nwhile\n˙rw=2γHa\n2/parenleftBig\n2+αϕr\nαϕϕ/parenrightBig\n+αrr. (30)\nFor isotropic and local Gilbert damping coefficients,22\n˙rw\nλw=αγHa\n1+α2. (31)\nInthe nextsection, weformulatehowthe Gilbert scatter-\ning tensor can be computed from time-dependent scat-\ntering theory.IV. SCATTERING THEORY OF MESOSCOPIC\nMAGNETIZATION DYNAMICS\nScattering theory of transport phenomena24has\nproven its worth in the context of magnetoelectronics.\nIt has been used advantageously to evaluate the non-\nlocal exchange interactions multilayers or spin valves,11\nthe giantmagnetoresistance,25spin-transfertorque,9and\nspin pumping.10We first review the scattering theory\nof equilibrium magnetic properties and anisotropy fields\nand then will turn to non-equilibrium transport.\nA. Conservative forces\nConsidering only the electronic degrees of freedom in\nour model, the thermodynamic (grand) potential is de-\nfined as\nΩ =−kBTlnTre−(ˆHALDA−µˆN), (32)\nwhileµis the chemical potential, and ˆNis the number\noperator. The conservative force\nF=−∂Ω\n∂ξ. (33)\ncan be computed for an open systems by defining a scat-\nteringregionthat isconnectedby idealleadstoreservoirs\nat common equilibrium. For a two-terminal device, the\nflow of charge, spin, and energy between the reservoirs\ncan then be described in terms of the S-matrix:\nS=/parenleftbigg\nr t′\nt r′/parenrightbigg\n, (34)\nwhereris the matrix of probability amplitudes of states\nimpinging from and reflected into the left reservoir, while\ntdenotes the probability amplitudes of states incoming\nfrom the left and transmitted to the right. Similarly,\nr′andt′describes the probability amplitudes for states\nthat originate from the right reservoir. r,r′,t, andt′are\nmatricesin the space spanned by eigenstates in the leads.\nWe areinterested in the free magnetic energymodulation\nby the magnetic configuration that allows evaluation of\nthe forces Eq. (33). The free energy change reads\n∆Ω =−kBT/integraldisplay\ndǫ∆n(ǫ)ln/bracketleftBig\n1+e(ǫ−µ)/kBT/bracketrightBig\n,(35)\nwhere ∆n(ǫ)dǫis the change in the number of states at\nenergyǫand interval dǫ, which can be expressed in terms\nof the scattering matrix45\n∆n(ǫ) =−1\n2πi∂\n∂ǫTrlnS(ǫ). (36)\nCarrying out the derivative, we arrive at the force\nF=−1\n2πi/integraldisplay\ndǫf(ǫ)Tr/parenleftbigg\nS†∂S\n∂ξ/parenrightbigg\n,(37)6\nwheref(ǫ) is the Fermi-Dirac distribution function with\nchemical potential µ. This established result will be re-\nproducedandgeneralizedtothedescriptionofdissipation\nand fluctuations below.\nB. Gilbert damping as energy pumping\nHere we interpretGilbert damping asan energypump-\ning process by equating the results for energy dissipa-\ntion from the microscopic adiabatic pumping formalism\nwith the LLG phenomenology in terms of collective co-\nordinates, Eq. (9). The adiabatic energy loss rate of a\nscattering region in terms of scattering matrix at zero\ntemperature has been derived in Refs. 26,27. In the ap-\npendices, we generalize this result to finite temperatures:\n˙E=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ,t)\n∂t∂S†(ǫ,t)\n∂t/bracketrightbigg\n.(38)\nSince we employ the adiabatic approximation, S(ǫ,t) is\nthe energy-dependent scattering matrix for an instanta-\nneous (“frozen”)scattering potential at time t. In a mag-\nnetic system, the time dependence arises from its magne-\ntization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the\ncollective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)})\n∂S[m(t)]\n∂t≈/summationdisplay\na∂S\n∂ξa˙ξa, (39)\nwhere the approximate sign has been discussed in the\nprevious section. We can now identify the dissipation\ntensor (10) in terms of the scattering matrix\nΓab=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂ξa∂S†(ǫ)\n∂ξb/bracketrightbigg\n.(40)In the macrospin model the Gilbert damping tensor can\nthen be expressed as\n˜αij=γ/planckover2pi1\n4πMs/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂mi∂S†(ǫ)\n∂mj/bracketrightbigg\n,(41)\nwheremiis a Cartesian component of the magnetization\ndirection..\nC. Gilbert damping and fluctuation-dissipation\ntheorem\nAt finite temperatures the forces acting on the mag-\nnetization contain thermal fluctuations that are related\nto the Gilbert dissipation by the fluctuation-dissipation\ntheorem, Eq. (14). The dissipation tensor is therefore ac-\ncessible via the stochastic forces in thermal equilibrium.\nThe time dependence of the force operators\nˆF(t) =−∂ˆHALDA(m)\n∂ξ(42)\nis caused by the thermal fluctuations of the magneti-\nzation. It is convenient to rearrange the Hamiltonian\nˆHALDAinto an unperturbed part that does not de-\npend on the magnetization and a scattering potential\nˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering\nwave functions of the leads, the force operator reads\nˆF=−/integraldisplay\ndǫ/integraldisplay\ndǫ′/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫ′β/an}b∇acket∇i}htˆa†\nα(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43)\nwhere ˆaβannihilates an electron incident on the scatter-\ning region, βlabels the lead (left or right) and quantum\nnumbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ-\nated scatteringeigenstateat energy ǫ′. We takeagainthe\nleft and rightreservoirsto be in thermal equilibrium with\nthe same chemical potentials, such that the expectation\nvalues\n/angbracketleftbig\nˆa†\nα(ǫ)ˆaβ(ǫ′)/angbracketrightbig\n=δαβδ(ǫ−ǫ′)f(ǫ).(44)\nTherelationbetweenthematrixelementofthescattering\npotential and the S-matrix\n/bracketleftbigg\nS†(ǫ)∂S(ǫ)\n∂ξ/bracketrightbigg\nαβ=−2πi/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as\nwell as unitarity of the S-matrix,S†S= 1. Taking these\nrelationsintoaccount,the expectationvalueof ˆFisfound\nto be Eq. (37). We now consider the fluctuations in the\nforceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation\nvalues\n/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)ˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n−/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)/angbracketrightbig/angbracketleftbig\nˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n=δα1β2δ(ǫ1−ǫ′\n2)δβ1α2δ(ǫ′\n1−ǫ2)f(ǫ1)[1−f(ǫ2)],\n(46)\nwhere we invoked Wick’s theorem. Putting everything7\ntogether, we finally find\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47)\nwhere Γ abhas been defined in Eq. (40). Comparing with\nEq. (14), we conclude that the dissipation tensor Γ ab\ngoverningthe fluctuationsisidentical tothe oneobtained\nfrom the energy pumping, Eq. (40), thereby confirming\nthe fluctuation-dissipation theorem.\nV. KUBO FORMULA\nThe quality factor of the magnetization dynamics of\nmost ferromagnets is high ( α/lessorsimilar0.01). Damping can\ntherefore often be treated as a small perturbation. In\nthe presentSectionwedemonstratethat the dampingob-\ntained from linear response (Kubo) theory agrees28with\nthat ofthe scattering theory ofmagnetization dissipation\nin this limit. At sufficiently low temperatures or strong\nelastic disorder scattering the coupling to phonons may\nbe disregarded and is not discussed here.\nThe energy dissipation can be written as\n˙E=/angbracketleftBigg\ndˆH\ndt/angbracketrightBigg\n, (48)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non-\nequilibrium state. We are interested in the adiabatic\nresponse of the system to a time-dependent perturba-\ntion. In the adiabatic (slow) regime, we can at any time\nexpand the Hamiltonian around a static configuration at\nthe reference time t= 0,\nˆH=ˆHst+/summationdisplay\naδξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r).(49)\nThe static part, ˆHst, is the Hamiltonian for a magneti-\nzation for a fixed and arbitrary initial texture mst, as,\nwithout loss of generality, described by the collective\ncoordinates ξa. Since we assume that the variation of\nthe magnetization in time is small, a linear expansion in\nterms of the small deviations of the collective coordinate\nδξi(t) is valid for sufficiently short time intervals. We can\nthen employ the Kubo formalism and express the energy\ndissipation as\n˙E=/summationdisplay\naδ˙ξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r),(50)\nwhere the expectation value of the out-of-equilibrium\nconservative force\n/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear\nin the perturbed magnetization direction:\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n(t) =/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/summationdisplay\nb/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t′).\n(52)\nHere, we introduced the retarded susceptibility\nχab(t−t′) =−i\n/planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig\n∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig\nst,(53)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions\nof the static configuration. Focussing on slow modula-\ntions we can further simplify the expression by expand-\ning\nδξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54)\nso that\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n=/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t)+\n/integraldisplay∞\n−∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55)\nThe first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞\n−∞dt′χab(t−t′)δξb(t),correspond to the energy vari-\nation with respect to a change in the static magnetiza-\ntion. These terms do not contribute to the dissipation\nsince the magnetic excitations are transverse, ˙ m·m= 0.\nOnly the last term in Eq. (55) gives rise to dissipation.\nHence, the energy loss reduces to29\n˙E=i/summationdisplay\nijδ˙ξaδ˙ξb∂χS\nab\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0, (56)\nwhereχS\nab(ω) =/integraltext∞\n−∞dt[χab(t)+χba(t)]eiωt/2. The\nsymmetrized susceptibility can be expanded as\nχS\nab=/summationdisplay\nnm(fn−fm)\n2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b)\n/planckover2pi1ω+iη−(ǫn−ǫm),\n(57)\nwhere|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith\neigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri-\nbution function at energy ǫ, andηis a positive infinites-\nimal constant. Therefore,8\ni/parenleftbigg∂χS\nab\n∂ω/parenrightbigg\nω=0=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58)\nand the dissipation tensor\nΓab=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59)\nWe nowdemonstratethatthe dissipationtensorobtained\nfrom the Kubo linear response formula, Eq. (59), is\nidentical to the expression from scattering theory, Eq.\n(40), following the Fisher and Lee proof of the equiv-\nalence of linear response and scattering theory for the\nconductance.36\nThe static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be\ndecomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m\nand a scattering potential ˆV(ξ). The eigenstates of ˆH0\nare denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=±\ndenotes the longitudinal propagation direction along the\nsystem (say, to the left or to the right), and qa trans-\nverse quantum number determined by the lateral con-\nfinement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The\noutgoing (+) and incoming ( −) scattering eigenstates\nof the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n,\nwhichform anothercomplete basiswith orthogonalityre-\nlations/angbracketleftBig\nψ(±)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±)\ns′,q′(ǫ′)/angbracketrightBig\n=δs,s′δq,q′δ(ǫ−ǫ′).33These\nwave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n= [1 +\nˆG(±)\nstˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −)\nGreen’s functions read ˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1. By\nexpanding Γ abin the basis of outgoing wave functions,\n|ψ(+)\ns,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes\nΓab=π/summationdisplay\nsq,s′q′/integraldisplay\ndǫ/parenleftbigg\n−∂fs,q\n∂ǫ/parenrightbigg/angbracketleftBig\nψ(+)\ns,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns,q/angbracketrightBig\n, (60)\nwhere wave functions should be evaluated at the energy ǫ.\nLet us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation,\nEq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the\nT-matrix is defined recursively by ˆT=ˆV[1+ˆG(+)\nstˆT]. We then find\n∂ˆT\n∂ξa=/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n.\nThe change in the scattering matrix appearing in Eq. (40) is then\n∂Ss′q′,sq\n∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n|ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig\nψ(−)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig\n. (61)\nSince\n/angbracketleftBig\nψ(−)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay\ns′q′Ssq,s′q′/angbracketleftBig\nψ(+)\ns′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62)\nandSS†= 1, we can write the linear response result,\nEq. (60), as energy pumping (40). This completes our\nproof of the equivalence between adiabatic energy pump-\ningintermsofthe S-matrixandtheKubolinearresponse\ntheory.VI. CONCLUSIONS\nWe have shown that most aspects of magnetization\ndynamics in ferromagnets can be understood in terms of\nthe boundary conditions to normal metal contacts, i.e.\na scattering matrix. By using the established numerical\nmethods to compute electron transport based on scatter-\ning theory, this opens the way to compute dissipation in\nferromagnets from first-principles. In particular, our for-9\nmalism should work well for systems with strong elastic\nscattering due to a high density of large impurity poten-\ntials or in disordered alloys, including Ni 1−xFex(x= 0.2\nrepresents the technologically important “permalloy”).\nThe dimensionless Gilbert damping tensors (41) for\nmacrospin excitations, which can be measured directly\nin terms of the broadening of the ferromagnetic reso-\nnance, havebeen evaluated for Ni 1−xFexalloysby ab ini-\ntiomethods.42Permalloy is substitutionally disordered\nand damping is dominated by the spin-orbit interaction\nin combination with disorder scattering. Without ad-\njustable parameters good agreement has been obtained\nwith the available low temperature experimental data,\nwhich is a strong indication of the practical value of our\napproach.\nIn clean samples and at high temperatures, the\nelectron-phonon scattering importantly affects damping.\nPhonons are not explicitly included here, but the scat-\ntering theory of Gilbert damping can still be used for\na frozen configuration of thermally displaced atoms, ne-\nglecting the inelastic aspect of scattering.12\nWhile the energy pumping by scattering theory has\nbeen applied to described magnetization damping,15it\ncan be used to compute other dissipation phenomena.\nThis has recently been demonstrated for the case of\ncurrent-induced mechanical forces and damping,43with\na formalism analogous to that for current-induced mag-\nnetization torques.13,14\nAcknowledgments\nWe would like to thank Kjetil Hals, Paul J. Kelly, Yi\nLiu, Hans Joakim Skadsem, Anton Starikov, and Zhe\nYuan for stimulating discussions. This work was sup-\nported by the EC Contract ICT-257159 “MACALO,”\ntheNSFunderGrantNo.DMR-0840965,DARPA,FOM,\nDFG, and by the Project of Knowledge Innovation Pro-\ngram(PKIP) of Chinese Academy of Sciences, Grant No.\nKJCX2.YW.W10\nAppendix A: Adiabatic Pumping\nAdiabatic pumping is the current response to a time-\ndependent scattering potential to first order in the time-\nvariation or “pumping” frequency when all reservoirsare\nat the same electro-chemical potential.38A compact for-\nmulation of the pumping charge current in terms of the\ninstantaneous scattering matrix was derived in Ref. 39.\nIn the same spirit, the energy current pumped out of the\nscattering region has been formulated (at zero tempera-\nture) in Ref. 27. Some time ago, we extended the charge\npumping concept to include the spin degree of free-\ndomandascertainedits importancein magnetoelectronic\ncircuits.10More recently, we demonstrated that the en-\nergyemitted byaferromagnetwith time-dependentmag-\nnetizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy\nloss by spin-flip processes inside the ferromagnet15and\ntherefore Gilbert damping. Here we derive the energy\npumping expressions at finite temperatures, thereby gen-\neralizing the zero temperature results derived in Ref. 27\nand used in Ref. 15. Our results differ from an earlier ex-\ntension to finite temperature derived in Ref. 40 and we\npoint out the origin of the discrepancies. The magneti-\nzation dynamics must satisfy the fluctuation-dissipation\ntheorem, which is indeed the case in our formulation.\nWe proceed by deriving the charge, spin, and energy\ncurrentsintermsofthetimedependenceofthescattering\nmatrix of a two-terminal device. The transport direction\nisxand the transverse coordinates are ̺= (y,z). An\narbitrary single-particle Hamiltonian can be decomposed\nas\nH(r) =−/planckover2pi12\n2m∂2\n∂x2+H⊥(x,̺), (A1)\nwhere the transverse part is\nH⊥(x,̺) =−/planckover2pi12\n2m∂2\n∂̺2+V(x,̺).(A2)\nV(̺) is an elastic scattering potential in 2 ×2 Pauli\nspin space that includes the lattice, impurity, and\nself-consistent exchange-correlation potentials, including\nspin-orbit interaction and magnetic disorder. The scat-\nteringregionisattachedtoperfect non-magneticelectron\nwave guides (left α=Land rightα=R) with constant\npotential and without spin-orbit interaction. In lead α,\nthe transverse part of the 2 ×2 spinor wave function\nϕ(n)\nα(x,̺) and its corresponding transverse energy ǫ(n)\nα\nobey the Schr¨ odinger equation\nH⊥(̺)ϕ(n)\nα(̺) =ǫ(n)\nαϕ(n)\nα(̺), (A3)\nwherenis the spin and orbit quantum number. These\ntransverse wave guide modes form the basis for the ex-\npansion of the time-dependent scattering states in lead\nα=L,R:\nˆΨα=/integraldisplay∞\n0dk√\n2π/summationdisplay\nnσϕ(n)\nα(̺)eiσkxe−iǫ(nk)\nαt//planckover2pi1ˆc(nkσ)\nα,(A4)\nwhere ˆc(nkσ)\nαannihilates an electron in mode nincident\n(σ= +) or outgoing ( σ=−) in leadα. The field opera-\ntors satisfy the anticommutation relation\n/braceleftBig\nˆc(nkσ)\nα,ˆc†(n′k′σ′)\nβ/bracerightBig\n=δαβδnn′δσσ′δ(k−k′).\nThe total energy is ǫ(nk)\nα=/planckover2pi12k2/2m+ǫ(n)\nα. In the leads\nthe particle, spins, and energy currents in the transport10\ndirection are\nˆI(p)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†∂ˆΨ\n∂x−∂ˆΨ†\n∂xˆΨ/parenrightBigg\n,(A5a)\nˆI(s)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†σ∂ˆΨ\n∂x−∂ˆΨ†\n∂xσˆΨ/parenrightBigg\n,(A5b)\nˆI(e)=/planckover2pi1\n4mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†H∂ˆΨ\n∂x−∂ˆΨ†\n∂xHˆΨ/parenrightBigg\n+H.c.,\n(A5c)\nwhere we suppressed the time tand lead index α,σ=\n(σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes\nthe trace in spin space. Note that the spin current Is\nflows in the x-direction with polarization vector Is/Is.\nTo avoid dependence on an arbitrary global potential\nshift, it is convenient to work with heat ˆI(q)rather than\nenergy currents ˆI(ǫ):\nˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6)\nwhereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current\nreads41\nˆI(p)\nα=/planckover2pi1\n4πm/integraldisplay∞\n0dkdk′/summationdisplay\nnσσ′(σk+σ′k′)×\nei(σk−σ′k′)xe−i/bracketleftBig\nǫ(nk)\nα−ǫ(nk′)\nα/bracketrightBig\nt//planckover2pi1ˆc†(nk′σ′)\nαˆc(nkσ)\nα.(A7)\nWeareinterestedinthelow-frequencylimitoftheFourier\ntransforms I(x)\nα(ω) =/integraltext∞\n−∞dteiωtI(x)\nα(t). Following Ref.\n41 we assume long wavelengths such that only the inter-\nvals withk≈k′andσ=σ′contribute. In the adiabatic\nlimitω→0 this approach is correct to leading order in\n/planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the\n(current-normalized) operator\nˆc(nσ)\nα(ǫ(nk)\nα) =1/radicalBig\ndǫ(nkσ)\nα\ndkˆc(nkσ)\nα, (A8)\nwhich obey the anticommutation relations\n/braceleftBig\nˆc(nσ)\nα(ǫα),ˆc†(n′σ′)\nβ(ǫβ)/bracerightBig\n=δαβδnn′δσσ′δ(ǫα−ǫβ). (A9)\nThe charge current can be written as\nˆI(c)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\nǫ(n)\nαdǫdǫ′/summationdisplay\nnσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ)\nα(ǫ′)ˆc(nσ)\nα(ǫ). (A10)\nWeoperateinthe linearresponseregimeinwhichapplied\nvoltages and temperature differences as well as the exter-\nnally induced dynamics disturb the system only weakly.\nTransport is then governed by states close to the Fermi\nenergy. We may therefore extend the limits of the en-\nergy integration in Eq. (A10) from ( ǫ(n)\nα,∞) to (−∞\nto∞). We relabel the annihilation operators so that\nˆa(nk)\nα= ˆc(nk)\nα+denotes particles incident on the scattering\nregion from lead αandˆb(nk)\nα= ˆc(nk)\nα−denotes particles\nleavingthe scatteringregionbylead α. Using the Fourier\ntransforms\nˆc(nσ)\nα(ǫ) =/integraldisplay∞\n−∞dtˆc(nσ)\nα(t)eiǫt//planckover2pi1, (A11)\nˆc(nσ)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫˆc(nσ)\nα(ǫ)e−iǫt//planckover2pi1,(A12)\nwe obtain in the low-frequency limit41\nˆI(p)\nα(t) = 2π/planckover2pi1/bracketleftBig\nˆa†\nα(t)ˆaα(t)−ˆb†\nα(t)ˆbα(t)/bracketrightBig\n,(A13)\nwhereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n)\nα}. Analogouscalculations lead\nto the spin current\nˆI(s)\nα= 2π/planckover2pi1/parenleftBig\nˆa†\nασˆaα−ˆb†\nασˆbα/parenrightBig\n(A14)\nand the energy current\nˆI(e)\nα=iπ/planckover2pi12/parenleftBigg\nˆa†\nα∂ˆaα\n∂t−ˆb†\nα∂ˆbα\n∂t/parenrightBigg\n+H.c..(A15)\nNext, we express the outgoing operators ˆb(t) in terms\nof the incoming operators ˆ a(t) via the time-dependent\nscattering matrix (in the space spanned by all waveguide\nmodes, including spin and orbit quantum number):\nˆbα(t) =/summationdisplay\nβ/integraldisplay\ndt′Sαβ(t,t′)ˆaβ(t′).(A16)\nWhen the scattering region is stationary, Sαβ(t,t′) only\ndepends on the relative time difference t−t′, and its\nFourier transform with respect to the relative time is\nenergy independent, i.e.transport is elastic and can11\nbe computed for each energy separately. For time-\ndependent problems, Sαβ(t,t′) also depends on the total\ntimet+t′and there is an inelastic contribution to trans-\nport as well. An electron can originate from a lead with\nenergyǫ, pick up energy in the scattering region and end\nup in the same or the other lead with different energy ǫ′.\nThe reservoirs are in equilibrium with controlled lo-\ncal chemical potentials and temperatures. We insert the\nS-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value\nat thermal equilibrium\n/angbracketleftBig\nˆa†(n)\nα(t2)ˆa(m)\nβ(t1)/angbracketrightBig\neq=δnmδαβfα(t1−t2)/2πℏ,(A17)\nwherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext\ndǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and\nfα(ǫ) is the Fermi-Dirac distribution of electrons with\nenergyǫin theα-th reservoir. We then find\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18)\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)σˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19)\n2π/planckover2pi1/angbracketleftBig\n/planckover2pi1∂tˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2/bracketleftbig\n/planckover2pi1∂tS∗\nαβ(t,t2)/bracketrightbig\nSαβ(t,t1)fβ(t1−t2). (A20)\nNext, we use the Wigner representation (B1):\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫS/parenleftbiggt+t′\n2,ǫ/parenrightbigg\ne−iǫ(t−t′)//planckover2pi1, (A21)\nand by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞\nn=0∂n\ntS(t,ǫ)(t′−t)n/(2nn!), we find\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22)\nand\n/planckover2pi1∂tS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1\n2/planckover2pi1∂t−iǫ/parenrightbigg\nS(t,ǫ). (A23)\nThe factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated\nas\nI(c)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−fα(ǫ)/bracketrightBig\n(A24a)\nI(s)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig\nσ/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)/bracketrightBig\n(A24b)\nI(ǫ)\nα(t) =−1\n4π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S†\nβα(ǫ,t)/parenrightBig/parenleftBig\ne+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n−1\n4π/planckover2pi1/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n,(A24c)\nwhere the adjoint of the S-matrix has elements S†(n′,n)\nβα=S∗(n,n′)\nαβ.\nWe are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration\nover energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext\nαIα,\nwhen the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12\nenergy currents out of the system in a time interval τcan be written compactly as\nI(c)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−f(ǫ)/bracerightbigg\n, (A25a)\nI(s)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg\nσ/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†/bracerightbigg\n, (A25b)\nI(ǫ)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n+1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg/parenleftbigg\n−i/planckover2pi1∂S†\n∂t/parenrightbigg/bracerightbigg\n, (A25c)\nwhere Tr is the trace over all waveguide modes (spin\nand orbital quantum numbers). As shown in Ap-\npendix C the charge pumped into the reservoirs vanishes\nfor a scattering matrix with a periodic time dependence\nwhen,integrated over one cycle:\nI(p)\nout= 0. (A26)\nThis reflects particle conservation; the number of elec-\ntrons cannot build up in the scattering region for peri-\nodic variations ofthe system. We can showthat a similar\ncontribution to the energy current, i.e.the first line in\nEq. (A25c), vanishes, leading to to the simple expression\nI(e)\nout=−i\n2π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg∂S†\n∂t/bracerightbigg\n.\n(A27)\nExpanded to lowest order in the pumping frequency the\npumped spin current (A25b) becomes\nI(s)\nout=1\n2π/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nSS†f−i/planckover2pi1\n2∂S\n∂tS†∂ǫf/parenrightbigg\nσ/bracerightbigg\n(A28)\nThis formula is not the most convenient form to com-\npute the current to specified order. SS†also contains\ncontributions that are linear and quadratic in the pre-\ncession frequency since S(t,ǫ) is theS-matrix for a time-\ndependent problem. Instead, wewouldliketoexpressthe\ncurrent in terms of the frozenscattering matrix Sfr(t,ǫ).\nThe latter is computed for an instantaneous, static elec-\ntronic potential. In our case this is determined by a mag-\nnetization configuration that depends parametrically on\ntime:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex-\npand it to lowest order in the pumping frequency, and\ninsert it into (A28) leads to39\nI(s)\nout=i\n2π/summationdisplay\nβ/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftbigg∂Sfr\n∂tS†\nfrσ/bracerightbigg\n.\n(A29)\nWe evaluate the energy pumping by expanding (A27)\nto second order in the pumping frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\n−ifS∂S†\n∂t−(∂ǫf)1\n2∂S\n∂t∂S†\n∂t/bracerightbigg\n.\n(A30)\nAs a consequence of unitarity of the S-matrix (see Ap-\npendix C), the first term vanishes to second order in the\nprecession frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftBigg\n∂Sfr\n∂t∂S†\nfr\n∂t/bracerightBigg\n,(A31)\nwhere,at this point , we may insert the frozen scattering\nmatrix since the current expression is already propor-\ntional to the square of the pumping frequency. Further-\nmore, since there is no net pumped charge current in\none cycle (and we are assuming reservoirs in a common\nequilibrium), the pumped heat current is identical to the\npumped energy current, I(q)\nout=I(e)\nout.\nOur expression for the pumped energy current (A31)\nagrees with that derived in Ref. 27 at zero temperature.\nOur result (A31) differs from Ref. 40 at finite tempera-\ntures. The discrepancy can be explained as follows. In-\ntegration by parts over time tin Eq. (A27), using\n/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\ni/planckover2pi1∂S\n∂t/bracketrightbigg\nS†= 2/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−2/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†,(A32)\nand the unitarity condition from Appendix C,\n/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†=/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫǫf(ǫ), (A33)13\nthe DC pumped energy current can be rewritten as\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n. (A34)\nNext, we expand this to the second order in the pumping frequency and find\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\nǫf(ǫ)/parenleftbig\nSS†−1/parenrightbig\n−ǫ(∂ǫf)i/planckover2pi1\n2∂S\n∂tS†−ǫ(∂2\nǫf)/planckover2pi12\n8∂2S\n∂t2S†/bracerightbigg\n. (A35)\nThis form of the pumped energy current, Eq. (A35),\nagrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as-\nsumesSS†= 1. Although for the frozen scattering ma-\ntrix,SfrS†\nfr= 1, unitarity does not hold for the Wigner\nrepresentation of the scattering matrix to the second or-\nder in the pumping frequency. ( SS†−1) therefore does\nnot vanish but contributes to leading order in the fre-\nquency to the pumped current, which may not be ne-\nglected at finite temperatures. Only when this term is\nincluded our new result Eq. (A31) is recovered.\nAppendix B: Fourier transform and Wigner\nrepresentation\nThere is a long tradition in quantum theory to trans-\nform the two-time dependence of two-operator correla-\ntion functions such as scattering matrices by a mixed\n(Wigner)representationconsistingofaFouriertransform\nover the time difference and an average time, which has\ndistinct advantages when the scattering potential varies\nslowlyintime.44Inordertoestablishconventionsandno-\ntations, we present here a short exposure how this works\nin our case.\nThe Fourier transform of the time dependent annihi-\nlation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1\nandt2,A=A(t1,t2). The Wigner representation with\nt= (t1+t2)/2 andt′=t1−t2is defined as:\nA(t1,t2) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1)\nA(t,ǫ) =/integraldisplay∞\n−∞dt′A/parenleftbigg\nt+t′\n2,t−t′\n2/parenrightbigg\neiǫt′//planckover2pi1.(B2)\nWe also need the Wigner representation of convolutions,\n(A⊗B)(t1,t2) =/integraldisplay∞\n−∞dt′A(t1,t′)B(t′,t2).(B3)\nBy a series expansion, this can be expressed as44\n(A⊗B)(t,ǫ) =e−i(∂A\nt∂B\nǫ−∂B\nt∂A\nǫ)/2A(t,ǫ)B(t,ǫ) (B4)\nwhich we use in the following section.\nAppendix C: Properties of S-matrix\nHere we discuss some general properties of the two-\npoint time-dependent scattering matrix. Current conser-\nvation is reflected by the unitarity of the S-matrix which\ncan be expressed as\n/summationdisplay\nβn′s′/integraldisplay\ndt′S(α1β)\nn1s1,n′s′(t1,t′)S(α2β)∗\nn2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1)\nPhysically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the\nscattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over\nthe local time variable, this implies (using Eq. (B4))\n1 =/parenleftbig\nS⊗S†/parenrightbig\n(t,ǫ) =e−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ), (C2)\nwhere 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum\nnumbern). Similary, we find\n1 =/parenleftbig\nS†⊗S/parenrightbig\n(t,ǫ) =e+i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S†(t,ǫ)S(t,ǫ). (C3)\nTo second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give\nTr/braceleftbigg∂S\n∂t∂S†\n∂ǫ−∂S\n∂ǫ∂S†\n∂t/bracerightbigg\n= 0 (C4)14\nand\nTr/braceleftbig\nSS†−1/bracerightbig\n= Tr/braceleftbigg∂2S\n∂t2∂2S†\n∂ǫ2−2∂2S\n∂t∂ǫ∂2S†\n∂t∂ǫ+∂2S\n∂ǫ2∂2S†\n∂t2/bracerightbigg\n. (C5)\nFurthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse\nwaveguide modes Y, Eq. (C2) implies\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg\nY/bracerightbigg\n= 0. (C6)\nIntegration by parts with respect to tandǫgives\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S\nt∂ZS†\nǫ/parenrightBig\n/2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg\nY/bracerightbigg\n= 0, (C7)\nwhich can be simplified to\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg/bracketleftbigg\nZ/parenleftbigg\nǫ+i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\nS†(t,ǫ)−Z(ǫ)/parenrightbigg\nY/bracerightbigg\n= 0. (C8)\nSimilarly from (C3), we find\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nS†(t,ǫ)/bracketleftbigg\nZ/parenleftbigg\nǫ−i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\n−1/parenrightbigg\nY/bracerightbigg\n= 0. (C9)\nUsing this result for Y= 1 andZ(ǫ) =f(ǫ) in the\nexpression for the DC particle current (A25a), we see\nthat unitarity indeed implies particle current conserva-\ntion,/summationtext\nαI(c)\nα= 0 for a time-periodic potential. 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In case the system-reservoir coupling break s the time reversal\nsymmetry the magnetic moments perform a damped precession a round an effective field which is\nself-organized by the mutual interaction of the moments. Th e resulting evolution equation has the\nform of the Landau-Lifshitz-Gilbert equation. In case the b ath variables are constant vector fields\nthe moments mfulfill the reversible Landau-Lifshitzequation. Applying Noether’s theorem we find\nconserved quantities under rotation in space and within the configuration space of the moments.\nPACS numbers: 75.78.-n, 11.10.Ef, 75.10.Hk\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nThe dynamics of magnetic systems is described in a wide range of time a nd length scales\nfrom a quantum approach up to a macroscopic thermodynamic acce ss. On a coarse-grained\nmesoscopic level the relevant electronic degrees of freedom are g rouped into effective mag-\nnetic moments. As the consequence the magnetization is characte rized by a spatiotemporal\nvector field m(r,t). Owing to the mutual interaction between the magnetic moments t hey\nperform a precession motion around a local effective field giving rise t o the propagation of\nspin-wave excitations. Due to a system-reservoir coupling the pre cession of the moments\nshould be a damped one. To analyze this situation one has to specify t he coupling between\nthe system and the bath. The most popular approach to incorpora te dissipation is the em-\nbedding of the relevant system into a quantum-statistical environ ment which is assumed to\nremain in thermal equilibrium. The reservoir is often represented by harmonic oscillators or\nspin moments which are analyzed by path integral techniques [1–3]. A specification of the\npath integral approach for spin systems can be found in [4–7]. A mor e generic description\nconcerning dissipative semiclassical dynamics is presented in [8]. Altho ugh the application\nof path integrals can be considered as an intuitive formalism analytica l calculations are often\nimpossible and numerical schemes are necessary. In the present p aper we propose an alter-\nnative way to include dissipative effects for mesoscopic magnetic sys tems. On this level the\nanalysis of magnetodynamics is performed properly by applying the L andau-Lifshitz-Gilbert\nequation designated as LLG [9, 10]. A comprehensive survey of magn etization dynamics is\ngiven in [11]. Our investigation can be grouped in the effort to underst and dissipative mech-\nanisms in magnets. So, a non-linear dissipative model for magnetic sy stems was discussed in\n[12]. On the relation between fluctuation-dissipation theorems and d amping terms like that\none occurring in the LLG was reported in [13]. The dynamical respons e of ferromagnetic\nshape memory alloy actuators can be modeled by means of a dissipativ e Euler-Lagrange\nequation as performed in [14]. Likewise, the pinning of magnetic domain walls in multifer-\nroics is discussed in terms of the EL equations in [15]. An alternative a nsatz is introduced in\n[16], where a Lagrangian density is obtained based on a projection on to the complex plane.\nThe procedure gives rise to a dynamical equation which is equivalent t o the Landau-Lifshitz\nequation. Different to the mentioned approaches the present pap er is aimed to derive an\nequation of motion for a magnetic system on a mesoscopic scale unde r the influence of a\n2bath which likewise consists of mesoscopic moments. Following this idea we propose a La-\ngrangian comprising both fields, m(r,t) as the system variables and σas the bath variables.\nThe bath becomes dynamically active by the coupling to the system. I n case the coupling\nbetween system and reservoir breaks the time reversal symmetr y the motion of the moments\nm(r,t) is damped. The Lagrangian is modified in such a manner that dissipatio n can occur.\nII. THE LAGRANGIAN\nAs indicated we are interested to construct a Lagrangian describin g the motion of a magne-\ntization vector field within a bath of spins. This reservoir should influe nce the measurable\nmagnetization due to the mutual interaction. Let us formulate the general assumptions for\nthe underlying model. The magnetic order is originated by single magne tic atoms which\noccupy equivalent crystal positions. Here we refer to a continuou s description in terms of\na field vector denoted as m(r,t). Because the ferromagnet is considered below the Curie\ntemperature a sufficient number of microscopic spins preferring a p arallel alignment are in-\ncluded in m, i.e. the effective magnetic moment is given by m(r,t) =/summationtext\niµiwhere the\nsum is extended over all microscopic moments within a small volume aro und the spatial\ncoordinate rat timet. As each axial vector the moment fulfills m(−t) =−m(t). The bath\nin which the moments are embedded consists likewise of mesoscopic sp ins. They are denoted\nasσand are also composed of microscopic moments ηi. This bath moments which play\nthe role of ’virtual’ moments are also axial vectors changing their sig n by time inversion.\nA further new aspect is that the coupling between the real and the virtual moments is not\nassumed to be weak. As the result the complete system consists of two subsystems. One of\nthem abbreviated as L1 is occupied exclusively by the real spins with t he moments mand\nthe other one denoted as L2 is occupied by the bath spins σ. The situation is illustrated in\nFIG. 1. Now let us introduce the action\nS[{qα}] =/integraldisplay\ndt/integraldisplay\nd3xL[{qα}], (1)\nwhere the set {qα}consists of the set of both moments σandm. The Lagrange density\ncomprises three terms\nL[m,˙m,∇m;σ,∇σ] =L(m)[m,˙m,∇m]+L(σ)[∇σ]+L(mσ)[σ,˙m], (2)\n3µi−1 µi µi+1 ηj−k ηj ηj+1 ηj+1+kJµηJηηJµµ\nFIG. 1. (Color online) Schematic illustration of the basic m odel. The red spins represent the\nmagnetic moments µiand refer to the lattice L1 introduced in the text. The green s pin vectors\nηibuild the bath lattice L2. Interactions are possible betwee n theµiandηj,µiandµjandηi\nandηj. The respective coupling strengths correspond to the coupl ing parameters in Eqs. (3)-(5)\nas follows: Jµη↔J(mσ),Jµµ↔J(m)andJηη↔J(σ).\nwhereL(m)indicates the Lagrangian of the magnetic system, L(σ)represents the reservoir\nand the interaction term is denoted as L(mσ). To be more specific the magnetic moments of\nthe system interact via exchange coupling defined by the Lagrangia n\nL(m)=1\n2J(m)\nαβ∂mν\n∂xα∂mν\n∂xβ+Aν(m) ˙mν, (3)\nwhereJ(m)\nαβisthecouplingparameter, diagonalintheisotropiccase. Thefirstt ermrepresents\nthe energy density of the magnetic system. Because we are not co nsidering the acceleration\nof magnetic moments a term of the order ˙m2is missing. Moreover, the magnetic moments\nperformaprecessionaroundaneffective magneticfield, whichisself -organizedbythemutual\ninteraction. Thereforethevectorpotential Adependsonthemoments, i. e. A=A(m(r,t)).\nThe coupling has the same form as the minimal coupling in electrodynam ics. The bath\nLagrangian is defined in a similar manner as\nL(σ)=1\n2J(σ)\nαβ∂σν\n∂xα∂σν\n∂xβ, (4)\nwith the coupling constant J(σ)\nαβ. Eventually, the interacting part between system and bath\nis written as\nL(mσ)=J(mσ)\nαβ∂mν\n∂xα∂σν\n∂xβ+Bν(σ) ˙mν, (5)\n4with the coupling strength J(mσ)\nαβ. The second term is constructed in the same manner as\nin Eq. (3), where the potential B(σ) will be specified below, see Eq. (8). The dynamics\nof the bath variable σremains unspecified for the present, i.e. the Lagrangian does not\ninclude a term of the form ∝˙σ. Owing to the constraint, introduced in the next section,\nthe dynamically passive bath is sensitive to a change of the system va riablesmin such a\nmanner that small variations of the system variables mare related to small variations of\nσ. This procedure leads to a coupling between bath and system so tha t the time reversal\nsymmetry is broken.\nIII. RELATION TO THE LANDAU-LIFSHITZ-GILBERT EQUATION\nIn this section we find the equation of motion for the magnetization m(r,t) from Eq. (2)\ncombined with Eqs. (3)-(5). Using the principle of least action it follow s\n/bracketleftbigg∂L\n∂σβ−∂\n∂xα∂L\n∂/parenleftBig\n∂σβ\n∂xα/parenrightBig/bracketrightbigg\nδσβ+/bracketleftbigg∂L\n∂mβ−∂\n∂t∂L\n∂˙mβ−∂\n∂xα∂L\n∂/parenleftBig\n∂mβ\n∂xα/parenrightBig/bracketrightbigg\nδmβ= 0,(6)\nwhereδmβandδσβare the small variations which drive the value for the action out of th e\nstationary state. In general, one derives a system of coupled par tial differential equations.\nHowever, to proceed further let us impose a constraint on the sys tem. A small variation of\nσβshould be related to a small variation of mβ. Thus, we make the ansatz\nδσβ=−κδmβ,withκ= const>0. (7)\nNotice that this condition should be valid only locally but not globally. Ins ofar Eq. (7) is\ncomparable to an anholonom condition in mechanics. Moreover relatio n (7) is in accordance\nwith thebehavior of themoments mandσunder timeinversion. Physically the last relation\nmeans that the bath reacts to a change of the system only tempor arily. Because the system-\nreservoir coupling should typically break the time reversal symmetr y the expansion of the\nfunctionBν(σ) in terms of σincludes only odd terms. In lowest order we get from Eq. (5)\nBν(σ) =−cσν,withc= const. (8)\nDue to Eqs. (7) and (8) the second term in Eq. (5) is of the form ∝σ·˙m. Such a term is not\ninvariant under time reversal symmetry t→ −t. As demonstrated below the broken time\n5inversion invariance gives rise to damping effects. Inserting Eqs. (7 ) and (8) into Eq. (2)\nand performing the variation according to Eq. (6) we get\n0 =/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν+c˙σβ+κc˙mβ\n−[J(m)−κJ(mσ)]∇2mβ−[J(mσ)−κJ(σ)]∇2σβ.(9)\nHere we have assumed for simplicity that all coupling tensors Jare diagonal: Jαβ=Jδαβ.\nThe first term on the right hand side in Eq. (9) reminds of the field str ength tensor in\nelectrodynamics [17]. Thus, we rewrite\n/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν≡Fβν˙mν=/bracketleftBig\n˙m×(∇m×A(m))/bracketrightBig\nβ. (10)\nAs mentioned above the vector function A(m) is regarded as vector potential which depends\non space-time coordinates via the magnetic moment m(r,t). In vector notation the last\nequation reads\n˙m×(∇m×A) = [J(m)−κJ(mσ)]∇2m−κc˙m−c˙σ+[Jmσ−κJ(σ)]∇2σ.(11)\nIf one is interested in weak excited states of a ferromagnet it is rea sonable to assume that\nthe direction of the magnetization in space changes slowly while its abs olute value is fixed,\nthat ism2= 1. Without loss of generality we have set the amplitude of mto unity. In order\nto proceed it is necessary to specify the condition which should be fu lfilled by the function\nA(m). Having in mind the LLG then we make the ansatz ∇m×A(m) =gm,g= const.\nBased on these assumptions we get from Eq. (11)\n∂m\n∂t=1\ng/parenleftBigg\nm×Heff/parenrightBigg\n−κc\ng/parenleftBigg\nm×∂m\n∂t/parenrightBigg\n. (12)\nHere the effective field is given by the expression\nHeff=/parenleftBig\nJ(m)−κJ(mσ)/parenrightBig\n∇2m−c∂σ\n∂t+/parenleftBig\nJ(mσ)−κJ(σ)+/parenrightBig\n∇2σ. (13)\nEq. (12) is nothing else than the Gilbert equation [10] by relating the prefactors as follows\nγ=−1\ng, α=−κc\ng=κcγ, (14)\nwhereγandαare the gyromagnetic ratio and the Gilbert damping parameter, res pectively.\nSince bothparameters arepositive quantities itfollows that g<0aswell as κc>0. Further,\n6Eq. (12) can be converted into the form of the equivalent and widely used Landau-Lifshitz-\nGilbert equation which reads\n∂m\n∂t=−γ\n(1+α)2(m×Heff)−αγ\n(1+α2)/bracketleftBig\nm×(m×Heff)/bracketrightBig\n, (15)\nBoth quantities γandαare still related to the model parameters by the expressions in\nEq. (14) whereas the the effective field Heffis given by Eq. (13). Now we want to analyze\nthis expression and in particular, to assign a physical meaning to the more or less ad hoc\nintroduced quantity σ. In doing so one can distinguish four different cases:\n(i) The bath is not included which corresponds formally to σis a constant vector depend-\ning neither on coordinates nor on time. Then obviously all derivatives with respect to the\ncoordinates and the time of σdisappear in Eq. (13) and consequently, the set of {qα}in\nEq. (1) does not include σ. From here we conclude that the variation fulfills δσ= 0 in\nEq. (6) which can be easily realized setting κ= 0, cf. Eq. (7). Thus, the effective field\nin Eq. (13) comprises the pure exchange interaction J(m)between the magnetic moments\nand the damping term in Eq. (12) is absent due to α= 0 in Eq. (14). A constant bath\nfieldσlead to the Landau-Lifshitz equation in the exchange interaction ap proach without\ndamping, compare [18]. It describes the precession of magnetic mom ents of an effective field\nwhich is self-organized by the mutual interaction of the moments.\n(ii)σ=σ(t) depends only on the time and not on the spatial coordinates. Rega rding\nEq. (13) the effective field is modified by two additional contributions , namely one propor-\ntional to ∇2m, originated in the exchange interaction of the magnetic moments, a nd the\nother one ∝˙σ. The latter one could be associated with an external time dependen t field\nor, ifσpoints into a fixed direction, gives rise to magnetic anisotropy. In th at case the\nanisotropy axis is spatially constant but the amount of the anisotro py is changing in time.\nSuch a situation could be realized for instance when the ferromagne tic sample is excited by\nthe irradiation with electromagnetic waves. As already mentioned th e exchange coupling\nJ(m)is supplemented by a term −κJ(mσ). In this manner the exchange interaction is influ-\nenced by the coupling between mandσalthough the spatial dependence of σis not taken\ninto account explicitly.\n(iii)σ=σ(r) depends only on the spatial coordinates and not on the time. In th is case we\nfirst recognize that the coupling strength J(m)in the term ∝ ∇2mis influenced in the same\nmanner as in case when σ=σ(t), see the previous point. Different to the former cases the\n7expression ∝ ∇2σbecomes important for the effective field in Eq. (13). The appearan ce of\nthis term suggests that spatial inhomogeneities of the surroundin gs of the magnetic system\nrepresented by mhave to be incorporated into the effective field. It seems to be reas onable\nthat the origin of this term is an inherent one and should not be led bac k to external fields.\nAs possible sources we have in mind local varying fields like inner and out er demagnetization\nfields as well as accessible fields created for instance by different loc al temperatures.\n(iv)σ=σ(r,t) isthemost general case. Then external aswell asinternal fields arecaptured\nin the model. Thus, the effective field in Eq. (13) can be rewritten as\nHeff(r,t) =Hexch(r)+h(r,t), (16)\nwhereHeffconsists of two parts. The term Hexch= (J(m)−κJ(mσ))∇2mis due to the\nexchange interaction between the magnetic moments whereas h(r,t) represents other possi-\nble influences as discussed under the points (ii) and (iii). The function his related to the\nquantity σby\nh(r,t) =−c∂σ(r,t)\n∂t+/bracketleftbig\nJ(mσ)−κJ(σ)/bracketrightbig\n∇2σ(r,t). (17)\nRemark that the formerly introduced quantity σis related to the physically relevant effec-\ntive field by the first derivation with respect to the time and the seco nd derivation with\nrespect to the spacial coordinates via Eq. (17). This equation is an inhomogeneous diffusion\nequation which can be generally solved by means of the expansion into Fourier series and the\nassumption of accurate initial and boundary conditions which depen d on the actual physical\nproblem.\nIV. SYMMETRY AND CONSERVATION\nAfter regarding the special example of the LLG we proceed with the investigation of more\ngeneral aspects. The Lagrangian density allows to discuss the beh avior under space-time\ndependent group transformation. For this purpose we apply Noet her’s theorem [19] to our\nmodel. To be more precise we consider the conservation equation [20 ]\n∂\n∂Xα/bracketleftbigg/parenleftBig\nLδαβ−∂L\n∂(∂αΨγ)∂βΨγ/parenrightBig\n∆Xβ+∂L\n∂(∂αΨγ)∆Ψγ/bracketrightbigg\n= 0. (18)\nHere, the expression in the square brackets are the components of the Noether current Iα.\nThe term∂/∂Xαin front ofIαshould be interpreted as an implicit derivative with respect\n8to time and three spatial coordinates. The symmetry operations ∆ Xαand ∆Ψ αwill be\nspecified below. With regard to the Lagrangian in Eq. (2) we introduc e the components\nΨα= (mx,my,mz,σx,σy,σz) and their partial derivatives with respect to the independent\nvariables∂βΨα=∂Ψα/∂Xβ. Since we examine an Euclidean field theory a distinction\nbetween upper and lower indices is not necessary. Eq. (18) can be r ewritten by using\nEq. (6). This yields\n∂\n∂tL∆t+∂\n∂xαL∆xα+∂L\n∂Ψα/parenleftbig\n∆Ψα−∂\n∂tΨα∆t−∂\n∂xβΨα∆xβ/parenrightbig\n= 0. (19)\nIn this equation we distinguish between the time and space variables tandxαexplicitly.\nEq. (19) is the basis for the application of the following symmetry ope rations. Now we study\nthe rotation around a certain axis as a relevant one. Here we select for instance the z-axis.\nPerforming a rotation in coordinate space with the infinitesimal angle ∆Θ the change of the\nxandy-coordinates obeys\n∆t= 0,∆xα= ∆Rαβxβ,∆R=\n0 ∆Θ\n−∆Θ 0\n. (20)\nIn the same manner one can perform the rotation in the configurat ion space of the moments\nmandσsymbolized by the before introduced vector Ψ α={mx,my,mz,σx,σy,σz}. The\ntransformation reads ∆Ψ α= ∆Sαβ(∆Φ)Ψ β, where the rotation matrix is a 6 ×6-matrix\ndetermined by the rotation angle ∆Φ. Because both rotations in coo rdinate space and\nconfiguration space, respectively, are in general independent fr om each other we find two\nconserved quantities. Using Eq. (19) it results\nˆDzL= 0,ˆΓzL= 0. (21)\nHere the two operators ˆDzandˆΓzare expressed by\nˆDz=ˆLz−/parenleftBig\nˆLzψα/parenrightBig∂\n∂Ψα,\nˆΓz=ˆS(m)\nz+ˆS(σ)\nz.(22)\nThe quantity ˆLzis the generator of an infinitesimal rotation around the z-axis in the coor-\ndinate space\nˆLz=y∂\n∂x−x∂\n∂y, (23)\n9and therefore, it is identical with the angular momentum operator. The other quantities\nˆS(m)\nzandˆS(σ)\nzare the corresponding generators in the configuration space of t he moments.\nThey are defined as\nˆS(m)\nz=my∂\n∂mx−mx∂\n∂my,\nˆS(σ)\nz=σy∂\n∂σx−σx∂\n∂σy.(24)\nThese operators reflect the invariance of the total magnetic mom entm+σunder rotation.\nMoreover the system is invariant under the combined transformat ion expressed by ˆDzand\nˆΓz, where ˆDzoffers due to the coupling between system and bath variables as well as the\nbreaking of time reversal invariance a coupling between magnetic mo ments and the angular\nmomentum.\nV. CONCLUSION\nIn this paper we have presented an approach for a mesoscopic mag netic system with dissi-\npation. The Lagrangian consists of two interacting subsystems ch aracterized by the active\nmagnetic moments of the system mand the dynamically inactive moments of the bath\ndenoted as σ. Both systems are in contact so that a small local alteration of the system\nvariables mis related as well to a small change of the bath variables σand vice versa.\nDue to this constraint we are able to describe the system by a commo n Lagrangian which\nincorporates both degrees of freedom and their coupling. In case the bath variables are\nconstant then the coupling between both systems is absent and th e whole system decays\ninto two independent subsystems. The magnetic moments mperform a precession around\nan effective field which is self-organized by the mutual interaction of the moments. If the\ncoupling between both subsystems breaks the time reversal symm etry the related evolution\nequation of the moments mis associated with the Landau-Lifshitz-Gilbert equation which\ndescribes both the precession of magnetic moments as well as their damping. It turned\nout that the bath variable σcan be linked to the effective magnetic field which drives the\nmotion of the magnetic moments. As consequence the motion of the moments is influenced\nby the additional bath degrees of freedom. This influence is formula ted mathematically and\nis described by an inhomogeneous diffusion equation. Finally, we have f ound conservation\nlaws by means of symmetry considerations based on Noether’s theo rem. Aside from the\n10expected symmetry transformation in the coordinate space and t he configuration space of\nthe moments, the analysis offers in a non-relativistic Euclidean field th eory an unexpected\ncoupling between both. This point deserves further consideration . Our approach could be\nalso considered as starting point for a further analysis in magnetic a nd multiferroic systems.\nEspecially, we are interested in more refined models which include for in stance higher order\ncouplings or anisotropy in the Lagrangian. In multiferroic systems o ne could study the case\nthat the magnetic and the polar subsystem have their own reservo irs.\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which\nis supported by the Saxony-Anhalt State, Germany.\n11[1] R. Feynman, A. Hibbs, and D. Styer, Quantum Mechanics and Path Integrals: Emended\nEdition(Dover Publications, 2010).\n[2] H. Kleinert, Path integrals in quantum mechanics, statistics, polymer p hysics, and financial\nmarkets (World Scientific, 2009).\n[3] U. Weiss, Quantum dissipative systems , Series in modern condensed matter physics (World\nScientific, 1999).\n[4] L. Schulman, Phys. Rev. 176, 1558 (1968).\n[5] D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, J. Phys. A30, 2699 (1997).\n[6] V. V. Smirnov, J. Phys. A 32, 1285 (1999).\n[7] H. Grinberg, Phys. Lett. A 311, 133 (2003).\n[8] W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold ,\nPhys. Rev. Lett. 100, 230402 (2008).\n[9] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[10] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[11] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halp erin, Rev. Mod. Phys. 77, 1375\n(2005).\n[12] P. Durand and I. Paidarov ˜A¡, EPL 89, 67004 (2010).\n[13] V. L. Safonov and H. N. Bertram, Phys. Rev. B 71, 224402 (2005).\n[14] P. Weetman and G. Akhras, J. Appl. Phys. 105, 023917 (2009).\n[15] Z. V. Gareeva and A. K. Zvezdin, EPL 91, 47006 (2010).\n[16] I. V. Ovchinnikov and K. L. Wang, Phys. Rev. B 82, 024410 (2010).\n[17] J. Jackson, Classical electrodynamics (Wiley, 1999).\n[18] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Statistical Physics Part 2: Theory of the Con-\ndensed State (Pergamon Press, Oxford, 1980).\n[19] E. Noether, Nachr. Ges. Wiss. G¨ ottingen , 235 (1918).\n[20] E. L. Hill, Rev. Mod. Phys. 23, 253 (1951).\n12"    },    {        "title": "1105.4148v2.Magnetization_Dissipation_in_the_Ferromagnetic_Semiconductor__Ga_Mn_As.pdf",        "content": "Magnetization Dissipation in the Ferromagnetic Semiconductor (Ga,Mn)As\nKjetil M. D. Hals and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nWe compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization\nrelaxation. The disorder scattering is included non-perturbatively. In the clean limit, spin-pumping\nfrom the localized d-electrons to the itinerant holes dominates the relaxation processes. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is balanced by the e\u000bects of interband scattering,\nwhich cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system\nshape induces a large anisotropy in the Gilbert damping.\nI. INTRODUCTION\nThe magnetization dynamics of a ferromagnet can be\ndescribed phenomenologically by the Landau-Lifshitz-\nGilbert (LLG) equation:1,2\n1\n\rdM\ndt=\u0000M\u0002He\u000b+M\u0002\"~G(M)\n\r2M2sdM\ndt#\n:(1)\nHere,\ris the gyromagnetic ratio, He\u000bis the e\u000bective\nmagnetic \feld (which is the functional derivative of the\nfree energy He\u000b=\u0000\u000eF[M]=\u000eM),Mis the magnetiza-\ntion andMsis its magnitude. The Gilbert damping con-\nstant ~G(M) parameterizes the dissipative friction process\nthat drives the magnetization towards an equilibrium\nstate.3In the most general case, ~G(M) is a symmetric\npositive de\fnite matrix that depends on the magnetiza-\ntion direction; however, it is often assumed to be inde-\npendent of Mand proportional to the unit matrix, as-\nsumptions which are valid for isotropic systems. Gilbert\ndamping is important in magnetization dynamics. It de-\ntermines the magnitudes of the external magnetic \felds4\nand the current densities1that are required to reorient\nthe magnetization direction of a ferromagnet. Therefore,\na thorough understanding of its properties is essential for\nmodeling ferromagnetic systems.\nThe main contribution to the Gilbert damping process\nin metallic ferromagnets is the generation of electron-\nhole pairs.1,2,5,6A model that captures this process was\ndeveloped by Kambersky.5In this model, the electrons\nare excited by a time-varying magnetization via electron-\nmagnon coupling. If the ferromagnet is in metallic con-\ntact with other materials, the spin-pumping into the ad-\njacent leads provides an additional contribution to the\nmagnetization relaxation.7A general theory that cap-\ntures both of these e\u000bects was recently developed.8The\nmodel expresses the ~G(M) tensor in terms of the scatter-\ning matrix Sof the ferromagnetic system ( m\u0011M=Ms):\n~Gij(m) =\r2\u0016h\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (2)\nThe expression is evaluated at the Fermi energy. Instead\nof~G(M), one often parameterizes the damping by the\ndimensionless Gilbert damping parameter ~ \u000b\u0011~G=\rM s.\nEq. (2) allows studying both the e\u000bects of the systemshape and the disorder dependency of the magnetization\ndamping beyond the relaxation time approximation.9\nIn anisotropic systems, the Gilbert damping is ex-\npected to be a symmetric tensor with non-vanishing o\u000b-\ndiagonal terms. We are interested in how this tensor\nstructure in\ruences the dynamics of the precessing mag-\nnetization in (Ga,Mn)As. Therefore, to brie\ry discuss\nthis issue, let us consider a homogenous ferromagnet in\nwhich the magnetization direction m=m0+\u000empre-\ncesses with a small angle around the equilibrium direc-\ntionm0that points along the external magnetic \feld\nHext.10For clarity, we neglect the anisotropy in the\nfree energy and choose the coordinate system such that\nm0= (0 0 1) and \u000em= (mxmy0). For the lowest order\nof Gilbert damping, the LLG equation can be rewritten\nas:_m=\u0000\rm\u0002Hext+\rm\u0002(~\u000b[Hext\u0002m]), where ~\u000b[:::]\nis the dimensionless Gilbert damping tensor that acts on\nthe vector Hext\u0002m. Linearizing the LLG equation re-\nsults in the following set of equations for mxandmy:\n\u0012\n_mx\n_my\u0013\n=\u0000\rHext \n\u000b(0)\nyy (1\u0000\u000b(0)\nxy)\n\u0000(1 +\u000b(0)\nxy)\u000b(0)\nxx!\u0012\nmx\nmy\u0013\n:\n(3)\nHere,\u000b(0)\nijare the matrix elements of ~ \u000bwhen the tensor\nis evaluated along the equilibrium magnetization direc-\ntionm0. For the lowest order of Gilbert damping, the\neigenvalues of (3) are \u0015\u0006=\u0006i\rH ext\u0000\rHext\u000b, and the\neigenvectors describe a precessing magnetization with a\ncharacteristic life time \u001c= (\u000b\rH ext)\u00001. The e\u000bective\ndamping coe\u000ecient \u000bis:10\n\u000b\u00111\n2\u0010\n\u000b(0)\nxx+\u000b(0)\nyy\u0011\n: (4)\nThe value of \u000bis generally anisotropic and depends on\nthe static magnetization direction m0. The magnetiza-\ntion damping is accessible via ferromagnetic resonance\n(FMR) experiments by measuring the linear relation-\nship between the FMR line width and the precession fre-\nquency. This linear relationship is proportional to \u001c\u00001\nand thus depends linearly on \u000b. Therefore, an FMR ex-\nperiment can be used to determine the e\u000bective damping\ncoe\u000ecient\u000b. In contrast, the o\u000b-diagonal terms, \u000b(0)\nxy\nand\u000b(0)\nyx, do not contribute to the lowest order in the\ndamping and are di\u000ecult to probe experimentally.\nIn this paper, we use Eq. (2) to study the anisotropy\nand disorder dependency of the Gilbert damping in thearXiv:1105.4148v2  [cond-mat.mtrl-sci]  2 Nov 20112\nferromagnetic semiconductor (Ga,Mn)As. Damping co-\ne\u000ecients of this material in the range of \u000b\u00180:004\u00000:04\nfor annealed samples have been reported.11{14The damp-\ning is anisotropically dependent on the magnetization\ndirection.11,12,14The few previous calculations of the\nGilbert damping constant in this material have indicated\nthat\u000b\u00180:003\u00000:04.11,15{17These theoretical works\nhave included the e\u000bects of disorder phenomenologically,\nfor instance, by applying the relaxation time approxima-\ntion. In contrast, Eq. (2) allows for studying the disor-\nder e\u000bects fully and non-perturbatively for the \frst time.\nIn agreement with Ref. 15, we show that spin-pumping\nfrom the localized d-electrons to the itinerant holes dom-\ninates the damping process in the clean limit. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is bal-\nanced by e\u000bects of the interband transitions, which cause\nthe damping to saturate. In determining the anisotropy\nof the Gilbert damping tensor, we \fnd that the shape of\nthe sample is typically more important than the e\u000bects\nof the strain and the cubic symmetry in the GaAs crys-\ntal.18This shape anisotropy of the Gilbert damping in\n(Ga,Mn)As has not been reported before and provides\na new direction for engineering the magnetization relax-\nation.\nII. MODEL\nThe kinetic-exchange e\u000bective Hamiltonian approach\ngives a reasonably good description of the electronic\nproperties of (Ga,Mn)As.19The model assumes that the\nelectronic states near the Fermi energy have the character\nof the host material GaAs and that the spins of the itin-\nerant quasiparticles interact with the localized magnetic\nMn impurities (with spin 5/2) via the isotropic Heisen-\nberg exchange interaction. If the s-d exchange interac-\ntion is modeled by a mean \feld, the e\u000bective Hamiltonian\ntakes the form:19,20\nH=HHoles +h(r)\u0001s; (5)\nwhereHHoles is the k\u0001pKohn-Luttinger Hamiltonian de-\nscribing the valence band structure of GaAs and h(r)\u0001s\nis a mean \feld description of the s-d exchange interac-\ntion between the itinerant holes and the local magnetic\nimpurities ( sis the spin operator). The exchange \feld\nhis antiparallel to the magnetization direction m. The\nexplicit form of HHoles that is needed for realistic model-\ning of the band structure of GaAs depends on the doping\nlevel of the system. Higher doping levels often require\nan eight-band model, but a six- or four-band model may\nbe su\u000ecient for lower doping levels. In the four-band\nmodel, the Hamiltonian is projected onto the subspace\nspanned by the four 3/2 spin states at the top of the\nGaAs valence band. The six-band model also includes\nthe spin-orbit split-o\u000b bands with spin 1/2. The spin-\norbit splitting of the spin 3/2 and 1/2 states in GaAs\nis 341 meV.21We consider a system with a Fermi level\nof 77 meV when measured from the lowest subband. Inthis limit, the following four-band model gives a su\u000ecient\ndescription:\nH=1\n2m\u0014\n(\r1+5\n2\r2)p2\u00002\r3(p\u0001J)2+h\u0001J\u0015\n+\n\r3\u0000\r2\nm(p2\nxJ2\nx+c:p:) +Hstrain +V(r): (6)\nHere, pis the momentum operator, Jiare the spin 3/2\nmatrices22and\r1,\r2and\r3are the Kohn-Luttinger pa-\nrameters.V(r) =P\niVi\u000e(r\u0000Ri) is the impurity scat-\ntering potential, where Riis the position of the impurity\niandViare the scattering strengths of the impurities23\nthat are randomly and uniformly distributed in the in-\nterval [\u0000V0=2;V0=2].Hstrain is a strain Hamiltonian and\narises because the (Ga,Mn)As system is grown on top\nof a substrate (such as GaAs).24The two \frst terms in\nEq. (6) have spherical symmetry, and the term propor-\ntional to\r3\u0000\r2represents the e\u000bects of the cubic sym-\nmetry of the GaAs crystal. Both this cubic symmetry\nterm25and the strain Hamiltonian24are small compared\nto the spherical portion of the Hamiltonian. A numeri-\ncal calculation shows that they give a correction to the\nGilbert damping on the order of 10%. However, the un-\ncertainty of the numerical results, due to issues such as\nthe sample-to-sample disorder \ructuations, is also about\n10%; therefore, we cannot conclude how these terms in-\n\ruence the anisotropy of the Gilbert damping. Instead,\nwe demonstrate that the shape of the system is the dom-\ninant factor in\ruencing the anisotropy of the damping.\nTherefore, we disregard the strain Hamiltonian Hstrain\nand the term proportional to \r3\u0000\r2in our investigation\nof the Gilbert damping.\nGaAs GaAs (Ga,Mn)As \nyx\nFIG. 1: We consider a (Ga,Mn)As system attached along the\n[010] direction to in\fnite ballistic GaAs leads. The scattering\nmatrix is calculated for the (Ga,Mn)As layer and one lattice\npoint into each of the leads. The magnetization is assumed to\nbe homogenous. In this paper, we denote the [100] direction\nas the x-axis, the [010] direction as the y-axis and the [001]\ndirection as the z-axis\nWe consider a discrete (Ga,Mn)As system with\ntransverse dimensions Lx2 f17;19;21gnm,Lz2\nf11;15;17gnm andLy= 50 nm and connected to in\fnite\nballistic GaAs leads, as illustrated in Fig. 1. The leads\nare modeled as being identical to the (Ga,Mn)As system,\nexcept for the magnetization and disorder. The lattice\nconstant is 1 nm, which is much less than the Fermi wave-\nlength\u0015F\u001810 nm. The Fermi energy is 0.077 eV when3\nmeasured from the lowest subband edge. The Kohn-\nLuttinger parameters are \r1= 7:0 and\r2=\r3= 2:5,\nimplying that we apply the spherical approximation for\nthe Luttinger Hamiltonian, as mentioned above.25We\nusejhj= 0:032 eV for the exchange-\feld strength. To\nestimate a typical saturation value of the magnetization,\nwe useMs= 10j\rj\u0016hx=a3\nGaAs withx= 0:05 as the doping\nlevel andaGaAs as the lattice constant for GaAs.26\nThe mean free path lfor the impurity strength V0is\ncalculated by \ftting the average transmission probability\nT=hGi=GshtoT(Ly) =l=(l+Ly),27whereGshis the\nSharvin conductance and hGiis the conductance for a\nsystem of length Ly.\nThe scattering matrix is calculated numerically us-\ning a stable transfer matrix method.28The disorder ef-\nfects are fully and non-perturbatively included by the\nensemble average h\u000bi=PNI\nn=1\u000bn=NI, whereNIis the\nnumber of di\u000berent impurity con\fgurations. All the\ncoe\u000ecients are averaged until an uncertainty \u000eh\u000bi=r\u0010\nh\u000b2i\u0000h\u000bi2\u0011\n=NIof less than 10% is achieved. The\nvertex corrections are exactly included in the scattering\nformalism.\nIII. RESULTS AND DISCUSSION\nWithout disorder, the Hamiltonian describing our sys-\ntem is rotationally symmetric around the axis parallel\ntoh. Let us brie\ry discuss how this in\ruences the\nparticular form of the Gilbert damping tensor ~ \u000b.29For\nclarity, we choose the coordinate axis such that the ex-\nchange \feld points along the z-axis. In this case, the\nHamiltonian is invariant under all rotations Rzaround\nthe z-axis. This symmetry requires the energy dissipa-\ntion _E/_mT~\u000b_m(_mTis the transposed of _m) of the\nmagnetic system to be invariant under the coordinate\ntransformations r0=Rzr(i.e., ( _m0)T~\u000b0_m0=_mT~\u000b_m\nwhere m0=Rzmand ~\u000b0is the Gilbert damping ten-\nsor in the rotated coordinate system). Because ~ \u000bonly\ndepends on the direction of m, which is unchanged\nunder the coordinate transformation Rz, ~\u000b= ~\u000b0and\nRT\nz~\u000bRz= ~\u000b. Thus, ~\u000bandRzhave common eigenvectors\b\nj\u0006i\u0011 (jxi\u0006ijyi)=p\n2;jzi\t\n, and the spectral decompo-\nsition of ~\u000bis ~\u000b=\u000b+j+ih+j+\u000b\u0000j\u0000ih\u0000j +\u000bzjzihzj.\nRepresenting the damping tensor in the fjxi;jyi;jzig\ncoordinate basis yields \u000b\u0011~\u000bxx= ~\u000byy= (\u000b++\u000b\u0000)=2,\n~\u000bzz=\u000bz, ~\u000byx= ~\u000bxy= 0, and\u000b+=\u000b\u0000. The last\nequality results from real tensor coe\u000ecients. However,\n\u000bzzcannot be determined uniquely from the energy dis-\nsipation formula _E/_mT~\u000b_mbecause _mis perpendicu-\nlar to the z-axis. Therefore, \u000bzzhas no physical signi\f-\ncance and the energy dissipation is governed by the single\nparameter\u000b. For an in\fnite system, this damping pa-\nrameter does not depend on the speci\fc direction of the\nmagnetization, i.e., it is isotropic because the symmetry\nof the Hamiltonian is not directly linked to the crystallo-\ngraphic axes of the underlying crystal lattice (when the\n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n44.5 55.5 6x 10 −3 \n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n4.5 55.5 6x 10 −3 a\nbFIG. 2: ( a) The dimensionless Gilbert damping parameter\n\u000bas a function of the magnetization direction for a system\nwhereLx= 17 nm,Ly= 50 nm and Lz= 17 nm. ( b) The\ndimensionless Gilbert damping parameter \u000bas a function of\nthe magnetization direction for a system where Lx= 21 nm,\nLy= 50 nm and Lz= 11 nm. Here, \u0012and\u001eare the polar and\nazimuth angles, respectively, that describe the local magne-\ntization direction m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). In both\nplots, the mean free path is l\u001822 nm\ncubic symmetry term in Eq. (6) is disregarded). For a\n\fnite system, the shape of the system induces anisotropy\nin the magnetization damping. This e\u000bect is illustrated\nin Fig. 2, which plots the e\u000bective damping in Eq. (4) as\na function of the magnetization directions for di\u000berent\nsystem shapes. When the cross-section of the conductor\nis deformed from a regular shape to the shape of a thin-\nner system, the anisotropy of the damping changes. The\nmagnetization damping varies from a minimum value of\naround 0.004 to a maximum value of 0.006, e.g., the\nanisotropy is around 50%. The relaxation process is\nlargest along the axis where the ballistic leads are con-\nnected, i.e., the y-axis. This shape anisotropy is about\nfour- to \fve-times stronger than the anisotropy induced\nby the strain and the cubic symmetry terms in the Hamil-\ntonian (6), which give corrections of about 10 percent.\nFor larger systems, we expect this shape e\u000bect to be-4\ncome less dominant. In these systems, the anisotropy\nof the bulk damping parameter, which is induced by the\nanisotropic terms in the Hamiltonian, should play a more\nsigni\fcant role. The determination of the system size\nwhen the strain and cubic anisotropy become comparable\nto the shape anisotropy e\u000bects is beyond the scope of this\npaper because the system size is restricted by the com-\nputing time. However, this question should be possible to\ninvestigate experimentally by measuring the anisotropy\nof the Gilbert damping as a function of the \flm thickness.\n0 1 2 3 4 52345678910 x 10 −3 \nLy/l ααmin\nαmax \nαmean \nFIG. 3: The e\u000bective dimensionless Gilbert damping (4) as a\nfunction of the disorder. Here, lis the mean free path and\nLyis the length of the ferromagnetic system in the transport\ndirection.\u000bminand\u000bmaxare the minimum and maximum val-\nues of the anisotropic Gilbert damping parameter and \u000bmean is\nthe e\u000bective damping parameter averaged over all the magne-\ntization directions. The system dimensions are Lx= 19 nm,\nLy= 50 nm and Lz= 15 nm.\nWe next investigate how the magnetization relaxation\nprocess depends on the disorder. Ref. 15 derives an ex-\npression that relates the Gilbert damping parameter to\nthe spin-\rip rate T2of the system: \u000b/T2(1 + (T2)2)\u00001.\nIn the low spin-\rip rate regime, this expression scales\nwithT2as\u000b/T\u00001\n2, while the damping parameter is\nproportional to \u000b/T2in the opposite limit . As ex-\nplained in Ref. 15, the low spin-\rip regime is dominated\nby the spin-pumping process in which angular momen-\ntum is transferred to the itinerant particles; the trans-\nferred spin is then relaxed with a rate proportional to\nT\u00001\n2. This process appears inside the ferromagnet itself,\ni.e., the spin is transferred from the magnetic system to\nthe itinerant particles in the ferromagnet, which are then\nrelaxed within the ferromagnet. Therefore, this relax-\nation mechanism is a bulk process and should not be\nconfused with the spin-pumping interface e\u000bect across\nthe normal metal jferromagnet interfaces reported in\nRef. 7. In (Ga, Mn)As, this bulk process corresponds to\nspin-pumping from the d-electrons of the magnetic Mn\nimpurities to the itinerant spin 3/2 holes in the valence\nband of the host compound GaAs. The transfer of spin\nto the holes is then relaxed by the impurity scatteringwithin the ferromagnet. By contrast, the opposite limit\nis dominated by the breathing Fermi-surface mechanism.\nIn this mechanism, the spins of the itinerant particles are\nnot able to follow the local magnetization direction adia-\nbatically and lag behind with a delay time of T2. In our\nsystem, which has a large spin-orbit coupling in the band\nstructure, we expect the spin-\rip rate to be proportional\nto the mean free path ( l/T2).30The e\u000bective dimen-\nsionless Gilbert damping (4) is plotted as a function of\ndisorder in Fig. 3. The damping ( \u000bmean) partly shows\nthe same behavior as that reported in Ref. 15. For clean\nsystems (i.e., those with a low spin-\rip rate regime), the\ndamping increases with disorder. In such a regime, the\ntransfer of angular momentum to the spin 3/2 holes is the\ndominant damping process, i.e., the bulk spin-pumping\nprocess dominates. \u000bmean starts to decrease for smaller\nmean free paths, implying that the main contribution\nto the damping comes from the breathing Fermi-surface\nprocess. Refs. 11,16,17 have reported that the Gilbert\ndamping may start to increase as a function of disorder in\ndirtier samples. The interband transitions become more\nimportant with decreasing quasi-particle life times and\nstart to dominate the intraband transitions (The intra-\nband transitions give rise to the breathing Fermi-surface\ne\u000bect). We do not observe an increasing behavior in the\nmore di\u000busive regime, but we \fnd that the damping sat-\nurates at a value of around 0.0046 (See Fig. 3). In this\nregime, we believe that the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions. The damping\ndoes not vanish in the limit 1 =l= 0 due to scattering\nat the interface between the GaAs and (Ga,Mn)As lay-\ners in addition to spin-pumping into the adjacent leads\n(an interface spin-pumping e\u000bect, as explained above).\nFig. 3 shows that the shape anisotropy of the damp-\ning is reduced by disorder because the di\u000berence between\nthe maximum ( \u000bmax) and minimum ( \u000bmin) values of the\ndamping parameter decrease with disorder. We antici-\npate this result because disorder increases the bulk damp-\ning e\u000bect, which is expected to be isotropic for an in\fnite\nsystem.\nIV. SUMMARY\nIn this paper, we studied the magnetization damping\nin the ferromagnetic semiconductor (Ga,Mn)As. The\nGilbert damping was calculated numerically using a\nrecently developed scattering matrix theory of mag-\nnetization dissipation.8We conducted a detailed non-\nperturbative study of the e\u000bects of disorder and an inves-\ntigation of the damping anisotropy induced by the shape\nof the sample.\nOur analysis showed that the damping process is\nmainly governed by three relaxation mechanisms. In the\nclean limit with little disorder, we found that the magne-\ntization dissipation is dominated by spin-pumping from\nthe d-electrons to the itinerant holes. For shorter mean\nfree paths, the breathing Fermi-surface e\u000bect starts to5\ndominate, which causes the damping to decrease. In\nthe di\u000busive regime, the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions and the e\u000bective\ndamping parameter saturates at a value on the order of\n0.005.\nFor the small samples considered in this study, we\nfound that the shape of the system was typically more\nimportant than the anisotropic terms in the Hamiltonian\nfor the directional dependency of the damping parame-\nter. This shape anisotropy has not been reported beforeand o\u000bers a new way of manipulating the magnetization\ndamping.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by the European\nUnion FP7 Grant No. 251759 \\MACALO\".\n1For a review, see D. C. Ralph and M. Stiles, J. Magn.\nMagn. Mater. 320, 1190 (2008), and reference therein.\n2B. Heinrich, D. Fraitov\u0013 a, and V. Kambersky, Phys. Status\nSolidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48,\n2906 (1970); V. Korenman and R.E. Prange, Phys. Rev.\nB6, 2769 (1972); V.S. Lutovinov and M.Y. Reizer, Zh.\nEksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355\n(1979)]; V.L. Safonov and H.N. Bertram, Phys. Rev. B 61,\nR14893 (2000); J. Kunes and V. Kambersky, Phys. Rev. B\n65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416\n(2007).\n3T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4J.A.C. Bland and B. Heinrich, Ultrathin Magnetic Struc-\ntures III Fundamentals of Nanomagnetism (Springer Ver-\nlag, Heidelberg, 2004).\n5V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n6K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n7Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n8A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); Phys. Rev. B 84, 054416\n(2011).\n9A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601\n(2010); Y. Liu, Z. Yuan, A. A. Starikov, and P. J. Kelly,\narXiv:1102.5305.\n10A similar analysis is presented in J. Seib, D. Steiauf, and\nM. F ahnle, Physical Review B 79, 092418 (2009).\n11J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n12Y.H. Matsuda, A. Oiwa, K. Tanaka, and H. Munekata,\nPhysica B 376-377 , 668 (2006).\n13A. Wirthmann et al. , Appl. Phys. Lett. 92, 232106 (2008).\n14Kh. Khazen et al. , Phys. Rev. B 78, 195210 (2008).\n15Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n16I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403(2009).\n17I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n18This shape anisotropy in the Gilbert damping should not\nbe confused with the shape anisotropy (in the anisotropy\n\feld) caused by surface dipoles in non-spherical systems.\n19T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809864 (2006).\n20M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n21P.Y. Yu and M. Cardona, Fundamentals of Semicon-\nductors: Physics and Materials Properties , 3rd Edition\n(Springer Verlag, Berlin, 2005).\n22Note that in Eq. (6) the spin operator s(in the p-d ex-\nchange term) is represented in the basis consisting of the\nfour spin 3/2 states ( s=J=3). The factor 1 =3 is absorbed\nin the exchange \feld h.\n23In the discrete version of Eq. (6), as used in the numerical\ncalculation, we have one impurity at each lattice site.\n24A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y.\nLyanda-Geller, and L.P. Rokhinson, Nature Physics 5, 656\n(2009).\n25A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697\n(1973).\n26The prefactor of 10 comes from 4 Ga atoms per unit cell\ntimes spin 5/2 per substitutional Mn, which are assumed to\nbe fully polarized. The reduction of the net magnetization\ndue to the interstitial Mn ions and p holes are disregarded\nin our estimate.\n27S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, Cambridge, England, 1995).\n28T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N.\nYokoyama, Phys. Rev. B 52, 8244 (1995).\n29D. Steiauf and M. F ahnle, Physical Review B 72, 064450\n(2005).\n30The spin relaxation time of holes in GaAs is on the scale of\nthe momentum relaxation time. See D.J. Hilton and C.L.\nTang, Phys. Rev. Lett. 89, 146601 (2002), and references\ntherein."    },    {        "title": "1106.4861v1.Ratchet_effect_on_a_relativistic_particle_driven_by_external_forces.pdf",        "content": "arXiv:1106.4861v1  [nlin.PS]  23 Jun 2011Ratchet effect on a relativistic particle driven by\nexternal forces\nNiurka R. Quintero1, Renato Alvarez-Nodarse2and Jos´ e A.\nCuesta3\n1Departamento de F´ ısica Aplicada I, E. S. P., Universidad de Sevilla, C/ Virgen de\n´Africa 7, E-41011, Sevilla, Spain\n2Departamento de An´ alisis Matem´ atico, Universidad de Sevilla, apdo . 1160,\nE-41080, Sevilla, Spain\n3Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamen to de\nMatem´ aticas, Universidad Carlos III de Madrid, avda. de la Univers idad 30, E-28911\nLegan´ es, Madrid, Spain\nE-mail:niurka@us.es ,ran@us.es ,cuesta@math.uc3m.es\nAbstract. We study the ratchet effect of a damped relativistic particle driven b y\nboth asymmetric temporal bi-harmonic and time-periodic piecewise c onstant forces.\nThis system can be formally solved for any external force, providin g the ratchet\nvelocity as a non-linear functional of the driving force. This allows us to explicitly\nillustrate the functional Taylor expansion formalism recently propo sed for this kind of\nsystems. The Taylor expansion reveals particularly useful to obta in the shape of the\ncurrent when the force is periodic, piecewise constant. We also illust rate the somewhat\ncounterintuitiveeffectthatintroducingdampingmayinducearatch eteffect. When the\nforce is symmetric under time-reversal and the system is undampe d, under symmetry\nprinciples no ratchet effect is possible. In this situation increasing da mping generates\na ratchet current which, upon increasing the damping coefficient ev entually reaches a\nmaximum and decreases toward zero. We argue that this effect is no t specific of this\nexample and should appear in any ratchet system with tunable dampin g driven by a\ntime-reversible external force.\nPACS numbers:\nSubmitted to: J. Phys. A: Math. Gen.Ratchet effect on a relativistic particle 2\n1. Introduction\nThe ratchet effect is identified with the motion of particles or solitons induced by\nzero-average periodic forces [1, 2], sometimes in the presence of t hermal fluctuations.\nThe effect arises as a subtle interplay between nonlinearities in the sy stem and broken\nsymmetries. Ratchets appear in many fields of physics, where net m otion is generated\neither by an asymmetric, periodic, spatial potential [3, 4, 5, 6, 7, 8 , 9, 10], or by an\nasymmetric temporal forcing [10, 11, 12, 13, 14, 15, 16, 17, 18]. I n bothcases the ratchet\neffect can be regarded as an application of Curie’s symmetry principle , which states that\na symmetry transformation of the cause (forces) is also a symmet ry transformation of\nthe effect (ratchet velocity) [19, 20].\nMost studies of ratchets driven by temporal forces employ a bi-ha rmonic forcing\nf(t) =ǫ1cos(qωt+φ1)+ǫ2cos(pωt+φ2), (1)\nwherepandqare positive integers which, without loss of generality, can be taken co-\nprime (otherwise common factors can be absorbed in the frequenc yω) andǫ1,ǫ2are\nsmall non-zero parameters. If both pandqare odd, the force (1) exhibits the shift\nsymmetry (Sf)(t) =f(t+T/2) =−f(t), where T= 2π/ω. In systems invariant under\ntime translations this implies that both, f(t) and−f(t) generate the same ratchet\ncurrent (or velocity) defined as [21, 22]\nv= lim\nt→+∞1\nt/integraldisplayt\n0˙x(τ)dτ= lim\nt→+∞x(t)\nt, (2)\nwherex(t) is the position of the particle, soliton, or localized structure. If re versing\nthe force changes the sign of the current, this current must be z ero. So shift-symmetric\nbi-harmonic forces cannot induce a ratchet effect. In contrast, ifpandqhave different\nparity, shift symmetry is broken by f(t) so the force can induce a nonzero net current\n[23].\nMany attempts have been made to determine quantitatively the dep endence of the\nratchetvelocity, v,ontheparametersofthebi-harmonicforce(1)[11,24,25]. Inva riably,\nthe analysis performed in these works rests on the so-called method of moments, where\nit is assumed that the average ratchet velocity can be expanded as a series of the odd\nmoments of f(t), i.e./summationtext∞\nk=1/angbracketleft[f(t)]2k+1/angbracketrightwith/angbracketlefth(t)/angbracketright=/integraltextT\n0dth(t). This method seemed\nto work for some systems but not for others without a clear reaso n and with no known\ncriterion to tell ones from the others. We have recently shown tha t the moment method\nrelies on an assumption that almost never holds, and have provided a n alternative\nprocedure that yields the correct result regardless of the syste m [23].\nThe aimofthispaper istoprovide explicit examples which illustratesthis otherwise\nabstractmethod—thefunctionalexpansionof vintermsof f—usingaworkingexample\nfor which an analytic solution can be found. The system represents the motion of a\ndamped, relativistic particle under the effect of two different force s: a bi-harmonic forceRatchet effect on a relativistic particle 3\nlike (1), and a time-periodic piecewise constant force like\nf(t) =\n\nǫ1if 0< t < T l,\n0 if Tl< t < T−Tl,\n−ǫ1ifT−Tl< t < T.(3)\nTo this purpose we introduce the model as well as its analytic solution in section 2.\nIn section 3 we discuss the phenomenon of damping-induced ratche ts. The formalism\ndeveloped in [23] is fully illustrated for this problem in section 4. For the se two specific\ndriving forces it is also shown that the method of moments is valid only w hen the\ndynamics of the relativistic particle is overdamped, and fails otherwis e. Conclusions are\nsummarized in section 5.\n2. Motion of a relativistic particle driven by a bi-harmonic force\nThe equation of motion of a relativistic particle with mass M >0, whose position and\nvelocity at time tare denoted x(t) andu(t), respectively, is\ndx\ndt=u(t), x (0) =x0, (4a)\nMdu\ndt=−f(t)(1−u2)3/2−γu(1−u2), u(0) =u0, (4b)\nwherex0andu0are the initial conditions, γ >0 represents the damping coefficient\nandf(t) is aT-periodic driving force. Notice that if the force f(t) satisfied ( Sf)(t) =\nf(t+T/2) =−f(t), then (4 b) would be invariant under a combination of shift symmetry\n(S:t/mapsto→t+T/2) and the sign change x/mapsto→ −x.\nChanging the variable u(t) by the momentum\nP(t) =Mu(t)/radicalbig\n1−u2(t)(5)\ntransforms (4 b) into the linear equation\ndP\ndt=−βP−f(t), P(0) =P0=Mu0/radicalbig\n1−u2\n0, (6)\nwhereβ=γ/M. Equation (6) is easily solved to give\nP(t) =P0e−βt−/integraldisplayt\n0dzf(z)e−β(t−z). (7)\nFrom (5) one obtains\nu(t) =∞/summationdisplay\nk=0/parenleftbigg\n−1\n2/parenrightbiggk(2k−1)!!\nk!/parenleftbiggP(t)\nM/parenrightbigg2k+1\n. (8)\nLet usnowfocusourattentiononthe T-periodicdriving force f(t)givenby (1)with\np= 2 and q= 1 (the most common choice of parameters [26, 22, 14, 27]). Subst ituting\n(1) into (7) leads to\nP(t) =/tildewideP0e−βt−˜ǫ1cos(ωt+φ1−χ1)−˜ǫ2cos(2ωt+φ2−χ2),Ratchet effect on a relativistic particle 4\nwith\n/tildewideP0=P0+˜ǫ1cos(φ1−χ1)+˜ǫ2cos(φ2−χ2),\n˜ǫ1=ǫ1(β2+ω2)−1/2, χ 1= tan−1(ω/β),\n˜ǫ2=ǫ2(β2+4ω2)−1/2, χ 2= tan−1(2ω/β).\nAst→ ∞the momentum P(t) behaves, for any β >0, as\nP(t)∼ −˜ǫ1cos(ωt+φ1−χ1)−˜ǫ2cos(2ωt+φ2−χ2),\nthus the term P(t)2k+1in (8) isO(ǫr\n1ǫs\n2) withr+s= 2k+1. Since the time average of\nP(t) is zero, the leading term of (2) in powers of ǫ1andǫ2will be\n−1\n2M3lim\nt→∞1\nt/integraldisplayt\n0P(τ)3dτ=3\n2M3T˜ǫ2\n1˜ǫ2/integraldisplayT\n0cos(ωτ+φ1−χ1)2cos(2ωτ+φ2−χ2)dτ\n=3\n8M3˜ǫ2\n1˜ǫ2cos(2φ1−φ2+χ2−2χ1).\nTherefore, the rachet velocity (2), for small amplitudes ǫ1andǫ2, is given by\nv=Bǫ2\n1ǫ2cos(2φ1−φ2+θ0), (9)\nwith\nB=3\n8M3(β2+ω2)/radicalbig\nβ2+4ω2, θ 0=χ2−2χ1=−tan−1/parenleftbigg2ω3\nβ(β2+3ω2)/parenrightbigg\n,(10)\nin agreement with the result reported in [23].\nNotice that in the undamped limit γ→0 (equivalently β→0) the parameters (10)\nbecome\nB=3\n16M3ω3, θ 0=−π\n2,\nwhereas in the overdamped limit M→0 (and therefore β→ ∞withMβ=γ) we get\nB=3\n8γ3, θ 0= 0,\nboth limits agree with the predictions of [23].\n3. Ratchet induced by damping\nThe depence of von parameters of the system like the damping coefficient (through\nβ=γ/M) shown in (9) and (10) reveals an interesting effect. If we take φ1=φ2= 0 in\nf(t) and do some algebra, the ratchet velocity (for small amplitudes of the force) turns\nout to be\nv=ǫ2\n1ǫ2\nω3M3V(β/ω), V(x) =3x(x2+3)\n8(x2+1)2(x2+4). (11)\nFunction V(x) is depicted in Figure 1. The most remarkable observation is that the\ncurrentincreases up to a maximum with increasing damping before it begins to show\nthe expected decay. Intuition dictates that the current should d ecrease with dampingRatchet effect on a relativistic particle 5\n0 1 2 3 4 5\nβ/ω00.020.040.060.080.1v (Mω)3/ε12ε2\nFigure 1. Plot of the current velocity v, in units of ǫ2\n1ǫ2/(Mω)3, vs. the damping\ncoefficient β, in units of the frequency, ω, induced by a biharmonic force like (1) with\nφ1=φ2= 0. Notice that this force is time reversible, i.e., f(−t) =f(t).\nbecause friction opposes movement, so the fast increase it revea ls for small damping is\ncounterintuitive.\nThe cause of this effect is the interplay between the breaking of the time-reversal\nsymmetry R:t/mapsto→ −tthat generates the ratchet current, and the damping that hinde rs\nit [27]. In the limit β→0 the system (2) is invariant under Rand a sign change of\nu, because for φ1=φ2= 0 the force (1) satisfies f(−t) =f(t). Accordingly v= 0 in\nthis limit. But introducing damping breaks the symmetry of the equat ion and induces\na net movement of the particle. For small damping, the higher the da mping coefficient\nβthe larger v. If we keep on increasing βeventually the friction it introduces in the\nmovement of the particle causes the decay of vasβ−3.\nThis argument makes it clear that in any ratchet system with a tunab le damping\nand undergoing the action of a time-reversible bi-harmonic force, t he ratchet effect can\nbe generated upon increasing damping above zero.\n4. Ratchet velocity as a functional of the force\nThe starting point to obtain formula (9) for a ratchet system is to r ealize that vis a\nfunctional of f(t) and that, under certain regularity assumptions, one such funct ional\ncan be expanded as a functional Taylor series [28, 29, 30] as\nv[f] =/summationdisplay\nnodd/integraldisplayT\n0dt1\nT···/integraldisplayT\n0dtn\nTcn(t1,...,t n)f(t1)···f(tn), (12)\nwhere the kernels cn(t1,...,t n) are proportional to the nth functional derivatives of\nthe functional v[f]. These kernels can be taken T-periodic in each variable and totally\nsymmetric under any exchange of variables. Only odd terms appear in this expansion\nas a cosequence of the symmetry v[−f] =−v[f] that these systems have.Ratchet effect on a relativistic particle 6\nThatvis indeed a functional of f(t) in this example is obvious from equations (2)–\n(7). The aim of this section is to determine explicitly the expansion (12 ) for this exactly\nsolvable example.\nLet us start off by rewriting the integral in (7) as\n/integraldisplayt\n0dzf(z)e−β(t−z)=I1(t)+I2(t), (13)\nI1(t) =n(t)/summationdisplay\nk=1/integraldisplayT\n0dzf(z)e−β(t−z−(k−1)T),\nI2(t) =/integraldisplayα(t)\n0dzf(z)e−β(α(t)−z),\nwhereα(t) =t−n(t)Tandn(t) = [t/T] ([X] denoting the integer part of X). Notice\nthatα(t+T) =α(t). Now, since\nS(t)≡n(t)/summationdisplay\nk=1eβ(k−1)T=eβnT−1\neβT−1, (14)\nthenI1(t) = e−βtCS(t), with\nC=/integraldisplayT\n0dzf(z)[eβz−1]. (15)\nUsing this form in (7) we can write\nP(t) =Ae−βt+/tildewideP(t), (16)\nwhereA=P(0)+C(eβT−1)−1and/tildewideP(t) is theT-periodic function\n/tildewideP(t) =−1\neβT−1/integraldisplayT\n0dyf(y)e−βα(t)[eβy−1]−/integraldisplayα(t)\n0dyf(y)e−β(α(t)−y).\nIt is thus enough to obtain /tildewideP(t) in the interval 0 ≤t < T, where it adopts the compact\nform\n/tildewideP(t) =−/integraldisplayT\n0dyf(y)e−β(t−y)χ(y,t), (17)\ndefining\nχ(y,t) =1−e−βy\neβT−1+Θ(t−y) (18)\n(as it is customary, Θ( x) denotes the Heaviside function, which is 1 if x >0 and 0\notherwise).\nEquations (17)–(18) have a well defined β→0+limit, namely\n/tildewideP(t) =−/integraldisplayT\n0dyf(y)χ1(y,t), χ 1(y,t) =y\nT+Θ(t−y). (19)\nOntheotherhand, forzero-averageforces f(t)thekernel χ(t,z)canbefurthersimplified\nto\n/tildewideP(t) =−/integraldisplayT\n0dyf(y)e−β(t−y)χ2(y,t), χ 2(y,t) =1\neβT−1+Θ(t−y). (20)Ratchet effect on a relativistic particle 7\nWhatever the form, it should be periodically extended beyond the int erval [0,T).\nIt is then clear that (2) and (8) boil down to\nv=∞/summationdisplay\nk=0/parenleftbigg\n−1\n2/parenrightbiggk(2k−1)!!\nk!M2k+1/integraldisplayT\n0dτ\nT/tildewideP(τ)2k+1. (21)\nA direct comparison of this equation with the functional Taylor serie s (12) yields\nc2k(t1,...,t 2k) = 0, (22a)\nc2k+1(t1,...,t 2k+1) =/parenleftbigg\n−1\n2/parenrightbiggk(2k−1)!!\nk!M2k+1T2ka2k+1(t1,...,t 2k+1),(22b)\nwhere\nam(t1,...,t m) =/integraldisplayT\n0dτe−βm(τ−¯t)m/productdisplay\nk=1χ(tk,τ),¯t=1\nmm/summationdisplay\nk=1tk.(23)\nAs expected [23], functions am(t1,...,t m) are, by construction, T-periodic in each\nvariable and symmetric under any exchange of their arguments.\nThe integral in (23) can be performed integrating by parts and tak ing into account\nthatd\ndτχ(y,τ) =δ(τ−y) (a Dirac delta). The result is\nam(t1,...,t m) =eβm¯t\nβm/braceleftBiggm/productdisplay\nk=1χ(tk,0)−m/productdisplay\nk=1χ(tk+T,0)+m/summationdisplay\nj=1e−βmtjm/productdisplay\nk=1, k/negationslash=jχ(tk,tj)/bracerightBigg\n,(24)\nwhere we have used the fact that χ(tk,T)e−βT=χ(tk+T,0). As usual, empty products\nare assumed to be 1 (the case of the last term for m= 1).\nThe limit β→0 of this expression is better obtained by replacing χ(y,t) byχ1(y,t)\nin (23) and integrating by parts again. This results in\nam(t1,...,t m) =Tm/productdisplay\nk=1χ1(tk,T)−m/summationdisplay\nj=1tjm/productdisplay\nk=1, k/negationslash=jχ1(tk,tj). (25)\nFinally, in the overdamped case ( M→0,β→ ∞), instead of (6) the evolution of\nPis given by P(t) =−(1/β)f(t), sovcan be expressed simply as\nv=−∞/summationdisplay\nk=0/parenleftbigg\n−1\n2/parenrightbiggk(2k−1)!!\nk!γ2k+11\nT/integraldisplayT\n0dtf(t)2k+1. (26)\nFrom (12) and (26) it follows that c2k(t1,...,t 2k) = 0 and\nc2k+1(t1,...,t 2k+1) =−/parenleftbigg\n−T2\n2/parenrightbiggk(2k−1)!!\nk!γ2k+1δ(t1−t2)···δ(t2k−t2k+1). (27)\n4.1. Forcing with a time-periodic piecewise constant force\nTheexpansion(12)withkernels (22 b)and(23)turnsouttobeuseful toanalyzedifferent\ntypes of forcing. For instance, another standard choice in the lite rature (see [1] and\nreferences therein), alongside with the bi-harmonic force (1), ha s been the time-periodic\npiecewise constant force defined in (3). This force is shift-symmet ric only for Tl=T/2,\nso any other value Tl< T/2 breaks this symmetry and induces a ratchet current.Ratchet effect on a relativistic particle 8\nIn order to ascertain the effect of this force in system (4 b) for small amplitudes\nǫ1≪1, we will compute the first nonzero term in the expansion (12). To t hat purpose\nwe need to evaluate (c.f. equation (23))\nKm≡ /angbracketleftam(t1,...,t m)f(t1)···f(tm)/angbracketright=/integraldisplayT\n0/bracketleftbig\ne−βτI(τ)/bracketrightbigmdτ, (28a)\nI(τ)≡1\nT/integraldisplayT\n0eβtχ2(t,τ)f(t)dt, (28b)\nwhere the choice χ2(t,τ) instead of χ(t,τ) is made because f(t) in (3) has zero average.\nAccording to (20) χ2(t,τ) = (1−e−βT)−1¯χ2(t,τ), where\n¯χ2(t,τ) =/braceleftBigg\n1 if t < τ,\ne−βTift > τ.(29)\nSubstitution into (28 b) yields\nI(τ) =ǫ1\nβT/bracketleftbigg4\n1−e−βTsinh2/parenleftbiggβTl\n2/parenrightbigg\n+Q(τ)/bracketrightbigg\n, (30a)\nQ(τ) =\n\neβτ−eβTl if 0< τ < T l,\n0 if Tl< τ < T −Tl,\neβ(T−Tl)−eβτifT−Tl< τ < T.(30b)\nIt is straightforward to check that K1in (28a) vanishes, so the first term that may\nnot be zero is K3. Lengthy calculations lead to\nK3=−32ǫ3\n1\nβ4T3eβT\n(eβT−1)2sinh2/parenleftbiggβ(T−2Tl)\n2/parenrightbigg\nsinh4/parenleftbiggβTl\n2/parenrightbigg\n, (31)\nthat is to say\nv=4ǫ3\n1\n(βM)3βTsinh2/parenleftbiggβ(T−2Tl)\n2/parenrightbiggsinh4(βTl/2)\nsinh2(βT/2)+o(ǫ3\n1). (32)\nIt is interesting to noticing that K3= 0 ifTl=T/2 because in that case the time-\nperiodic piecewise constant force (3) is shift-symmetric. On the ot her hand, we can\ndetermine the value of Tlfor which the ratchet effect is maximum by differetiating (32).\nThis leads to\nsinh/parenleftbiggβ(T−3Tl)\n2/parenrightbigg\nsinh/parenleftbiggβ(T−2Tl)\n2/parenrightbigg\nsinh3/parenleftbiggβTl\n2/parenrightbigg\n= 0. (33)\nThe only three solutions to this equation are Tl= 0,Tl=T/2 andTl=T/3. The first\ntwo do not produce any ratchet current (with Tl= 0f= 0 whereas for Tl=T/2 the\nforce is shift-symmetric), therefore the last one provides its max imum value.\nAs a final remark, expression (32) has well defined overdamped ( M→0,β→ ∞,\nwith finite γ=βM) and undamped ( β→0) limits. In fact, the undamped limit of (32)\nyields\nv=ǫ3\n1\n4(MT)3T4\nl(T−2Tl)2+o(ǫ3\n1), (34)Ratchet effect on a relativistic particle 9\nwhereas the overdamped produces v=o(ǫ3\n1). Indeed, since f(t)2k+1=ǫ2k\n1f(t), in the\noverdamped case Eq. (26) immediately implies v= 0. This is in marked contrast with\nthe overdamped deterministic dynamic of a particle in a sinusoidal pot ential driven by\na bi-harmonic force [31]. In this case, the zero ratchet velocity ca n be explained as a\nsymmetryeffect. Indeed, noticethat f(t) =−f(−t)whenf(t)isgivenby(3)(something\nthat only happens for the bi-harmonic force (1) for specific choice s of the phases), and\nthat the overdamped limit of Eq. (4 b) remains invariant under a simultaneously action\nof time-reversal and a sign change of xandu(see [23] for further details).\n5. Discussion\nWe have studied the dynamics of a damped relativistic particle under t wo zero\naverageT-periodic forces which breaks the shift-symmetry f(t+T/2) =−f(t). This\nnonlinearsystem canbeexplicitly solvedthroughatransformationt hatrenders itlinear.\nTherefore, theratchetaveragevelocity, v, isexactlyobtainedforanyarbitraryforce f(t).\nThis result allows us to show, first of all, that the ratchet velocity ca nnot be obtained in\ngeneral by using the method of moments (according to which vis obtained as a series of\nthe odd moments of f(t)). And secondly, that vis a functional of f(t), i.e.v[f]. Indeed,\nfor anyT-periodic force we have explicitly found the coefficients of the funct ional Taylor\nexpansion (12). In particular, this expansion shows that the meth od of moments is only\njustified in the strict overdamped limit (see Eqs. (26) and (27)). Du e to the symmetry\nv[f] =−v[−f] only odd terms contribute to the Taylor expansion. Besides, since the\nratchet velocity is translationally invariant, the kernel c1(t1) must be a constant. So the\nfirst order term vanishes because the force has zero average. T herefore, the first term in\nthe expansion (12) that is not necessarily zero is the third one, irre spective of the kind\nof nonlinearity of the system.\nWe have chosen to illustrate this functional representation the bi- harmonicforce(1)\n(withp= 2 andq= 1) as well as a time-periodic piecewise constant force (3). We have\nobtained the leading term of the average velocity for both these fo rces. They are given\nby equations (9)–(10) and (32), respectively. It is worth emphas izing that the method\nof moments always predicts a zero ratchet velocity when the syste m is driven by a time-\nperiodic piecewise constant. This is to be compared with the result (3 2) obtained here.\nWe have discussed the two limiting dynamics: undamped and overdamp ed. In these two\nlimits the system remains invariant if the driving force has the symmet riesf(t) =f(−t)\nandf(t) =−f(−t), respectively. If the relativistic particle is driven by a bi-harmonic\nforcev∼ǫ2\n1ǫ2cos(2φ1−φ2)intheoverdampedlimit, whereas v∼ǫ2\n1ǫ2sin(2φ1−φ2)inthe\nundamped limit. In the latter case this means no ratchet current if w e setφ1=φ2= 0.\nThe unexpected consequence of this is that introducing damping generates a ratchet\ncurrent, whose intensity grows up to a maximum before it drops to z ero upon a further\nincrease of the damping. This effect is a result of a trade off between symmetry effects\nand friction and our prediction is that should be observed in any syst em with damping\nand forced with a time-reversible external force.Ratchet effect on a relativistic particle 10\nOn its side, if an overdamped relativistic particle is driven by a time-per iodic\npiecewise constant force like (3), the ratchet velocity is always zer o as a consequence of\nthe symmetry f(t) =−f(−t) exhibited by this force.\nSummarizing, we hope to have illustrated the predictive power of the Taylor\nfunctional expansion method introduced in [23]. This working examp le also shows\nthat this is the only reliable method to analyze the ratchet current a s a function of\nthe parameters of the external force. The most widely used alter native so far, the\nmethod of moments, is shown to work only in the overdamped limit of th e dynamics of\na relativistic particle driven by a periodic force. When damping is finite a nd the forcing\nof the system is bi-harmonic, the rathet current predicted by the method of moments\nstill retains some relevant features of the exact one (9). Howeve r, it dramatically fails\nif the system is driven by a piecewise constant force, because it alwa ys predicts a zero\nratchet current, in marked constrast with the result (32) predic ted by the functional\nTaylor expansion.\nAcknowledgments\nWe acknowledge financial support through grants MTM2009-1274 0-C03-02 (R.A.N.),\nFIS2008-02380/FIS (N.R.Q.), and MOSAICO (J.A.C.) (from Ministerio d e Educaci´ on\ny Ciencia, Spain), grants FQM262 (R.A.N.), FQM207 (N.R.Q.), and P09-F QM-4643\n(N.R.Q., R.A.N.) (from Junta de Andaluc´ ıa, Spain), and project MODEL ICO-CM\n(J.A.C.) (from Comunidad de Madrid, Spain).\nReferences\n[1] P. Reimann. Phys. Rep. , 361:57, 2002.\n[2] M. Salerno and N. R. Quintero. Soliton ratchets. Phys. Rev. E , 65:025602(R), 2002.\n[3] M. O. Magnasco. Phys. Rev. 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Symme tries shape the current in ratchets\ninduced by a bi-harmonic force. Phys. Rev. E , 81:030102R, 2010.\n[24] C. E. Skov and E. Pearlstein. Rev. Sci. Inst. , 35:962, 1964.\n[25] S. Denisov, S. Flach, A. A. Ovchinnikov, O. Yevtushenko, and Y . Zolotaryuk. Phys. Rev. E ,\n66:041104, 2002.\n[26] O. H. Olsen and M. R. Samuelsen. Phys. Rev. B , 28:210, 1983.\n[27] L. Morales-Molina, N. R. Quintero, A. S´ anchez, and F. G. Mert ens.Chaos, 16:013117, 2006.\n[28] R.F.CurtainandA.J.Pritchard. Functional Analysis in Modern Applied Mathematics . Academic\nPress, London, 1977.\n[29] J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman. The Theory of Critical\nPhenomena: An Introduction to the Renormalization Group . Oxford University Press, Oxford,\n1992.\n[30] J.P.HansenandI. R.McDonald. Theory of Simple Liquids . AcademicPress,London, 3rdedition,\n2006.\n[31] V. Lebedev D. Cubero and F. Renzoni. Current reversals in a ro cking ratchet: Dynamical versus\nsymmetry-breaking mechanisms. Phys. Rev. E , 82:041116, 2010."    },    {        "title": "1106.5808v2.Stability_of_precessing_domain_walls_in_ferromagnetic_nanowires.pdf",        "content": "arXiv:1106.5808v2  [cond-mat.mtrl-sci]  6 Oct 2011Stability of precessing domain walls in ferromagnetic nano wires\nYan Gou1, Arseni Goussev1,2, JM Robbins1, Valeriy Slastikov1\n1School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom\n2Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Straße 38, D-01187 Dresden, Germany\n(Dated: November 21, 2018)\nWe show that recently reported precessing solution of Landa u-Lifshitz-Gilbert equations in ferro-\nmagnetic nanowires is stable under small perturbations of i nitial data, applied field and anisotropy\nconstant. Linear stability is established analytically, w hile nonlinear stability is verified numerically.\nPACS numbers: 75.75.-c, 75.78.Fg\nI. INTRODUCTION\nThe manipulation and control of magnetic domain\nwalls (DWs) in ferromagnetic nanowires has recently be-\ncome a subject of intense experimental and theoretical\nresearch. The rapidly growing interest in the physics of\nthe DW motion can be mainly explained by a promising\npossibility of using DWs as the basis for next-generation\nmemoryandlogicdevices1–5. However,inordertorealize\nsuch devices in practice it is essential to be able to posi-\ntion individual DWs precisely along magnetic nanowires.\nGenerally, this can be achieved by either applying ex-\nternal magnetic field to the nanowire, or by generating\npulses of spin-polarized electric current. The current\nstudy is concerned with the former approach.\nEventhoughthephysicsofmagneticDWmotionunder\nthe influence of external magnetic fields has been studied\nfor more than half a century6–9, currentunderstanding of\nthe problem is far from complete and many new phenom-\nena have been discovered only recently10–14. In particu-\nlar, a new regime has been reported13,14in which rigid\nprofile DWs travel along a thin, cylindrically symmetric\nnanowire with their magnetization orientation precessing\naround the propagation axis. In this paper we address\nthe stability of the propagation of such precessing DWs\nwith respect to perturbations ofthe initial magnetization\nprofile, some anisotropy properties of the nanowire, and\napplied magnetic field.\nLetm(x) = (cosθ(x),sinθ(x)cosφ(x),sinθ(x)sinφ(x))\ndenote the magnetization along a one-dimensional wire.\nWith easy magnetization axis along ˆ xand hard axis\nalongˆ y, the micromagnetic energy is given by15\nE(m) =1\n2/integraldisplay/parenleftBig\nAm′2+K1(1−m2\n1)+K2m2\n2/parenrightBig\ndx\n=1\n2/integraldisplay/parenleftBig\nAθ′2+sin2θ(Aφ′2+K1+K2cos2φ)/parenrightBig\ndx(1)\nwhereAis the exchange constant and K1,K2the\nanistropy constants. Here and in what follows, integrals\nare taken between −∞and∞(for the sake of brevity,\nlimits of integration will be omitted).\nWe consider here the case of uniaxial anisotropy, K2=0. Minimizers of Esubject to the boundary conditions\nlim\nx→±∞m(x) =±ˆ x, (2)\ndescribe optimal profiles for a domain wall separating\ntwo magnetic domains with opposite orientation. The\noptimal profiles satisfy the Euler-Lagrange equation\nm×H= 0, (3)\nwhere\nH=−δE\nδm=Am′′+K1(m·ˆ x)ˆ x=−e0m+e1n+e2p.\n(4)\nHerem,n=∂m/∂θandp=m×nformanorthonormal\nframe, and the components of Hin this frame are given\nby\ne0=Aθ′2+sin2θ(K1+Aφ′2)\ne1=Aθ′′−1\n2sin2θ(K1+Aφ′2),\ne2=Asinθφ′′+2Acosθθ′φ′. (5)\nIn terms of these components, the energy Eq. (1) (with\nK2= 0) is given by\nE(m) =1\n2/integraldisplay\ne0dx, (6)\nand the Euler-Lagrange equation becomes e1=e2= 0.\nWhile the energy Eis invariant under translations\nalong and rotations about the x-axis, the optimal pro-\nfiles cannot be so invariant (because of the boundary\nconditions). Instead, the optimal profiles form a two-\nparameter family obtained by applying translations, de-\nnotedT(s), and rotations, denoted R(σ), to a given op-\ntimal profile m∗. We denote the family by T(s)R(σ)m∗.\nIn polar coordinates, T(s)R(σ)m∗is given by φ(x) =σ\n(the optimal profile lies in a fixed half-plane), and θ(x) =\nθ∗((x−s)/d0), where d0=/radicalbig\nA/K1and\nθ∗(ξ) = 2tan−1(e−ξ). (7)\nIt is clear that θ∗(ξ) satisfies\nθ′\n∗=−sinθ∗,sinθ∗(ξ) = sechξ. (8)2\nThe dynamics of the magnetization in the presence of\nan applied magnetic field is described by the Landau-\nLifschitz-Gilbert equation16, which for convenience we\nwrite in the equivalent Landau-Lifschitz (LL) form,\n˙m=m×(H+Ha)−αm×(m×(H+Ha)).(9)\nHereα >0 is the damping parameter, and we take the\napplied field to lie along ˆ x,\nHa=H1(t)ˆ x. (10)\nIn polar coordinates, the LL equation is given by\n˙θ=αe1−e2−αH1sinθ, (11)\nsinθ˙φ=e1+αe2−H1sinθ. (12)\nTheprecessingsolutionisatime-dependenttranslation\nand rotation of an optimal profile, which we write as\nT(x0(t))R(φ0(t))m∗. The centre x0(t) and orientation\nφ0(t)ofthedomainwallfortheprecessingsolutionevolve\naccording to\n˙x0=−αd0H1,˙φ0=−H1. (13)\nIt was shown13,14thatT(x0)R(φ0)m∗satisfies the LL\nequation.\nIt is important to note that the precessing so-\nlution is fundamentally different from the so-called\nWalker solution8. Indeed, the latter is defined\nonly for K2>0 (the fully anisotropic case) and\ntime-independent H1less than the breakdown field\nHW=αK2/2. The Walkersolutionisgivenby m(x,t) =/parenleftbig\ncosθW(x,t),sinθW(x,t)cosφW,sinθW(x,t)sinφW/parenrightbig\nwith\nθW(x,t) =θ∗/parenleftbig\nγ−1(x−VWt)/parenrightbig\n, (14)\nsin2φW=H1/HW, (15)\nand\nVW=γ(α+α−1)d0H1, (16)\nγ=/parenleftbiggK1\nK1+K2cos2φW/parenrightbigg1\n2\n. (17)\nEquations (14)-(17) describe a DW traveling with a\nconstant velocity VWwhose magnitude cannot exceed\nγ(α+α−1)d0HW; note that VWdoes not depend lin-\nearly on the applied field H1. In contrast, the velocity ˙ x0\nof the precessing solution is proportional to H1, and can\nbe arbitrarily large. Also, while for the Walker solution\nthe plane of the DW remains fixed, for the precessing\nsolution it rotates about the nanowire at a rate propor-\ntional to H1. Finally, for the Walker solution, the DW\nprofile contracts ( γ <1) in response to the applied field,\nwhereas for the precessing solution the DW profile prop-\nagates without distortion.\nIn this paper we consider the stability of the precess-\ning solution. We establish linear stability with respectto perturbations of the initial optimal profile (Sec. II),\nsmall hard-axis anisotropy (Sec. III), and small trans-\nverse applied magnetic field (Sec. IV); specifically, we\nshow, to leading order in the perturbation parameter,\nthat up to translation and rotation, the perturbed solu-\ntion converges to the precessing solution (in the case of\nperturbed initial conditions) or stays close to it for all\ntimes (for small hard-axis anisotropy and small trans-\nverse magnetic field). The argument is based on consid-\nerations of energy, and depends on the fact that for all\nt, the precessing solution belongs to the family of global\nminimizers. The analytic argument establishes only lin-\near stability. Nonlinear stability is verified numerically\nfor all three cases in Sec. V. For convenience we choose\nunits so that A=K1= 1.\nII. PERTURBED INITIAL PROFILE\nLetmǫ(x,t) denote the solution of the LL equation\nwith initial condition m∗+ǫµ, a perturbation of an opti-\nmal profile. Let T(xǫ(t))R(φǫ(t))m∗denote the optimal\nprofile which, at time t, is closest to mǫ; that is, the\nquantity\n||mǫ−T(s)R(σ)m∗||2=/integraldisplay/parenleftbig\nmǫ(x,t)−R(σ)m∗(x−s)/parenrightbig2dx\n(18)\nis minimized for s=xǫ(t) andσ=φǫ(t). Then the\nfollowing conditions must hold:\n/integraldisplay\nmǫ·/parenleftbigg\nT(xǫ(t))R(φǫ(t))∂m∗\n∂x/parenrightbigg\ndx= 0,\n/integraldisplay\nmǫ·(ˆ x×T(xǫ(t))R(φǫ(t))m∗)dx= 0.(19)\nIt is clear that xǫ(t) =x0(t) +O(ǫ) andφǫ(t) =\nφ0(t) +O(ǫ), but we shall not explicitly calculate the\nO(ǫ) corrections produced by the perturbation. Rather,\nour approach is to show that to leading order O(ǫ2),\n||mǫ−T(xǫ)R(φǫ)m∗||2decays to zero with t. This will\nimply that the precessingsolution is linearlystable under\nperturbations of initial conditions up to translations and\nrotations.\nLetθǫ(x,t) andφǫ(x,t) denote the spherical coordi-\nnates of mǫ(x,t). We expand these in an asymptotic\nseries,\nθǫ(x,t) =θ∗(x−xǫ(t)) +ǫθ1(x−xǫ(t),t)+···,\nφǫ(x,t) =φ∗(t) + ǫφ1(x−xǫ(t),t)+···(20)\nwhere the correction terms θ1(ξ,t),φ1(ξ,t), etc are ex-\npressedin areferenceframe movingwith the domain wall\nThen to leading order O(ǫ2),\n||mǫ−T(xǫ)R(φǫ)m∗||2=ǫ2/integraldisplay\n(θ2\n1+sin2θ∗φ2\n1)dξ\n=ǫ2/angbracketleftθ1|θ1/angbracketright+ǫ2/angbracketleftsinθ∗φ1|sinθ∗φ1/angbracketright,(21)3\nwhere for later convenience we have introduced Dirac no-\ntation, expressing the integral in Eq. (21) in terms of\ninner products. It is straightforward to show that the\nconditions Eq. (19) imply (using θ′\n∗=−sinθ∗)) that\n/angbracketleftsinθ∗|θ1/angbracketright=/angbracketleftsinθ∗|sinθ∗φ1/angbracketright= 0,(22)\nwhich expressesthe fact that the perturbations described\nbyθ1andφ1are orthogonal to infinitesimal translations\n(described by sin θ∗) along and rotations about ˆ x.\nSince the difference between mǫandT(xǫ)R(φǫ)m∗\nisO(ǫ), the difference in their energies is O(ǫ2) (as\nT(xǫ)R(φǫ)m∗satisfies the Euler-Lagrange equation\nEq. (3)), and is given to leading order by the second\nvariation of Eaboutm∗,\n∆Eǫ=E(mǫ)−E(T(xǫ)R(φǫ)m∗) =\nE(mǫ)−E(m∗) =ǫ2\n2/integraldisplay\nf0dξ, (23)\nwheref0=θ′\n12+cos2θ∗θ2\n1+sin2θ∗φ′\n12.\nUsing the relations Eq. (8) and performing some integra-\ntions by parts, we can write\n/integraldisplay\nf0dξ=/angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright,(24)\nwhereHistheSchr¨ odingeroperator −d2/dξ2+V(ξ) with\npotential given by\nV(ξ) = 1−2sech2ξ. (25)\nV(ξ) is a particular case of the P¨ oschl-Teller potential,\nforwhichthespectrumof Hisknown17.Hhastwoeigen-\nstates, namely sin θ∗(ξ) = sechξwith eigenvalue λ0= 0,\nand cosθ∗(ξ) = tanh ξwith eigenvalue λ1= 1, and its\ncontinuous spectrum is bounded below by λ= 1. This\nis consistent with the fact that the optimal profiles are\nglobal minimizers of E(subject to the boundary condi-\ntions Eq. (2)), which implies that the second variation\nofEaboutm∗is positive for variations transverse to\ntranslations and rotations of m∗. It follows that, for any\n(smooth) square-integrable function f(ξ) orthogonal to\nsinθ∗, we have that\n/angbracketleftf|Hj+1|f/angbracketright ≥ /angbracketleftf|Hj|f/angbracketright (26)\nforj≥0(wewillmakeuseofthisfor j= 0andj= 1). In\nparticular, since θ1and sinθ∗φ1are orthogonal to sin θ∗\n(cf Eq. (22)), it follows that\n/angbracketleftθ1|H|θ1/angbracketright ≥ /angbracketleftθ1|θ1/angbracketright, (27)\n/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright ≥ /angbracketleftsinθ∗φ1|sinθ∗φ1/angbracketright.(28)\nTherefore, from the preceding Eqs. (27)–(28) and\nEqs. (21) and (23)–(24), we get, to leading order O(ǫ2),\nthat\n||mǫ−T(xǫ)R(φǫ)m∗||2≤2∆Eǫ.(29)Below we show that, to leading order O(ǫ2), for small\nenoughH1(it turns out that |H1|<1/2 is sufficient), we\nhave the inequality\nd\ndt∆Eǫ≤ −γ∆Eǫ (30)\nfor some γ >0. Taking Eq. (30) as given, it follows from\nthe Gronwall inequality that\n∆Eǫ≤1\n2Cǫ2e−γt(31)\nfor some C >0 (which depends only on the form of the\ninitial perturbation). From Eq. (29), it follows that\n||mǫ−T(xǫ)R(φǫ)m∗||2≤Cǫ2e−γt.(32)\nThe result Eq. (32) shows that, to O(ǫ2),mǫconverges\ntoanoptimalprofilewithrespecttothe L2-norm. Infact,\nwith a small extension of the argument, we can also show\nthat, to O(ǫ2),mǫconverges to an optimal profile uni-\nformly (that is, with respect to the L∞-norm). Indeed,\nmaking use of the preceding estimates, one can obtain\na bound on ||m′\nǫ−T(xǫ)R(φǫ)m′\n∗||, theL2-norm of the\ndifference in the spatial derivatives of the perturbed so-\nlution and the optimal profile. To O(ǫ2),\n||m′\nǫ−T(xǫ)R(φǫ)m′\n∗||2\n=ǫ2(/angbracketleftθ′\n1|θ′\n1/angbracketright+/angbracketleftsinθ∗φ′\n1|sinθ∗φ′\n1/angbracketright+/angbracketleftsinθ∗θ1|sinθ∗θ1/angbracketright)\n≤ǫ2(3(/angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright)\n≤6ǫ2∆Eǫ.(33)\nArguing as in Eqs. (29)–(32), we may conclude that\n||m′\nǫ−T(xǫ)R(φǫ)m′\n∗||decays exponentially with t.\nThus,mǫconverges to an optimal profile with respect\nto the Sobolev H1-norm (where ||f||2\nH1=||f||2+||f′||2).\nIt is a standard result that this implies that the conver-\ngence is also uniform (again, to O(ǫ2)).\nIt remainsto establish Eq. (30). From Eq. (9), we have\nthat for any solution m(x,t) of the LL equation,\nd\ndtE(m) =−/integraldisplay\nH·˙ mdx\n=/integraldisplay\n(m×H)·Hadx−\n−α/integraldisplay/parenleftbig\nm×H)2+(m×H)·(m×Ha/parenrightbig\ndx\n=−α/integraldisplay/parenleftbig\ne2\n1+e2\n2+H1sinθe1/parenrightbig\ndx,(34)\nwheree1ande2are given by Eq. (5), and we have used\nthe fact that the term ( m×H)·Havanishes on inte-\ngration. Substituting the perturbed solution mǫinto\nEq.(34)andnotingthatthe E(T(xǫ)R(φǫ)m∗) =E(m∗)\ndoes not vary in time, we obtain after some straightfor-4\nward manipulation that\nd\ndt∆Eǫ=\n−αǫ2/parenleftbig\n/angbracketleftθ1|H2|θ1/angbracketright+/angbracketleftsinθ∗φ1|H2|sinθ∗φ1/angbracketright+H1F/parenrightbig\n(35)\nto leading O(ǫ2), where\nF=/integraldisplay/parenleftbig\ncosθ∗f0+cosθ∗sin2θ∗θ2\n1/parenrightbig\ndξ.(36)\nFor the first two terms on the rhs of Eq. (35), we have,\nfrom Eq. (26) and Eqs. (23)–(24), that\n/angbracketleftθ1|H2|θ1/angbracketright+/angbracketleftsinθ∗φ1|H2|sinθ∗φ1/angbracketright\n≥ /angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright\n=2\nǫ2∆Eǫ.(37)\nThe term H1Fin Eq. (35) is not necessarily positive,\nasH1can have arbitrary sign. But for sufficiently\nsmall|H1|, it is smaller in magnitude than the preceding\ntwo terms. Indeed, we have, again using Eq. (26) and\nEqs. (23)–(24), that\n|F| ≤/integraldisplay/parenleftbig\n|f0|+θ12/parenrightbig\ndξ≤2\nǫ2∆Eǫ+/angbracketleftθ1|θ1/angbracketright\n≤2\nǫ2∆Eǫ+/angbracketleftθ1|H|θ1/angbracketright ≤4\nǫ2∆Eǫ.(38)\nSubstituting Eqs. (37) and (38) into Eq. (35), we get that\nd\ndt∆Eǫ≤ −2α(1−2|H1|)∆Eǫ, (39)\nfrom which the required estimate (30) follows for |H1|<\n1/2.\nIt is to be expected that the stability of the pre-\ncessing solution depends on the applied field not being\ntoo large. Indeed, it is easily shown that, for H1>1\n(resp.H1<−1), the static, uniform solution m=−ˆ x\n(resp. m= +ˆ x) becomes linearly unstable. As the\nprecessing solution is nearly uniform away from the do-\nmain wall, one would expect it to be similarly unstable\nfor|H1|>1. The numerical results of Sec. VA bear this\nout. Finally, we remark that the stability criterion ob-\ntained here, namely |H1|<1/2, is certainly not optimal.\nIII. SMALL HARD-AXIS ANISOTROPY\nNext we suppose the hard-axis anisotropy is small but\nnonvanishing, taking K2=ǫ >0. Letmǫ(x,t) denote\nthe solution of the LL equation with initial condition\nmǫ(x,0) =m∗(x). As above, let T(xǫ(t))R(φǫ(t))m∗\ndenote the translated and rotated optimal profile closest\ntomǫat timet. Adapting the argument of the preceding\nsection, we show below that, to leading order O(ǫ2),\n||mǫ−T(xǫ)R(φǫ)m∗||2≤C2ǫ2for allt >0 (40)for some constant C2>0. In contrast to the preceding\nresult Eq. (32) for perturbed initial conditions, here we\ndo not expect mǫto converge to T(xǫ)R(φǫ)m∗. Indeed,\nwhile an explicit analytic solution of the LL equation is\nnot available for small K2(the Walker solution is valid\nonly forK2>2|H1|/α), it is easily verified that there are\nno exact solutions of the form T(xǫ(t))R(φǫ(t))m∗. The\nresultEq.(40)demonstratesthat, throughlinearorderin\nǫ, the solution for K2=ǫremains close to the precessing\nsolution, up to translation and rotation.\nTo proceed, let ∆ Eǫdenote, as above, the difference in\ntheuniaxialmicromagnetic energy, i.e. the energy given\nby Eq. (1) with K2= 0, between mǫandT(xǫ)R(φǫ)m∗.\nThen, as in Eq. (29), we have that\n||mǫ−T(xǫ)R(φǫ)m∗||2≤2∆Eǫ.(41)\nAsE(T(xǫ)R(φǫ)m∗) =E(m∗) is constant in time, we\nhave that\nd\ndt∆Eǫ=d\ndtE(mǫ). (42)\nThe hard-axis anisotropy affects the rate of change of\nthe uniaxial energy through additional terms in ˙ m. In-\ndeed, foranysolution m(x,t)oftheLLequation, wehave\nthat\nd\ndtE(m) =d\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nK2=0E(m)+G(m),(43)\nwhered/dt|K2=0E(m) denotes the rate of change when\nK2= 0, as given by Eq. (34), and\nG(m) =−ǫ/integraldisplay\nR(m·ˆy)(m×H(m))·ˆydx\n+ǫα/integraldisplay\n(m×H(m))·(m׈y)(m·ˆy)dx.(44)\nTakingm=mǫ, we recall from the preceding section\n(c.f. Eq. (30)) that, for |H1|<1/2,\nd\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nK2=0E(mǫ)≤ −γ∆Eǫ (45)\nfor some γ >0. Below we show that there exists con-\nstantsC1,γ1withγ1< γsuch that\n|G(mǫ)| ≤γ1∆Eǫ+C1ǫ2. (46)\nTaking Eq. (46) as given and substituting it along with\nEq. (45) into Eqs. (42)–(43), we get that\nd\ndt∆Eǫ≤ −(γ−γ1)∆Eǫ+C1ǫ2.(47)\nFrom Gronwall’s equality it follows that\n∆Eǫ≤C1\nγ−γ1ǫ2, (48)5\nwhich together with Eq. (41) yields the required result\nEq. (40).\nIt remains to show Eq. (46). Substituting the asymp-\ntotic expansion Eq. (20), we obtain after straightforward\ncalculations that, to leading order O(ǫ2),\nG(mǫ) =−ǫ2cos2φ∗(t)\n×/integraldisplay/parenleftbig\nsin4θ∗φ′\n1+4/3αsin3θ∗θ′\n1/parenrightbig\ndξ.(49)\nThis can be estimated using the elementary inequality\n2|ab| ≤βa2+b2\nβ, (50)\nwhich holds for any β >0. Indeed, recalling Eqs. (8),\n(23), (27), and using integration by parts where neces-\nsary, we have that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nsin4θ∗φ′\n1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β\n2/integraldisplay\nsin2θ∗φ′\n12dξ+1\n2β/integraldisplay\nsin6θ∗dξ\n≤β\nǫ2∆Eǫ+8\n15β,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nsin3θ∗θ′\n1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β\n2/integraldisplay\nθ′\n12dξ+1\n2β/integraldisplay\nsin6θ∗dξ\n≤β\nǫ2∆Eǫ+8\n15β. (51)\nFrom Eqs. (49)–(51), it is clear that β,γ1andC1can be\nchosen so that Eq. (46) is satisfied.\nIV. SMALL TRANSVERSE APPLIED FIELD\nSuppose the applied magnetic field has a small trans-\nverse component, so that Ha=H1ˆ x+H2ˆ y, where\nH2=ǫh2(x) (52)\n(h2depends on xbut not t). For simplicity, let K2=\n0. Letmǫ(x,t) denote the solution of the LL equation\nwith initial condition mǫ(x,0) =m∗(x). As above, let\nT(xǫ(t))R(φǫ(t))m∗denote the translated and rotated\noptimal profile closest to mǫat timet.\nWe first note that, unless h2vanishes as x→ ±∞,\nmǫwillnotremain close to T(xǫ(t))R(φǫ(t))m∗. For\nexample, if h2is constant, then away from the domain\nwall,mǫwill relax to one of the local minimizers of the\nhomogeneous energy K1(1−m2\n1)−Ha·m, and these\ndo not lie along ±ˆ xforH2/negationslash= 0. It follows that ||mǫ−\nT(xǫ(t))R(φǫ(t))m∗||will diverge with time.\nPhysically, this divergence is spurious. It stems from\nthe fact that we are taking the wire to be of infinite ex-\ntent. One way to resolve the issue, of course, would be to\ntake the wire to be of finite length. However, one would\nthen no longer have an explicit analytic solution of the\nLL equation.Here we shall take a simpler approach, and assume\nthat the transverse field h2(x) approaches zero as xap-\nproaches ±∞. In fact, for technical reasons, it will be\nconvenienttoassume that the integralof h2\n2+h′\n22, i.e. the\nsquared Sobolev norm ||h2||H1, is finite. Then without\nloss of generality, we may assume\n||h2||2\nH1=/integraldisplay\n(h2\n2+h′\n22)dξ= 1. (53)\nUnder this assumption, the main result of this section is\nthatmǫstaysclosetoanoptimalprofileuptotranslation\nand rotation. That is, for some C1>0,\n||mǫ−T(xǫ)R(φǫ)m∗||2≤C1ǫ2.(54)\nThe demonstration proceeds as in the preceding sec-\ntion, so we will discuss only the points at which the\npresent case is different. The main difference is that,\nin place of Eq. (49), we get (by considering the LL equa-\ntion with H2/negationslash= 0 rather than K2/negationslash= 0) the following\nexpression for G(mǫ) to leading order O(ǫ2):\nG(mǫ) =ǫ2/parenleftBig\nαcosφ∗(t)/integraldisplay\ncosθ∗(θ′′\n1−cos2θ∗θ1)h2dξ\n−αsinφ∗(t)/integraldisplay\nsinθ∗(φ′′\n1−2cosθ∗φ′\n1)h2dξ\n−sinφ∗(t)/integraldisplay\n(θ′′\n1−cos2θ∗θ1)h2dξ\n−cosφ∗(t)/integraldisplay\nsinθ∗cosθ∗(φ′′\n1−2cosθ∗φ′\n1)h2dξ/parenrightBig\n.\n(55)\nAfter some straightforward manipulations including in-\ntegration by parts, and making use of the inequality\nEq. (50), one can show that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ncosθ∗(θ′′\n1−cos2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β\n2/bardblθ1/bardbl2\nH1+1\n2β,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nsinθ∗(φ′′\n1−2cosθ∗φ′\n1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β\n2||sinθ∗φ′\n1||2+1\n2β,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n(θ′′\n1−cos2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β\n2/bardblθ1/bardbl2\nH1+1\n2β,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nsinθ∗cosθ∗(φ′′\n1−2cosθ∗φ′\n1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤β\n2||sinθ∗φ′\n1||2+1\n2β.(56)\nFrom Eqs. (23), (24) and (27) it follows that\n/integraldisplay/parenleftBig\nθ′\n12+sin2θ∗φ′\n12/parenrightBig\ndξ≤4\nǫ2∆Eǫ,(57)\nand\n/integraldisplay\nθ2\n1dξ≤2\nǫ2∆Eǫ. (58)6\nSubstituting Eqs. (56)–(58) into Eq. (55), we get that\n|G(mǫ)| ≤(1+α)/parenleftbigg\n3β∆Eǫ+1\nβǫ2/parenrightbigg\n.(59)\nThis estimate is of the same form as (46), and the ar-\ngument given there, with βchosen appropriately, estab-\nlishes Eq. (54).\nV. NUMERICAL STUDIES\nIn the preceding Sections II–IV we have shown that\nthe precessing solution is linearly stable; to leading or-\nderO(ǫ), a perturbed solution either approaches or stays\nclose to the precessing solution up to a translation and\nrotation, according to whether the perturbation is to the\ninitial conditions or to the anistropy and transverse ap-\nplied magnetic field in the LL equation. Here we present\nnumerical results which verify nonlinear stability for the\nprecessing solution under small perturbations. To this\nend, weinvestigatetheenergy,∆ Eǫ=E(mǫ)−E(m∗), of\nthe numerically computed perturbed DW, mǫ(x,t), rela-\ntive to the minimum energy E(m∗) of an optimal profile,\nas a function of time t. Throughout, Eis taken to be\ntheuniaxial micromagnetic energy given by Eq. (1) with\nK2= 0. As in the preceding sections, we choose units so\nthatA=K1= 1. In these units, E(m∗) = 2. In typical\nferromagnetic microstructures, the value of the Gilbert\ndamping parameter αis known to lie between 0.04 and\n0.22 (see e.g. Ref.18and references within), so we take\nα= 0.1 throughout our numerical study.\nA. Perturbed initial profile\nWe first investigate the evolution of a DW, mǫ(x,t),\nfrom an initial perturbation of an optimal profile. We\ntake the initial condition in polar coordinates to be given\nby\nθǫ(x,0) =θ∗/parenleftbiggx\n1+ǫ1/parenrightbigg\n, φǫ(x) =φ0+ǫ2x,(60)\nwhich corresponds to stretching the unperturbed profile\nalong and twisting it around the axis of the nanowire.\nThe applied field is directed along the nanowire, Ha=\nH1ˆx, and we take K2= 0.\nFigure 1 shows the dependence of the relative energy\n∆Eǫon timetfor different values of the applied field H1.\nThe figure presents 13 curves corresponding, from top to\nbottom, to H1varying from −1.2 to 0 at the increment\nof 0.1. In the initial condition given by Eq. (60), we take\nǫ1= 0.1 andǫ2=π/50.\nFigure 1 clearly indicates that ∆ Eǫ(t) decays exponen-\ntially for weak applied fields, |H1| ≤1/2, in accord with\nthe analytic result Eq. (31). However, for |H1| ∼1, devi-\nations from exponential decay are evident, and the pre-\ncessing solutionappears to become unstable for |H1|/greaterorsimilar1.0 50 100 150 20010−810−610−410−2100\nt∆Eǫ\nFIG. 1: (Color online) Relative energy, ∆ Eǫ(t), of the per-\nturbed DW for 13 different values of the applied field H1. See\ntext for discussion.\nB. Small hard-axis anisotropy\nWe consider next the evolution of a DW from an opti-\nmal profile at t= 0 when the hard-axis anisotropy K2is\nnonvanishing. We fix H1=−0.5.\n0102030405060708000.0010.0020.0030.0040.0050.0060.0070.0080.0090.01\nt∆Eǫ\nFIG. 2: (Color online) Relative energy, ∆ Eǫ(t), of the per-\nturbed DW for 5 different values of the hard-axis anisotropy\nconstant K2. See text for discussion.\nFigure 2 shows the dependence of the relative energy\n∆Eǫon time tfor different values of K2. The figure\npresents 5 curves corresponding, from top to bottom, to\nK2varying from 0 .1 to 0.02 at the decrement of 0 .02.\n(The blue and red colorings alternate to make adjacent\ncurves more easily distinguishable.) It is evident that the\nrelative energy remains small, verifying the linear analy-\nsis of Sec. III.\nFigure 3 shows the maximum value of the relative en-\nergy ∆Eǫ(over the interval 0 ≤t≤80) as a func-\ntion ofK2. Red squares represent numerically computed\nvalues. The black solid curve is the parabola CKK2\n2,\nwithCK= 1.3207 fitted by the method of least squares7\n0 0.02 0.04 0.06 0.08 0.100.0010.0020.0030.0040.0050.0060.0070.0080.0090.01\nK2max(∆ Eǫ)\nFIG. 3: (Color online) Maximum value of the relative en-\nergy ∆Eǫof the perturbed DW as a function of the hard-\naxis anisotropy K2. Numerically computed values are repre-\nsented by (red) squares. The (black) solid curve is a parabol a,\nmax(∆Eǫ) =CKK2\n2withCK= 1.3207, fitted by the method\nof least squares through the data points with K2≤0.04.\nthrough the data points with K2≤0.04. We obtain con-\nvincing confirmation of the leading-order analytical re-\nsult Eq. (48). For larger values of K2, we see departures\nfrom quadratic dependence; for sufficiently large values\nofK2(not shown), the Walker solution was recovered.\nC. Small transverse applied field\nFinally, we address the stability of the precessing solu-\ntion under an applied magnetic field, Ha=H1ˆx+H2ˆy,\nwith a small transverse component, H2(x). As discussed\nin Sec. IV, we want H2(x) to vanish as x→ ±∞. Here\nwe take\nH2(x) =¯H2w(x), (61)\nwherew(x) is equal to one inside the window 0 ≤x≤20\nandvanishesoutside(the argumentofSectionIViseasily\nmodified to establish the linear stability result Eq. (48)\nin this case). We consider the evolution of a DW given\natt= 0 by the optimal profile m∗centred at x= 0. We\ntakeH1=−0.5, so that in the absence of the transverse\nfield, the DW velocity is positive (cf. Eq. (13)) and the\nDW crosses the window. We take K2= 0.\nFigure 4 shows the dependence of the relative energy\n∆Eǫon timetfor different values of the transverse field\namplitude ¯H2. The figure presents 5 curves correspond-\ning, from top to bottom, to ¯H2varying from 0 .1 to 0.02\nat the decrement of 0 .02. (The blue and red colorings al-\nternate to make adjacent curves more easily distinguish-\nable.) The relative energy ∆ Eǫ(t) is presented over the\ntime interval 0 ≤t≤400, which, for small values of\n¯H2, is sufficient for the DW to traverse the spatial win-\ndow 0≤x≤20 (cf. Eq. (13)). The results confirm that05010015020025030035040010−310−210−1100\nt∆Eǫ\nFIG. 4: (Color online) Relative energy, ∆ Eǫ(t), of the per-\nturbed DW for 5 different values of the transverse field am-\nplitude¯H2. See text for discussion.\nthe relative energy of the perturbed magnetization pro-\nfile remains small for small values of ¯H2, in accord with\nthe leading-order results of Section IV.\n0 0.02 0.04 0.06 0.08 0.100.20.40.60.81\n¯H2max(∆ Eǫ)\nFIG. 5: (Color online) Maximum value of the relative energy\n∆Eǫof the perturbed DW as a function of the amplitude\nof the transverse applied field, ¯H2. Numerically computed\nvalues are represented by (red) squares. The (black) solid\ncurve is a parabola, max(∆ Eǫ) =CH¯H2\n2withCH= 99.6586,\nfitted by the method of least squares through the data points\nwith¯H2≤0.04.\nFigure 5 shows the maximum value of the relative en-\nergy ∆Eǫ(over the interval 0 ≤t≤400) as a function of\n¯H2. Red squares represent numerically computed values.\nThe black solid curve corresponds to the parabola CH¯H2\n2\nwithCH= 99.6586 fitted by the method of least squares\nthrough the data points with ¯H2≤0.04. The figure\nprovides a confirmation of the leading-order analytical\nresult of Sec. IV that the maximum relative energy de-\npends quadratically on ¯H2for small ¯H2. Deviations from\nthe parabolic dependence can be seen for ¯H2/greaterorsimilar0.08.8\nVI. CONCLUSIONS\nThe precessing solution is a new, recently reported ex-\nact solution of the Landau-Lifschitz-Gilbert equation. It\ndescribes the evolution of a magnetic domain wall in a\none-dimensional wire with uniaxial anisotropy subject to\na spatially uniform but time-varying applied magnetic\nfield alongthe wire. We haveanalysedthe stabilityofthe\nprecessing solution. We have proved linear stability with\nrespect to small perturbations of the initial conditions aswellastosmallhard-axisanisotropyandsmalltransverse\napplied fields, provided the applied magnetic field along\nthe wire is not too large. We have also carried out nu-\nmerical calculations that confirm full nonlinear stability\nunder these perturbations.\nNumerical calculations suggest that, for sufficiently\nlarge perturbations and applied longitudinal fields, the\nprecessing solution becomes unstable, and new stable so-\nlutions appear. It would be interesting to analyse these\nbifurcationsandstudythesenewregimesforDWmotion.\n1D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson,\nD. Petit, R. P. Cowburn, Science 309, 1688 (2005).\n2R. P. Cowburn, Nature (London) 448, 544 (2007).\n3S. S. P. Parkin, M. Hayashi, L. Thomas, Science 320, 190\n(2008).\n4M. Hayashi, L. Thomas, R. Moriya, C. Rettner,\nS. S. P. Parkin, Science 320, 209 (2008).\n5L. Thomas, R. Moriya, C. Rettner, S. S. P. Parkin, Science\n330, 1810 (2010).\n6L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu-\nnion8, 153 (1935).\n7T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMag.40, 3443 (2004).\n8N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n9A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys.\nRep.194, 117 (1990).\n10M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008).11X. R. Wang, P. Yan, J. Lu, Europhys. Lett. 86, 67001\n(2009).\n12X. R. Wang, P. Yan, J. Lu, C. He, Ann. Phys. (N.Y.) 324,\n1815 (2009).\n13Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104, 037206\n(2010).\n14A. Goussev, J. M. Robbins, V. Slastikov, Phys. Rev. Lett.\n104, 147202 (2010).\n15V. Slastikov and C. Sonnenberg, IMA J. Appl. Math.\nXXX, XXX (2011), doi:10.1093/imamat/hxr019\n16A. Hubert and R. Sch¨ afer, Magnetic Domains: The Anal-\nysis of Magnetic Microstructures (Springer, Berlin, 1998).\n17P.M. Morse and H. Feshbach, Methods of Theoretical\nPhysics, Part I McGraw-Hill, New York, 1953\n18Y. Tserkovnyak and A. Brataas, Phys. Rev. Lett. 88,\n117601 (2002)."    },    {        "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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B 55, 3050 (1997).\n20"    },    {        "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": "1108.5758v1.Magnetization_Dynamics__Throughput_and_Energy_Dissipation_in_a_Universal_Multiferroic_Nanomagnetic_Logic_Gate_with_Fan_in_and_Fan_out.pdf",        "content": " 1Magnetization Dynamics, Throughput and Ener gy Dissipation in a Universal Multiferroic \nNanomagnetic Logic Gate with Fan-in and Fan-out  \n \nMohammad Salehi Fashami1, Jayasimha Atulasimha1* and Supriyo Bandyopadhyay2 \n1Department of Mechanical and Nuclear Engineering, 2Department of Electrical and Computer \nEngineering, Virginia Commonwealth University, Richmond VA 23284, USA. \n \nAbstract  \nThe switching dynamics of a multiferroic nanomagnetic NAND gate with fan-in/fan-out is simulated by \nsolving the Landau-Lifshitz-Gilbert (LLG) equation while  neglecting thermal fluctuation effects. The gate \nand logic wires are implemented with dipole-coupled 2-phase (magnetostrictive/piezoelectric) \nmultiferroic elements that are clocked with electrosta tic potentials of ~50 mV applied to the piezoelectric \nlayer generating 10 MPa stress in the magnetostrictive layers for switching. We show that a pipeline bit \nthroughput rate of ~ 0.5 GHz is achievable with pr oper magnet layout and sinusoidal four-phase clocking. \nThe gate operation is completed in 2 ns with a late ncy of 4 ns. The total (internal + external) energy \ndissipated for a single gate operation at this throughput rate is found to be only ~ 1000 kT in the gate and \n~3000 kT in the 12-magnet array comprising two input and two output wires for fan-in and fan-out. This \nmakes it respectively 3 and 5 orders of magnitude more energy-efficient than complementary-metal-\noxide-semiconductor-transistor (CMOS) based and sp in-transfer-torque-driven nanomagnet based NAND \ngates. Finally, we show that the dissipation in th e external clocking circuit can always be reduced \nasymptotically to zero using increasingly slow adiaba tic clocking, such as by designing the RC time \nconstant to be 3 orders of magnit ude smaller than the clocking period. However, the internal dissipation in \nthe device must remain and cannot be eliminated if we  want to perform fault-tolerant classical computing. \n \nKeywords:  Nanomagnetic logic, multiferroics, straintr onics and spintronics, Landau-Lifshitz-Gilbert . \n----------------------------------------------------------------------------- \n* Corresponding author. E-mail: jatulasimha@vcu.edu   2I. Introduction \n \n A major challenge in designing digital computi ng machinery is to reduce energy dissipation during \nthe execution of a computational st ep since excessive dissipation is the primary impediment to device \ndownscaling envisioned in Moore’s law. A state-of-t he-art CMOS nanotransistor presently dissipates over \n50,000 kT (2×10-16 Joules) of energy at room temperature  to switch in isolation , and over 106 kT (4.2×10-\n15 Joules) to switch in a circuit at a clock rate  of few GHz [1], which makes further downscaling \nproblematic. Therefore, nanomagnet-based computing and signal processing are attracting increasing \nattention. Single-domain nanomagnets are intrinsica lly more energy-efficient than transistors as logic \nswitches and do not suffer from cu rrent leakage that results in  standby power dissipation.  Consequently, \nmagnetic architectures hold the promise of outpacing tr ansistor-centric architect ures in energy-efficient \ncomputing. \nA nanomagnetic logic switch is typically implemen ted with a shape-anisotropic nanomagnet that \nhas two stable magnetization orientations along the easy axis. These two magnetizations encode the logic \nbits 0 and 1. The nanomagnet’s advantage over the transistor accrues from the fact that when the \nmagnetization is flipped, all the spins (information carriers) in a single-domain nanomagnet rotate in \nunison  like a giant classical spin (because of exchange interaction between spins), so that ideally the \nmagnet has but a single  degree of freedom [2]. Accordingly, the minimum energy dissipated in switching \nit non-adiabatically is ~ kTln(1/p)  where p is the static error probability  associated with random switching \ndue to thermal fluctuations. In contrast, all the di fferent electrons or holes (information carriers) in a \ntransistor act independently, so that the minimum  energy dissipated in switching a transistor non-\nadiabatically is NkTln(1/p)  [2], where N is the number of information carri ers (electrons or holes) in the \ntransistor. The advantage of the single-domain nanomagnet thus accrues not from any innate  advantage of \nspin over charge as an information carrier, but from the fact that spins mutually interact in a way which \nreduces the degrees of freedom, and hence the energy dissipation.   3There are numerous ways of impl ementing nanomagnetic logic (NML) [3], [4]. In one type of \narchitecture, termed ‘magnetic quantum cellular auto mata’ [5], logic gates are configured by placing \nnanomagnets in specific geometric patterns on a su rface so that dipole interactions between neighbors \nelicit the desired logic operations on the bits en coded in the magnetization orientations of the \nnanomagnets[4] ，[5]. This approach is the same as that envisioned in the Single Spin Logic (SSL) \nparadigm, where exchange interaction between spin s played the role of dipole interaction between \nmagnets, while up- and down-spin pol arizations encoded logic bits [6]. However, SSL required cryogenic \noperation while NML can operate at room temperature. Curiously, the minimum  energy dissipated per bit \nflip in NML is the same as that in SSL because a si ngle spin and a giant classi cal spin have the same \ndegree of freedom. This is a remarkable feature that makes NML particularly attractive.  Of course, the \nactual energy dissipated per bit flip in NML will always be somewhat higher than kTln(1/p)  because of \ninternal dynamics such as sp in-orbit interaction giving rise to Gilb ert damping within the magnet, but that \nis still a small price to pay for room temperature operation. \na. Magnet switching schemes  \nUnfortunately, the nanomagnet’s advantage over th e transistor will be squandered if the method \nemployed to switch the nanomagnet becomes so ener gy-inefficient that the energy dissipated in the \nswitching circuit vastly exceeds the energy dissipated in the nanomagnet. In the end, this can make \nmagnetic architectures less energy-efficient than tran sistor based architectures, thereby defeating the \nentire purpose of using magnetic switches. Therefore, the switching scheme is vital. \nMagnets are typically switched with either a magnetic  field generated by a current [7], or with spin \ntransfer torque [8], or with domain wall motion indu ced by a spin polarized current [9]. In the first \napproach, a local magnetic field is generated by a local current based on Ampere’s law: \n \ncI Hd l\n    (1) \nwhere the integral is taken around the current loop.  4Now, the minimum magnetic field minH\nrequired to flip a magnet can be estimated by equating the \nmagnetic energy in the field to the energy barrier bEseparating the two magnetization directions \nencoding the bits 0 and 1, i.e.  \n 0m i n s b MHE          (2). \nHere0is the permeability of free space, sMis the saturation magnetization which we assume is 105 A/m \n(typical value for nickel or cobalt), and is the nanomagnet’s volume which we assume is \n100 nm  100 nm  10 nm since such a volume results in a single domain ferromagnet at room \ntemperature. The energy barrier Eb is determined by the static bit error probabilitybEk Tethat we can \ntolerate. For reasonable error probability (< e-30), we will need that 30kTbE . This yields from \nEquations (1) and (2) that minI= 6 mA if we assume that the loop radius is 100 nm so that it can \ncomfortably encircle the magnet. The resistance of loop  will be ~50 ohms if we assume that it is made of \nsilver (lowest resistivity among metals; 2.6 S-cm ) and has a wire radius of 10nm, so that the \nenergy dissipated – assuming that the magnet flips in 1 ns – is at least 1.8 pJ, or ~ 4 108 kT, which \nmakes the magnet switch ~400 times more dissipative than  a state-of-the-art transistor switch because of \nthe inefficient switching scheme. There is also a nother disadvantage; the magnetic field cannot be \nconfined to small spaces, which means that indi vidual magnets cannot be addressed unless the magnet \ndensity is sparse (magnet separation  0.5 m). That not only reduces device density, but might make \ndipole interaction between magnets so weak as to ma ke magnetic quantum cellular automata inoperable. \nTherefore, this method is best adapted to addr essing not individual magnets, but groups of (closely \nspaced) magnets together, as envisioned in ref. [7]. However, that approach makes magnetic quantum \ncellular automata architecture non-pipelined  and hence very slow [10]. In the end, this is clearly a sub-\noptimal method of switching magnetic switches. \nSpin transfer torque (STT) is  better adapted to addressing indi vidual magnets since it switches \nmagnets with a spin polarized current passed dir ectly through the magnet. It dissipates about 108 kT of  5energy to switch a single-domain nanomagnet in ~ 1 ns , even when the energy barrier within the magnet \nis only ~ 30 kT [11]. Thus, it is hardly better than th e first approach in terms of energy efficiency. A more \nefficient method of switching a magnet is by induci ng domain wall motion by passing a spin polarized \ncurrent through the magnet. There is at least one report of switching a multi-domain nanomagnet  in 2 ns \nby this approach while dissipating 104 kT – 105 kT of energy [12]. This makes it 1-2 orders of magnitude \nmore energy-efficient than a transistor in a circuit. \nRecently, we devised a much more efficient magne t switching scheme. We showed that a 2-phase \nmultiferroic nanomagnet, consisting of a piezoelectric  layer elastically coupled with a magnetostrictive \nlayer, can be switched by applying a small voltage of few mV to the piezoelectric layer [13] ，[14]. This \nvoltage generates uniaxial strain in the piezoelectric layer that is transferred almost entirely to the \nmagnetostrictive layer by elastic coupling if the latter layer is much thinner than the former. Uniaxiality \ncan be enforced in two ways: either by applying the electric field in the direction of expansion and \ncontraction ( d33 coupling) or by mechanically clamping th e multiferroic in one direction and allowing \nexpansion/contraction in the pe rpendicular direction through d31 coupling when the voltage is applied \nacross  the piezoelectric layer. The substrate is assumed to  be a soft material (e.g. a polymer) that allows \nuniaxial expansion/contraction. The uniaxial strain/stress will cause the magnetization of the \nmagnetostrictive layer to rotate by a large angle. Su ch rotations can be used for Bennett clocking of NML \ngates for logic bit propagation [13]. In ref. [14] ，[15], we showed that the energy dissipated in the \nmagnet and clock together is a few hundreds of kT for a switching delay of 1 ns or less. This makes it one \nof the most energy-efficient magnet switching schemes. \nIn this paper, we have studied the switching d ynamics of a NAND gate with fan-in/fan-out wires \nimplemented with multiferroic elements and calcu lated the energy dissipation in the entire block \nassuming low enough temperature when effects of thermal fluctuations can be neglected. At room \ntemperature, thermal fluctuations will act as a random magnetic field that will increase the switching error  6probability and mandate higher stress le vels (along with larger energy  dissipation) for reliable gate \noperation. This study is deferred to a future date.  \nThe present paper is organized as follows: In S ection II, the theoretical framework for studying \nmagnetization dynamics in the NAND ga te with fan-in and fan-out wires is discussed. Section III presents \nand discusses simulation results. Section IV discusses energy dissipation and strategies to reduce the \nenergy overhead in the clocking circuit. Fina lly in section V, we present our conclusions. \n \nII. Magnetization Dynamics in a Multiferroic Nano magnetic Logic Gate with Fan-in and Fan-\nout \nConsider an all-multiferroic NAND gate with fan- in and fan-out wires as shown in Fig 1. Each \nmultiferroic element has the shape of an ellip tical cylinder with homogeneous magnetization ()Mrin the \nmagnetostrictive layer. We assume that the piezoelectric layer has a thickness of 40 nm and the \nmagnetostrictive layer has a thickness of 10 nm, wh ich will ensure that strain generated in the \npiezoelectric layer is mostly transferred to the magne tostrictive layer through elastic coupling. There are \nmechanical clamps along the minor axis of the magne t (not shown) that prevent expansion/contraction \nalong that sideward direction, so that application of a voltage across the piezoelectric layer generates \nuniaxial stress along the major axis via the 31dcoupling. The magnets are fabricated on a soft substrate \nthat does not hinder expansion and contraction of the elements along any direction by clamping from \nbelow.  \nIf the planar dimensions of each element are ~ 101. 75 nm × 98.25 nm, the ex change coupling penalty \nprecludes the formation of multi-domain states in the magnetostrictive layer [16], [17] so that we can \nmodel it as a single-domain nanomagnet. The shape anis otropy of the element gi ves rise to an energy \nbarrier of 32 kT (at room temperature) between the two stable orientations along the easy axis (major \naxis) of the ellipse.   7The magnetization dynamics of any nanomagnet under the influence of an effective fieldeffH\n acting \non it is described by the Landau-Lifs hitz -Gilbert (LLG) equation [17]: \n                             eff eff\nsdM tMtHt M t M tHtdt M          \n     (3).                                 \nHere i\neffH\n is the effective magnetic field on the ith nanomagnet, which is the partial derivative of its total \npotential energy ( Ui) with respect to its magnetization ( iM\n), is the gyromagnetic ratio, sMis the \nsaturation magnetization of the magnetostrictive layer and is the Gilbert damping factor [18] associated \nwith internal dissipation in the magnet when  its magnetization rotates. Accordingly, \n                 \n\n0011 i i\neff m i\ns iUtH tU tM Mt      ,          (4).                                 \nwhere  is the volume of any nanomagnet (only that of th e magnetostrictive layer) in the chain shown in \nFig 1. The total potential energy of any element in this chain is given by:          \n       \n2 222 0\n___\n22sin cos sin sin cos2\n3sin sin2dipole dipole\nshape anisotropy\ndipoleij\nis d x x i i d y y i d z z i i\nji\nE\nE\nsi iUt E t M N t t N t t N t\nt  \n  \n\n              \n\n\nother terms\nstress anisotropyi\nEt\n\n   (5).  \nHere \ndipole dipoleijE\nis the dipole-dipole interaction energy due to in teraction between the i-th and j-th magnets, \nshape anisotropyE   is the shape anisotropy ener gy due to the elliptical shape of  the multiferroic element, and \nstress anisotropyE  is the stress anisotropy energy caused by the stress  transferred to the magnetostrictive \nlayer of the multiferroic upon application of an elect rostatic potential to the piezoelectric layer. The \nquantities dk kNare the demagnetization factors in the k-th direction. We assume that the \nmagnetostrictive layer is polycrystalline so that we can neglect magnetocrystalline anisotropy. The \nquantity sis the magnetostrictive coefficient and the quantities and are the polar and azimuthal  8angles of the magnetization vector iM\n. We assume that the major axis ( easy axis) of the ellipse is in the \ny-direction, the minor axis (in-plane hard axis) is in the x-direction and the out-of-p lane hard axis is in the \nz-direction as shown in Fig. 1(b). The ‘other term s’ in the above equation are time-dependent (they \ninclude the chemical potential, the mechanical potential due to stress/strain etc.), but do not depend on \niM\n (or ,ii) and hence do not affect i\neffH\n. \nConsider two adjacent multiferroic elem ents in the chain (labeled as the ith and jth element), whose \nmagnetizations subtend an angle of ,ii and ,j j respectively with the positive z-direction and x-\ndirection in the x-y plane. The dipole-dipole interaction energy is [19]: \n    22\n0\n323.. .4| | | |ij s\ndipole dipole i j i i j j i j\nij ijMEt m t m t m t r m t rrr\n\n \n \n             (6).                                \nwhere  ijr is the vector distance between the  ith and jth magnet and kmis the magnetization of the k-th \nmagnet normalized to Ms. \nFor two neighboring magnets whose in-plane hard  axes are collinear with the line joining their \ncenters, the dipole coupling energy is: \n    \n   \n 22\n0\n32 sin cos sin cos\nsin sin sin sin4\ncos cosii j j\nij s\ndipole dipole i j j i\nj\nijtt t t\nMEt t t t tr\ntt \n \n\n\n \n\n                 (7).   \nIf the line joining the centers subtends an angle γ with their hard axes as shown in Fig 1 (b), the \ndipole coupling energy is:  \n     \n    \n        \n   22\n2222\n0\n3sin cos )(sin cos 2(cos ) (sin )\nsin sin sin sin 2(sin ) (cos )\n4 sin cos sin sin sin cos sin sin\n3sin cos cos cosii j j\nii j jij s\ndipole dipole\nii jj j j i i\nijtt tt\ntt ttMEtr tt tt tt tt\ntt   \n   \n    \n  \n \n\n  \n\n j\n\n\n\n  (8).       \nwhere r is the separation between their centers.   9The total potential energy of a magnet given in Equa tion (5) is used to find the effective field \neffH\nacting on it in accordance with Equation (4): \n \n0\n0 0() 1() s i n () c o s ()()\n() 13() s i n () s i n () () s i n () s i n ()()\n1()ij\ndipole dipole i\neff x s d xx i i\nsx\nji\nij\ndipole dipole i\neff y s d yy i i s i i i bias\nsy s\nji\ni\neff zEtHt M N t tMm t\nEtHt M N t t t t t HMm t M\nHt\n  \n\n\n\n\n\n\n\n \n      \n\n\n\n0()cos ( )()ij\ndipole dipole\nsd z z i\nsz\njiEtMN tMm t\n\n\n (9). \nwhere  biasH is any external time-invariant magnetic field a pplied in the y-direction (see later discussion \nas to when and why it might be needed).  \nFurther, it appears from equation (9) that a stre ss applied in the y-direct ion, produces only a H eff \nalong the y-direction and could not  rotate the magnetization along th e x or z-direction. However, \na deeper analysis of its effect on the magnetization shows this is not true. We will explain this by \nwriting the stress anisotropy ener gy term and the associated Heff in Cartesian co-ordinates as:  \n   2\n_\n033( )        H ( )2stress anisotropy s i y eff y s i y\nsE tm t t m tM    \nNow suppose a compressive stress (negative ) is applied to a material with positive \nmagnetostriction ( λ) and the magnetization is initially direct ed close but not parallel to +y-axis so \nthat m y ~ 1. The Heff  is along the -yaxis and would make th e magnetization rotate away from +y \ntowards the x-direction (also lifting the magnetizati on vector  a little out of  plane; the lifting is \nnegligible when N d_ZZ  is very large) . However, when the magnetization rotates close to the hard \n(x-axis), m y ~0, and the Heff  due to stress vanishes. If the magnetization had initially pointed \nalong the -y-axis, Heff would have been direct ed along the +y-axis, which would have again \nrotated the magnetization away from the -y-axi s towards the in plane hard axis (x-axis). \n \nThe effective field Heff for each nanomagnet described by Equation (9) includes the effect of dipole \ncoupling with neighboring na nomagnets, shape anisotropy and stress anisotropy. For dipole coupling, the \nsummation is performed over nearest, second-nearest and sometimes third-nearest neighbors since the  10interaction with remote neighbors may not be negligib le compared to that with the nearest neighbor. We \nelucidate this below: \nIn Fig 1 (a), consider a magnet marked \"I\". The nearest neighbor marked “II” is at a distance r, but \nthe second nearest neighbor marked 'III' is at a distance of 52r which results in a dipole field \n(proportional to 31r) that is ~25 % of the dipole field due to the magnet marked 'II'. Hence both have to \nbe considered. Similarly, for a magnet marked \"III\", in teractions with magnets marked 'I\", \"II\" and \"IV\", \nwhose centers are respectively at distances of 52r, 32rand r, are considered. For magnets marked \n\"IV\" , \"V\", and \"VII\" both nearest and second nearest  neighbor dipole coupling terms were considered. \nHowever, for the magnet marked \"VI\" only the nearest  neighbors' dipole coupling was considered since \nthe second nearest neighbor is at a distance 2 r away and contributes only 12.5% of the interaction caused \nby the nearest neighbor.  \nThe effective magnetic field Heff evaluated from Equation (9) is used in the scalar version of equation \n(3) which is described in ref. [15] to determine th e magnetization state of any magnet at any instant of \ntime. Equation (3) leads to two coupled ordinary differential equations (ODEs) for each nanomagnet. \nThus, for 12 nanomagnets, 24 coupled ODEs have to be solved simultaneously. This allows us to \ncompute the temporal evolution of the magnetization orientations  ,iitt  of all 12 magnets in Fig \n1.  \nIn order to flip the magnetization of any magnet,  it is first subjected to stress with a voltage V applied \nacross the piezoelectric layer. The si gn of the stress (compressive or tensile) depends on the sign of the \nmagnetostrictive coefficient sof the material. For a material with positives, such as Terfenol-D, we \napply a compressive (negative) stress to rotate th e magnetization away from the easy axis (y-axis) \ntowards the in-plane hard axis (x-axis). If the str ess is of sufficient magnitude to overcome the shape \nanisotropy energy barrier, the magnetization will ultima tely align along the in-plane hard axis (x-axis). \nOnce that happens, the stress is reversed to tensile so  that the magnetization relaxes back to the easy axis  11(y-axis), but in a direction anti-parallel to the orig inal orientation along the easy axis. What ensures the \nanti-parallel orientation is that stress not only causes in-plane rotation of the magnetization vector, but \nalso lifts it slightly out of plane in such a way that the resulting y-component ofeffH\n prefers the anti-\nparallel orientation over the parallel orientation. This results in a magnetization flip or bit flip.  \nThe time evolution of the magnetization during this  process is tracked by solving the LLG equation \n(Equation (3)) starting with the initial condition. If we  choose the initial orientation of the magnetization \nto be exactly  along the easy axis 090 , then the effective field on the magnet due to stress \nvanishes [see Equation (9)] and therefore stress becom es ineffective in causing any rotation. Hence, we \nalways assume that the initial state of any magnetizati on is never exactly along an easy axis but deflected \nby 0.1 degrees from the easy axis 0090 , 89.9  . Such deflections ~ 5º can easily occur because \nof thermal fluctuations [20] but we are conserva tive and assume only 0.1º deflection to ensure our \nsimulation results are applicable over a greater confidence interval . \nThe energy dissipated in flipping a bit has two components: \ni) Energy dissipated while applying, reversing a nd removing a voltage on the piezoelectric layer for \ngenerating stress. This is the energy dissipated in the clocking circuit and is given by [21]: \n     2\n21\n2 1clockRCEC V\nRC\n\n           (10a). \nwhere C is the capacitance of the piezoelectric laye r, R is the resistance of the wires and V is the voltage \napplied across it. We assume that the volta ge waveform is sinusoidal with a period 2. However, the \nproblem with this RC circuit is that the en ergy stored in the capacitor (piezoelectric \nlayer), when it is fully charged, minus the dissipati on in the resistor, will be dissipated in the power \nsource in each cycle. A better scheme is to use an  LCR circuit where energy is merely transferred \nbetween the capacitive and inductive elements and the onl y energy lost per cycle is the energy dissipated \nin the resistive element [22]. In such a clocking circuit, the energy dissipated is: 212 ( ) CV 12                                                                         \n          (10b). \nii) Internal energy dissipated in the magnet during magnetization rotation [14, 15, 17]. This energy \ndEis calculated as: \n    \n0 . d\neffdE t dMHddt dt\n \n                 (11).                     \nBy substituting Equation (3) for dM\ndt\n  in equation (11) and integrating one obtains: \n   2 0\n2\n00  ||  (1 )d\nde f f\nsdEE dt H t M t dtdt M    \n           (12).           \nThis expression clearly shows that this dissipation is  associated with damping in the magnet because it \ndisappears when 0. \n           \n \nIII. Results and Discussions \n \nWe have used 4th order Runge -Kutta  method as described in [15] to solve the system of 24 coupled \nordinary differential equations for a chain of twelve dipole coupled mu ltiferroic elements shown in Fig. \n1(a). These 12 magnets comprise the NAND-gate and wiri ng for fan-in of 2 and fan-out of 3. The stress \napplied on the four nanomagnets comprising the actual gate follows a 4-phase si nusoidal clocking scheme \nshown in Fig 1(c). The magnets are grouped into 8 gr oups I through VIII. The sinusoidal clocks applied \nto each group and the relative phase lags between the clock signals for different groups is shown in Fig. \n1(c). Clearly, a 4-phase clock is required. When the phase for the clock on magnets marked \"I\" goes past \n90º so that the compressive stress on these magnets be gins to decrease, the compressive stress on magnets \nmarked II just begins to increase.  Thus, when the stress on magnets \"I\" has decreased to 12of the 2\n2()clockVE RCR 13maximum applied compression, the magnets marked \"II\" are at a state of 12of the maximum \ncompression and have been sufficiently rotated away from the easy direction. Consequently, as \ncompressive stress decreases to a point where the shap e anisotropy begins to dominate and therefore the \nmagnetizations of magnets marked \"I\" rotate towards their easy axes, their orientation is influenced \nstrongly and ultimately uniquely determined by the orientations of the \"input\" magnets ensuring uni-\ndirectionality of information propagation [15].   \nFrom the time-dependent voltages on any magnet, we  derive the time-depende nt stresses and hence \nthe time dependent effective fields i\neffHt\non each magnet. These are used to solve the LLG equation \n(24 coupled ODEs). The solutions yield the orientation ()  ,  ()iitt of each element. The in-plane \nmagnetization orientation (  ()it ) of each of the 12 nanomagnets (Fig 2 a-d) is plotted to demonstrate: (i) \nsuccessful NAND operation for any arbitrary input combina tion [(1,1), (0,0), (1,0), (0,1)] starting with the \ninitial input state (1, 1), and (ii) the complete ma gnetization dynamics showing that the primitive gate \noperation is always completed in 2 ns and the latency is 4 ns. \nIn this study, we assumed that the magnetostric tive layers were made of polycrystalline Terfenol-D \nwith material properties and dimensi ons given in Table I. The piezoelectric layer is assumed to be lead-\nzirconate-titanate (PZT) that has a reasonably large d31 coefficient (10-10m/V[27]), albeit also a large \nrelative dielectric constant of 1000.  Terfenol-D was chosen for its high magnetostriction [23] (even in the \nnanoscale [24]). The geometric parameters for the i ndividual magnets and the array were chosen to \nensure: (i) the shape anisotropy energy of the elements was sufficiently high (~0.8 eV or ~32 kT at room \ntemperature) so that the bit error probability due to spontaneous magnetizatio n flipping was very low \n(~32 1410 e ), (ii) the dipole interaction energy was limited to 0.26 eV which was significantly lower \nthan the shape anisotropy energy to prevent spontan eous flipping of magnetization, but large enough to \nensure that the magnetization of the multiferroic elemen ts always flipped to the correct orientation when \nstress was applied, even under the influence of ra ndom thermal fluctuations, and (iii) the maximum  14applied stress of 10 MPa corresponded to a stress-anisotropy energy 3\n2s= 172 kT that was \nsignificantly larger than the shape anisotropy energy barrier of 32 kT.  The reason why such large stress \nwas required are: (1) some magnets (for example th e magnet marked \"III\") had to overcome significant \namount of dipole coupling from interaction with multiple neighbors to rotate close to the hard axis; (2) the \nstress anisotropy is least effective close to Φ=0 and hence the stress had to  be large to ensure fast \nmagnetization rotation for angles close to the hard axis. \nIn all our simulations (Fig 2 a-d), the initial ma gnetizations of the nanoma gnets always correspond \nto the ground state of the array corresponding to input bits “1” and “1”. When a new input stream arrives, \nthe input bits are changed to conform to the new inputs. Thus, at time t = 0, the magnetizations of input-1 \nand input-2 are respectively set to (1, 1) [Fig 2(a)],   (0 , 0) [Fig 2 (b)],   (1, 0) [Fig 2 (c)],   (0, 1) [Fig \n2(d)].   We then consider the time evolution of  the in-plane magnetization orientations of every \nmultiferroic nanomagnet when a 4-phase stress cycle is a pplied, as shown in Fig 1 (c), to clock the array. \nIn Fig 2(a), the inputs are unchanged as input-1 = 1 a nd input-2 = 1. This is a trivial case as the ground \nstate already corresponds to the correct output. But it is still important to simulate the magnetization \ndynamics to verify that the gate works correctly. As  seen in Fig 2(a), all magnetizations rotate through \n±90º to the hard axis under compressive stress and then  rotate back to their initial (correct) orientations \nunder the influence of dipole coupling as the stresses are reversed to tensile. This results in a logical \nNAND output of \"0\". As expected there is a phase (a nd time) lag between instants when the compressive \nstress reaches a maximum and the magnetization is closes t to the hard axis. This is because magnetization \ntakes a finite time to respond to the applied st ress, as is evident from the LLG equations.  \nIn Fig 2(b), the inputs are both changed so that input-1 = 0 and input-2 = 0. Therefore, all the \nmagnets in the input wire, gate and output wire f lip through 180º, rotating first through ±90º on \napplication of a compressive stress and then furthe r rotating through ±90º under the influence of dipole \ncoupling. The phasing of the clock not only ensures the correct logical NAND output of \"1\" is reached \nbut that the information is propagated unidirectionally  through the input branches as well as the three  15output branches. The 4-phase clock achieves the fo llowing: As the compressive stress on a magnet is \nlowered to a point where the shape an isotropy barrier is about to be re stored, the compressive stress on its \nright (subsequent) neighbor has already rotated it toward s its hard axis. Therefore, the state of its left \n(previous) neighbor determines the easy direction to wards which the stressed magnet will relax as the \nstress in lowered. This ensures unidirectional logic bi t propagation as in the case of Bennett clocking [28].  \nFinally, Figs. 2(c) and Fig 2(d) show magnetization dy namics for the cases when one of the inputs is set \nto \"1\" while the other is set to \"0\". Here again the correct logical NAND output of \"1\" is achieved and \npropagated to the three fan-out branches. \nIn summary, we have proved through simulation that the NAND gate, fan-in and fan-out work \ncorrectly for all four input combinations for a given initial state of the nanomagnets. This is repeated in \nthe supplementary material accompanying this paper for different initial ground states – (0, 0), (0, 1) and \n(1, 0) – in order to be exhaustive.  \n \nIV Energy Considerations: Power dissipated in the magnets and the clock \n \nThere are two important sources of energy dissipation: (i) internal energy dissipated in the magnets \ndue to Gilbert damping and (ii) energy dissipated in  the clock while charging the capacitance of the PZT \nlayer that can be modeled as a parallel-plate capacitor.     \n \n(i) Internal energy dissipation \nThe internal energy dEdissipated in the magnets during magnetization rotation under stress was \nestimated using Equation (12). The energy dissipated in  the 4 nanomagnets (magnets 3, 4, 5 and 8) that \ncomprise the NAND gate over one clock cycle vari ed depending on the operation performed. For \nexample, when both inputs were \"1\" (Fig 2(a)) the energy dissipated was 767 kT while for both inputs set \nto \"0\" (Fig 2(b)), the energy dissipated was 948 kT.  On the average, the energy  dissipated in these four  16nanomagnets over one clock cycle is ~1000 kT. When  all 12 nanomagnets are considered, the average \nenergy dissipated over one clock cycle is ~3000 kT. Ultimately, this energy (~250kT/nanomagnet/bit) is \nwell over the Landauer limit of kTln(2) [29] but consider ably less than that dissipated in a transistor, or a \nnanomagnet switched with spin transfer torque, or  domain wall motion or current-generated magnetic \nfield when the gate operation is completed in 2 ns. The dissipation is governed by the (i) strength of \ndipole coupling needed to ensure the nanomagnets switc h to the correct state with low dynamic error [30] \nunder thermal noise and (ii) the large stress anisot ropy needed to ensure that the switching is \naccomplished in ~ 2 ns.  \n \n(ii) Energy dissipated in the clock \nThe energy dissipated in the clock is governed by th e electrostatic potential that needs to be applied \nacross the PZT layer to generate the stress (10 MP a) that can beat the sh ape anisotropy to flip \nmagnetization and complete the gate op eration in 2 ns. In this paper, we assume that the PZT layer is 40 \nnm thick (to ensure it is stiff compared to the magneto strictive layer, ensuring most of the strain in it is \ntransferred). On application of an electrostatic poten tial of ~50 mV across the PZT layer, an electric field \nof  1.25 MV/m is generated. Since the d31 coefficient of PZT is ~ -10-10m/V [27], a strain of ~1250610  \nis generated and transferred to the Terfenol-D layer resulting in a stress ~10 MPa since the Young’s \nmodulus of Terfenol-D is 1081 0 Pa.  \nNext, we estimate the capacitance of the ~40 nm thick PZT layer of surface area 101.75 nm × \n98.25 nm and thickness 40 nm as 1.74 fF (relative dielectric constant ~ 1000 [27]). Thus the energy \ndissipated in applying 50 mV, then switching it to -50 mV and discharging to zero (to generate the stress \ncycle shown in Fig 1 c) is ~ 3200 kT per nanomagnet if this is done abruptly (the energy dissipated in \ncharging the capacitor abruptly with a square wave pulse is 2 1\n2CV so that charging it up to + V from 0, \nreversing it to – V, and then discharging it back to 0 dissipates an energy of 3CV2. In contrast, driving an \n2\n2()clockVER CR 17LCR circuit with a sinusoidal source dissipates , resulting in an energy saving by a factor3RC.  \nAbrupt (non-adiabatic) switching with  a square wave pulse will cause a total energy dissipation of ~ \n40,000 kT in the clocking circuit (or more than 10 times the internal energy dissipated in the nanomagnets). However, if the LCR circuit is driven  with a sinusoidal voltage of low frequency (large \ntime period), then clocking becomes quasi-adiabatic. Th is reduces dissipation considerably because of the \nlarge energy saving factor. From Equation 10 (b), we can  see that if the time period is much larger than \nRC, the dissipation is greatly reduced. In our case,  we assume that the PZT layer is electrically accessed \nwith a silver wire of resistivity 2.6 μΩ-cm [31] so that an access line of length10 μm and cross section 50 \nnm×50 nm has resistance ~100 Ω. Hence, the RC time constant is ~0.174 ps. The clock period is 2 ns, so \nthat the reduction factor \n3RC= 45.47 10 . This makes the dissipation in the clock only about 22 \nkT, which is negligible compared to the internal energy dissipation of 3000 kT. \n \nV Conclusions \n We have modeled the nonlinear magnetization dyna mics of an all-multiferroic nanomagnetic NAND \ngate with fan-in/fan-out and shown that a throughput  of ~ 1 bit per 2 ns and latency ~4 ns can be \nachieved, so that the clock rate can be 0.5 GHz. Su ch a gate circuit is estimated to dissipate ~ 3000 \nkT/clock cycle internally in the 12 nanomagnets combin ed and much less energy (20 kT/clock cycle) in \nthe external access circuitry for the clock signal, if we use a 4-phase clocking scheme with a sinusoidal \nvoltage source driving an LCR circuit.   \nAll this begs the question as to whether it is po ssible to reduce the internal energy dissipation by \nsome appropriate scheme. This was discussed in ref. [2 ]. Imagine a magnet made of a material that has no \nGilbert damping \n 0 . If we remove the shape anisotropy barrier and make the magnet isotropic \n(circular disk), then a magnetic field applied pe rpendicular to the magnet’s plane will make the  18magnetization vector precess around it without any damping in accordance with the first term in the right \nhand side of Equation (3). There is now no internal  dissipation. The magnetic field however must be \nremoved precisely  at the juncture when the magnetization completes 1800 rotation if we wish to flip the \nbit. This requires exact precision; otherwis e, the magnet will either not have completed 1800 rotation, or \novershot, resulting in more than 1800 rotation. This error will continue  to build up with time and finally \nbecome too large to endure. In other words, there is  no fault tolerance. This is a well-known problem that \nhas been discussed by numerous authors starting from the Fredkin billiard ball computer which can \ncompute without dissipating energy [32], but cannot to lerate any error. Clearly, if we require fault \ntolerance, we must have damp ing, and hence some internal  dissipation. In the presence of damping, \nfluctuations can deviate the magnetization from the de sired orientation (minimum energy state), but the \nlatter will return to the correct orientation (minimum energy state) by dissipatin g energy. Therefore, the \ndissipation in the clocking circuit can be eliminat ed by adiabatic approaches (increasingly slow \nswitching), but the internal dissipation must remain for the sake of fault tolerance. \nThe internal energy dissipated in the magnet must be provided by the power source driving the clock. \nThis source need not dissipate any energy to raise and lower the barrier separating the logic bits as long as \nwe raise and lower the barrier adiabatically, but it mu st dissipate some energy internally in the logic \ndevice to maintain fault tolerance.  \n \nAcknowledgement:  This work is supported by the US  National Science Foundation under the \n“Nanoelectronics Beyond the Year 2020” grant 1124714.  \n \n \n  \n \n  19References :  \n[1] ITRS Report CORE9GPLL_HCMOS9_TEC_4.0 Databook, STMicro (2003). \n \n[2] Salahuddin S and Datta S 2007 Appl. Phys. Lett.  90, 093503. \n \n[3] Ney A,  Pampuch C , Koch R &. Ploog K. H 2003 Nature  425, 485. \n \n[4] Cowburn R.P , Welland M.E 2000  Science  287  1466. \n \n[5]  Casba G, Imre A, Bernstein G.H , Porod W, Metlushko V , 2002 IEEE Transaction on \nNanotechnology , 1, 209. \n \n[6] Bandyopadhyay S, Das B and Miller A E 1994 Nanotechnology 5 113.  \n \n[7] Alam M. T, Siddiq M. J,Bernstein G. H, Niemier M, Porod W and Hu X. S 2010 IEEE Trans. \nNanotech.  9  348 . \n \n[8] Ralph D C and Stiles M D 2008 J. Magn. Magn. Mater. 320 1190. \n \n[9] Yamanouchi M, Chiba D, Matsukura F and Ohno H, 2004 Nature (London) 428 539.  \n \n[10] Bandyopadhyay S and Cahay M 2009 Nanotechnology 20 412001. \n [11] Roy K, Bandyopadhyay S and Atulasimha J, 2011, Phys. Rev. B,  83, 224412. \n  20[12] Fukami S, Suzuki T, Nagahara K, Ohshima N, Ozaki Y, Saito S, Nebashi R, Sakimura N, Honjo H, \nMori K, Igarashi C, Miura S, Ishiwata N, and Sugibayashi T 2009 Symposium on VLSI Technology , \nSymposium on VLSI Technology, Digest of Technical Papers , 230-231.  \n \n[13] Atulasimha J, Bandyopadhyay S 2010  Appl. Phys. Lett.  97 173105  . \n \n[14] Roy K, Bandyopadhyay S and Atulasimha J 2011  Appl. Phys. Lett.  99, 063108.   \n \n[15 ] Salehi Fashami M, Roy K, Atulasimha J, Bandyopadhyay S 2011 Nanotechnology  22 155201. \n \n[16] Cowburn R. P, Koltsov D. K, Adeyeye A. O, Welland M. E and Tricker D. M 1999 Phys. Rev. Lett.  \n83 1042. \n \n[17] Bertotti G, Serpico C, Isaak and Mayergoyz I D 2008 Nonlinear Magnetization Dynamics in \nNanosystems  (Elsevier Series in Electromagnetism ) (Oxford: Elsevier). \n   [18] Gilbert L  2004  IEEE Transaction on magnetics, Vol 40, No 6 . \n \n[19] Chikazumi S 1964 Physics of Magnetism (New York: Wiley) \n \n[20] Nikonov, D. E., Bourianoff, G. I., Rowlands, G., Krivorotov, I. N  2010 J. of Appl. Phys , \n107, \n113910.  \n[21] Boechler G. P,. Whitney J.M, Lent C. S, Orlov A. O., Snider G. L., 2010\n Appl. Phys. Lett. , 97, \n103502.  21 \n[22] A. J. G. Hey, Feynman and Computing: Exploring the limits of computers, (c) 1999 Perseus Book \nPublishing,  L.L.C, Chapter 15 Computing Machines in th e future (transcript of 1985 Nishira Memorial \nLecture by R.P. Feynman).   \n[23] Abbundi R and Clark A E 1977 IEEE Trans. Mag.  \n13 1547. \n \n[24] Ried K, Schnell M, Schatz F, Hirscher  M, Ludescher B, Sigle W and Kronmuller H 1998 Phys. Stat. \nSol. (a)  167, 195. \n \n[25] Kellogg R A and Flatau A B 2008 J. of Intelligent Material Systems and Structures  19 583. \n [26] Walowski J, Djorjevic Kaufman M, Lenk B, Hamann C, McCord J and Münzenberg M 2008 J. Phys. \nD: Appl. Phys.  \n41 164016. \n \n[27] http://www.piezo.com/prodmaterialprop.html \n \n[28] Bennett, C.H 1982  Int. J. Theoretical Physics  21 905. \n \n[29] Landauer R  1982  Inter, J. Theoretical Physics  21  283. \n [30] Likharev K. K 1982 Int. J. Theor. Phys.  \n21 311. \n \n[31] http://www.itrs.net/Links/2009ITRS/2009Chapters 2009Tables/2009 Interconnect.pdf  \n [32] Fredkin E and Toffoli T  1982  Int. J. Theor. Phys . \n21 219 .  22 \nTable.1 Material parameters a nd geometric design for Terfenol-D \n3()2s 491 0  [23]，[24] \nsM 610.8 10  A m  \nYoung’s modulus  1081 0 P a [25] \n 0.1 [26]\nDimension  abt  101.75 98.25 10 nm nm nm   \nr 200 nm\n \n \n \n  \n \n  \n \n \n \n  \n \n   23 \n Figure captions \n Fig 1(a):\n Design of all multiferroic NAND gate w ith input \"logic wires\" and fan-out. The \n magnetization directions shown depict the corr ect initial (ground) state corresponsing to input-\n 1 = 1 and input-2 = 1. \n \nFig 1 (b):  Two nanomagnets whose hard axes are at an angle γ to the line joining their centers  . \n \n     Fig 1 (c): 4 phase clock showing sinusoidal stress applied to the nanomagnets. \n \n      Fig  2:  LLG simulation of magnetization dynamics of a ll magnets in the chain with initial (ground)  \n  states corresponding to the input-1=1 and input-2=1. Thereafter, inputs are changed to:  \n  (a) Input-1=1 and input-2=1 followed by applying  4-phase clock, which results in output=0.  \n     (b) Input-1=0 and input-2=0 followed by applying  4-phase clock, which results in output=1. \n     (c) Input-1=0 and input-2=1 followed by applying  4-phase clock, which results in output=1. \n    (d) Input-1=1 and input-2=0 followed by applying  4-phase clock, which results in output=1. \n \n \n  \n \n  \n \n  24 \n   \n \n \n \n \n       Fig 1(a) \n \n \n \n \n  25 \n \n \nFig 1 (b)  \n \n \n  \n \n \n \n  26 \n \n \n Fig 1 (c): 4 phase clock \n \n \n \n \n \n \n \n  27 \n \nFig.2 (a) \n \n \n \n \n \n \n  \n \n  28 \n(b) \n \n \n  \n \n \n \n  \n  29 \n \n(c) \n \n \n  \n \n \n \n   30 \n \n \n(d)  \n \n \n \n SUPPLEMENTARY INFORMATION  \n \nMagnetization Dynamics, Throughput  and Energy Dissipation in a \nUniversal Multiferroic Na nomagnetic Logic Gate  \n \nMohammad Salehi-Fashamia, Jayasimha Atulasimhaa, Supriyo Bandyopadhyayb  \nEmail: {salehifasham, jatulasimha, sbandy}@vcu.edu  \n \n(a) Department of Mechanical Engineering,  \n(b) Department of Electrical a nd Computer Engineering,  \nVirginia Commonwealth Univer sity, Richmond, VA 23284, USA. \n \n \nIn the main paper we theoreti cally demonstrated the switchi ng dynamics for the case when \nthe ground state (initial conditon) of all nanomagnets in th e input logic wire s, NAND gate, and \nfan-out correspond to the correct states for input-1 =1 and input-2=1. Thereafte r,  different inputs  \ncombinations as shown in Table-1 were applied and it was theoretically demonstrated that the \nmagnetizations switch (from the above ground state) to give the correct output states. Thus, we \nproved that this device works infallibly for all combinations of inputs wh en the initial (ground) \nstate of the nanomagnetic logic wires, NAND gate , and fan-out correspond to the correct states \nfor input-1=1 and input-2=1. (NOTE: That Case  1 is trivial as the ground state already \ncorresponds to the correct output, bu t it is still important to verify  that the gate works correctly \nfor this state.) \n \n \nTable 1 . Different initial conditions. \n \nDifferent Inputs to \nNAND gate INPUT-1 INPUT-2 OUTPUT \n1. 1 1 0 \n2. 0 0 1 \n3. 0 1 1 \n4. 1 0 1 \n \n In the supplement we present 12 more switch ing diagrams to show that the NAND gate \nworks for 3 other initial conditions of the na nomagnetic logic wires, NAND gate, and fan-out: \n \nCASE I. Initial (groud) state correspo nding to input-1=0 and input-2=0  \nFig S1: shows the correct initial (gr ound) state for input-1=0 and input-2=0. \nFig S2 (a)-(d):show the switchi ng diagrams for four different combinations of input-1 and \n input-2. In each case the co rrect output corresponsing to a NAND gate is acheived.   \n \nCASE II. Initial (groud) state correspo nding to input-1=0 and input-2=1   \nFig S3: shows correct initial (ground)  state for input-1=0 and input-2=1. \nFigures S4 (a)-(d):show the switching diagrams for four different combinations of input-1 \nand input-2. In each case the correct output  corresponsing to a NAND gate is acheived. \n   \nCASE III. Initial (groud) state correspo nding to input-1=1 and input-2=0   \nFig S5: shows correct initial (ground)  state for input-1=1 and input-2=0. \nFigures S6 (a)-(d):show the switching diagrams  for four different combinations of input-1 \nand  input-2. In each case th e correct output corresponsing to  a NAND gate is acheived.   \n These simulations in the supplemant, taken together  with the simulations in the main paper,   \nprove that the NAND gate works for different combin ations of inputs for a ll possible initial    \nconditions. \n \n \n \n       \n \n Figure S1:  Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=0 and \ninput-2=0. \n \n Figure S2:  LLG simulation of magnetization dynamics of a ll magnets in the chain with initial (ground) \nstates corresponding to the input-1=0 and inpu t-2=0. Thereafter, inputs are changed to: \n (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0.  \n   (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n   (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. \n   (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n \n Figure S3:  Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=0 and \ninput-2=1. \n \n Figure S4:  LLG simulation of magnetization dynamics of a ll magnets in the chain with correct initial \n(ground) state corresponding to the input-1=0 and input-2=1. Thereafter, inputs are changed to: \n (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0.  \n   (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n   (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. \n   (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n \n \n  Figure S5:  Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=1 and \ninput-2=0. \n \n Figure S6:  LLG simulation of magnetization dynamics of a ll magnets in the chain with correct initial \n(ground) state corresponding to the input-1=1 and input-2=0. Thereafter, inputs are changed to: \n (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0.  \n   (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n   (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. \n   (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. \n \n \n   \n \n           Supplementary Figures : \n Case I.  Fig.S1 \n \n \n       Fig.S2(a) \n \n \n        Fig.S2(b) \n \n \n        Fig.S2(c). \n \n \n       Fig.S2(d) \n \n \n       Case II \nFig.S3.  \n \n \n     \n \n   \n   \n \n      \n Fig.S4(a) \n \n \n        Fig.S4(b). \n \n \n        Fig.S4(c). \n \n \n        Fig.S4(d). \n \n \n        Case III:  \nFig.S5. \n \n \n          \n   \n \n      \n Fig.S6(a) \n \n \n \n \n \n \n \n  Fig.S6(b) \n \n \n \n \n \n \n  \n Fig.S6(c) \n \n \n \n \n \n  \n \n  \n Fig.S6(d) \n \n \n \n \n \n \n "    },    {        "title": "1109.2465v1.Externally_driven_transmission_and_collisions_of_domain_walls_in_ferromagnetic_wires.pdf",        "content": "arXiv:1109.2465v1  [cond-mat.mes-hall]  12 Sep 2011Externally-driven transmission and collisions of domain w alls in ferromagnetic wires\nAndrzej Janutka∗\nInstitute of Physics, Wroclaw University of Technology,\nWybrze˙ ze Wyspia´ nskiego 27, 50-370 Wroc/suppress law, Poland\nAnalytical multi-domain solutions to the dynamical (Landa u-Lifshitz-Gilbert) equation of a one-\ndimensional ferromagnet including an external magnetic fie ld and spin-polarized electric current are\nfound using the Hirota bilinearization method. A standard a pproach to solve the Landau-Lifshitz\nequation (without the Gilbert term) is modified in order to tr eat the dissipative dynamics. I estab-\nlish the relations between the spin interaction parameters (the constants of exchange, anisotropy,\ndissipation, external-field intensity, and electric-curr ent intensity) and the domain-wall parameters\n(width and velocity) and compare them to the results of the Wa lker approximation and micromag-\nnetic simulations. The domain-wall motion driven by a longi tudinal external field is analyzed with\nespecial relevance to the field-induced collision of two dom ain walls. I determine the result of such\na collision (which is found to be the elastic one) on the domai n-wall parameters below and above\nthe Walker breakdown (in weak- and strong-field regimes). Si ngle-domain-wall dynamics in the\npresence of an external transverse field is studied with rele vance to the challenge of increasing the\ndomain-wall velocity below the breakdown.\nPACS numbers: 75.78.Fg, 85.70.Kh\nKeywords: domain wall, Landau-Lifshitz equation, magneti c dissipation, spin-transfer torque, soliton colli-\nsion\nI. INTRODUCTION\nDescription of the magnetic-field- and electric-current-\ninduced motionsof domain walls(DWs) in nanowireshas\nbecame a hot topic because of novel methods of storing\nand switching the (magnetically encoded) binary infor-\nmation. These proposals offer a progress in the minia-\nturization of memory and logic elements, utilizing crucial\nadvantages of magnetic information encoding, (when a\nbit is identified with a single magnetic domain). Such in-\nformation is insensitive to the voltage fluctuations while\nits maintenance does not cost any energy, which enables\nthe data processing with the production of a small heat\namount. Currentlyinvestigatedrandom-accessmemories\nare built of metallic nanowires, formed into a parallel-\ncolumn structure, which store magnetic domains sepa-\nrated by DWs. Such a three-dimensional (3D) magnetic\nsystemhasthepotentialofstoringmoreinformationthan\ndevices based on 2D systems, like hard-disk drives or\nelectronic memories, in a given volume [1, 2]. Also, an\ninteresting concept of logical operation via transmitting\nmagnetic DWs through nanowires of specific geometries\nis being developed [3, 4]. I mention that ferroelectric\nnanosystems offer similar capabilities while their basic\npropertiesarestudied with thesamedynamical(Landau-\nLifshitz-Gilbert) equation even though the effects of elec-\ntroelastic coupling are strong [5, 6].\nIn orderto write and switchinformation, one can move\nthe DWs via the application of an external magnetic\nfield (parallel to the easy-axis) or via the application of\na voltage which induces the spin-polarized electric cur-\nrent through the DW. The directions of the field-driven\n∗Andrzej.Janutka@pwr.wroc.plmotion are different for the tail-to-tail and head-to-head\nDWs; thus the magnetic field induces the DW collisions,\nwhile the direction of the voltage-drivenmotion uniquely\ncorresponds to the current direction. Field-driven mo-\ntion and current-driven (below the Walker breakdown)\nmotion are possible due to the magnetic dissipation and\nitsdescriptiondemands inclusionofthe Gilbert terminto\nthe Landau-Lifshitz (LL) equation. However, existing\nmany-domain analytical solutions to the LL equation do\nnot include the dissipation [7, 8], while approximate so-\nlutions using the Walker ansatz describe a single DW\nonly [9]. Since the parameters of the DW solutions to\nthe Landau-Lifshitz-Gilbert (LLG) equation determine\naccessible values of technological characteristics (e.g. the\nminimal domain length, the bit-switching time, etc.), it\nis of interest to know the analytic DW solution to the\nLLG equation. Knowledge of the many-domain solution\nis of importance for preventing unwanted DW collisions\nwhich can result in an instability of the record and it en-\nables verification of DW-collision simulations regardless\nof the internal structure or the geometryof the simulated\nsystem.\nIn the present paper, I perform an analytical study of\nthe dynamics of multi-domain systems including the dis-\nsipation. The dynamical LLG system is bilinearized fol-\nlowing the Hirota method of solving nonlinear equations\nand it is extended, via doubling the number of freedom\ndegreesand the number of equations, into a time-reversal\ninvariant form. The field solving the extended system\ncontains proper and virtual (unphysical) dynamical vari-\nables and the physical components of the solution are\nshown to satisfy the pimary LLG system in a relevant\ntime regime. In particular, aiming to analytically de-\nscribe the field-induced collision, I establish asymptotic\nthree-domainmagnetizationprofiles(relevant in the time\nlimitst→ ±∞). With connection to the phenomenon of2\nthe Walker breakdown, (a cusp in the dependence of the\nDW velocity on the external field and in its dependence\non the current intensity), I modify the LLG model in a\nway to make it applicable below the breakdown. I re-\nduce it into a model of plane rotators. In this weak-field\nregime, the spin alignment in the DW area is saturated\nto a plane while, above the breakdown (the strong-field\nregime), the spins rotate about the magnetic-field axis\n(the easy axis). These considerations supplement micro-\nmagnetic simulations of the DW collisions in terms of the\nstudied DW-parameter regimes [10]. Within the present\nmethod, I verify the Walker-ansatz predictions on the\ncurrent-driven DW motion above and below the break-\ndown,(includingadiabaticandnon-adiabaticpartsofthe\nspin-transfer torque) [11]-[15], thus, showing the applica-\nbility of the present formalism to the field-driven motion\nof multi-domain systems with an additional voltage ap-\nplied. With relevance to the challenge of controlling the\nmaximum DW velocity below the Walker breakdown, we\nanalyze the longitudinal-field-driven motion and current-\ndrivenmotionoftheDWsinthepresenceofanadditional\nperpendicular (with respect to the easy axis) field.\nA complementary study of the DW collision of mag-\nnetic DWs in the subcritical regime is performed in a\nseparate paper [16].\nIn Sec. II, I extend the dissipative equations of motion\nof the ferromagnet in such a way as to make equations\napplicable to the unlimited range of time t∈(−∞,∞).\nSection III is devoted to the analysis of its single- and\ndouble-DW solutions in the presence of a magnetic field\nandan electriccurrent. Istudy the field-induced collision\nin detail. The plane-rotatorapproachto the DW dynam-\nics is described in Sec. IV. In Sec. V, consequences of\nthe application of an external field perpendicular to the\neasy axis for the DW statics and dynamics are consid-\nered. Conclusions are given in Sec. VI.\nII. DYNAMICAL EQUATIONS\nThe dynamics of the magnetization vector m(|m|=\nM) in the 1D ferromagnet is described with the LLG\nequation\n∂m\n∂t=J\nMm×∂2m\n∂x2+γm×H+β1\nM(m·ˆi)m׈i\n−β2\nM(m·ˆj)m׈j−δ∂m\n∂x−δβ\nMm×∂m\n∂x\n−α\nMm×∂m\n∂t.(1)\nThe first term of the right-hand side (RHS) of (1) re-\nlates to the exchange spin interaction while the second\nterm depends on the external magnetic field H; thus,γ\ndenotes the giromagnetic factor (up to its sign). The\nconstant β1(2)determines the strength of the easy axis\n(plane) anisotropy and ˆi≡(1,0,0),ˆj≡(0,1,0). Note\nthat the long axis of the magnetic nanowire is an easy\naxis for the majority of real systems; however, anotherchoice of the anisotropy axes does not influence the mag-\nnetization dynamics. The constant δis proportional to\nthe intensity of the electric current through the wire and\nδchanges its sign under time-arrow reversal (the inver-\nsion ofthe electronflow) [17, 18]. The non-adiabaticpart\nof the current-induced torque (which depends on β) is of\ndissipative origin; thus, β→0 with decreasing Gilbert\ndamping constant α→0 (one takes α,β≪1). Its inclu-\nsionisnecessaryifonedescribesanobservedmonotonous\nmotion of the DW below the Walker breakdown [11]-[14].\nNotice that including the magnetic-dissipation term fol-\nlowing the original LL approach (changing the last term\nof (1) into −αm×[m×heff], where heffj=−δH/δmj,\nandHdenotes the Hamiltonian) would lead to changing\nthe constant βintoβ−α, [13]. Although the discussion\nof the relevance of both approaches to the magnetic dis-\nsipation remains open [13, 19], I believe, the clinching\nargument for the Gilbert approach is the expectation for\nthe proper dissipative term to be dependent on the time\nderivative of the dynamical parameter m. In other case,\nthe dissipative term could influence static solutions to\nthe LL equation while one expects the magnetic friction\nto be the kinetic.\nSince (1) is valid only when the constraint |m|=Mis\nsatisfied, I intend to write equations of the unconstrained\ndynamics equivalent to (1). Introducing the complex dy-\nnamical parameters m±=my±imz, I represent the\nmagnetization components using a pair of complex func-\ntionsg(x,t),f(x,t). This way I reduce the number of\nindependent degrees of freedom. The relation between\nthe primary and secondary dynamical variables\nm+=2M\nf∗/g+g∗/f, m x=Mf∗/g−g∗/f\nf∗/g+g∗/f(2)\n[where (·)∗denotes the complex conjugate (c.c.)] ensures\nthat|m|=Mwhile there areno constraintson g,f. The\ntransform (2) enables bilinearization (”trilinearization”)\nof (1) following the Hirota method of solving nonlinear\nequations [7, 8]. In the particular case Hy=Hz= 0,\nfrom (1) and (2), we arrive at the trilinear equations for\nf,g\nf/bracketleftbig\n−iDt+JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗·g\n+Jg∗D2\nxg·g−/parenleftbigg\nγHx+β1+β2\n2/parenrightbigg\n|f|2g\n−β2\n2f∗2g∗= 0,\ng∗/bracketleftbig\n−iDt−JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗·g\n−JfD2\nxf∗·f∗+/parenleftbigg\n−γHx+β1+β2\n2/parenrightbigg\n|g|2f∗\n+β2\n2g2f= 0,(3)\nwhereDt,Dxdenote Hirota operators of differentiation\nwhich are defined by\nDm\ntDn\nxb(x,t)·c(x,t)≡(∂/∂t−∂/∂t′)m\n×(∂/∂x−∂/∂x′)nb(x,t)c(x′,t′)|x=x′,t=t′.3\nThe inclusion of the dissipation into the LLG equation\nis connected to breaking the symmetry with relevance\nto the time reversal. Therefore, neither (1) nor (3) can\ndescribe the magnetization evolution on the whole time\naxis. In particular, the application of an external mag-\nnetic field or current to the DW system (or creation of\na domain in the presence of an external field) initiates a\nnonequilibrium process of the DW motion. Such a mo-\ntioncannotbepresentinthedistantpast( t→ −∞)since\nanonzerovalueofthedissipativefunction[relevanttothe\nGilbert term in (1)] would indicate unlimited growth of\nthe energy with t→ −∞. Thus,solitary-wave solutions\nto (1) are relevant only in the limit of large positive val-\nues of time [11, 13]. This fact makes impossible an exact\nanalysis of the DW collisions using (3) and motivates\nextension of the dynamical system within a formalism\napplicable to the whole length of the time axis.\nWe write modified equations of motion using a similar\ntrick to the one proposed by Bateman with application\nto the Lagrangian description of the damped harmonic\noscillator [20]. It is connected to the concept by Laksh-manan and Nakamura of removing the dissipative term\nfromtheevolutionequationofferromagnetsviamultiply-\ning the time variable by a complex constant [21], how-\never, it demands an improvement in the spirit of Bate-\nman’s idea [22]. The concept is to extend the dynami-\ncal system doubling the number of freedom degrees and\nadding the equations which differ from the original ones\nby the sign of the dissipation constants. Resulting ex-\ntended system is symmetric with relevance to the time-\narrow reversal, however, its solution consists of physical\nand virtual fields. Let us mention that different quantum\ndissipative formalisms (non-equilibrium Green functions,\nthermo-field dynamics, rigged Hilbert space) are based\non the Bateman’s trick [23].\nWe extend the system of secondary dynamical equa-\ntions (3) since it describes unconstrained dynamics un-\nlike the primary LLG equation (1). We replace g,g∗,f,\nf∗in (3) with novel fields of the corresponding set g1,g∗\n2,\nf2,f∗\n1and of the set of their c.c.. For α,β= 0, in the\nabsence of dissipation, g1=g2=g,f1=f2=f. For the\ncaseHy=Hz= 0, the secondary dynamical equations\ntransform into\nf2/bracketleftbig\n−iDt+JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗\n1·g1+Jg∗\n2D2\nxg1·g1−/parenleftbigg\nγHx+β1+β2\n2/parenrightbigg\nf2f∗\n1g1−β2\n2f∗\n2f∗\n1g∗\n1= 0,\ng∗\n2/bracketleftbig\n−iDt−JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗\n1·g1−Jf2D2\nxf∗\n1·f∗\n1+/parenleftbigg\n−γHx+β1+β2\n2/parenrightbigg\ng∗\n2g1f∗\n1+β2\n2g2g1f1= 0.(4)\nWriting (4), we have replaced the last terms on the lhs of (3) in a way t o be linear in g(∗)\n2,f(∗)\n2, which ensures that\nthey vanish (diverge) with time in the presence of Hx∝negationslash= 0 with similar damping (exploding) rates as all other terms\nof these equations, (in particular, their damping does not modify th e anisotropy). The additional equations of the\ndynamical system differ from (4) by the sign of the dissipation const antsα,β\nf1/bracketleftbig\n−iDt+JD2\nx+δ(−β−i)Dx−αDt/bracketrightbig\nf∗\n2·g2+Jg∗\n1D2\nxg2·g2−/parenleftbigg\nγHx+β1+β2\n2/parenrightbigg\nf1f∗\n2g2−β2\n2f∗\n1f∗\n2g∗\n2= 0,\ng∗\n1/bracketleftbig\n−iDt−JD2\nx+δ(−β−i)Dx−αDt/bracketrightbig\nf∗\n2·g2−Jf1D2\nxf∗\n2·f∗\n2+/parenleftbigg\n−γHx+β1+β2\n2/parenrightbigg\ng∗\n1g2f∗\n2+β2\n2g1g2f2= 0.(5)\nThough the previous dynamical variables g(f),g∗(f∗)\nwere mutually independent, they had to be c.c. to each\nother in order that the system of equations (3) and their\nc.c. was closed. In the system of eight equations; (4)-\n(5) and their c.c., g1(f1) is not a c.c. to g2(f2), while\ncomparing (4) and (5), one sees that g2(x,t) (f2(x,t))\ncan be obtained from g1(x,t) (f1(x,t)) via changing the\nsign of its parameters α,β.\nUnder the time-arrow inversion, the system of the\nnovel equations transforms into itself if one accompanies\nthis operation by the transform of the novel dynamical\nvariables g1(2)→f2(1),f1(2)→ −g2(1). The equations\n(4) and their c.c., which determine the magnetization\ndynamics for large positive values of time (in particular,\nfort→ ∞), contain the differentials of the functions g1,\ng∗\n1,f1,f∗\n1. Therefore, the magnetization vector shouldbe expressed with these functions in the relevant time\nregime. Writing the magnetization in the form\nm+=2M\nf∗\n1/g1+g∗\n1/f1, mx=Mf∗\n1/g1−g∗\n1/f1\nf∗\n1/g1+g∗\n1/f1(6)\nensures that their components satisfy |m|=M,mx=\nm∗\nx, and they reproduce (2) for α=β= 0. In the regime\nof large negative values of time, in particular, for t→\n−∞, we can analyze the evolution of the magnetization\nwith the inversed time arrow. It is described with the\nreversed magnetization vector\n˜m+=−2M\nf∗\n2/g2+g∗\n2/f2,˜mx=−Mf∗\n2/g2−g∗\n2/f2\nf∗\n2/g2+g∗\n2/f2.(7)4\nIII. DOMAIN-WALL MOTION\nLet us analyze the multi-domain solutions to (4)-(5) in\nthe absence of any external magnetic field, H= 0. We\nsearch for the solutions which describe a single DW and\ntwo DWs in the forms f∗\n1= 1,g1=w1ek1x−l1t, andf∗\n1=\n1+v∗ek1x−l1tek2x−l2t,g1=w1ek1x−l1t+w2ek2x−l2t, re-\nspectively, where kj= Re(kj), and sign( k1) =−sign(k2).\nInserting these ansatz into (4)-(5), one finds\nlj=1\n1+iα/braceleftbigg\n−/radicalBig\n−/parenleftbig\nJk2\nj−β1/parenrightbig/bracketleftbig\nJk2\nj−(β1+β2)/bracketrightbig\n+δ(1+iβ)kj/bracerightbigg\n,\nkj=/radicalBigg\nβ1+β2(wj+w∗\nj)2/(4|wj|2)\nJ.(8)\nThe single-wall (two-domain) solutions with |k1| ∈\n(/radicalbig\nβ1/J,/radicalbig\n(β1+β2)/J) describe moving solitary waves\n(topological solitons), [7, 24]. One sees the correspon-\ndence between the wall width and the spin deviation\nfrom the easy plain since the dependence of kjonwj.\nWhenδ= 0, static two-domain solutions represent the\nBloch DW, ( |k1|=/radicalbig\nβ1/J,w1=−w∗\n1), or Neel DW,\n(|k1|=/radicalbig\n(β1+β2)/J,w1=w∗\n1), respectively (the\nnomenclature of [24] which differs from Neel and Bloch\nwall definition of e.g. [26]). These two solutions corre-\nspond to the ones found in [25, 27] within the XY model.\nThe DW profiles can be described with the functions\nm+(x,t) =Mw1e−iIml1t\n|w1|sech[k1x−Rel1t+log|w1|],\nmx(x,t) =−Mtanh[k1x−Rel1t+log|w1|],(9)\n(k1>0relatestothehead-to-headstructurewhile k1<0\nto the tail-to-tail structure). At the time points of the\ndiscrete set t=πn/Iml1,n= 0,±1,±2,..., one finds\ng1=g2=g,f1=f2=fand (4) coincide with (3).\nTherefore, (9) is a solitary-wave solution to (1), (in par-\nticular, it coincides with the one representing a Bloch\nDWoraNeelDw for δ∝negationslash= 0, [12]). Throughoutthe paper,\nwe focus our attention on the externally-driven dynam-\nics of the Bloch and Neel DWs, since these initially-static\nstructures are the most important with relevance to the\nmagnetic data storage.\nWeestablishthatstaticdouble-wall(three-domain)so-\nlutions to the LLG equation cannot be written with the\nabove Hirota expansion when k1=−k2. In this case the\ncoefficient v;\nv=−β2Jk2\n1w∗\n1w∗\n2\n(Jk2\n1−β1)(Jk2\n1−β1−β2)(10)\ndiverges with |k1| →/radicalbig\nβ1/Jor|k1| →/radicalbig\n(β1+β2)/J.\nAnalogously to the XY model, the Hirota expansion\nis inapplicable to static three-domain configurations of\nthe Bloch walls or Neel walls, while there exists staticsolution to (4)-(5) which describes a pair of different-\ntype (Neel and Bloch) walls [28]. In particular, for\nk1=/radicalbig\nβ1/J,k2=−/radicalbig\n(β1+β2)/J, andw1=−w∗\n1,\nw2=w∗\n2, one finds\nv=−β2w1w2\n2β1+β2−2/radicalbig\nβ1(β1+β2)(11)\nLet us emphasize that we have not excluded the coex-\nistence of a pair of Neel walls or Bloch walls in a mag-\nnetic wire. However, the overlap of both the topological\nsolitons induces their interaction which leads to an insta-\nbility of their parameters and, unlike for nontopological\nsolitons, is not a temporal one [29].\nSolving (4)-(5) in the presence of a longitudinal mag-\nnetic field Hx∝negationslash= 0, we apply the ansatz\nf∗\n1=/parenleftbig\n1+v∗ek1x−l1tek2x−l2t/parenrightbig\neγHxt/(−2i+2α),\ng1=/parenleftbig\nw1ek1x−l1t+w2ek2x−l2t/parenrightbig\ne−γHxt/(−2i+2α)(12)\nat the discrete time points t=tn≡4πn(1+α2)/(γHx),\nwheren= 0,±1,±2..., (letδ= 0forsimplicity). Behind\nthesetimepoints,inthepresenceofthelongitudinalfield,\nthe last terms on the lhs of (4)-(5) change faster (they\noscillate with the three-times higher frequency) than the\nother ones. Therefore, taking the above ansatz, we apply\nan approach similar to the ’rotating wave approxima-\ntion’ in the quantum optics. Since this ansatz describes\nthe spin structure rotation about the x-axis, it is appli-\ncable when the external field exceeds Walker-breakdown\ncritical value, |Hx|> HW. From the single-wall solu-\ntion, (the case of w2= 0 orw1= 0), for k1=/radicalbig\nβ1/J,\nk2=−/radicalbig\n(β1+β2)/J, we establish that applying the\nmagnetic field in the easy-axis direction drives the DW\nmotion with the velocity\nc1(2)=γ|Hx|α/[|k1(2)|(1+α2)]. (13)\nCorrespondingly, for Hx= 0, applying the electric cur-\nrent through the initially static wall drives it to move\nwith the velocity\nc=δ(1+αβ)\n1+α2(14)\nwhich is independent of the DW width.\nThe essential difference between both kinds of the\ndriven motion emerges from the analysis of the three-\ndomain solutions. Under the external field, the two con-\nsecutive DWs move in the opposite directions. The walls\nwhich are closing up to each other collide and eventually\nthey can annihilate or wander off each other. The ap-\nplication of the electric current along the magnetic wire\ndrives both the DWs to move in the same direction with\nthe samevelocity. Analyzinglong-timelimits ofthe mag-\nnetization vector in different regions of the coordinate x,\nwe establish the consequences of the field-induced colli-\nsion of the complex of a Bloch DW interacting with a\nNeel DW. We use the ansatz (12) and assume δ= 0.\nLetηj≡kj(x−x0j)−γHxαt/(1 +α2), ˜ηj≡kj(x−\nx0j) +γHxαt/(1 +α2). ForHx>0, att=tn(within5\nthe above ’rotating wave approximation’), we find the\ndistant-future limit of the magnetization (6)\nm+≈/braceleftBigg\nm(1)\n+η2≪η1∼0\nm(2)\n+η1≪η2∼0= lim\nt→∞m+,\nmx≈/braceleftBigg\nm(1)\nxη2≪η1∼0\nm(2)\nxη1≪η2∼0= lim\nt→∞mx,(15)\nwhere\nm(j)\n+= 2Mv/w∗\nje˜ηke−iγHxt/(1+α2)\n1+|v|2/|wj|2e2˜ηk,\nm(j)\nx=−M1−|v|2/|wj|2e2˜ηk\n1+|v|2/|wj|2e2˜ηk,(16)\nandj∝negationslash=k. Identifying the parameters x0jwith the\nDW-center positions, we introduce the restriction on wj,\n|v|/|wj|= 1. We notice that m(1)\n+,m(1)\nxas well as m(2)\n+,\nm(2)\nxare the Walker single-DW solutions to the primary\nLLG equation which describe the motion of well sepa-\nrated DWs [9, 30]. Thus, our three-domain profiles of\nthe fields (6) tend to satisfy (1) in the limit t→ ∞ac-\ncording to the requirement formulated in the previous\nsection.\nIn the distant-past limit, we describe the magnetiza-\ntion evolution with the reversed time arrow. Following\n(7),\n˜m+≈/braceleftBigg\n˜m(1)\n+˜η1≪˜η2∼0\n˜m(2)\n+˜η2≪˜η1∼0= lim\nt→−∞˜m+,\n˜mx≈/braceleftBigg\n˜m(1)\nx˜η1≪˜η2∼0\n˜m(2)\nx˜η2≪˜η1∼0= lim\nt→−∞˜mx,(17)\nwhere\n˜m(j)\n+=−2Mv/w∗\nkeηje−iγHxt/(1+α2)\n1+|v|2/|wk|2e2ηj,\n˜m(j)\nx=M1−|v|2/|wk|2e2ηj\n1+|v|2/|wk|2e2ηj,(18)\nandj∝negationslash=k. In orderto consider the collision of the pair of\nDWs which are infinitely distant from each other at the\nbeginning of their evolution, we determine the magneti-\nzation dynamics in the limit t→ −∞. For this aim, one\nhas to invert the propagation direction of the kinks of ˜m\nandtoreversethearrow’sheadofthefield vector ˜m. Uti-\nlizing the properties ˜ m(j)\n+(x+x0k,0) = ˜m(j)\n+(−x+x0k,0),\n˜m(j)\nx(x+x0k,0) =−˜m(j)\nx(−x+x0k,0), we arrive at\nm+(x,t) =/braceleftBigg\n−˜m(1)\n+(−x+2x01,t)η1≫η2∼0\n−˜m(2)\n+(−x+2x02,t)η2≫η1∼0,\nmx(x,t) =/braceleftBigg\n−˜m(1)\nx(−x+2x01,t)η1≫η2∼0\n−˜m(2)\nx(−x+2x02,t)η2≫η1∼0.(19)The applicability of the above procedure to study the\nasymptotic evolution of a single DW is easy to verify\nsince any single-DW solution satisfies\nm+(x,t) =−˜m+(−x+2x01,t),\nmx(x,t) = ˜mx(−x+2x01,t). (20)\nTypically, one should consider the formulas (15), (19)\nwith relevancetothe case β1≫β2, thusk1≈ −k2, which\ncorresponds to commonly studied crystalline magnetic\nnanowires, e.g. for Fe, FePt, β2/β1∼10−1, [31]. For\nnoncrystalline(permalloy)nanowiresFe 1−xNixdeposited\non a crystalline substrate, the easy-axis anisotropy con-\nstant determined from uniform-resonance measurements\nwasfound to be, unexpectedly, asbig asin the crystalline\nnanowires [32]. Therefore, even when neglect structural\neffects in real systems which lead to the saturation of\nthe spin alignment in the DW area to the easy plain\nor hard plain (the Walker breakdown) which suppresses\ntheirspontaneousmotion, the spectrumofspontaneously\npropagatingDWsin nanowireswouldbeverynarrowand\ntheir velocities would be very small.\nAccording to (15), (19), two initially closing up DWs\nhave to diverge after the collision. If one of the colliding\nDWs that was initially, for t→ −∞, described with the\nfield ingredient ˜ m(j)\n+, ˜m(j)\nx, it is finally, for t→ ∞, de-\nscribed with the field ingredient m(j)\n+,m(j)\nx. Therefore,\nreflecting DWs exchange their parameters x01↔x02,\nw1↔w2,k1↔ −k2. It is connected to exchanging the\ndirections of the spin orientation in the yz-plane in the\nwall areas, (after the collision the Neel wall changes into\nthe Bloch wall and vice versa as shown in Fig. 1). Our\nprediction corresponds to the result of the collision anal-\nysis performed for spontaneously propagating DWs (in\nabsence of external field, electric current, and dissipa-\ntion). According to findings of [24, 33], the DWs reflect\nduring the collision in a way that one can say they pass\nthrough each other without changing their widths and\nvelocities, however, with changing their character from\nthe head-to-head one into the tail-to-tail one and vice\nversa.\nUp till now, we have considered systems of infinite do-\nmains whose energy cannot be defined. However, the\nsmaller a domain is the bigger percentage of the Zeeman\npart ofits energy is lost per time unit due to the DW mo-\ntion. The condition of the domain-energy minimization\ndetermines the direction of this motion. The domains\naligned parallelly to the external field grow while the do-\nmains aligned antiparallelly to the field diminish. Any\nDW reflection induces a motion which contradicts this\nrule. Such a motion has to be decelerated and, eventu-\nally, it has to be suppressedwhen the decreaseofthe DW\ninteraction energy equals the increase of the Zeeman en-\nergy. The outcome of a many-collisionprocess in a finite-\nsize system is the appearance of a 1D magnetic-bubble\nstructure similar to widely known 2D bubble structures\n[34]. The bubble size and concentration depend on the\nmagnetic field intensity. Each bubble is ended with a6\nmmz\nyx\nt=0\nt= t/c68\nt=2 t/c68\nt=3 t/c68v2\nv1mx\nx\nt=0\nt= t/c68\nt=2 t/c68\nt=3 t/c68v1v2\nv2\nv2v1\nv1v1\nv1 v1\nv1v2\nv2\nv2\nv2\nFIG. 1. The magnetization dynamics of a system of one\nNeel DW and one Bloch DW in a longitudinal field above\nthe Walker breakdown. Their reflection takes place in the\ntime region (∆ t,2∆t) and it is accompanied by a change of\nthe Bloch wall (of the velocity v2) into the Neel wall (of the\nvelocity v1) and vice versa. Since |Hx|> HW, the spin struc-\nture monotonously rotates about the x-axis.\nNeel DW at one of its sides and with a Bloch DW at the\nother side. Some analogy with a complex boundary of\nhard (quasi-2D) magnetic bubbles can be noticed since\nsuch a border contains alternating Neel and Bloch points\nin its structure[35]. Let us mention, that interestingcon-\ncepts of storing and transforming the binary information\nhave had been developed several decades ago with rele-\nvance to 2D magnetic bubble systems, though, they have\nbeen abandoned because of technologicalproblems of the\ntime [36].\nNumerical analyses of the field-induced DW collision\n(micromagnetic studies) have been performed with rele-\nvance to flattened nanowires (quasi 1D nanostripes) be-\nlow and just above the Walker breakdown using the dis-\nsipative LL equation [10]. The systems below the thresh-\nold correspond to a plane-rotator model studied in sec-\ntion 4, while the systems above the threshold are qual-\nitatively described with the present model. The men-\ntioned simulations focus on the collisions of similar-type\n(Neel or Bloch) DWs neglecting the anisotropy. They\nhave predicted mutual annihilation or reflection of the\nwalls depending on (parallel or anti-parallel) spin align-\nments in both the DW centers. This result is partially\nsupported by a perturbation analysis of Bloch-wall in-\nteractions within the XY model which has shown such\nDWs to repel or attract each other depending on their\nchiralities [29]. The method of present study cannot be\nappliedtothesecollisionssinceoneisunabletodetermine\nneither the double-Neel nor double-Bloch wall analytical\nsolutionsto the dynamical equations. However, except in\nthe case of a periodically distributed DWs, multi-Bloch\nor multi-Neel structures are unstable because of unbal-\nanced DW interactions, thus, they seem to be less suit-\nable for the information-storing purposes than the Neel-\nBloch DW structures. To the best of the author knowl-\nedge, the field-induced collision of the Neel DW with the\nBloch DW has not been simulated.IV. PLANE-ROTATOR MODEL\nIn order to describe the DW dynamics below the\nWalker breakdown, we consider a system of plane rota-\ntors. Let us reduce the primary (LLG) dynamical system\nto its single component. Saturating the magnetization\ndynamics to the easy plain ( my= 0), we neglect the spin\nrotation about the x-axis and z-axis since the relevant\ntorque components are equal to zero. For H= (Hx,0,0),\ninserting\nmx=M1−a2\n1\n1+a2\n1, my= 0, mz= 2Ma1\n1+a2\n1(21)\n(wherea1takes real values) into the y-component of (1),\none arrives at a nonlinear diffusion equation\n/parenleftbigg\n−α∂a1\n∂t−γHxa1−δβ∂a1\n∂x+J∂2a1\n∂x2/parenrightbigg/parenleftbig\n1+a2\n1/parenrightbig\n−2Ja1/parenleftbigg∂a1\n∂x/parenrightbigg2\n−β1a1/parenleftbig\n1−a2\n1/parenrightbig\n= 0.(22)\nWe use another ansatz describing the dynamics con-\nstrained to the xy-plain (a hard plain)\nmx=M1−a2\n2\n1+a2\n2, my= 2Ma2\n1+a2\n2, mz= 0.(23)\nThen, we insert it into the z-component of (1) and we\narrive at\n/parenleftbigg\n−α∂a2\n∂t−γHxa2−δβ∂a2\n∂x+J∂2a2\n∂x2/parenrightbigg/parenleftbig\n1+a2\n2/parenrightbig\n−2Ja2/parenleftbigg∂a2\n∂x/parenrightbigg2\n−(β1+β2)a2/parenleftbig\n1−a2\n2/parenrightbig\n= 0(24)\nwhich differs from (22) by a constant at the anisotropy\nterm. With relevance to the case δ= 0, one finds the\ntwo-domain solution\na1(2)=wek1(2)x−γHxt/α,\n|k1|=/radicalbigg\nβ1\nJ,|k2|=/radicalbigg\nβ1+β2\nJ,(25)\nwhich correspond to the Bloch DW (to the Neel DW).\nWhenHx∝negationslash= 0, the DW propagates with the velocity\nc1(2)=γ|Hx|\n|k1(2)|α. (26)\nThe applicability of the plane-rotator model is limited\nby the Walker-breakdown condition. The magnetic field\n|Hx|cannot exceed a critical value HW(see Fig. 2a)\nwhich corresponds to the spin deviation from the basic\nmagnetization plane (a canting) at the center of the DW\nabout a limit angle equal or smaller than π/4. We esti-\nmate an upper limit of the Walker critical field consid-\nering the x-component of the LLG equation at the DW\ncenter\n∂mx\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=x01(2)≈/bracketleftbiggβ2\nMmymz−δ∂mx\n∂x/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=x01(2),(27)7\nwherex01(2)≡ −log(w)/k1(2). Letϕdenotes the an-\ngle of the spin deviation (a canting) at DW center. In-\nserting (21) and transforming mymz→m2\nzsin(ϕ)cos(ϕ)\nin (27) or inserting (23) and transforming mzmy→\nm2\nysin(ϕ)cos(ϕ) in (27), one arrives at\n∂a1(2)\n∂t=β2\n2sin(2ϕ)a1(2)−δ∂a1(2)\n∂x(28)\nand finally, assuming |ϕ| ≤π/4, at\n|Hx| ≤HW≤maxHW≡αβ2\n2γ+αδ\nγ|k2|.(29)\nThis expression corresponds to the one given in [30, 37,\n38]. However,wenoticethat, fortypicalnanowireswhose\nwidth-to-thickness ratio is bigger than 20 (double-atomic\nor triple-atomic layers of a submicrometer width), mea-\nsuringHW, one has estimated the canting angle to take\na value of a few degrees at most, [14, 30, 39].\nForH= 0,δ∝negationslash= 0, the two-domainsolution to(22)-(24)\ntakes the form\na1(2)=wek1(2)(x−δβt/α),\n|k1|=/radicalbigg\nβ1\nJ,|k2|=/radicalbigg\nβ1+β2\nJ.(30)\nIt is seen that only the non-adiabatic part of the spin-\ntransfer torque contributes to (22),(24) since the current-\ndependent term is proportional to β. From (28), the\ncurrent-induced Walker breakdown corresponds to the\ncritical current intensity\nδW≤β2/[2|k1|(1+β/α)] (31)\nifHx= 0, (see Fig. 2b). It has been observed that\nHW,δWdecreasewith decreasingthe nanowirewidth-to-\nthicknessratio, [39, 40], becausethis ratiodeterminesthe\nstrength of the easy-plain anisotropy while HW,δW→0\nwithβ2→0, [41]. Notice that analytical calculations us-\ning 2D XY model, experimental observations, and simu-\nlations of the spin ordering in nanostripes show this or-\ndering to vary alongthe cross-sectionwidth ofthe nanos-\ntripe in the DW area, thus, revealing a complex topo-\nlogical structure [10, 42]. Therefore, our plane-rotator\ndescription is valid only for a qualitative analysis of the\nDW dynamics in the nanostripes.\nNeither finding nonstationary double-Bloch nor\ndouble-Neel solutions in the form of the Hirota expan-\nsion (including its second order) does not manage. In\nparticular, inserting\na1(2)=w1ek1(2)x+w2ek′\n1(2)x\n1+v1(2)ek1(2)x+k′\n1(2)xe−γHxt/α,(32)\ninto (22)-(24), for k1=−k′\n1=±/radicalbig\nβ1/J,k2=−k′\n2=\n±/radicalbig\n(β1+β2)/J, leads to the divergence of v1(2)as it fol-\nlows from the approach of section 3. In order to describe\nthe collision of Bloch and Neel walls below the Walker\nbreakdown, I propose to apply an effective 1D model as-\nsuming the magnetization precession to be overdamped,|H□|c\nHW1(2)\nx /c100c\n/c100Wa b\nFIG. 2. a) A scheme of the longitudinal-field dependence\nof the DW velocity for: a wire with single-axis anisotropy\n(solid line), a wire with double-axis anisotropy (dashed li ne).\nb) A scheme of the current-intensity dependence of the DW\nvelocity for: a wire with single-axis anisotropy (solid lin e), a\nwire with double-axis anisotropy; β > α(dashed line), β < α\n(dotted line), β= 0 (dash-dotted line).\nthus, taking the lhs of (1) to be equal to zero. Inclusion\nof the constraint |m|=Mleads to the modified (by ne-\nglectingthefirsttermsonthelhs)system(4)-(5). Solving\nit, we predict the field-induced DW collision below the\nWalker breakdown to result in their reflection similar to\nthe one described in section 3. The reflection is accom-\npanied by the change of the Bloch wall into the Neel wall\nand vice versa. Let us emphasize that there is no spon-\ntaneous DW motion below the Walker breakdown when\nneglect magnetostatic effects [43].\nThe technological challenge of increasing the DW\nspeed is especially important below the Walker break-\ndown, where the driving-field is relatively weak. Refer-\nring to this purpose, we mention an attempt utilizing an\nincrease of the nanostripe-edge roughness, thus, an in-\ncrease of the damping constant α, [44]. This approach\nfails since, accordingto simulationsof [39], the maximum\nof the field-induced DW velocity is insensitive to αbe-\nlow the breakdown. It is because c1(2)∝α−1|Hx|while\n|Hx| ≤HW∝α. On the other hand, since βgrows with\nα, (thenon-adiabaticpartofthespin-transfertorqueisof\na dissipative origin), the velocity of the current-induced\nDW motion\nc=δβ/α (33)\ncan be insensitive to the increase of the nanostripe-edge\nroughness as well. One has attributed some reported\nDW-velocity increase due to the nanostripe-edge rough-\nness to a decrease ofits effective cross-sectionwidth. An-\nother attempt utilized an increase of HWdue to the in-\ncreaseofthe anisotropyconstant β2. It hasbeen donevia\nthenanowiredepositiononaspecificcrystallinesubstrate\n[45]. However, the most efficient method of influencing\nthe maximum DW velocity below the Walker breakdown\nis the application of the transverse magnetic field, which\nis considered in the next section [46].8\nV. DOMAIN WALL IN PERPENDICULAR TO\nEASY AXIS FIELD\nLet us define H±≡Hy±iHz. ForHx= 0, we search\nfor a two-domain solution to (1) using a different ’multi-\nlinearization’ than used in the previous sections\nf2/bracketleftbig\n−iDt+JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗\n1·g1\n+γH+\n2f∗\n1(f∗\n1f2+g∗\n2g1)−/parenleftbigg\nβ1+β2\n2/parenrightbigg\nf2f∗\n1g1\n−β2\n2f∗2\n1g∗\n2= 0,\ng∗\n2/bracketleftbig\n−iDt−JD2\nx+δ(β−i)Dx+αDt/bracketrightbig\nf∗\n1·g1\n−γH−\n2g1(f∗\n1f2+g∗\n2g1)+/parenleftbigg\nβ1+β2\n2/parenrightbigg\ng∗\n2g1f∗\n1\n+β2\n2g2\n1f2= 0,\nf2g1D2\nxf∗\n1·f∗\n1−f∗\n1g∗\n2D2\nxg1·g1= 0.(34)\nLet us focus our attention on the case α,β,δ= 0 for\nsimplity. Then one has f1=f2=f,g1=g2=g,\nwhile in the general case the relations (6), (7) apply. We\nanalyze the two cases of the external-field direction; the\none parallel to the easy plane H+= iHz, and the one\nperpendicular to the easy plane H+=Hy.\nIn the case of H+= iHz, we apply the ansatz\nf∗\n1=f2=q1+s1ek1x−l1t,\ng1=−g∗\n2= i/parenleftbig\ns1+q1ek1x−l1t/parenrightbig\n, (35)\nwherek1,q1,s1denote real constants, (the parameter\nl1can take complex values when α∝negationslash= 0). The solution\nin the form (35) describes the wall between two domains\nwhose spins are deviated from the easy axis onto the\nexternal-field direction about an angle which grows with\n|H+|. Inserting this ansatz into (34), one finds\ns1=β1−/radicalbig\nβ2\n1−γ2H2z\nγHzq1. (36)\nConsidering the solutions which are static in the absence\nof the electric current, l1= 0 forδ= 0, one arrives at\n|k1|=/radicalBigg\nβ2\n1−γ2H2z\nβ1J. (37)\nIn the case of H+=Hy, the ansatz relating to the\ndeviationofthe domainmagnetizationfromthe easyaxis\nonto the external-field direction takes the form\nf∗\n1=f2=q2+s2ek2x−l2t,\ng1=g∗\n2=s2+q2ek2x−l2t. (38)\nwith real k2,q2,s2. From (34), we find\ns2=β1+β2−/radicalBig\n(β1+β2)2−γ2H2y\nγHyq2.(39)The static solutions correspond to\n|k2|=/radicalBigg\n(β1+β2)2−γ2H2y\n(β1+β2)J. (40)\nThe transverse external field does not drive the DW\nmotion even in the presence of the magnetic dissipation\n(α∝negationslash= 0). When the current through the wire and the dis-\nsipation are applied, under the transverse magnetic field,\nthe DW moves with the velocity cgiven by (14), which is\nindependent of the value of this field. Then the solution\nto (34) satisfies the bilinearized LLG system (Eqs. (3)\nwith additional H+-dependent terms) at the time points\nof the discrete set t=πn/Iml1, wheren= 0,±1,±2,...,\nsincef1=f2=f,g1=g2=gat these points. In-\ncluding an additional to H+longitudinal component of\nthe magnetic field Hxdrives the DW motion. For the\nrealistic case HW∼ |Hx|<|H+| ≪ |β1/γ|, neglect-\ning small contributions to the Hx-dependent part of the\ntorque, one finds the velocity of such a DW propagation\n(13) or (26) above and below the Walker breakdown, re-\nspectively, with |k1(2)|given by (37), (40). This velocity\nnonlinearly increases with |H+|, [47]. Searching for c1(2),\nin the case |Hx|< HW, additionally, I have taken the lhs\nof (1) equal to zero as discussed in section 4. The ma-\nnipulation of c1(2)via the application of the transverse\nmagnetic field is potentially useful for speeding up the\nprocessing with a magnetically-encoded information. We\nalso notice that the transverse-field dependence of |k1(2)|\nenables influencing the magnitude of the critical current\nof the Walker breakdown δW, following (31).\nVI. CONCLUSIONS\nWe have analytically studied the DW dynamics in the\npresence of the external magnetic field and the electric\ncurrent along the magnetic wire within the LLG ap-\nproach. It has demanded overcoming the difficulty aris-\ning from breaking the time-reversal symmetry by inclu-\nsion of the magnetic dissipation. We have removed this\nasymmetry of the dynamical system by introducing ad-\nditional (virtual) dynamical variables, which is a similar\ntrick to the Lagrangian approach to the damped har-\nmonic oscillator. Determining a connection of the addi-\ntional dynamical variables to the evolution of the mag-\nnetization vector in specific ranges of time, we have ana-\nlyzed the dynamics of a single DW and of a pair of DWs.\nThe magnetic-field-induced velocities of the DWs, the\nformulas (13) and (26), and the current-induced veloci-\nties (14) and (33) are found to correspond to the ones\nof the Walker approach above and below the breakdown,\nrespectively. According to [28], static three-domain so-\nlutions to 1D LLG equation describe pairs of Neel and\nBloch walls. For the purposes of the qualitative dynam-\nics analysis of a number of DWs below the Walker break-\ndown, especially of the Neel-Bloch pairs, we have pro-\nposed a dynamical equation which differs from the LLG9\none by neglecting the lhs in (1). Below and above the\nbreakdown, the neighboring Neel and Bloch walls move\nin the presence of the longitudinal external field in the\nopposite directions. Their collision results in the DW\nreflection accompanied by the reorientation of the Neel\nwall into the Bloch wall and vice versa. In other words,\nthe DWs pass trough each other without changing their\nwidths and velocities, however, the head-to-head DW\nstructure changes into the tail-to-tail one and vice versa.\nOur method is useful for the analysis of two-domain\nsystems under the transverse (with respect to the easy\naxis) external field, which enables a verification of nu-\nmericalandexperimentalresults[46–48]. Areorientationof the magnetic domains due to the transverse field in-\nduces a widening of the DW area up to the infinity when\napproach with the field intensity to the coercivity value.\nThe consequence of the transverse-field application is an\nincrease of the DW mobility (the ratio c1(2)/|Hx|) and\nan increase of the critical current (a shift of the current-\ndriven Walker breakdown).\nACKNOWLEDGEMENTS\nThis work was partially supported by Polish Govern-\nment Research Founds for 2010-2012in the framework of\nGrant No. N N202 198039.\n[1] S. S. P. Parkin, M. Hayashi, L. 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Kulagin, JETP\n85, 516 (1997)."    },    {        "title": "1109.4964v1.Hole_spin_relaxation_and_coefficients_in_Landau_Lifshitz_Gilbert_equation_in_ferromagnetic_GaMnAs.pdf",        "content": "arXiv:1109.4964v1  [cond-mat.mtrl-sci]  22 Sep 2011Hole spin relaxation and coefficients in Landau-Lifshitz-Gi lbert equation in\nferromagnetic GaMnAs\nK. Shen and M. W. Wu∗\nHefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\n(Dated: April 19, 2022)\nWe investigate the temperature dependence of the coefficient s in the Landau-Lifshitz-Gilbert\nequation in ferromagnetic GaMnAs by employing the Zener mod el. We first calculate the hole spin\nrelaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation\ntime is typically several tens femtoseconds and can present a nonmonotonic temperature dependence\ndue to the variation of the interband spin mixing, influenced by the temperature related Zeeman\nsplitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-\nLifshitz-Gilbert equation, such as the Gilbert damping, no nadiabatic spin torque, spin stiffness and\nvertical spin stiffness coefficients. We find that the nonadiab atic spin torque coefficient βis around\n0.1∼0.3 at low temperature, which is consistent with the experimen t [Adam et al., Phys. Rev. B\n80, 193204 (2009)]. As the temperature increases, βmonotonically increases and can exceed one\nin the vicinity of the Curie temperature. In the low temperat ure regime with β <1, the Gilbert\ndamping coefficient αincreases with temperature, showing good agreement with th e experiments\n[Sinovaet al., Phys. Rev. B 69, 085209 (2004); Khazen et al.,ibid.78, 195210 (2008)]. Furthermore,\nwe predict that αdecreases with increasing temperature once β >1 near the Curie temperature.\nWe also find that the spin stiffness decreases with increasing temperature, especially near the Curie\ntemperature due to the modification of the finite β. Similar to the Gilbert damping, the vertical\nspin stiffness coefficient is also found to be nonmonotonicall y dependent on the temperature.\nPACS numbers: 72.25.Rb, 75.50.Pp, 72.25.Dc, 75.30.Gw\nI. INTRODUCTION\nThe ferromagnetic semiconductor, GaMnAs, has been\nproposed to be a promising candidate to realize all-\nsemiconductor spintronic devices,1,2where the existence\nof the ferromagnetic phase in the heavily doped sample\nsustains seamless spin injection and detection in normal\nnon-magneticsemiconductors.3,4Oneimportant issuefor\nsuch applications lies in the efficiency of the manipula-\ntion of the macroscopic magnetization, which relies on\nproperties of the magnetization dynamics. Theoretically,\nthe magnetization dynamics can be described by the ex-\ntended Landau-Lifshitz-Gilbert (LLG) equation,5–10\n˙n=−γn×Heff+αn×˙n−(1−βn×)(vs·∇)n\n−γ\nMdn×(Ass−Av\nssn×)∇2n, (1)\nwithnandMdstanding for the direction and magni-\ntude of the magnetization, respectively. Heffis the ef-\nfective magnetic field and/or the external field. The sec-\nond term on the right hand side of the equation is the\nGilbert damping torque with αdenoting the damping\ncoefficient.5,6The third one describes the spin-transfer\ntorque induced by the spin current vs.7,8As reported,\nthe out-of-plane contribution of the spin-transfer torque,\nmeasured by the nonadiabatic torque coefficient β, can\nsignificantly ease the domain wall motion.7,8In Eq.(1),\nthespinstiffnessandverticalspinstiffnesscoefficientsare\nevaluated by AssandAv\nssrespectively, which are essen-\ntially important for the static structure of the magnetic\ndomain wall.10Therefore, for a thorough understandingof properties of the magnetization dynamics, the exact\nvalues of the above coefficients are required.\nIn the past decade, the Gilbert damping and nonadia-\nbatic torque coefficients have been derived via many mi-\ncroscopic approaches, such as the Blotzmann equation,11\ndiagrammatic calculation,12,13Fermi-surface breathing\nmodel14–16andkineticspinBlochequations.10,17Accord-\ning to these works, the spin lifetime of the carriers was\nfound to be critical to both αandβ. However, to the\nbest of our knowledge, the microscopic calculation of the\nhole spin lifetime in ferromagnetic GaMnAs is still ab-\nsent in the literature, which prevents the determination\nofthe values of αandβfrom the analyticalformulas. Al-\nternatively, Sinova et al.18identified the Gilbert damp-\ning from the susceptibility diagram of the linear-response\ntheory and calculated αas function of the quasiparticle\nlifetime and the hole density. Similar microscopic calcu-\nlation on βwas later given by Garate et al..19In those\nworks, the quasiparticle lifetime was also treated as a\nparameter instead of explicit calculation. Actually, the\naccurate calculation of the hole spin and/or quasiparti-\ncle lifetime in ferromagnetic GaMnAs is difficult due to\nthe complex band structure of the valence bands. In the\npresentwork,weemploythe microscopickineticequation\ntocalculatethespinlifetimeoftheholegasandtheneval-\nuateαandβin ferromagnetic GaMnAs. For the velocity\nof the domain-wall motion due to the spin current, the\nratioβ/αisanimportantparameter,whichhasattracted\nmuch attention.12,19,20Recently, a huge ratio ( ∼100) in\nnanowire was predicted from the calculation of the scat-\ntering matrix by Hals et al..20By calculating αandβ, we2\nare able to supply detailed information of this interesting\nratio in bulk material. Moreover, the peak-to-peak fer-\nromagnetic resonance measurement revealed pronounced\ntemperature and sample preparation dependences of the\nGilbert damping coefficient.18,21,22For example, in an-\nnealed samples, αcan present an increase in the vicinity\nof the Curie temperature,18,21which has not been stud-\nied theoretically in the literature. Here, we expect to\nuncover the underlying physics of these features. In ad-\ndition, the nonadiabatic torque coefficient βin GaMnAs\nhas been experimentally determined from the domain-\nwall motion and quite different values were reported by\ndifferent groups, from 0.01 to 0.36,23,24which need to be\nverified by the microscopic calculation also. Moreover,\nto the best of our knowledge, the temperature depen-\ndence of βhas not been studied theoretically. We will\nalso address this issue in the present work.\nIn the literature, the spin stiffness in GaMnAs was\nstudied by K¨ onig et al.,25,26who found that Assincreases\nwith hole density due to the stronger carrier-mediated\ninteraction between magnetic ions, i.e., Ass=Nh/(4m∗)\nwithNhandm∗being the density and effective mass\nof hole gas, separately. However, as shown in our pre-\nvious work, the stiffness should be modified as Ass∼\nNh/[4m∗(1+β2)] in ferromagnetic GaMnAs with a finite\nβ.10As a result, Assas well as the vertical spin stiffness\nAv\nss=βAssmay show a temperature dependence intro-\nduced by β. This is also a goal of the present work.\nFor a microscopic investigation of the hole dynamics,\nthe valence band structure is required for the descrip-\ntion of the occupied carrier states. In the literature,\nthe Zener model27based on the mean-field theory has\nbeen widely used for itinerant holes in GaMnAs,28–31\nwhere the valence bands split due to the mean-field p-\ndexchange interaction. In the present work, we utilize\nthis model to calculate the band structure with the ef-\nfective Mn concentration from the experimental value of\nthe low-temperaturesaturatemagnetization in GaMnAs.\nThe thermal effect on the band structure is introduced\nviathe temperaturedependence ofthe magnetizationfol-\nlowing the Brillouin function. Then we obtain the hole\nspin relaxation time by numerically solving the micro-\nscopic kinetic equations with the relevant hole-impurity\nand hole-phonon scatterings. The carrier-carrier scatter-\ning is neglected here by considering the strongly degen-\nerate distribution of the hole gas below the Curie tem-\nperature. We find that the hole spin relaxation time\ndecreases/increases with increasing temperature in the\nsmall/large Zeeman splitting regime, which mainly re-\nsults from the variation of the interband spin mixing.\nThen we study the temperature dependence of the co-\nefficients in the LLG equation, i.e., α,β,AssandAv\nss,\nby using the analytical formulas derived in our previous\nworks.10,17Specifically, we find that βincreases with in-\ncreasing temperature and can exceed one in the vicinity\nof the critical point, resulting in very interesting behav-\niors of other coefficients. For example, αcan present an\ninteresting nonmonotonic temperature dependence withthe crossoveroccurringat β∼1. Specifically, αincreases\nwithtemperatureinthelowtemperatureregime,whichis\nconsistent with the experiments. Near the Curie temper-\nature, an opposite temperature dependence of αis pre-\ndicted. Similar nonmonotonic behavior is also predicted\nin the temperature dependence of Av\nss. Our results of β\nandAssalso show good agreement with the experiments.\nThis work is organized as follows. In Sec.II, we setup\nour model and lay out the formulism. Then we show\nthe band structure from the Zener model and the hole\nspin relaxation time from microscopic kinetic equations\nin Sec.III. The temperature dependence of the Gilbert\ndamping, nonadiabatic spin torque, spin stiffness and\nvertical spin stiffness coefficients are also shown in this\nsection. Finally, we summarize in Sec.IV.\nII. MODEL AND FORMULISM\nInthesp-dmodel, theHamiltonianofholegasinGaM-\nnAs is given by31\nH=Hp+Hpd, (2)\nwithHpdescribing the itinerant holes. Hpdis thesp-d\nexchange coupling. By assuming that the momentum k\nis still a good quantum number for itinerant hole states,\none employsthe Zenermodel and utilizes the k·ppertur-\nbation Hamiltonian to describe the valence band states.\nSpecifically, we take the eight-band Kane Hamiltonian\nHK(k) (Ref.32) in the present work. The sp-dexchange\ninteraction reads\nHpd=−1\nN0V/summationdisplay\nl/summationdisplay\nmm′kJmm′\nexSl·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc†\nmkcm′k,\n(3)\nwithN0andVstanding for the density of cation sites\nand the volume, respectively. The cation density N0=\n2.22×1022cm−3. The eight-band spin operator can be\nwritten as ˆJ= (1\n2σ)⊕J3/2⊕J1/2, where1\n2σ,J3/2and\nJ1/2represent the total angular momentum operators of\nthe conduction band, Γ 8valence band and Γ 7valence\nband, respectively. Jmm′\nexstands for the matrix element\nof the exchange coupling, with {m}and{m′}being the\nbasis defined as the eigenstates of the angular momen-\ntum operators ˆJ. The summation of “ l” is through all\nlocalized Mn spins Sl(atrl).\nThen we treat the localized Mn spin in a mean-field\napproximation and obtain\n¯Hpd=−xeff∝an}b∇acketle{tS∝an}b∇acket∇i}ht·/parenleftBigg/summationdisplay\nmm′kJmm′\nex∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc†\nmkcm′k/parenrightBigg\n,\n(4)\nwhere∝an}b∇acketle{tS∝an}b∇acket∇i}htrepresents the average spin polarization of\nMn atoms with uncompensated doping density NMn=\nxeffN0. Obviously, ¯Hpdcan be reduced into three blocks\nasˆJ, i.e.,¯Hmm′\npd(k) = ∆mmn·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htwith the Zee-\nman splitting of the m-band ∆mm=−xeffSdJmm\nexM(T)\nM(0).3\nHere,nis the direction of ∝an}b∇acketle{tS∝an}b∇acket∇i}ht. For a manganese ion, the\ntotal spin Sd= 5/2. The temperature-dependent spon-\ntaneous magnetization M(T) can be obtained from the\nfollowing equation of the Brillouin function33\nBSd(y) =Sd+1\n3SdT\nTcy, (5)\nwherey=3Sd\nSd+1M(T)\nM(0)Tc\nTwithTcbeing the Curie\ntemperature. Here, BSd(y) =2Sd+1\n2Sdcoth(2Sd+1\n2Sdy)−\n1\n2Sdcoth(1\n2Sdy).\nThe Schr¨ odinger equation of the single particle Hamil-\ntonian is then written as\n/bracketleftbig\nHK(k)+¯Hpd(k)/bracketrightbig\n|µ,k∝an}b∇acket∇i}ht=Eµk|µ,k∝an}b∇acket∇i}ht.(6)\nOne obtains the band structure and wave functions from\nthe diagonalization scheme. In the presence of a finite\nZeemansplitting, thestructureofthe valencebandsdevi-\nates from the parabolic dispersion and becomes strongly\nanisotropicaswewillshowinthenextsection. Moreover,\nthe valence bands at Fermi surface are well separated in\nferromagnetic GaMnAs because of the high hole density\n(>1020cm−3) and Zeeman splitting, suggesting that the\nFermi golden rule can be used to calculate the lifetime of\nthe quasiparticlestates. For example, the contribution of\nthe hole-impurity scattering on the µth-band state with\nenergyǫcan be expressed by\n[τhi\nµ,p(ǫ)]−1= 2π/summationdisplay\nνni\nDµ(ǫ)/integraldisplayd3k\n(2π)3/integraldisplayd3q\n(2π)2δ(ǫ−ǫµk)\n×δ(ǫµk−ǫνq)U2\nk−q|∝an}b∇acketle{tµk|νq∝an}b∇acket∇i}ht|2f(ǫµk)[1−f(ǫνq)],(7)\nwhereDµ(ǫ) stands for the density of states of the µth\nband.f(ǫµk) satisfies the Fermi distribution in the equi-\nlibrium state. The hole-impurity scattering matrix ele-\nmentU2\nq=Z2e4/[κ0(q2+κ2)]2withZ= 1.κ0and\nκdenote the static dielectric constant and the screening\nconstant under the random-phase approximation,34re-\nspectively. Similar expression can also be obtained for\nthe hole-phonon scattering.\nHowever, it is very complicated to carry out the multi-\nfold integrals in Eq.(7) numerically for an anisotropic\ndispersion. Also the lifetime of the quasiparticle is not\nequivalent to the spin lifetime of the whole system, which\nis required to calculate the LLG coefficients according\nto our previous work.10,17Therefore, we extend our ki-\nnetic spin Bloch equation approach35to the current sys-\ntem to study the relaxation of the total spin polarization\nas follows. By taking into account the finite separation\nbetween different bands, one neglects the interband co-\nherence and focuses on the carrier dynamics of the non-\nequilibrium population. The microscopic kinetic equa-\ntion is then given by\n∂tnµ,k=∂tnµ,k/vextendsingle/vextendsinglehi+∂tnµ,k/vextendsingle/vextendsinglehp, (8)\nwithnµ,kbeing the carrier occupation factor at the µth\nband with momentum k. The first and second terms onthe right hand side stand for the hole-impurity and hole-\nphonon scatterings, respectively. Their expressions can\nbe written as\n∂tnµ,k/vextendsingle/vextendsinglehi=−2πni/summationdisplay\nν,k′U2\nk−k′(nµk−nνk′)|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2\n×δ(Eµk−Eνk′), (9)\nand\n∂tnµ,k/vextendsingle/vextendsinglehp=−2π/summationdisplay\nλ,±,ν,k′|Mλ\nk−k′|2δ(Eνk′−Eµk±ωλ,q)\n×[N±\nλ,q(1−nνk′)nµk−N∓\nλ,qnνk′(1−nµk)]|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2,(10)\nwithN±\nλ,q= [exp(ωλ,q/kBT)−1]−1+1\n2±1\n2. Thedetailsof\nthe hole-phonon scattering elements |Mλ\nq|2can be found\nin Refs.36–38. From an initial condition with a small\nnon-equilibrium distribution, the temporal evolution of\nthe hole spin polarization is carried out by\nJ(t) =1\nNh/summationdisplay\nµ,k∝an}b∇acketle{tµk|ˆJ|µk∝an}b∇acket∇i}htnµ,k(t),(11)\nfrom the numerical solution of Eq.(8). The hole spin\nrelaxation time can be extracted from the exponential\nfitting of Jwith respect to time. One further calculates\nthe concerned coefficients such as α,β,AssandAv\nss.\nIII. NUMERICAL RESULTS\nIn the Zener model, the sp-dexchange interaction con-\nstantsJmm\nexareimportant parametersfor the band struc-\nture. In the experimental works, the p-dexchange cou-\npling constant Jpp\nexwas reported to vary from −1 eV\nto 2.5 eV, depending on the doping density.39–41In\nferromagnetic samples, Jpp\nexis believed to be negative,\nwhich was demonstrated by theoretical estimation Jpp\nex≈\n−0.3 eV (Ref.42). In our calculation, the antiferromag-\nneticp-dinteraction Jpp\nexis chosen to be −0.5 eV or\n−1.0 eV. The ferromagnetic s-dexchange coupling con-\nstant is taken to be Jss\nex= 0.2 eV (Ref.31).\nAnother important quantity for determining the Zee-\nman splitting is the macroscopic magnetization or the\neffective concentration of the Mn atoms. As deduced\nfrom the low-temperature saturate magnetization, only\naround 50% Mn atoms can contribute to the ferromag-\nnetic magnetization, which hasbeen recognizedasthe in-\nfluence of the compensation effect due to the deep donors\n(e.g., Asantisites)ortheformationofsixfold-coordinated\ncenters defect Mn6As(Ref.43). As only the uncompen-\nsated Mn atoms can supply holes and contribute to the\nferromagneticmagnetization,44one can also estimate the\ntotal hole density from the saturate magnetization.45\nHowever, the density of the itinerant hole can be smaller\nthan the effective Mn concentration because of the local-\nized effect in such disordered material. It was reported4\nTc Ms NMn\n(K) (emu ·cm−3) (1020cm−3)\nAa130 38 8\nBa157 47 10\nCb114 33 6.9\nDc110 – –\nEd139 53.5 11.5\naRef. 21,bRef. 23,cRef. 18,dRef. 45\nTABLE I: The parameters obtained from the experiments\nfordifferentsamples: A:Ga 0.93Mn0.07As/Ga 0.902In0.098As; B:\nGa0.93Mn0.07As/GaAs; C: Ga 0.93Mn0.07As/Ga 1−yInyAs; D:\nGa0.92Mn0.08As; E: Ga 0.896Mn0.104As0.93P0.07.Msstands for\nthe saturate magnetization at zero temperature M(0).\nthat the hole density is only 15-30% of the total concen-\ntration of the Mn atoms.43\nIn our calculation, the magnetization lies along the\nprinciple axis chosen as [001]-direction.31The conven-\ntional parameters are mainly taken from those of GaAs\nin Refs.46 and 47. Other sample-dependent parame-\nters such as the Curie temperature and effective Mn\nconcentration are picked up from the experimental\nworks.18,21,23,45For sample A, B and E (C), only the\nsaturate magnetization at 4 (104) K was given in the\nreferences. Nevertheless, one can extrapolate the zero\ntemperature magnetization Msfrom Eq.(5). The effec-\ntive Mn concentrations listed in TableI are derived from\nNMn=Ms/(gµBSd). It is clearto see that all of these ef-\nfectiveMnconcentrationsaremuchsmallerthanthedop-\ning density ( ≥1.5×1021cm−3) due to the compensation\neffect as discussed above. Since the saturate magneti-\nzation of sample D is unavailable, we treat the effective\nMn concentration as a parameter in this case. More-\nover, since the exact values of the itinerant hole densities\nare unclear in such strongly disordered samples, we treat\nthem as parameters. Two typical values are chosen in\nour numerical calculation, i.e., Nh= 3×1020cm−3and\n5×1020cm−3. The effective impurity density is taken to\nbe equal to the itinerant hole density.\nFor numerical calculation of the hole spin dynamics,\nthe momentum space is partitioned into blocks. Com-\npared to the isotropic parabolic dispersion, the band\nstructure in ferromagnetic GaMnAs is much more com-\nplex as we mentioned above [referred to Figs.1(b) and\n4]. Therefore, we need to extend the partition scheme\nused in isotropic parabolic dispersion48into anisotropic\ncase. In our scheme, the radial partition is still carried\nout with respect to the equal-energy shells, while the an-\ngular partition is done by following Ref.48. In contrast\nto the isotropic case, the number of states in one block is\ngenerally different from that in another block even both\nof them are on the same equal-energy shell. We calculate\nthe number of states of each block from its volume inmomentum space.\n 0 10 20 30 40 50 60 0.2  0.4  0.6  0.8  1∆pp (meV)T/Tc\n(a)\n6.9×1020 cm-3\n8×1020 cm-3\n1×1021 cm-3\n-0.4-0.3-0.2-0.1 0 0.1-0.1  0  0.1  0.2\nE (eV)k (2π/a ) [111] [001]\n(b)\n 0 51015\n-0.1 -0.05  0 0.05  0.1  0.15  0.2DOS (1020/eVcm3)\nE (eV)T = 0.1 TcNMn = 8×1020 cm-3\n(c)\n-0.05  0 0.05  0.1  0.15  0.2 0 5 10 15\nDOS (1020/eVcm3)\nE (eV)T = 0.99 Tc\n(d)\nFIG. 1: (Color online) (a) Zeeman energy as function of tem-\nperature. (b)Thevalencebandstructurewith∆pp= 45meV.\nThe blue dashed curve illustrates the Fermi level for the hol e\ndensityNh= 3×1020cm−3, while the green dotted one gives\nNh= 5×1020cm−3. The density of states as function of\nenergy at (c) T/Tc= 0.1 and (d) T/Tc= 0.99 for the uncom-\npensated Mn density NMn= 8×1021cm−3. In (d), the blue\ndashed curve stands for the upper heavy hole band from the\nspherical approximation and the corresponding DOS from the\nanalytical formula (√\n2E[√\nm∗/(2π/planckover2pi1)]3) is given as the green\ndotted curve. Here, Jpp\nex=−0.5 eV.\nA. Density of states\nBy solving Eq.(5), one obtains the magnetization at\nfinite temperature M(T) and the corresponding Zeeman\nenergy ∆pp. In Fig.1(a), the Zeeman splitting from\nJpp\nex=−0.5 eV is plotted as function of the temperature.\nIt is seen that the Zeeman energy is tens of milli-electron\nvolts at low temperature and decreases sharply near the\nCurie temperature due to the decrease of the magnetiza-\ntion. To show the anisotropicnonparabolicfeature of the\nband structure in the presence of the Zeeman splitting,\nwe illustrate the valence bands along [001]- and [111]-\ndirections in Fig.1(b), which are obtained from Eq.(6)\natT/Tc= 0.1 forNMn= 8×1020cm−3. In this case, the\nZeeman splitting ∆pp= 45 meV. The Fermi levels for the\nhole densities Nh= 3×1020cm−3and 5×1020cm−3are\nshown as blue dashed and green dotted curves, respec-\ntively. As one can see that all of the four upper bands\ncan be occupied and the effective mass approximation\nobviously breaks down.\nBy integrating over the volume of each equal-energy\nshell, one obtains the density of states (DOS) of each\nband as function of energy in Fig.1(c) and (d). Here the\nenergy is defined in the hole picture so that the sign of5\nthe energy is opposite to that in Fig.1(b). It is seen that\nthe DOS of the upper two bands are much larger than\nthose of the other bands, regardless of the magnitude of\nthe Zeeman splitting. For T/Tc= 0.99, the systems ap-\nproaches the paramagnetic phase and the nonparabolic\neffect is still clearly seen from the DOS in Fig.1(d), es-\npecially in the high energy regime. Moreover, the pro-\nnounced discrepancy of the DOS for the two heavy hole\nbands suggests the finite splitting between them. We\nfind that these features are closely connected with the\nanisotropy of the valence bands, corresponding to the\nLuttinger parameters γ2∝ne}ationslash=γ3in GaAs.49In our calcu-\nlation, we take γ1= 6.85,γ2= 2.1 andγ3= 2.9 from\nRef.47. As a comparison, we apply a spherical approx-\nimation ( γ1= 6.85 andγ2=γ3= ¯γ= 2.5) and find\nthat the two heavy hole bands become approximately\ndegenerate.38The DOS of the upper heavy hole band\nis shown as the blue dashed curve in Fig.1(d), where\nwe also plot the corresponding DOS from the analyti-\ncal expression, i.e.,√\n2E[√\nm∗/(2π/planckover2pi1)]3, as the green dot-\nted curve. Here, we use the heavy-hole effective mass\nm∗=m0/(γ1−2¯γ) withm0denoting the free electron\nmass. The perfect agreement between the analytical and\nour numerical results under the spherical approximation\nsuggests the good precision of our numerical scheme.\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  10  20  30  40  50  60Equilibrium Hole Polarization\n∆pp (meV)A: Nh=3×1020 cm-3\n5×1020 cm-3\nB: Nh=3×1020 cm-3\n5×1020 cm-3\nFIG. 2: (Color online) The equilibrium hole spin polarizati on\nas function of Zeeman splitting for sample A and B. Here,\nJpp\nex=−0.5 eV.\nB. Hole spin relaxation\nIn this part, we investigate the hole spin dynamics by\nnumericallysolvingthe microscopickinetic equation, i.e.,\nEq.(8). By taking into account the equilibrium hole spin\npolarization, we fit the temporal evolution of the total\nspin polarization along [001]-direction by\nJz(t) =J0\nz+J′\nze−t/τs, (12)\nwhereJ0\nzandJ′\nzcorrespondto the equilibrium and non-\nequilibrium spin polarizations, respectively. τsisthe hole\nspin relaxation time.In all the cases of the present work, the equilibrium\nhole spin polarization for a fixed hole density is found\nto be approximately linearly dependent on the Zeeman\nsplitting. In Fig.2, J0\nzin samples A and B (similar be-\nhaviorforothers) areplotted asfunction ofZeemansplit-\nting, where the exchange coupling constant Jpp\nexis taken\nto be−0.5 eV. One notices that the average spin polar-\nizationbecomessmallerwith the increaseofthe holeden-\nsity, reflecting the large interband mixing for the states\nin the high energy regime.\n 30 40 50 60 70 80\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1τs (fs)\nT/Tc(a)Jexpp = -0.5 eV\nA: Nh=3×1020 cm-3\n5×1020 cm-3\nB: Nh=3×1020 cm-3\n5×1020 cm-3\n 40 50 60 70 80\n 0  0.2  0.4  0.6  0.8  1  1.2 0  20  40  60  80  100  120τs (fs)\nT/Tc∆pp (meV)\n(b)\nJexpp = -1 eV0.4 Tc0.99 Tc\nB: Nh=3×1020 cm-3\n5×1020 cm-3\nFIG. 3: (Color online) (a) Spin relaxation time as function o f\ntemperaturewith Jpp\nex=−0.5eVfor sampleAandB.(b)Spin\nrelaxation time as function of temperature and Zeeman split -\nting obtained from the calculation with Jpp\nex=−1 eV for sam-\nple B. The inset at the left (right) upper corner illustrates the\nband structure from [001]-direction to [111]-direction [r efer to\nFig.1(b)] for T/Tc= 0.4 (0.99) and ∆pp= 105 (16.7) meV.\nThe Fermi levels of Nh= 3×1020cm−3and 5×1020cm−3\nare shown as the blue dashed and green dotted curves in the\ninsets, separately.\nThe temperature dependence of the hole spin relax-\nation time in samples A and B with Jpp\nex=−0.5 eV is\nshown in Fig.3(a), where the spin relaxation time mono-\ntonically decreases with increasing temperature. This\nfeature can be understood from the enhancement of\nthe interband mixing as the Zeeman splitting decreases\n(shownbelow).50Togainacompletepicture ofthe roleof\nthe Zeeman splitting on the hole spin relaxation in fer-6\nromagnetic GaMnAs, we also carry out the calculation\nwith the exchange constant Jpp\nex=−1 eV.31,39Very in-\nterestingly, onefinds that the holespin relaxationtime at\nlow temperature increases with increasing temperature,\nresulting in a nonmonotonic temperature dependence of\nthe hole spin relaxation time in sample B. The results\nin this case are shown as solid curves in Fig.3(b), where\nwe also plot the Zeeman splitting dependence of the hole\nspin relaxation time as dashed curves. It is seen that\nthe hole spin relaxation time for the hole density Nh=\n3×1020cm−3first increases with increasing temperature\n(alternativelyspeaking,decreasingZeemansplitting)and\nstarts to decrease at around 0 .8Tcwhere the Zeeman\nsplitting ∆pp= 70 meV. To understand this feature, we\nshowthetypicalbandstructureinthe increase(decrease)\nregime of the hole relaxation time at T/Tc= 0.4 (0.99),\ncorresponding to ∆pp= 105 (16.7) meV, in the inset at\nthe left (right) upper corner. The Fermi levels of the\nhole density 3 ×1020cm−3are labeled by blue dashed\ncurves. One finds that the carrier occupations in the\nincrease and decrease regimes are quite different. Specif-\nically, all of the four upper bands are occupied in the\ndecrease regime while only three valence bands are rele-\nvant in the increase regime.\nOne may naturally expect that the increase regime\noriginates from the contribution of the fourth band via\nthe inclusion of the additional scattering channels or the\nmodification of the screening. However, we rule out this\npossibilitythroughthecomputationwiththefourthband\nartificially excluded, where the results are qualitatively\nthe same as those in Fig.3(b). Moreover, the variations\nof the screening and the equilibrium distribution at fi-\nnite temperature are also demonstrated to be irrelevant\nto the present nonmonotonic dependence by our calcula-\ntion (not shown here). Therefore, the interesting feature\nhas to be attributed to the variations of the band dis-\ntortion and spin mixing due to the exchange interaction.\nThis is supported by our numerical calculation, where\nthe nonmonotonic behavior disappears once the effect of\nthe interband mixing is excluded by removing the wave-\nfunction overlaps |∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2in Eqs.(9) and (10) (not\nshown here).\nFor a qualitative understanding of the nonmonotonic\ntemperature dependence of the hole spin relaxation time,\nwe plot the Fermi surface in the kx-kz(ky= 0) and kx-\nky(kz= 0) planes at Nh= 3×1020cm−3in Fig.4. We\nchoose typical Zeeman splittings in the increase regime\n(∆pp= 105meV), the decreaseregime(∆pp= 16.7meV)\nand also the crossover regime (∆pp= 70 meV). One no-\ntices that the Fermi surfacesin Fig.4(a) and (d) arecom-\nposed of three closed curves, meaning that only three\nbands are occupied for ∆pp= 105 meV [also see the in-\nset of Fig.3(b)]. For the others with smaller Zeeman\nsplittings, all of the four upper bands are occupied. The\nspin expectation of each state at Fermi surface is repre-\nsented by the color coding. Note that the spin expecta-\ntion of the innermost band for ∆pp= 70 meV is close to\n−1.5 [see Fig.4(b) and (e)], suggesting that this band is-1.5-1-0.5 0 0.5 1 1.5ξ∆pp = 105 meV  70 meV  16.7 meV\n(a)-0.2-0.10.00.10.2kz (2π/a)\n-1.5-1-0.5 0 0.5 1 1.5\n(b)\n-1.5-1-0.5 0 0.5 1 1.5\n(c)\n-1.5-1-0.5 0 0.5 1 1.5\n(d)\n-0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2ky (2π/a)-1.5-1-0.5 0 0.5 1 1.5\n(e)\n-0.2 -0.1 0.0 0.1 0.2\nkx (2π/a)-1.5-1-0.5 0 0.5 1 1.5\n(f)\n-0.2 -0.1 0.0 0.1 0.2\nFIG.4: (Color online)TheFermisurface inthe kx-kz(ky= 0)\nandkx-ky(kz= 0) planes with ∆pp=105 meV (a,d), 70 meV\n(b,e) and 16.7 meV (c,f). The color coding represents the\nspin expectation of each state, ξ=/angbracketleftµ|Jz|µ/angbracketright. Here, Nh=\n3×1020cm−3.\nthe spin-down heavy hole band and the mixing of other\nspin components in this band is marginal. Therefore,\nthe spin-flip scattering related to this band is weak and\ncan not result in the increase of the hole spin relaxation\ntime mentioned above. By comparing the results with\n∆pp= 105 meV and 70 meV, one notices that the spin\nexpectation of the Fermi surface of the outermost band\nis insensitive to the Zeeman splitting. Therefore, this\nband can not be the reason of the increase regime also.\nMoreover, for the second and third bands in Fig.4(a)\nand (b), from the comparable color coding between the\ntwo figures in this regime [also see Fig.4(d) and (e) with\nkz= 0], one finds that the spin expectation for the\nstates with small kzis also insensitive to the Zeeman\nsplitting. However, for the states with large kz, the spin\nexpectation of the spin-down states ( ξ <0) approaches\na large magnitude ( −1.5) with decreasing Zeeman split-\nting, suggestingthe decrease ofthe mixing from the spin-\nup states. As a result, the interband spin-flip scattering\nfrom/to these states becomes weak and the hole spin re-\nlaxation time increases. In the decrease regime of the\nhole spin relaxation time, Fig.4(c) and (f) show that the\ntwo outer/inner bands approach each other, leading to a\nstrong and anisotropic spin mixing. Therefore, the spin-\nflip scattering becomes more efficient in this regime and\nthe spin relaxation time decreases. One may suppose\nthat the nonmonotonic temperature dependence of the\nhole spin relaxation time can also arise from the varia-\ntionofthe shapeofthe Fermisurface, accordingtoFig.4.\nHowever, this variation itself is not the key of the non-\nmonotonic behavior, because the calculation with this\neffect but without band mixing can not recover the non-\nmonotonic feature as mentioned in the previous para-\ngraph. For the hole density Nh= 5×1020cm−3, the\nstructures of the Fermi surface at ∆pp= 105 meV are\nsimilartothoseinFig.4(b)and(e). Thisexplainstheab-\nsence of the increase regime for this density in Fig.3(b).\nMoreover,weshouldpoint outthat the increaseregime7\nof the hole spin relaxation time in sample A for Jpp\nex=\n−1 eV is much narrower than that in sample B. The\nreason lies in the fact of lower effective Mn density in\nsample A, leading to the smaller maximal Zeeman split-\nting∼90 meV, only slightly larger than the crossover\nvalue 70 meV at Nh= 3×1020cm−3.\nAs a summary of this part, we find different temper-\nature dependences of the hole spin relaxation time due\nto the different values of effective Mn concentration, hole\ndensity and exchange coupling constant Jpp\nex. In the case\nwith large coupling constant and high effective Mn con-\ncentration, the interband spin mixing can resultin a non-\nmonotonic temperature dependence of the hole spin re-\nlaxation time. Our results suggest a possible way to esti-\nmate the exchange coupling constant with the knowledge\nof itinerant hole density, i.e., by measuring the temper-\nature dependence of the hole spin relaxation time. Al-\nternatively, the discrepancy between the hole relaxation\ntime from different hole densities in Fig.3(b) suggests\nthat one can also estimate the itinerant hole density if\nthe exchange coupling constant has been measured from\nother methods.\nC. Gilbert damping and non-adiabatic torque\ncoefficients\nFacilitated with the knowledge of the hole spin re-\nlaxation time, we can calculate the coefficients in the\nLLG equation. According to our previous works,10,17\nthe Gilbert damping and nonadiabatic spin torque co-\nefficients can be expressed as\nα=Jh/[NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|(β+1/β)], (13)\nand\nβ= 1/(2τs∆pp), (14)\nrespectively. In Eq.(13), Jhrepresents the total equi-\nlibrium spin polarization of the itinerant hole gas, i.e.,\nJh=NhJ0\nzwithJ0\nzbeing the one defined in Eq.(12) in\nour study. The average spin polarization of a single Mn\nion is given by |∝an}b∇acketle{tS∝an}b∇acket∇i}ht|=SdM(T)/M(0).\nIn Fig.5(a), (c) and (e), the nonadiabatic spin torque\ncoefficients βin sample A-C are plotted as function of\ntemperature with Jpp\nex=−0.5 eV and −1.0 eV. Our re-\nsults in sample C show good agreement with the experi-\nmental data (plotted as the brown square) in Fig.5(e).23\nAt low temperature, the value of βis around 0.1 ∼0.3,\nwhich is also comparable with the previous theoretical\ncalculation.19Very interestingly, one finds that βsharply\nincreases when the temperature approaches the Curie\ntemperature. This can be easily understood from the\npronounced decreases of the spin relaxation time and\nthe Zeeman splitting in this regime [see Figs.1(a) and\n3]. By comparing the results with different values of\nthe exchange coupling constant, one finds that βfrom\nJpp\nex=−1 eV is generally about one half of that ob-\ntained from Jpp\nex=−0.5 eV because of the larger Zeeman 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80  100  120β\nT (K)Sample A(a)-0.5 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n-1.0 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n 0 0.01 0.02 0.03 0.04\n 20  40  60  80  100  120α\nT (K)(b)\n 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80 100  120  140  160β\nT (K)Sample B(c)\n 0 0.01 0.02 0.03 0.04\n 20  40  60  80 100  120  140  160α\nT (K)(d)\n 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80  100  120β\nT (K)Sample C(e)\n 0 0.01 0.02 0.03 0.04 0.05\n 20  40  60  80  100  120α\nT (K)(f)\nFIG. 5: (Color online) βandαas function of temperature\nwithJpp\nex=−0.5 eV and −1.0 eV in sample A-C. In (b) and\n(d), the dots represent the experimental data from ferromag -\nnetic resonance measurement for [001] (brown solid upper tr i-\nangles), [110] (orange solid circles), [100] (green open sq uares)\nand [1-10] (black open lower triangles) dc magnetic-field or i-\nentations (Ref.21). The brown solid square in (e) stands for\nthe experimental result from domain-wall motion measure-\nment (Ref.23).\nsplitting. Moreover, one notices that the nonmonotonic\ntemperature dependence of the hole spin relaxation time\nin Fig.3(b) is not reflected in βdue to the influence of\nthe Zeeman splitting. In all cases, the values of βcan\nexceed one very near the Curie temperature.\nThe results of the Gilbert damping coefficient from\nEq.(13) are shown as curves in Fig.5(b), (d) and (f).\nThe dots in these figures are the reported experimental\ndata from the ferromagnetic resonance along different\nmagnetic-field orientations.21Both the magnitude and\nthetemperaturedependenceofourresultsagreewellwith\nthe experimental data. From Fig.2, one can conclude\nthat the prefactor in Eq.(13), Jh/(NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|), is almost\nindependent of temperature. Therefore, the temperature\ndependence of αmainly results from the nonadiabatic\nspin torque coefficient β. Specifically, αis insensitive\nto the temperature in the low temperature regime and\nit gradually increases with increasing temperature due8\nto the increase of β. Moreover, we predict that αbe-\ngins to decrease with increasing temperature once βex-\nceeds one. This crossover lying at β≈1 can be expected\nfrom Eq.(13). By comparing the results with different\nvalues of Jpp\nex, one finds that the value of αis robust\nagainst the exchange coupling constant in the low tem-\nperature regime. In this regime, β≪1 and one can sim-\nplify the expression of the Gilbert damping coefficient as\nα≈Nh\nNMnSdJ0\nz\n(τs∆pp). Since the total hole spin polariza-\ntion is proportional to the Zeeman splitting (see Fig.2)\nandτsis only weakly dependent on the Zeeman split-\nting (see Fig.3) in this regime, the increase of Jpp\nexdoes\nnot show significant effect on α. However, at high tem-\nperature, the scenario is quite different. For example,\none has the maximum of the Gilbert damping coefficient\nαm≈Nh\n2NMn|/angbracketleftS/angbracketright|J0\nz∝Jpp\nexatβ= 1.\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\n 20  40  60  80  100β\nT (K)Sample D(a)-0.5 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n-1.0 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n 0 0.02 0.04 0.06 0.08 0.1\n 20  40  60  80  100α\nT (K)(b)\nFIG. 6: (Color online) βandαas function of temperature by\ntakingNMn= 5×1020cm−3withJpp\nex=−0.5 eV and −1.0 eV\nin sample D. The dots are from ferromagnetic resonance mea-\nsurement (Ref.18) for [001] (brown solid upper triangles) a nd\n[110] (orange solid circles) dc magnetic-field orientation s.\nSince the effective Mn concentration of sample D is\nunavailable as mentioned above, we here take NMn=\n5×1020cm−3. The results are plotted in Fig.6. It is\nseen that the Gilbert damping coefficients from our cal-\nculation with Jpp\nex=−1 eV agree with the experiment\nvery well. As reported, the damping coefficient in this\nsample is much larger ( ∼0.1) before annealing.18The\nlarge Gilbert damping coefficient in the as-grown sample\nmay result from the direct spin-flip scattering between\nthe holes and the random Mn spins, existing in low qual-\nity samples. In the presence of this additional spin-flip\nchannel, the hole spin relaxation time becomes shorter\nand results in an enhancement of αandβ(forβ <1).\nMoreover, in the low temperature regime, a decrease of\nthe Gilbert damping coefficient was observed by increas-\ning temperature,18which is absent in our results. This\nmay originate from the complicated localization or cor-\nrelation effects in such a disordered situation. The quan-\ntitatively microscopic study in this case is beyond the\nscope of the present work.\nIn addition, one notices that βin Ref.24 was deter-\nmined to be around 0.01, which is one order of magni-\ntude smaller than our result. The reason is because of\nthe incorrectparameterused in that work, aspointed outby Adam et al..23\n 0 0.2 0.4 0.6 0.8 1\n 0  20  40  60  80  100  120  140Ass(v) (pJ/m)\nT (K)Nh=3×1020 cm-3, 1.0m0\n1×1020 cm-3, 1.0m0\n1×1020 cm-3, 0.5m0\nFIG. 7: (Color online) Spin stiffness (vertical spin stiffnes s)\ncoefficient as function of temperature is plotted as curves\nwith (without) symbols. The calculation is carried out with\nJpp\nex=−0.5 eV in sample E. The effective mass is taken to be\n1.0 (0.5) m0as labeled in the figure. The brown solid (from\nthe period of the domains) and open (from the hysteresis cy-\ncle) squares are the experimental data of spin stiffness from\nRef.45.\nD. Spin stiffness and vertical spin stiffness\nIn this subsection, we calculate the spin stiffness and\nvertical spin stiffness coefficients according to our previ-\nous derivation10\nAss=Nh/[4m∗(1+β2)] (15)\nand\nAv\nss=Nhβ/[4m∗(1+β2)]. (16)\nSince the effective mass m∗is a rough description for the\nanisotropic valence bands in the presence of a large Zee-\nman splitting, it is difficult to obtain the accurate value\nof the stiffness coefficients from these formulas. Nev-\nertheless, one can still estimate these coefficients with\nthe effective mass taken as a parameter. The results are\nplotted in Fig.7. By fitting the DOS of the occupied hole\nstates,wefind m∗≈m0, whichisconsistentwiththepre-\nvious work.31The spin stiffness and vertical spin stiffness\ncoefficients with Nh= 3×1020cm−3(1×1020cm−3) are\nplotted as the red solid (blue dashed) curves with and\nwithout symbols, respectively. The sudden decrease of\nAssoriginates from the increase of βin the vicinity of\nthe Curie temperature (see Fig.5). Our results are com-\nparable with the previous theoretical work from 6-band\nmodel.26As a comparison, we take m∗= 0.5m0, which\nis widely used to describe the heavy hole in the low en-\nergy regime in the absence of the Zeeman splitting.51\nThe spin stiffness becomes two times larger. Moreover,9\nAv\nssis found to present a nonmonotonic behavior in the\ntemperature dependence as predicted by Eq.(16).\nIn Fig.7, we also plot the experimental data of the\nspin stiffness coefficient from Ref.45. It is seen that these\nvalues of Assare comparable with our results and show\na decrease as the temperature increases. However, one\nnotices that the experimental data is more sensitive to\nthe temperature especially for those determined from the\ndomain period in the low temperature regime. This may\noriginate from the strong anisotropic interband mixing\nand inhomogeneity in the real material.\nIn Ref.10, we have shown that the vertical spin stiff-\nness can lead to the magnetization rotated around the\neasy axis within the domain wall structure by ∆ ϕ=\n(/radicalbig\n1+β2−1)/βin the absence of the demagnetization\nfield. For β= 1, ∆ϕ≈0.13π, while ∆ ϕ=β/2→0 for\nβ≪1. As illustrated above, βis always larger than 0.1.\nTherefore, the vertical spin stiffness can present observ-\nable modification of the domain wall structure in GaM-\nnAs system.10\nIV. SUMMARY\nIn summary, we theoretically investigate the tempera-\nture dependence of the LLG coefficients in ferromagnetic\nGaMnAs, based on the microscopic calculation of the\nhole spin relaxation time. In our calculation, we employ\nthe Zener model with the band structure carried out by\ndiagonalizing the 8 ×8 Kane Hamiltonian together with\nthe Zeeman energy due to the sp-dexchange interaction.\nWe find that the hole spin relaxation time can present\ndifferent temperature dependences, depending on the ef-fective Mn concentration, hole density and exchangecou-\npling constant. In the case with high Mn concentra-\ntion and large exchange coupling constant, the hole spin\nrelaxation time can be nonmonotonically dependent on\ntemperature, resulting from the different interband spin\nmixings in the large and small Zeeman splitting regimes.\nThese features are proposed to be for the estimation of\nthe exchange coupling constant or itinerant hole density.\nBysubstituting the hole relaxationtime, we calculatethe\ntemperature dependence of the Gilbert damping, nona-\ndiabatic spin torque, spin stiffness, and vertical spin stiff-\nness coefficients. We obtain the nonadiabatic spin torque\ncoefficient around 0 .1∼0.3 at low temperature, which\nis consistent with the experiment. As the temperature\nincreases, this coefficient shows a monotonic increase. In\nthe low temperature regime, the Gilbert damping co-\nefficient increases with temperature, which shows good\nagreement with the experiments. We predict that the\nGilbert damping coefficient can decrease with increasing\ntemperatureoncethenonadiabaticspintorquecoefficient\nexceed one in the vicinity of the Curie temperature. 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In particular, it is shown that varying the dissipation\nratio of one of the two oscillators can signi\fcantly suppress the nonlinear\nbeat phenomenon.\n1 Introduction\nThe damped least squares is a simple but e\u000bective analytical manipulation that\nhelps to avoid singularity in practical minimization and control algorithms. It\nis also known as Levenberg-Marquardt method [11]. In order to illustrate the\nidea in simple terms, let us consider the minimization problem\nkE\u0000A\u000euk2!min (1)\nwhereE2Rnis a given vector, the notation k:::kindicates the Euclidean\nnorm inRn,Ais typically a Jacobian matrix of nrows andmcolumns, and\n\u000eu2Rmis an unknown minimization vector. Although a formal solution of this\nproblem is given by \u000eu= (ATA)\u00001ATE, the matrix product ATAmay appear\nto be singular so that no unique solution is possible. This fact usually points\nto multiple possibilities of achieving the same result unless speci\fc conditions\nare imposed on the vector \u000eu. The idea of damped least squares is to avoid\nsuch conditioning by adding one more quadratic form to the left hand side of\nexpression (1) as follows\nkE\u0000A\u000euk2+\u0015k\u000euk2!min (2)\n1arXiv:1110.2811v2  [math.OC]  14 Oct 2011where\u0015is a positive scalar number, which is often called damping constant ;\nnote that the term `damping' has no relation to the physical damping or energy\ndissipation e\u000bects in vibrating systems usually characterized by damping ratios .\nNow the inverse matrix includes the damping constant \u0015which can provide\nthe uniqueness of solution given by\n\u000eu= (ATA+\u0015I)\u00001ATE (3)\nwhereIisn\u0002nidentity matrix.\nDi\u000berent arguments are discussed in the literature regarding the use of\ndamped least squares and best choice for the damping parameter \u0015[1], [2],\n[3], [4], [6], [7], [9], [10], [15], [16], [17], [23], [24]. In particular, it was noticed\nthat the parameter \u0015may a\u000bect convergence properties of the corresponding\nalgorithms. The parameter \u0015can be used also for other reason such as shifting\nthe solution \u000euinto desired area in Rm. In this case, the meaning of \u0015is rather\nclose to that of Lagrangian multiplier imposing constraints on control inputs.\nIn case of dynamical systems, when all the quantities in (2) may depend on\ntime, a continuous time analogue of (2) can be written in the integral form\nmin\n\u000euZT\n0(kE\u0000A\u000euk2+\u0015k\u000euk2)dt (4)\nwhere the interval of integration is manipulated as needed, for instance, Tcan\nbe equal to sampling time of the controller [12].\nHowever, in the present work, a discrete time algorithm based on the damped\nleast squares solution (3), which is used locally at every sample time tn, is intro-\nduced. Such algorithm appears to be essentially discrete namely using di\u000berent\ntime stephmay lead to di\u000berent results. Nevertheless, if the parameters \u0015and\nhare coupled by some condition then the control input and system response\nshow no signi\fcant dependence on the time step.\nA motivation for the present work is as follows. In order to comply with\nthe standard tool of dynamical systems dealing with di\u000berential equations, the\nmethods of control are often formulated in continuous time by silently assuming\nthat a discrete time analogous is easy to obtain one way or another whenever it is\nneeded for practical reasons. For instance, data acquisition cards and on-board\ncomputers of ground vehicles usually acquire and process data once per 0.01 sec.\nTypically, based on the information, which is known about the system dynamic\nstates and control inputs by the time instance tn, the computer must calculate\ncontrol adjustments for the next active time instance, tn+1. The corresponding\ncomputational time should not therefore exceed tn+1\u0000tn= 0:01 sec. Generally\nspeaking, it is possible to memorize snapshots of the dynamic states and control\ninputs at some of the previous times f...,tn\u00002,tn\u00001g. However, increasing the\nvolume of input data may complicate the code and, as a result, slow down the\ncalculation process. Therefore, let us assume that updates for the control inputs\nare obtained by processing the system states, controls, and target states given\nonly at the current time instance, tn. The corresponding algorithm can be built\non the system model described by its di\u000berential equations of motion and some\n2rule for minimizing the deviation (error) of the current dynamic states from the\ntarget. Recall that, in the present work, such a rule will be de\fned according\nto the damped least squares (2). Illustrating physical example of two linearly\ncoupled Du\u000eng oscillators is considered. It is shown that the corresponding\nalgorithm, which is naturally designed and e\u000bectively working in discrete time,\nmay face a problem of transition to the continuous time limit.\n2 Problem formulation\nConsider the dynamical system\nx=f(x;_x;t;u ) (5)\nwherex=x(t)2Rnis the system position (con\fguration) vector, the overdot\nindicates derivative with respect to time t, the right-hand side f2Rnrepresents\na vector-function that may be interpreted as a force per unit mass of the system,\nandu=u(t)2Rmis a control vector, whose dimension may di\u000ber from that\nof the positional vector xso that generally n6=m.\nIn common words, the purpose of control u(t) is to keep the acceleration\nx(t) of system (5) as close as possible to the target  x\u0003(t). The term `close'\nwill be interpreted below through a speci\fcally designed target function of the\nfollowing error vector\nE(t) = x\u0003(t)\u0000x(t) (6)\nAs discussed in Introduction, for practical implementations, the problem\nmust be formulated in terms of the discrete time ftkgas follows. Let xk=\nx(tk), _xk= _x(tk), anduk=u(tk) are observed at some time instance tk. The\ncorresponding target acceleration,  x\u0003\nk= x\u0003(tk), is assumed to be known. Then,\ntaking into account (5) and (6), gives the following error at the same time\ninstance\nEk= x\u0003\nk\u0000f(xk;_xk;tk;uk) (7)\nNow the purpose of control is to minimize the following target function\nPk=1\n2ET\nkWkEk (8)\n=1\n2[x\u0003\nk\u0000f(xk;_xk;tk;uk)]TWk[x\u0003\nk\u0000f(xk;_xk;tk;uk)]\nwhereWkisn\u0002ndiagonal weight matrix whose elements are positive or at\nleast non-negative functions of the system states, Wk=W(xk;_xk;tk).\nNote that all the quantities in expression (8) represent a snapshot of the\nsystem att=tkwhile including no data from the previous time step tk\u00001.\nSince the control vector ukcannot be already changed at time tkthen quantity\nPkis out of control at time tk. In other words expression (8) summarizes all\nwhat is observed now, at the time instance tk. The question is how to adjust\nthe control vector ufor the next step tk+1based on the information included in\n3(8) while the system state at t=tk+1is yet unknown, and no information from\nthe previous times f...,tn\u00002,tn\u00001gis available.\nLet us represent such an update for the control vector in the form\nuk+1=uk+\u000euk (9)\nwere\u000eukis an unknown adjustment of the control input.\nReplacingukin (8) by (9) and taking into account that\nf(xk;_xk;tk;uk+1) =f(xk;_xk;tk;uk) +Ak\u000euk+O(k\u000eukk2) (10)\nAk=@f(xk;_xk;tk;uk)=@uk\ngives\nPk=1\n2(Ek\u0000Ak\u000euk)TWk(Ek\u0000Ak\u000euk) (11)\nwhereAkis the Jacobian matrix of nrows andmcolumns.\nAlthough the replacement ukbyuk+1in (10) may look arti\fcial, this is how\nthe update rule for the control vector uis actually de\fned here. Namely, if\nukdid not provide a minimum for Pk(x\u0003\nk;xk;_xk;tk;uk), then let us minimize\nPk(x\u0003\nk;xk;_xk;tk;uk+\u000euk) with respect to \u000eukand then apply the adjusted\nvector (9) at least the next next time, tn+1. Assuming that the variation \u000eukis\nsmall, in other words, ukis still close enough to the minimum, expansion (10)\nis applied. Now the problem is formulated as a minimization of the quadratic\nform (11) with respect to the adjustment \u000euk. However, what often happens\npractically is that function (11) has no unique minimum so that equation\ndPk\nd\u000euk= 0 (12)\nhas no unique solution. In addition, even if the unique solution does exist, it\nmay not satisfy some conditions imposed on the control input due to the physical\nspeci\fcs of actuators. As a result, some constraint conditions may appear to\nbe necessary to impose on the variation of control adjustment, \u000euk. However,\nthe presence of constraints would drastically complicate the problem. Instead,\nthe target function (11) can be modi\fed in order to move solution \u000eukinto the\nallowed domain. For that reason, let us generalize function (11) as\nPk=1\n2(Ek\u0000Ak\u000euk)TWk(Ek\u0000Ak\u000euk)\n+1\n2(Bk+Ck\u000euk)T\u0003k(Bk+Ck\u000euk) (13)\nwhere \u0003k= \u0003(xk;_xk;tk) is a diagonal regularization matrix, Bk=B(xk;_xk;tk)\nis a vector-function of nelements, and Ck=C(xk;_xk;tk) is a matrix of nrows\nandmcolumns.\nNote that the structure of new function (13) is a generalization of (2). Sub-\nstituting (13) in (12), gives a linear set of equations in the matrix form whose\nsolution\u000eukbrings relationship (9) to the form\n4uk+1=uk+ (AT\nkWkAk+CT\nk\u0003kCk)\u00001(AT\nkWkEk\u0000CT\nk\u0003kBk) (14)\nThe entire discrete time system is obtained by adding a discrete version\nof the dynamical system (5) to (14) . Assuming that the time step is \fxed,\ntk+1\u0000tk=h, a simple discrete version can be obtained by means of Euler\nexplicit scheme as follows\nxk+1=xk+hvk\nvk+1=vk+hf(xk;vk;tk;uk) (15)\nFinally, equations (14) and (15) represent a discrete time dynamical system,\nwhose motion should follow the target acceleration  x\u0003\nk= x\u0003(tk).\nIt will be shown in the next section that the structure of equation (14) does\nnot allow for the transition to continuous limit of the entire dynamic system (14)\nthrough (15), unless some speci\fc assumption are imposed on the parameters\nin order to guarantee that \u000euk=O(h) ash!0.\n3 The illustrating example\nThe algorithm, which is designed in the previous section, is applied now to a\ntwo-degrees-of-freedom nonlinear vibrating system for an active control of the\nenergy exchange (nonlinear beats) between the two oscillators. The problem of\npassive control of energy \rows in vibrating systems is of great interest [22], and\nit is actively discussed from the standpoint of nonlinear beat phenomena [14].\nThe beating phenomenon takes place when frequencies of the corresponding\nlinear oscillators are either equal or at least close enough to each other.\nFor illustrating purposes, let us consider two unit-mass Du\u000eng oscillators of\nthe same linear sti\u000bness Kcoupled by the linear spring of sti\u000bness \r. The system\nposition is described by the vector-function of coordinates, x(t) = (x1(t);x2(t))T.\nIntroducing the parameters \n = ( \r+K)1=2and\"=\r=(\r+K), brings the\ndi\u000berential equations of motion to the form\n_x1=v1\n_x2=v2\n_v1=\u00002\u0010\nv1\u0000\n2x1+\"(\n2x2\u0000\u000bx3\n1)\u0011f1(x1;x2;v1) (16)\n_v2=\u00002u\nv2\u0000\n2x2+\"(\n2x1\u0000\u000bx3\n2)\u0011f2(x1;x2;v2;u)\nwhere\u000bis a positive parameter, \u0010anduare damping ratios1of the \frst and the\nsecond oscillators, respectively; the damping ratio u, which is explicitly shown\nas an argument of the function f2(x1;x2;v2;u), will be considered as a control\ninput.\n1As mentioned in Introduction, the damping (dissipation) ratio should not be confused\nwith the damping coe\u000ecient \u0015.\n5The problem now is to \fnd such variable damping ratio u=u(t) under\nwhich the second oscillator accelerates as close as possible to the given (target)\nacceleration,  x\u0003\n2(t).\nFollowing the discussion of the previous section, let us consider the prob-\nlem in the discrete time ftkg. In order to avoid confusion, the iterator k\nwill be separated from the vector component indexes by coma, for instance,\nxk= (x1;k;x2;k)T. Since only the second mass acceleration is of interest and\nthe system under consideration includes only one control input u, then, assum-\ning the weights to be constant, gives\nWk=\u00140 0\n0 1\u0015\n,Ak=@\n@uk\u0014f1;k\nf2;k\u0015\nwheref1;k\u0011f1(x1;k;x2;k;v1;k) andf2;k\u0011f2(x1;k;x2;k;v2;k;uk), and other\nmatrix terms become scalar quantities, say, \u0003 k=\u0015,Bk=b, andCk= 1. The\nunities inWkandCkcan always be achieved by re-scaling the target function\nand parameters \u0015andb. Note that re-scaling the target function by a constant\nfactor has no e\u000bect on the solution of equation (12).\nAs a result, the target function (13) takes the form\nPk=1\n2\u0012\nx\u0003\n2;k\u0000f2;k\u0000@f2;k\n@uk\u000euk\u00132\n+\u0015\n2(b+\u000euk)2(17)\nIn this case, equation (12) represents a single linear equation with respect\nto the scalar control adjustment, \u000euk. Substituting the corresponding solution\nin (14) and taking into account (15), gives the discrete time dynamical system\nuk+1=uk\u0000(f2;k\u0000x\u0003\n2;k)(@f2;k=@uk) +\u0015b\n(@f2;k=@uk)2+\u0015(18)\nand\nx1;k+1=x1;k+hv1;k\nx2;k+1=x2;k+hv2;k\nv1;k+1=v1;k+hf1;k (19)\nv2;k+1=v2;k+hf2;k\nLet us assume now that the target acceleration  x\u0003\n2is zero, in other words,\nthe purpose of control is to minimize acceleration of the second oscillator at any\nsample time tkas much as possible. Let us set still arbitrary parameter balso\nto zero. Then the target function (17) and dynamical system (18) and (19) take\nthe form\nPk=1\n2\u0014\nf2(x1;k;x2;k;v2;k;uk) +@f2(x1;k;x2;k;v2;k;uk)\n@uk\u000euk\u00152\n+\u0015\n2(\u000euk)2(20)\n6uk+1=uk+2\nv2;k\n4\n2v2\n2;k+\u0015f2(x1;k;x2;k;v2;k;uk)\nx1;k+1=x1;k+hv1;k\nx2;k+1=x2;k+hv2;k (21)\nv1;k+1=v1;k+hf1(x1;k;x2;k;v1;k)\nv2;k+1=v2;k+hf2(x1;k;x2;k;v2;k;uk)\nwhere the functions f1andf2are de\fned in (16).\nAs follows from the \frst equation in (21), transition to the continuous time\nlimit for the entire system (21) would be possible under the condition that\n2\nv2;k\n4\n2v2\n2;k+\u0015=O(h), ash!0 (22)\nCondition (22) can be satis\fed by assuming that \n = O(h). Such an as-\nsumption, however, makes little if any physical sense. As an alternative choice,\nthe condition \u0015=O(h\u00001) can be imposed by setting, for instance,\n\u0015h=\u00150 (23)\nwhere\u00150remains \fnite as h!0.\nHowever, condition (23) essentially shifts the weight on control to the second\nterm of the target function (17) so that the function asymptotically takes the\nform\nPk'\u00150\n2h(\u000euk)2, ash!0 (24)\nSuch a target function leads to the solution \u000euk= 0, which e\u000bectively elim-\ninates the control equation. In other words, the iterative algorithm seems to\nbe essentially discrete. As a result, the control input uk, generated by the \frst\nequation in (21), depends upon sampling time interval h. Let us illustrate this\nobservation by implementing the iterations (21) under the \fxed set of parame-\nters,\"= 0:1, \n = 1:0,\u000b= 1:5,\u0010= 0:025, and initial conditions, u0= 0:025,\nx1;0= 1:0,x2;0= 0:1,v1;0=v2;0= 0. The values to vary are two di\u000berent sam-\npling time intervals, h= 0:01 andh= 0:001, and three di\u000berent values of the\ndamping constant, \u0015= 0:1,\u0015= 1:0, and\u0015= 10:0. For comparison reason, Fig.\n1 shows time histories of the system coordinates under the \fxed control vari-\nableu=\u0010. This (no control) case corresponds to free vibrations of the model\n(16) whose dynamics represent a typical beat-wise decaying energy exchange\nbetween the two oscillators. As mentioned at the beginning of this section, the\nbeats are due to the 1:1 resonance in the generating system ( \"= 0,u=\u0010= 0);\nmore details on non-linear features of this phenomenon, the related analytical\ntools, and literature overview can be found in [20] and [14]. In particular, the\nstandard averaging method was applied to the no damping case of system (16)\nin [20].\n7Now the problem is to suppress the beat phenomenon by preventing the\nenergy \row from the \frst oscillator into the second oscillator. As follows from\nFigs. 2 through 5, such a goal can be achieved by varying the damping ratio\nof the second oscillator, fukg, during the vibration process according to the\nalgorithm2(21). First, the diagrams in Figs. 2 and 3 con\frm that the sampling\ntime interval hrepresents an essential parameter of the entire control loop. In\nparticular, decreasing the sampling interval from h= 0:01 toh= 0:001 e\u000bec-\ntively increases the strength of the control; compare fragments (b) in Figs. 2\nand 3. However, if such decrease of the sampling time is accompanied by the\nincrease of \u0015according to condition (23), then the strength of control remains\npractically unchanged; compare now fragments (b) in Figs. 2 and 4. As follows\nfrom fragments (a) in Figs. 2 and 4, the above modi\fcation of both parameters,\nhand\u0015, also brings some di\u000berence in the system response during the interval\n80<t< 150, but this is rather due to numerical e\u000bect of the time step. Finally,\nanalyzing the diagrams in Figs. 3 and 5, shows that reducing the parameter \u0015\nas many as ten times under the \fxed time step hleads to a signi\fcant increase\nof the control input fukgwith a minor e\u000bect on the system response though.\nTherefore the parameter \u0015can be used for the purpose of satisfying some con-\nstraint conditions on the control inputs fukgin case such conditions are due\nto physical limits of the corresponding actuators. In addition, let us show that\nparameter\u0015may a\u000bect the convergence of algorithm (21) based on the following\nconvergence criterion [18]:\nFor a \fxed point z\u0003to be a point of attraction of the algorithm zk+1=G(zk)\na su\u000ecient condition is that the Jacobian matrix of Gat the point z\u0003has all\nits eigenvalues numerically less than 1, and a necessary condition is that they\nare numerically at most 1. The geometric rate of convergence is the numerically\nlargest eigenvalue of this Jacobian.\nApplying this criterion to the algorithm (21) at zero point, gives that one\nof the eigenvalues is always zero, q0= 0, whereas another four eigenvalues, qi\n(i= 1;:::;4) are proportional to the time step, qi=hpi, where the coe\u000ecients\npiare given by the roots of algebraic equation\np4+ 2\u0010\np3+ 2\n2p2+ 2\u0010\n3p+ (1\u0000\"2)\n4= 0 (25)\nAs follows from (25), the damping coe\u000ecient \u0015has no in\ruence on the con-\nvergence condition near the equilibrium point, and the convergence can always\nbe achieved under a small enough time step h. Nevertheless, the damping coef-\n\fcient may appear to a\u000bect the convergence away from the equilibrium point.\nIn this case, analytical estimates for eigen values of the Jacobian become tech-\nnically complicated unless \"= 0, when four of the \fve eigenvalues vanish as\nh!0, except one eigenvalue, which is estimated by\nq=\u0000\u0012\n1 +\u0015\n4\n2v2\n2\u0013\u00001\n(26)\n2Note that, although the algorithm is designed to suppress accelerations of the second\noscillator, acceleration and energy levels of vibrating systems are related.\n8This root gives q!q0= 0 asv2!0. However, when v26= 0, equation\n(26) gives the estimate 0 <q\u00141 as1>\u0015\u00150. Therefore, only the necessary\nconvergence condition is satis\fed for \u0015= 0.\n4 Conclusions\nIn this work, a discrete time control algorithm for nonlinear vibrating systems\nusing the damped least squares is introduced. It is shown that the corresponding\ndamping constant \u0015and sampling time step hmust be coupled by the condition\n\u0015h=constant in order to preserve the result of calculation when varying the\ntime step. In particular, the above condition prohibits a direct transition to\nthe continuos time limit. This conclusion and other speci\fcs of the algorithm\nare illustrated on the nonlinear two-degrees-of-freedom vibrating system in the\nneighborhood of 1:1 resonance. It is shown that the dissipation ratio of one of\nthe two oscillators can be controlled in such way that prevents the energy ex-\nchange (beats) between the oscillators. From practical standpoint, controlling\nthe dissipation ratio can be implemented by using devices based on the physical\nproperties of magnetorheological \ruids (MRF) [8], [19]. In particular, di\u000ber-\nent MRF dampers are suggested to use for semi-active ride controls of ground\nvehicles and seismic response reduction.\n9References\n[1] Chan, S. K., and Lawrence, P. D., \\General inverse kinematics with the\nerror damped pseudoinverse,\" Proc. IEEE International Conference on\nRobotics and Automation , 834-839, (1988).\n[2] Chiaverini, S., \\Estimate of the two smallest singular values of the Jacobian\nmatrix: Applications to damped least-squares inverse kinematics,\" Journal\nof Robotic Systems , 10, 991-1008, (1988).\n[3] Chiaverini, S. Siciliano, B., and Egeland, O., \\Review of damped least-\nsquares inverse kinematics with experiments on an industrial robot ma-\nnipulator,\" IEEE Transactions on Control Systems Technology , 2, 123-134,\n(1994).\n[4] Chung, C. Y., and Lee, B. H., \\Torque optimizing control with singularity-\nrobustness for kinematically redundant robots,\" Journal of Intelligent and\nRobotic Systems , 28, 231-258, (2000).\n[5] Colbaugh, R., Glass, K., and Seraji, H., \\Adaptive tracking control of\nmanipulators: theory and experiments,\" Robotics & Computer-Integrated\nManufacturing, 12(3), 209-216, (1996).\n[6] Deo, A. S., and Walker, I. D., \\Robot subtask performance with singularity\nrobustness using optimal damped least squares,\" Proc. IEEE International\nConference on Robotics and Automation , 434-441, (1992).\n[7] Deo, A. S., and Walker, I. D., \\Adaptive non-linear least squares for in-\nverse kinematics,\" Proc. IEEE International Conference on Robotics and\nAutomation , 186-193, (1993).\n[8] Dyke, S.J., Spencer Jr., B.F., Sain, M.K., and Carlson, J.D., \\Seismic\nresponse reduction using magnetorheological dampers,\" Proceedings of the\nIFAC World Congress, San Francisco, California, June 30-July 5 , Vol. L,\n145-150, (1996).\n[9] Hamada, C., Fukatani, K., Yamaguchi, K., and Kato, T., \\Development of\nvehicle dynamics integrated management,\" SAE Paper No. 2006-01-0922.\n[10] Hattori, Y., \\Optimum vehicle dynamics control based on tire driving and\nbraking forces,\" R&D Review of Toyota CRDL , 38 (5), 23-29, (2003).\n[11] Levenberg, K., \\A Method for the Solution of Certain Non-Linear Prob-\nlems in Least Squares,\" The Quarterly of Applied Mathematics , 2, 164{168,\n(1944).\n[12] Lee, J.H., and Yoo, W.S., \\Predictive control of a vehicle trajectory using\na coupled vector with vehicle velocity and sideslip angle,\" International\nJournal of Automotive Technology , 10 (2), 211-217, (2009).\n10[13] Maciejewski, A. A., and Klein, C. A., \\The singular value decomposi-\ntion: Computation and applications to robotics,\" International Journal\nof Robotic Research , 8 , 63-79, (1989).\n[14] Manevitch, L.I., Gendelman, O.V., Tractable Models of Solid Mechanics:\nFormulation, Analysis and Interpretation , Springer - Verlag, Berlin Heidel-\nberg, 2011, 297 p.\n[15] Mayorga, R. V., Milano, N., and Wong, A. K. C., \\A simple bound for\nthe appropriate pseudoinverse perturbation of robot manipulators,\" Proc.\nIEEE International Conference on Robotics and Automation , Vol. 2, 1485-\n1488, (1990).\n[16] Mayorga, R. V., Wong, A. K. C., and Milano, N., \\A fast procedure for\nmanipulator inverse kinematics evaluation and pseudoinverse robustness,\"\nIEEE Transactions on Systems, Man, and Cybernetics , 22, 790-798, (1992).\n[17] Nakamura, Y., and Hanafusa, H., \\Inverse kinematics solutions with sin-\ngularity robustness for robot manipulator control,\" Journal of Dynamic\nSystems, Measurement, and Control , 108 , 163-171, (1986).\n[18] Ostrowski, A M, Solution of Equations and Systems of Equations , Aca-\ndemic Press Inc., 1960, 352 p.\n[19] Phule, P.P., \\Magnetorheological (MR) \ruids: Principles and applica-\ntions,\" Smart Materials Bulletin , February, 7-10, (2001).\n[20] Pilipchuk, V.N., Nonlinear Dynamics: Between Linear and Impact Limits ,\nSpringer, 2010, 360 p.\n[21] Seraji, H., Colbaugh, R., \\Improved con\fguration control for redundant\nrobots,\" J. Robotic Systems , 7-6, 897-928, (1990).\n[22] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Ker-\nschen, G., Lee, Y.S., Nonlinear Targeted Energy Transfer in Mechanical\nand Structural Systems , Springer-Verlag, Berlin Heidelberg, 2009, 1032 p.\n[23] Wampler, C. W., \\Manipulator inverse kinematic solutions based on vector\nformulations and damped least squares methods,\" IEEE Transactions on\nSystems, Man, and Cybernetics , 16, 93-101, (1986).\n[24] Wampler, C. W., and Leifer, L. J., \\Applications of damped least-squares\nmethods to resolved-rate and resolved-acceleration control of manipula-\ntors,\" Journal of Dynamic Systems, Measurement, and Control , 110, 31-38,\n(1988).\n11Figure 1: No control beat dynamics with the decaying energy exchange\nbetween two Du\u000eng's oscillators; u=\u0010= 0:025.\n12Figure 2: Beat suppression under the time increment h= 0:01 and weight\nparameter\u0015= 1:0: (a) the system response, (b) control input - the damping\nratio of second oscillator.\n13Figure 3: Beat suppression under the reduced time increment h= 0:001 and\nthe same weight parameter \u0015= 1:0: (a) the system response, (b) control input\n- the damping ratio of second oscillator.\n14Figure 4: Beat suppression under the reduced time increment h= 0:001 but\nincreased weight parameter \u0015= 10:0: (a) the system response, (b) control input\n- the damping ratio of second oscillator.\n15Figure 5: Beat suppression under the reduced time increment h= 0:001 and\nvanishing weight parameter \u0015= 0:1: (a) the system response, (b) control input\n- the damping ratio of second oscillator.\n16"    },    {        "title": "1110.3387v2.Atomistic_spin_dynamic_method_with_both_damping_and_moment_of_inertia_effects_included_from_first_principles.pdf",        "content": "Atomistic spin dynamic method with both damping and moment of inertia e\u000bects\nincluded from \frst principles\nS. Bhattacharjee, L. Nordstr om, and J. Fransson\u0003\nDepartment of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden\n(Dated: October 28, 2018)\nWe consider spin dynamics for implementation in an atomistic framework and we address the\nfeasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the\nspirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized\nequation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in\nboth space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation,\nalso including the moment of inertia, and demonstrate how the exchange interaction, damping, and\nmoment of inertia, all can be calculated from \frst principles.\nPACS numbers: 72.25.Rb, 71.70.Ej, 75.78.-n\nIn recent years there has been a huge increase in the in-\nterest in fast magnetization processes on a femto-second\nscale, which has been initialized by important develop-\nments in experimental techniques [1{5], as well as po-\ntential technological applications [6]. From a theoretical\nside, the otherwise trustworthy spin dynamical (SD) sim-\nulation method fails to treat this fast dynamics due to\nthe short time and length scales involved. Attempts have\nbeen made to generalize the mesoscopic SD method to an\natomistic SD, in which the dynamics of each individual\natomic magnetic moment is treated [7, 8]. While this\napproach should in principle be well suited to simulate\nthe fast dynamics observed in experiments, it has not\nyet reached full predictive power as it has inherited phe-\nnomenological parameters, e.g. Gilbert damping, from\nthe mesoscopic SD. The Gilbert damping parameter is\nwell established in the latter regime but it is not totally\nclear how it should be transferred to the atomic regime.\nIn addition, very recently it was pointed out that the mo-\nment of inertia, which typically is neglected, plays an im-\nportant role for fast processes [9]. In this Letter we derive\nthe foundations for an atomistic SD where all the rele-\nvant parameters, such as the exchange coupling, Gilbert\ndamping, and moment of inertia, can be calculated from\n\frst principles electronic structure methods.\nUsually the spin dynamics is described by the phe-\nnomenological Landau-Lifshitz-Gilbert (LLG) equation\n[10, 11] which is composed of precessional and damping\nterms driving the dynamics to an equilibrium. By in-\ncluding the moment of inertia, we arrive at a generalized\nLLG equation\n_M=M\u0002(\u0000\rB+^G_M+^IM) (1)\nwhere ^Gand^Iare the Gilbert damping and the moment\nof inertia tensors, respectively. In this equation the e\u000bec-\ntive \feld Bincludes both the external and internal \felds,\nof which the latter includes the exchange coupling and\nanisotropy e\u000bects. Here, we will for convenience include\nthe anisotropy arising from the classical dipole-dipole in-\nteraction responsible for the shape anisotropy as a partof the external \feld. The damping term in the LLG\nequation usually consists of a single damping parame-\nter, which essentially means that the time scales of the\nmagnetization variables and the environmental variables\narewell separated . This separation naturally brings a\nlimitation to the LLG equation concerning its time scale\nwhich is restricting it to the mesoscopic regime.\nThe addition of a moment of inertia term to the LLG\nequation can be justi\fes as follows. A general process\nof a moment Munder the in\ruence of a \feld Fis al-\nways endowed with inertial e\u000bects at higher frequencies\n[12]. The \feld Fand moment Mcan, for example, be\nstress and strain for mechanical relaxation, electric \feld\nand electric dipole moment in the case of dielectric re-\nlaxation, or magnetic \feld and magnetic moment in the\ncase of magnetic relaxation. In this Letter we focus on\nthe latter case | the origin of the moment of inertia in\nSD. The moment of inertia leads to nutations of the mag-\nnetic moments, see Fig. 1. Its wobbling variation of the\nazimuthal angle has a crucial role in fast SD, such as fast\nmagnetization reversal processes.\nIn the case of dielectric relaxation the inertial e\u000bects\nare quite thoroughly mentioned in the literature [13, 14],\nespecially in the case of ferroelectric relaxors. Co\u000bey et\nal.[14] have proposed inertia corrected Debye's theory of\ndielectric relaxation and showed that by including inertial\nB\nprecessionĜ\ndampingÎ\nnutation\nFIG. 1: The three contributions in Eq. (1), the bare preces-\nsion arising from the e\u000bective magnetic \feld, and the super-\nimposed e\u000bects from the Gilbert damping and the moment of\ninertia, respectively.arXiv:1110.3387v2  [cond-mat.stat-mech]  1 Feb 20122\ne\u000bects, the unphysical high frequency divergence of the\nabsorption co-e\u000ecient is removed.\nVery recently Ciornei et al [9] have extended the LLG\nequation to include the inertial e\u000bects through a mag-\nnetic retardation term in addition to precessional and\ndamping terms. They considered a collection of uni-\nformly magnetized particles and treated the total angular\nmomentum Las faster variable. They obtained Eq. (1)\nfrom a Fokker-Plank equation where the number den-\nsity of magnetized particles were calculated by integrat-\ning a non-equilibrium distribution function over faster\nvariables such that faster degrees of freedom appear as\nparameter in the calculation.\nThe authors showed that at very short time scales the\ninertial e\u000bects become important as the precessional mo-\ntion of magnetic moment gets superimposed with nuta-\ntion loops due to inertial e\u000bects. It is pointed out that\nthe existence of inertia driven magnetization dynamics\nopen up a pathway for ultrafast magnetic switching [15]\nbeyond the limitation [16] of the precessional switching.\nIn practice, to perform atomistic spin dynamics simu-\nlations the knowledge of ^Gand^Iis necessary. There are\nrecent proposals [17, 18] of how to calculate the Gilbert\ndamping factor from \frst principles in terms of Kubo-\nGreenwood like formulas. Here, we show that similar\ntechniques may by employed to calculate the moment of\ninertia tensor ^I. Finally, we present a microscopical jus-\nti\fcation of Eq. (1), considering a collective magnetiza-\ntion density interacting locally with electrons constitut-\ning spin moments. Such a description would in principle\nbe consistent with the study of magnetization dynam-\nics where the exchange parameters are extracted from\n\frst-principles electronic structure calculations, e.g den-\nsity functional theory (DFT) methods. We \fnd that in\nan atomistic limit Eq. (1) actually has to be general-\nized slightly as both the damping and inertia tensors are\nnaturally non-local in the same way as the exchange cou-\npling included in the e\u000bective magnetic \feld B. From\nour study it is clear that both the damping and the mo-\nment of inertia e\u000bects naturally arise from the retarded\nexchange interaction.\nWe begin by considering the magnetic energy E=M\u0001\nB. Using that its time derivative is _E=M\u0001_B+_M\u0001B\nalong with Eq. (1), we write\n_E=M\u0001_B+1\n\r_M\u0001\u0010\n^G_M+^IM\u0011\n: (2)\nRelating the rate of change of the total energy to the\nHamiltonianH, through _E=hdH=dti, and expanding\ntheHlinearly around its static magnetization M0, with\nM(t) =M0+\u0016(t), we can writeH\u0019H 0+\u0016(t)\u0001r\u0016H0,\nwhereH0=H(M0). Then the rate of change of the total\nenergy equals _E= _\u0016\u0001hr\u0016Hito the \frst order. Following\nRef. [19] and assuming su\u000eciently slow dynamics such\nthat\u0016(t0) =\u0016(t)\u0000\u001c_\u0016(t) +\u001c2\u0016(t)=2,\u001c=t\u0000t0, we canwrite the rate of change of the magnetic energy as\n_E= lim\n!!0_\u0016i[\u001fij(!)\u0016j+i@!\u001fij(!) _\u0016j\u0000@2\n!\u001fij(!)\u0016j=2]:\n(3)\nHere,\u001fij(!) =R\n(\u0000i)\u0012(\u001c)h[@iH0(t);@jH0(t0)]iei!\u001cdt0,\n\u001c=t\u0000t0, is the (generalized) exchange interaction\ntensor out of which the damping and moments of in-\nertia can be extracted. Summation over repeated in-\ndices (i;j=x;y;z ) is assumed in Eq. (3). Equat-\ning Eqs. (2) and (3) results in an internal contribu-\ntion to the e\u000bective \feld about which the magnetiza-\ntion precesses Bint=\u0016lim!!0\u001f(!), the damping term\n^G=\rlim!!0i@!\u001f(!) as well as the moment of inertia\n^I=\u0000\rlim!!0@2\n!\u001f(!)=2.\nFor a simple order of magnitude estimate of the damp-\ning and inertial coe\u000ecients, ^Gand^I, respectively, we\nmay assume for a state close to a ferromagnetic state\nthat the spin resolved density of electron states \u001a\u001b(\")\ncorresponding to the static magnetization con\fguration\nH0is slowly varying with energy. At low temperatures\nwe, then, \fnd\n^G\u00182\r\u0019sp [h@iH0i\u001ah@jH0i\u001a]\"=\"F; (4)\nin agreement with previous results [19]. Here, sp denotes\nthe trace over spin 1/2 space. By the same token, the\nmoment of inertia is estimated as\n^I\u0018\u0000(\r=D) sp [h@iH0i\u001ah@jH0i\u001a]\"=\"F; (5)\nwhere 2Dis the band width of density of electron states\nof the host material. Typically, for metallic systems the\nband width 2 D\u00181|10 eV, which sets the time-scale\nof the inertial contribution to the femto second (10\u000015\ns) regime. It, therefore, de\fnes magnetization dynamics\non a time-scale that is one or more orders of magnitude\nshorter compared to e.g. the precessional dynamics of the\nmagnetic moment.\nNext, we consider the physics leading to the LLG equa-\ntion given in Eq. (1). As there is hardly any microscopical\nderivation of the LLG equation in the literature, we in-\nclude here, for completeness the arguments that leads to\nthe equation for the spin-dynamics from a quantum \feld\ntheory perspective.\nIn the atomic limit the spin degrees of freedom are\ndeeply intertwined with the electronic degrees of free-\ndom, and hence the main environmental coupling is the\none to the electrons. In this study we are mainly con-\ncerned with a mean \feld description of the electron\nstructure, as in the spirit of the DFT. Then a natural\nand quite general description of the magnetic interac-\ntion due to electron-electron interactions on the atomic\nsite around rwithin the material is captured by the s-d-\nlike modelHint=\u0000R\nJ(r;r0)M(r;t)\u0001s(r0;t)drdr0, where\nJ(r;r0) represents the interaction between the magneti-\nzation density Mand the electron spin s. From a DFT3\nperspective the interaction parameter J(r;r0) is related\nto the e\u000bective spin dependent exchange-correlation func-\ntionalBxc[M(r0)](r). For generality we assume a fully\nrelativistic treatment of the electrons, i.e. including the\nspin-orbit coupling. In this interaction the dichotomy of\nthe electrons is displayed, they both form the magnetic\nmoments and provide the interaction among them.\nOwing to the general non-equilibrium conditions in the\nsystem, we de\fne the action variable\nS=I\nCHintdt+SZ+SWZWN (6)\non the Keldysh contour [20{22]. Here, the ac-\ntionSZ=\u0000\rH\nCR\nBext(r;t)\u0001M(r;t)dtdrrepresents\nthe Zeeman coupling to the external \feld Bext(r;t),\nwhereas the Wess-Zumino-Witten-Novikov (WZWN)\ntermSWZWN =RH\nCR1\n0M(r;t;\u001c)\u0001[@\u001cM(r;t;\u001c)\u0002\n@tM(r;t;\u001c)]d\u001cdtjM(r)j\u00002drdescribes the Berry phase\naccumulated by the magnetization.\nIn order to acquire an e\u000bective model for the magne-\ntization density M(r;t), we make a second order [23] ex-\npansion of the partition function Z[M(r;t)]\u0011trTCeiS,\nand take the partial trace over the electronic degrees of\nfreedom in the action variable. The e\u000bective action \u000eSM\nfor the magnetization dynamics arising from the mag-\nnetic interactions described in terms of Hint, can, thus,\nbe written\n\u000eSM=\u0000I Z\nM(r;t)\u0001D(r;r0;t;t0)\u0001M(r0;t0)drdr0dtdt0;\n(7)\nwhereD(r;r0;t;t0) =R\nJ(r;r1)(\u0000i)hTs(r1;t)s(r2;t0)i\u0002\nJ(r2;r0)dr1dr2is a dyadic which describes the electron\nmediated exchange interaction.\nConversion of the Keldysh contour integrations into\nreal time integrals on the interval ( \u00001;1) results in\nS=Z\nM(fast)(r;t)\u0001[M(r;t)\u0002_M(r;t)]dtjM(r)j\u00002dr\n+Z\nM(fast)(r;t)\u0001Dr(r;r0;t;t0)\u0001M(r0;t0)drdr0dtdt0\n\u0000\rZ\nBext(r;t)\u0001M(fast)(r;t)dtdr; (8)\nwithM(fast)(r;t) =Mu(r;t)\u0000Ml(r;t) and M(r;t) =\n[Mu(r;t) +Ml(r;t)]=2 which de\fne fast and slow vari-\nables, respectively. Here, Mu(l)is the magnetization den-\nsity de\fned on the upper (lower) branch of the Keldysh\ncontour. Notice that upon conversion into the real time\ndomain, the contour ordered propagator Dis replaced by\nits retarded counterpart Dr.\nWe obtain the equation of motion for the (slow) mag-\nnetization variable M(r;t) in the classical limit by mini-\nmizing the action with respect to M(fast)(r;t), cross mul-\ntiplying by M(r;t) under the assumption that the totalmoment is kept constant. We, thus, \fnd\n_M(r;t) =M(r;t)\u0002\u0012\n\u0000\rBext(r;t)\n+Z\nDr(r;r0;t;t0)\u0001M(r0;t0)dt0dr0\u0013\n:(9)\nEq. (9) provides a generalized description of the semi-\nclassical magnetization dynamics compared to the LLG\nEq. (1) in the sense that it is non-local in both time and\nspace. The dynamics of the magnetization at some point\nrdepends not only on the magnetization locally at r,\nbut also in a non-trivial way on the surrounding magne-\ntization. The coupling of the magnetization at di\u000berent\npositions in space is mediated via the electrons in the\nhost material. Moreover, the magnetization dynamics is,\nin general, a truly non-adiabatic process in which the\ninformation about the past is crucial.\nHowever, in order to make connection to the magne-\ntization dynamics as described by e.g. the LLG equa-\ntion as well as Eq. (1) above, we make the following\nconsideration. Assuming that the magnetization dy-\nnamics is slow compared to the electronic processes in-\nvolved in the time-non-local \feld D(r;r0;t;t0), we ex-\npand the magnetization in time according to M(r0;t0)\u0019\nM(r0;t)\u0000\u001c_M(r0;t) +\u001c2M(r0;t)=2. Then for the inte-\ngrand in Eq. (9), we get\nDr(r;r0;t;t0)\u0001M(r0;t0) =\nDr(r;r0;t;t0)\u0001[M(r0;t)\u0000\u001c_M(r0;t) +\u001c2\n2M(r0;t)]:(10)\nHere, we observe that as the exchange coupling for the\nmagnetization is non-local and mediated through D, this\nis also true for the damping (second term) and the inertia\n(third term).\nIn order to obtain an equation of the exact same\nform as LLG in Eq. (1) we further have to assume\nthat the magnetization is close to a uniform ferromag-\nnetic state, then we can justify the approximations\n_M(r0;t)\u0019_M(r;t) and M(r0;t)\u0019M(r;t). When\nBint=\u0000R\nD(r;r0;t;t0)\u0001M(r0;t)dr0dt0=\ris included in\nthe total e\u000bective magnetic \feld B, the tensors of Eq. (1)\n^Gand^Ican be identi\fed with \u0000R\n\u001cD(r;r0;t;t0)dr0dt0\nandR\n\u001c2D(r;r0;t;t0)dr0dt0=2, respectively. From a \frst\nprinciples model of the host materials we have, thus, de-\nrived the equation for the magnetization dynamics dis-\ncussed in Ref. 9, where it was considered from purely\nclassical grounds. However it is clear that for a treatment\nof atomistic SD that allows for all kinds of magnetic or-\nders, not only ferromagnetic, Eq. (1) is not su\u000ecient and\nthe more general LLG equation of Eq. (9) together with\nEq. (10) has to be used.\nWe \fnally describe how the parameters of Eq. (1)\ncan be calculated from a \frst principles point of view.\nWithin the conditions de\fned by the DFT system, the\ninteraction tensor Dris time local which allows us to4\nwrite lim \"!0i@\"Dr(r;r0;\") =R\n\u001cDr(r;r0;t;t0)dt0and\nlim\"!0@2\n\"Dr(r;r0;\") =\u0000R\n\u001c2Dr(r;r0;t;t0)dt0, where\nDr(r;r0;\") = 4 spZ\nJr\u001aJ\u001a0r0f(!)\u0000f(!0)\n\"\u0000!+!0+i\u000e\n\u0002\u001bImGr\n\u001a0\u001a(!)\u001bImGr\n\u001a\u001a0(!0)d!\n2\u0019d!0\n2\u0019d\u001ad\u001a0:(11)\nHere,Jrr0\u0011J(r;r0) whereas Gr\nrr0(!)\u0011Gr(r;r0;!) is\nthe retarded GF, represented as a 2 \u00022-matrix in spin-\nspaces. We notice that the above result presents a general\nexpression for frequency dependent exchange interaction.\nUsing Kramers-Kr onig's relations in the limit \"!0, it\nis easy to see that Eq. (11) leads to\nDr(r;r0; 0) =\u00001\n\u0019sp ImZ\nJr\u001aJ\u001a0r0f(!)\n\u0002\u001bGr\n\u001a0\u001a(!)\u001bGr\n\u001a\u001a0(!)d!d\u001ad\u001a0;(12)\nin agreement with e.g. Ref. [24]. We can make connection\nwith previous results, e.g. Refs. 25, 26, and observe that\nEq. (11) contains the isotropic Heisenberg, anisotropic\nIsing, and Dzyaloshinsky-Moriya exchange interactions\nbetween the magnetization densities at di\u000berent points\nin space [22], as well as the onsite contribution to the\nmagnetic anisotropy.\nUsing the result in Eq. (11), we \fnd that the damping\ntensor is naturally non-local and can be reduced to\n^G(r;r0) =1\n\u0019spZ\nJr\u001aJ\u001a0r0f0(!)\n\u0002\u001bImGr\n\u001a0\u001a(!)\u001bImGr\n\u001a\u001a0(!)d!d\u001ad\u001a0;(13)\nwhich besides the non-locality is in good accordance with\nthe results in Refs. [17, 25], and is closely connected to\nthe so-called torque-torque correlation model [27]. With\ninclusion of the the spin-orbit coupling in Gr, it has been\ndemonstrated that Eq. (13) leads to a local Gilbert damp-\ning of the correct order of magnitude for the case of fer-\nromagnetic permalloys [17].\nAnother application of Kramers-Kr onig's relations\nleads, after some algebra, to the moment of inertia tensor\n^I(r;r0) = spZ\nJr\u001aJ\u001a0r0f(!)\u001b[ImGr\n\u001a0\u001a(!)\u001b@2\n!ReGr\n\u001a\u001a0(!)\n+ ImGr\n\u001a\u001a0(!)\u001b@2\n!ReGr\n\u001a0\u001a(!)]d!\n2\u0019d\u001ad\u001a0;(14)\nwhere we notice that the moment of inertia is not sim-\nply a Fermi surface e\u000bect but depends on the electronic\nstructure as a whole of the host material. Although the\nstructure of this expression is in line with the exchange\ncoupling in Eq. (12) and the damping of Eq. (13), it is\na little more cumbersome to compute due the presence\nof the derivatives of the Green's functions. Note that it\nis not possible to get completely rid of the derivatives\nthrough partial integration. These derivatives also makethe moment of inertia very sensitive to details of the elec-\ntronic structure, which has a few implications. Firstly the\nmoment of inertia can take large values for narrow band\nmagnetic materials, such as strongly correlated electron\nsystems, where these derivatives are substantial. For\nsuch systems the action of moment of inertia can be im-\nportant for longer time scales too, as indicated by Eq. (5).\nSecondly, the moment of inertia may be strongly depen-\ndent on the reference magnetic ordering for which it is\ncalculated. It is well known that already the exchange\ntensor parameters may depend on the magnetic order.\nIt is the task of future studies to determine how trans-\nferable the moment of inertia tensor as well as damping\ntensor are in-between di\u000berent magnetic ordering.\nIn conclusion, we have derived a method for atomistic\nspin dynamics which would be applicable for ultrafast\n(femtosecond) processes. Using a general s-d-like interac-\ntion between the magnetization density and electron spin,\nwe show that magnetization couples to the surrounding\nin a non-adiabatic fashion, something which will allow for\nstudies of general magnetic orders on an atomistic level,\nnot only ferromagnetic. By showing that our method\ncapture previous formulas for the exchange interaction\nand damping tensor parameter, we also derive a formula\nfor calculating the moment of inertia from \frst principles.\nIn addition our results point out that all parameters are\nnon-local as they enter naturally as bilinear sums in the\nsame fashion as the well established exchange coupling.\nOur results are straight-forward to implement in existing\natomistic SD codes, so we look on with anticipation to\nthe \frst applications of the presented theory which would\nbe fully parameter-free and hence can take a large step\ntowards simulations with predictive capacity.\nSupport from the Swedish Research Council is ac-\nknowledged. We are grateful for fruitful and encouraging\ndiscussions with A. Bergman, L. Bergqvist, O. Eriksson,\nC. Etz, B. Sanyal, and A. Taroni. J.F. also acknowledges\ndiscussions with J. -X. Zhu.\n\u0003Electronic address: Jonas.Fransson@physics.uu.se\n[1] E. Beaurepaire, J.-C. 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Titov, Phys.\nRev. E, 65, 032102 (2002)\n[15] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and Th. Rasing, Nature Phys. 5, 727 (2009).\n[16] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[17] H. Ebert, S. Mankovsky, D. Kodderitzsch, P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n[18] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n[19] A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).[20] J. -X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Bal-\natsky, Phys. Rev. Lett. 92, 107001 (2004).\n[21] J. Fransson and J. -X. Zhu, New J. Phys. 10, 013017\n(2008).\n[22] J. Fransson, Phys. Rev. B, 82, 180411(R) (2010).\n[23] If higher order spin interaction terms than bilinear are\nneeded it is straight forward to include higher order in\nthis expansion.\n[24] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde,\nand B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995).\n[25] V. P. Antropov, M. I. Katsnelson, and A. I. Liechtenstein,\nPhysica B, 237-238 , 336 (1997).\n[26] M. I. Katsnelson and A. I. Lichtenstein, J. Phys.: Con-\ndens. Matter, 16, 7439 (2004).\n[27] V. Kambersk\u0013 y, Phys. Rev. B, 76, 134416 (2007)."    },    {        "title": "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. The addition of B also helps in preventing  the diffusion between \nCo and Ni layers and enhances the stability of the multilayer structure and the \nperformance of our device. Using a 30 ns pulse as a writing current, the device could \nperform shift-register writing of a multidomain state in the wire with an average domain size of 900±35 nm (without field) and a minimum size of 630±25 nm (with field). \n \nAcknowledgments \nThis work was completed with the financial support from the Toyota School Foundation. \nWe thank Professor T. Kato and Professo r S. Iwata (Nagoya University) for AGM \nmeasurements. 12\nReferences \n[1] L. Berger: Phys. Rev. B 54 (1996) 9353. \n[2] J. C. Slonczewski: J. Magn. Magn. Mater. 159 (1996) L1. \n[3] J. Jaworowicz, N. Vernier, J. Ferré, A. Maziewski, D. Stanescu, D. Ravelosona, A. \nS. Jacqueline, C. Chappert, B. Rodmacq, and B. Diény: Nanotechnology 20 (2009) \n215401. \n[4] L. Leem and J. S. Harris: J. Appl. Phys. 105 (2009) 07D102. \n[5] S. S. P. 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Koyama, K. Ueda , H. Tanigawa, S. Fukani, T. Suzuki, N. \nOhshima, N. Ishwata, Y. Nakatani, and T. Ono: Appl. Phys. Express 3 (2010) \n073004. \n[14] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno: Nature 428 (2004) 539. \n[15] D. McGrouther, S. McVitie, J. N. Chap man, and A. Gentils: Appl. Phys. Lett. 91 \n(2007) 022506.  \n[16] K. J. Kirk, J. N. Chapman, and C. D. W. Wilkinson: Appl. Phys. Lett. 71 (1997) \n539. \n[17] J. Heinen, D. Hinzke, O. Boulle, G. Malinowski, H. J. M. Swagten, B. Koopmans, \nC. Ulysse, G. Faini, and M. Kläui: Appl. Phys. Lett. 99 (2011) 242501. \n[18] D.-T. Ngo, K. Ikeda, and H. Awano: J. Appl. Phys. 111 (2012) 083921. \n[19] L. Y. Jang, S. Yoon, K. Lee, S. Lee, C. Nam, and B.-K. Cho: Nanotechnology 20 \n(2009) 125401. \n[20] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine: Phys. Rev. Lett. \n97 (2006) 057203. \n[21] D.-T. Ngo, K. Ikeda, and H. Awano: Appl. Phys. Express 4 (2011) 093002. \n[22] D. Ravelosona, D. Lacour, J. A. Katine, B. 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": "1111.1219v1.Tunable_magnetization_relaxation_in_spin_valves.pdf",        "content": "arXiv:1111.1219v1  [cond-mat.mes-hall]  4 Nov 2011Tunable magnetization relaxation in spin valves\nXuhui Wang∗and Aurelien Manchon\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: June 24, 2018)\nIn spin values the damping parameters of the free layer are de termined non-locally by the entire\nmagnetic configuration. In a dual spin valve structure that c omprises a free layer embedded be-\ntween two pinned layers, the spin pumping mechanism, in comb ination with the angular momentum\nconservation, renders the tensor-like damping parameters tunable by varying the interfacial and dif-\nfusive properties. Simulations based on the Landau-Lifshi tz-Gilbert phenomenology for a macrospin\nmodel are performed with the tensor-like damping and the rel axation time of the free layer mag-\nnetization is found to be largely dependent on while tunable through the magnetic configuration of\nthe source-drain magnetization.\nPACS numbers: 75.70.Ak, 72.25.Ba, 75.60.Jk, 72.25.Rb\nA thorough knowledge of magnetization relaxation\nholds the key to understand magnetization dynamics in\nresponse to applied fields1and spin-transfer torques.2,3\nIn the framework of Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology, relaxation is well captured by the Gilbert\ndamping parameterthat is usuallycited asa scalarquan-\ntity. As pointed out by Brown half a century ago,4the\nGilbert damping for a single domain magnetic particle is\nin general a tensor.\nWhen a ferromagnetic thin film is deposited on a nor-\nmal metal substrate, an enhanced damping has been ob-\nserved ferromagnetic resonance experiments.5This ob-\nservation is successfully explained by spin pumping:6,7\nThe slow precession of the magnetization pumps spin\ncurrent into the adjacent normal metal where the dis-\nsipation of spin current provides a non-local mechanism\nto the damping. The damping enhancement is found\nto be proportional to spin mixing conductance, a quan-\ntity playing key roles in the magneto-electronic circuit\ntheory.7,8\nThepumped spincurrent Ip∝M×˙Misalwaysin the\nplaneformedbythefreelayermagnetizationdirection M\nand the instantaneous axis about which the magnetiza-\ntion precesses. Therefore, in a single spin valve, when M\nis precessingaround the source(drain) magnetization m,\nthe pumpingcurrentisalwaysinthe planeof mandM.9\nLet us assume an azimuth angle θbetween mandM. In\nsuch anin-plane configuration, the pumping current Ip\nhas a component Ipsinθthat is parallel to m. The spin\ntransfer torque acting on the source (drain) ferromagnet\nmis the component of spin current that is in the plane\nand perpendicular to m. To simplify the discussion, we\nconsider it to be completely absorbed by m. The lon-\ngitudinal (to m) component experiences multiple reflec-\ntion at the source (drain) contact, and cancels the damp-\ning torque by an amount proportional to Ipsin2θbut is\nstill aligned along the direction of M×˙M. Therefore\nthe total damping parameter has an angle θdependence\nbut still picks up a scalar (isotropic) form. This is the\nwell-known dynamic stiffness explained by Tserkovnyak\net al.9In the most general case, when the precessing axis\nof the free layer is mis-aligned with m, there is always anout-of-plane pumping torque perpendicular to the plane.\nIn the paradigmof Slonczewski, this out-of-plane compo-\nnent is not absorbed at the interface of the source (drain)\nferromagnetic nodes, while the conservation of angular\nmomentum manifests it as a damping enhancement that\nshows the tensor form when installed in the LLG equa-\ntion.\nStudies in lateral spin-flip transistors have suggested\na tensor form for the enhanced damping parameters.10\nIn spin valves, works based on general scattering the-\nory have discussed the damping in the framework of\nfluctuation-dissipation theorem11and shown that the\nGilbert dampingtensorcanbe expressedusingscattering\nmatrices,12thus enabling first-principle investigation.13\nBut explicit analytical expressions of the damping ten-\nsor, its dependence on the magnetic configuration as well\nas the material properties and particularly its impact on\nthe magnetization relaxation are largely missing.\nIn this paper, we investigate the Gilbert damping pa-\nrameters of the free layer in the so-called dual spin valve\n(DSV).14–16We analyze the origin of the damping tensor\nand derive explicit analytical expressions of its non-local\ndependence on the magnetic configuration and materials\nproperties. A generalization of our damping tensor to a\ncontinuous magnetic texture agrees well with the results\nin earlier works. Particularly, we show, in numeric sim-\nulations, that by tuning the magnetic configurations of\nthe entire DSV, the relaxation time of the free layer can\nbe increased or decreased.\nmLM mR\nReservoir Reservoir\nFIG. 1: A dual spin valve consists of a free layer (with magne-\ntization direction M) sandwiched by two fixed ferromagnetic\nlayers (with magnetization directions mLandmR) through\ntwo normal metal spacers. The fixed layer are attached to\nreservoirs.2\nTo analyze the spin and charge currents in a DSV, we\nemploy the magneto-electronic circuit theory and spin\npumping,7,8in combination with diffusion equations.17\nPillar-shaped metallic spin valves usually consist of\nnormal-metal ( N) spacers much shorter than its spin-flip\nrelaxation length, see for example Ref.[3,15]. To a good\napproximation, in the Nnodes, a spatially homogeneous\nspin accumulation is justified and the spin current ( Ii)\nconservation dictates/summationtext\niIi= 0 (where subscript iindi-\ncates the source of spin current).\nA charge chemical potential ( µ) and a spin accumula-\ntion (s) are assigned to every ForNnode. In a transi-\ntion metalferromagnet,astrongexchangefieldalignsthe\nspin accumulation to the magnetization direction. At ev-\neryF|Ninterface, the charge and spin currents on the N\nside are determined by the contact conductance and the\ncharge and spin distributions on both sides of the con-\ntact. For example, at the contact between the left lead\nferromagnet to the left normal metal N1, called L|N1\nthereafter, the currents are8\nIL=e\n2hGL[(µ1−µL)+PL(s1−sL)·mL],\nIL=−GL\n8π[2PL(µ1−µL)mL+(s1−sL)·mLmL\n+ηL(s1−s1·mLmL)]. (1)\nWe have used the notation G=g↑+g↓is the sum of\nthe spin- σinterface conductance gσ. The contact polar-\nisationP= (g↑−g↓)/(g↑+g↓). The ratio η= 2g↑↓/G\nis between the real part of the spin-mixing conductance\ng↑↓and the total conductance G. The imaginary part\nofg↑↓is usually much smaller than its real part, thus\ndiscarded.18The spin-coherence length in a transition\nmetalferromagnetisusuallymuchshorterthanthethick-\nness of the thin film,19which renders the mixing trans-\nmission negligible.7The precession of the free layer mag-\nnetization Mpumps a spin current Ip= (/planckover2pi1/4π)g↑↓\nFM×\n˙Minto the adjacent normal nodes N1andN2, which\nis given by the mixing conductance g↑↓\nFat theF|N1(2)\ninterface (normal metals spacers are considered identical\non both sides of the free layer).\nA back flow spin current at the F|N1interface reads\nI1=−GF\n8π[2PF(µ1−µF)M+(s1−sF)·MM\n+ηF(s1−s1·MM)] (2)\non theN1side. Therefore, a weak spin-flip scattering\ninN1demands IL+I1+Ip= 0, which is dictated by\nangularmomentum conservation. The sameconservation\nlaw rules in N2, whereIR+I2+Ip= 0.\nFor the ferromagnetic ( F) nodes made of transition\nmetals, the spin diffusion is taken into account properly.9\nIn a strong ferromagnet, any transverse components de-\ncay quickly due to the large exchange field, thus the\nlongitudinal spin accumulation sν=sνmν(withν=\nL,R,F) diffuses and decays exponentially at a length\nscale given by spin diffusion length ( λsd) as∇2\nxsν=sν/λsd. The difference in spin-dependent conductivty\nof majority and minority carriers is taken into account\nby enforcing the continuity of longitudinal spin current\nmν·Iν=−(D↑\nν∇xs↑\nν−D↓\nν∇xs↓\nν) at the every F|Nin-\nterface. We assume vanishing spin currents at the outer\ninterfaces to reservoirs.\nThe diffusion equations and current conservation de-\ntermine, self-consistently, the spin accumulations and\nspin currents in both NandFnodes . We are mainly\nconcerned with the exchange torque9T=−M×(IL+\nIR)×Macting on M. A general analytical formula is\nattainable but lengthy. In the following, we focus on two\nscenarios that are mostly relevant to the state-of-the-art\nexperiments in spin valves and spin pumping: (1) The\nfree layer has a strong spin flip (short λsd) and the thick-\nnessdF≥λsd, for which the permalloy (Py) film is an\nideal candidate;15(2) The free layer is a half metal, such\nas Co2MnSi studied in a recent experiment.20\nStrong spin flip in free layer. We assume a strong spin\nflip scattering in the free layer i.e., dF≥λsd. We leave\nthe diffusivity properties in the lead Fnodes arbitrary.\nThe total exchange torque is partitioned into two parts:\nAnisotropic part that is parallel to the direction of the\nGilbert damping M×˙Mand ananisotropic part that is\nperpendicular to the plane spanned by mL(R)andM(or\nthe projection of M×˙Mto the direction mL(R)×M),\ni.e.,\nT=/planckover2pi1g↑↓\nF\n4π(DL\nis+DR\nis)/parenleftBig\nM×˙M/parenrightBig\n+/planckover2pi1g↑↓\nF\n4πM×/bracketleftBig\n(DL\nanˆAL,an+DR\nanˆAR,an)˙M/bracketrightBig\n,(3)\nwhere the material-dependent parameters DL(R)\nisand\nDL(R)\nanare detailed in the Appendix A.\nMost interest is in the anisotropic damping described\nby a symmetric tensor with elements\nˆAij\nan=−mimj (4)\nwherei,j=x,y,z(we have omitted the lead index Lor\nR). The elements of ˆAanare given in Cartesian coordi-\nnates of the source-drain magnetization direction. The\nanisotropic damping appears as M׈Aan˙Mthat is al-\nways perpendicular to the free layer magnetization direc-\ntion, thus keeping the length of Mconstant.11It is not\ndifficult to show that when Mis precessing around m,\nthe anisotropic part vanishes due to ˆAan˙M= 0.\nWe generalizeEq.(4) toa continuousmagnetictexture.\nConsider here only one-dimensional spatial dependence\nand the extension to higher dimensions is straightfor-\nward. The Cartesian component of vector U≡M×\nˆAan˙MisUi=−εijkMjmkml˙Ml(whereεijkis the Levi-\nCivita tensor and repeated indices are summed). We as-\nsume the fixed layer and the free layer differ in space by\na lattice constant a0, which allows mk≈Mk(x+a0). A\nTaylor expansion in space leads to U=−a2\n0M×(ˆD˙M),\nwherethematrixelements ˆDkl= (∂xM)k(∂xM)landwe3\nhave assumed that the magnetization direction is always\nperpendicular to ∂xM. In this case, three vectors ∂xM,\nM×∂xMandMare perpendicular to each other. A\nrotation around Mbyπ/2 leavesMand˙Munchanged\nwhile interchanging ∂xMwithM×∂xM, we have\nˆDkl= (M×∂xM)k(M×∂xM)l, (5)\nwhich agrees with the so-called differential damping ten-\nsor Eq.(11) in Ref.[21].\nEq.(3) suggests that the total exchange torque on the\nfree layer is a linear combination of two independent ex-\nchange torques arsing from coupling to the left and the\nrightFnodes. This form arises due to a strong spin-\nflip scattering in the free layer that suppresses the ex-\nchange between two spin accumulations s1ands2in the\nNnodes. In the pursuit of a concise notation for the\nGilbert form, the exchange torque can be expressed as\nT=M×← →α˙Mwith a total damping tensor given by\n← →α=/planckover2pi1g↑↓\nF\n4π/parenleftBig\nDL\nis+DR\nis+DL\nanˆAL,an+DR\nanˆAR,an/parenrightBig\n.(6)\nThe damping tensor← →αis determined by the entire mag-\nnetic configuration of the DSV and particularly by the\nconductance of F|Ncontacts and the diffusive proper-\nties theFnodes.\nHalf metallic free layer . This special while experimen-\ntally relevant20case means PF= 1. Half-metallicity\nin combination with the charge conservation enforces a\nlongitudinal back flow that is determined solely by the\nbias current: The spin accumulations in Nnodes do\nnot contribute to the spin accumulation inside the free\nlayer, thus an independent contribution due to left and\nright leads is foreseen. We summarize the material spe-\ncific parameters in the Appendix A. When spin flip is\nweak in the source-drain ferromagnets, ξL≈0 leads to\nDis≈0. In this configuration, by taking a (parallel or\nanti-parallel)source-drainmagnetization direction as the\nprecessingaxis,thetotaldampingenhancementvanishes,\nwhich reduces to the scenario of ν= 1 in Ref.[9].\nMagnetization relaxation . To appreciate the impact of\nan anisotropic damping tensor on the magnetization re-\nlaxation, we perform a simulation, for the free layermag-\nnetization, using Landau-Lifshitz-Gilbert (LLG) equa-\ntion augmented by the tensor damping, i.e.,\ndM\ndt=−γM×Heff+α0M×dM\ndt\n+γ\nµ0MsVM×← →αdM\ndt.(7)\nα0is the (dimensionless) intrinsic Gilbert damping pa-\nrameter. Symbol γis the gyromagnetic ratio, Msis the\nsaturation magnetization, and Vis the volume of the\nfree layer. µ0stands for the vacuum permeability. The\ndynamics under the bias-driven spin transfer torque is\nnot the topic in this paper, but can be included in a\nstraightforward way.22We give in the Appendix B the\nexpressions of the bias-driven spin torques.We are mostly interested in the relaxation of the mag-\nnetization, instead of particular magnetization trajecto-\nries, in the presence of a tensor damping. The follow-\ning simulation is performed for the scenario where the\nfree layer has a strong spin flip, i.e., Case (1). We em-\nploy the pillar structure from Ref.[15] while consider-\ning the free layer (Py) to be 8nm thick (a thicker free\nlayer favors a better thermal stability.15) The source-\ndrain ferromagnets are cobalt (Co) and we expect the\nresults are valid for a larger range of materials selec-\ntions. The Py film is elliptic with three axes given by\n2a= 90 nm, 2 b= 35 nm,15andc= 8 nm. The de-\nmagnetizing factors Dx,y,zin the shape anisotropy en-\nergyEdem= (1/2)µ0M2\nsV/summationtext\ni=x,y,zDiM2\niareDx= 0.50,\nDy= 0.37 and Dz= 0.13. An external field Haleads\nto a Zeeman splitting EZee=−Vµ0MsHa·M. For Py\nfilms, we neglect the uniaxial anisotropy. The total free\nenergyET=EZee+Edemgives rise to an effective field\nHeff=−(1/VMsµ0)∂ET/∂M.\nThe spin-dependent conductivities in the bulk of Co\nandthe spin diffusion length λCo≈60nm aretaken from\nthe experimental data.24For Py, we take λPy≈4 nm.25\nTo have direct connection with experiments, the above\nmentioned bare conductance has to be renormalized by\nthe Sharvin conductance.26For Py/Cu the mixing con-\nductance, we take the value g↑↓\nFS−1≈15 nm−2,26which\ngivesRL(R)F≈1.0.\n56789100.950.960.970.980.991(a) Bz = 50 G; I.P. y−axis.\n  \n(y,y)(y,x)(x,x)(x,z)(y,z)(z,z)369(b) Relaxation time\n(x,x)\n(y,y)\n(z,z)\nFIG. 2: (Color online) Mzas a function of time (in ns) in\npresence of differentsource-drain magnetic configurations and\napplied fields. (a) The external magnetic field Bz= 50 Gauss\nis applied along z-axis. The blue (dashed), red (solid) and\nblack (dotted dash) curves correspond to source-drain magn e-\ntization in configurations ( y,y), (x,x), and (z,z) respectively.\n(b) Magnetization relaxation times (in the unit of ns)versu s\nsource-drain magnetic configurations at different applied fi eld\nalongz-axis.:Bz= 10 G (red /square),Bz= 50 G (blue /circlecopyrt),\nBz= 200 G (green ▽),Bz= 800 G (black ♦). Lines are a\nguide for the eyes. The initial position (I.P.) of the free la yer\nis taken along y-axis.\nThe relaxation time τris extracted from the sim-\nulations by demanding at a specific moment τrthe\n|Mz−1.0|<10−3, i.e., reaches the easy axis. In the\nabsence of bias, panel (a) of Fig.2 shows the late stage\nof magnetization relaxation from an initial position ( y-4\naxis) in the presence of an tensor damping, under various\nsource-drain (SD) magnetic configurations. The results\nare striking: Under the same field, switching the SD con-\nfigurations increases or decreases τr. In panel (b), the\nextracted relaxation times τrversus SD configurations\nunder various fields are shown. At low field Bz= 10 G\n(red/square), when switching from ( z,z) to (y,y),τris im-\nproved from 8.0 ns to 6.3 ns, about 21%. At a higher\nfieldBz= 800 G (black ♦), the improvement is larger\nfrom 5.2 at ( z,z) to 3.6 at ( y,y), nearly 31%. To a large\ntrend, the relaxation time improvement is more signifi-\ncant at higher applied fields.\nIn conclusion, combining conservation laws and\nmagneto-electronic circuit theory, we have analyzed the\nGilbert damping tensor of the free layer in a dual spin\nvalve. Analytical results of the damping tensor as func-\ntions of the entire magnetic configuration and material\nproperties are obtained. Numerical simulations on LLG\nequation augmented by the tensor damping reveal a tun-\nable magnetization relaxation time by a strategic selec-\ntion of source-drain magnetization configurations. Re-\nsults presented in this paper open a new venue to the\ndesign and control of magnetization dynamics in spin-\ntronic applications.\nX.Wang is indebted to G. E. W. Bauer, who has\nbrought the problem to his attention and offered invalu-\nable comments.\nAppendix A: Material dependent parameters\nIn this paper, RL(R)F≡g↑↓\nL(R)/g↑↓\nFis the mixing con-\nductance ratio and χL(R)≡mL(R)·M. The diffusivity\nparameter ξL(R)=φL(R)(1−P2\nL(R))/ηL(R), where for the\nleftFnode\nφL=1\n1+(σ↑\nL+σ↓\nL)λLe2\n4hSσ↑\nLσ↓\nLtanh(dL/λL)GL(1−P2\nL)(A1)\nwherehthe Planck constant, Sthe area of the thin film,\nethe elementary charge, λLthe spin diffusion length, dL\nthe thickness of the film, and σ↑(↓)the spin-dependent\nconductivity. φRisobtainedbysubstituting all LbyRin\nEq.(A1). Parameter ξFis given by ξF= (1−P2\nF)φF/ηF\nwith\nφF=1\n1+(σ↑\nF+σ↓\nF)λFe2\n4hSσ↑\nFσ↓\nFGF(1−P2\nF).(A2)\nThe material dependent parameters as appearing in the\ndamping tensor Eq.(6) are: (1) In the case of a strongspin flip in free layer,\nDL(R)\nis=RL(R)F\nLL(R)F/bracketleftBig\nξL(R)RL(R)F+ξL(R)ξF(1−χ2\nL(R))\n+ξF(1−χ2\nL(R))χ2\nL(R)/bracketrightBig\n,\nDL(R)\nan=RL(R)F\nLL(R)F(ξL(R)−1)[ξF(1−χ2\nL(R))+RL(R)F]\n1+RL(R)F,\nLL(R)F=(1+RL(R)Fχ2\nL)ξF(1−χ2\nL(R))\n+RL(R)F/bracketleftBig\n(1−χ2\nL(R))(1+ξL(R)ξF)\n+ξL(R)RL(R)F+ξFχ2\nL(R)/bracketrightBig\n; (A3)\n(2)In the case of a half metallic free layer\nDL(R)\nis=RL(R)FξL(R)\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F),\nDL(R)\nan=RL(R)F\n1+RL(R)F\n×ξL(R)−1\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F).(A4)\nAppendix B: Bias dependent spin torques\nThe full analytical expression of bias dependent spin\ntorques are rather lengthy. We give here the expres-\nsions, under a bias current I, for symmetric SD fer-\nromagnets (i.e., φL=φR=φthusξL=ξR=ξ)\nwith parallelor anti-parallelmagnetization direction. (1)\nWith a strong spin flip in the free layer, the parallel\nSD magnetization leads to vanishing bias-driven torque\nT(b)\n⇑⇑= 0; WhentheSDmagnetizationsareanti-parallelly\n(i.e.,mL=−mR≡m),\nT(b)\n⇑⇓=I/planckover2pi1Pφ\ne(1+R)L/bracketleftbig\n(ξF+RξFχ2+R)(1−χ2)\n+R(R+ξF(1−χ2)+χ2)/bracketrightbig\nmF×(m×mF).\n(B1)\n(2) When the free layer is half metallic, for symmetric\nSD ferromagnets , T(b)\n⇑⇑= 0 and\nT(b)\n⇑⇓=I/planckover2pi1\neφP\n(1−ξ)(1−χ2)+ξ(χ2+R)mF×(m×mF).\n(B2)\n∗Electronic address: xuhui.wang@kaust.edu.sa\n1L. D. Landau and E. M. Lifshitz, Statistical Physics ,Part\n2(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE. Trans.\nMag.40, 2443 (2004).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n3E. B. Myers, et al., Science 285, 867 (1999); J. A. 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Phys. Lett. 86, 152509 (2005).16P. Bal´ aˇ z, M. Gmitra, and J. Barna´ s, Phys. Rev. B 80,\n174404 (2009); P. Yan, Z. Z. Sun, and X. R. Wang, Phys.\nRev. B83, 174430 (2011).\n17T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993); A. A.\nKovalev, A. Brataas, and G. E. W. Bauer Phys. Rev. B\n66, 224424 (2002).\n18K. Xia,et al., Phys. Rev. B 65, 220401(R) (2002).\n19M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n20H. Chudo, et al., J. Appl. Phys. 109, 073915 (2011).\n21S. Zhang and S. -L. Zhang, Phys. Rev. Lett. 102, 086601\n(2010).\n22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72,\n014446 (2005).\n23J. Osborn, Phys. Rev. 67, 351 (1945).\n24J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274\n(1999).\n25A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338\n(1999).\n26G. E. W. Bauer, et al., Phys. Rev. B 67, 094421 (2003)."    },    {        "title": "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. In this formulation, the\nproperties of the barrier layer material are considered in t he local DOS, which can be easily\nobtainedbyfirst-principlescalculations.\n11/13J.Phys. Soc. Jpn. FULLPAPERS\nReferences\n1) T.Miyazaki andN.Tezuka: J. Magn.Magn.Mater. 139(L231)1995.\n2) J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey: Phy s. Rev. Lett. 74(1995)\n3273.\n3) S. Yuasa, A. Fukushima, T. Nagahama, K. Ando, and Y. Suzuki : Jpn. J. Appl. Phys. 43\n(2004)L588.\n4) S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando : Nat. Mater. 3(2004)\n862.\n5) S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughe s, M. Samant, and S.-H.\nYang:Nat. Mater. 3(2004)862.\n6) R. Scheuerlein, W. Gallagher, S. Parkin, A. Lee, S. Ray, R. Robertazzi, and W. Reohr:\nSolid-State Circuits Conference, 2000. Digest of Technica l Papers. ISSCC. 2000 IEEE\nInternational,2000,pp.128 –129.\n7) J. C. Slonczewski:J.Magn.Magn. Mater. 159(1996)L1.\n8) L. Berger: Phys.Rev.B 54(1996)9353.\n9) F. J. 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Bauer: Phys.Rev. B66(2002)224403.\n12/13J.Phys. Soc. Jpn. FULLPAPERS\n22) N. Umetsu,D. Miura, andA. Sakuma: J.Phys.:Conf. Ser. 266(2011)012084.\n23) N. Umetsu,D. Miura, andA. Sakuma: J.Appl.Phys. 111(2012)07D117.\n24) G. E.Volovik:J. Phys.C 20(1987)L83.\n25) P.-Q. Jin,Y.-Q. Li, andF.-C. Zhang:J. Phys.A 39(2006)7115.\n26) J. Oheand S. Maekawa: J.Appl.Phys. 105(2009)07C706.\n27) J. Ohe,S. E.Barnes, H.-W.Lee, andS. Maekawa:Appl.Phys .Lett.95(2009)123110.\n28) S. Zhang andS. S.-L. Zhang:Phys.Rev.Lett. 102(2009)086601.\n29) T.Moriyama,R.Cao,X.Fan,G.Xuan,B.K.Nikoli´ c,Y.Tse rkovnyak,J.Kolodzey,and\nJ. Q.Xiao: Phys.Rev.Lett. 100(2008)067602.\n30) J. Xiao,G. E.W. Bauer, and A.Brataas: Phys.Rev.B 77(2008)180407.\n31) Y. Tserkovnyak,T.Moriyama,and J.Q. Xiao:Phys.Rev. B 78(2008)020401.\n32) S.-H.Chen,C.-R.Chang,J.Q.Xiao,andB.K.Nikoli´ c:Ph ys.Rev.B 79(2009)054424.\n33) H. Kohno,G. Tatara, and J. Shibata:J. Phys.Soc. Jpn. 75(2006)113706.\n34) A. Takeuchiand G. Tatara: J. Phys.Soc. Jpn. 77(2008)074701.\n35) A. Takeuchi,K. Hosono,and G.Tatara: Phys.Rev. B 81(2010)144405.\n36) H.HaugandA.-P.Jauho: QuantumKineticsinTransportandOpticsofSemiconductors\n(Springer-Verlag, 2008)second,substantiallyreviseded .\n37) J. Rammer: Quantum Field Theory of Non-equilibrium States (Cambridge University\nPress, 2007).\n38) M.V. Berry: Proc. R. Soc. London,Ser. A 392(1984)45.\n39) A.I.Liechtenstein,M.I.Katsnelson,V.P.Antropov,an dV.A.Gubanov:J.Magn.Magn.\nMater.67(1987)65 .\n40) D. Miuraand A. Sakuma:J. Appl.Phys. 109(2011)07C909.\n13/13"    },    {        "title": "1111.4655v1.Null_controllability_of_the_structurally_damped_wave_equation_with_moving_point_control.pdf",        "content": "arXiv:1111.4655v1  [math.OC]  20 Nov 2011NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE\nEQUATION WITH MOVING POINT CONTROL\nPHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nAbstract. We investigate the internal controllability of the wave equ ation with structural\ndamping on the one dimensional torus. We assume that the cont rol is acting on a moving point\nor on a moving small interval with a constant velocity. We pro ve that the null controllability\nholds in some suitable Sobolev space and after a fixed positiv e time independent of the initial\nconditions.\n1.Introduction\nIn this paper we consider the wave equation with structural damping1\nytt−yxx−εytxx= 0 (1.1)\nwheretis time,x∈T=R/(2πZ) is the space variable, and εis a small positive parameter\ncorresponding to the strength of the structural damping. Th at equation has been proposed\nin [21] as an alternative model for the classical spring-mass-dam per PDE. We are interested\nin the control properties of ( 1.1). The exact controllability of ( 1.1) with an internal control\nfunction supported in the wholedomain was studied in [ 12,14]. With a boundary control, it\nwas proved in [ 22] that (1.1) is not spectrally controllable (hence not null controllab le), but that\nsome approximate controllability may be obtained in some ap propriate functional space.\nThebadcontrol propertiesfrom( 1.1)come fromtheexistenceof afiniteaccumulation pointin\nthe spectrum. Such a phenomenon was noticed first by D. Russel l in [25] for the beam equation\nwith internal damping, by G. Leugering and E. J. P. G. Schmidt in [15] for the plate equation\nwith internal damping, and by S. Micu in [ 19] for the linearized Benjamin-Bona-Mahony (BBM)\nequation\nyt+yx−ytxx= 0. (1.2)\nEven if the BBM equation arises in a quite different physical co ntext, its control properties share\nimportant common features with ( 1.1). Remind first that the full BBM equation\nyt+yx−ytxx+yyx= 0 (1.3)\nis a popular alternative to the Korteweg-de Vries (KdV) equa tion\nyt+yx+yxxx+yyx= 0 (1.4)\nas a model for the propagation of unidirectional small ampli tude long water waves in a uniform\nchannel. ( 1.3) is often obtained from ( 1.4) in the derivation of the surface equation by noticing\nKey words and phrases. Structuraldamping; waveequation; nullcontrollability; Benjamin-Bona-Mahony equa-\ntion; Korteweg-de Vries equation; biorthogonal sequence; multiplier; sine-type function.\n1The terminology internal damping is also used by some authors.\n12 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nthat, in the considered regime, yx∼ −yt, so thatyxxx∼ −ytxx. The dispersive term −ytxx\nhas a strong smoothing effect, thanks to which the wellposedne ss theory of ( 1.3) is dramatically\neasier than for ( 1.4). On the other hand, the control properties of ( 1.2) or (1.3) are very bad\n(compared to those of ( 1.4), see [23]) precisely because of that term. It is by now classical that\nan “intermediate” equation between ( 1.3) and (1.4) can be derived from ( 1.3) by working in a\nmoving frame x=ct,c∈R. Indeed, letting\nz(x,t) =y(x−ct,t) (1.5)\nwe readily see that ( 1.3) is transformed into the following KdV-BBM equation\nzt+(c+1)zx−czxxx−ztxx+zzx= 0. (1.6)\nIt is then reasonable to expect the control properties of ( 1.6) to be better than those of ( 1.3),\nthanks to the KdV term −czxxxin (1.6). In [24], it was proved that the equation ( 1.6) with\na forcing term supported in (any given) subdomain is locally exactly controllable in H1(T)\nprovided that T >(2π)/c. Going back to the original variables, it means that the equa tion\nyt+yx−ytxx+yyx=b(x+ct)h(x,t) (1.7)\nwith a moving distributed control is exactly controllable i nH1(T) in (sufficiently) large time.\nActually, this control time has to be chosen in such a way that the support of the control, which\nis moving at the constant velocity c, can visit all the domain T.\nThe concept of moving point control was introduced by J. L. Li ons in [17] for the wave\nequation. One important motivation for this kind of control is that the exact controllability of\nthe wave equation with a pointwise control and Dirichlet bou ndary conditions fails if the point is\na zero of some eigenfunction of the Dirichlet Laplacian, whi le it holds when the point is moving\nunder some (much more stable) conditions easy to check (see e .g. [2]). The controllability of\nthe wave equation (resp. of the heat equation) with a moving p oint control was investigated in\n[17,9,2] (resp. in [ 10,4]). See also [ 27] for Maxwell’s equations.\nAs the bad control properties of ( 1.1) come from the BBM term −εytxx, it is natural to ask\nwhether better control properties for ( 1.1) could be obtained by using a moving control, as for\nthe BBM equation in [ 24]. The aim of this paper is to investigate that issue.\nThroughout the paper, we will take ε= 1 for the sake of simplicity. All the results can be\nextended without difficulty to any ε>0. Letysolve\nytt−yxx−ytxx=b(x+ct)h(x,t). (1.8)\nThenv(x,t) =y(x−ct,t) fulfills\nvtt+(c2−1)vxx+2cvxt−vtxx−cvxxx=b(x)˜h(x,t) (1.9)\nwhere˜h(x,t) =h(x−ct,t). Furthermore the new initial condition read\nv(x,0) =y(x,0), vt(x,0) =−cyx(x,0)+yt(x,0). (1.10)\nAs for the KdV-BBM equation, the appearance of a KdV term (nam ely−cvxxxin (1.9))\nresults in much better control properties. We shall see that\n(i) there is no accumulation point in the spectrum of the free evolution equation ( ˜h= 0 in\n(1.9));CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 3\n(ii) the spectrum splits into one part of “parabolic” type, a nd another part of “hyperbolic”\ntype.\nIt follows that one can expect at most a null controllability result in large time. We will see that\nthis is indeed the case. Throughout the paper, we assume that c=−1 for the sake of simplicity.\nLet us now state the main results of the paper. We shall denote by (y0,ξ0) an initial condition\n(taken in some appropriate space) decomposed in Fourier ser ies as\ny0(x) =/summationdisplay\nk∈Zckeikx, ξ0(x) =/summationdisplay\nk∈Zdkeikx. (1.11)\nWe shall consider several control problems. The first one rea ds\nytt−yxx−ytxx=b(x−t)h(t), x∈T,t>0, (1.12)\ny(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T (1.13)\nwherehis the scalar control.\nTheorem 1.1. Letb∈L2(T)be such that\nβk=/integraldisplay\nTb(x)e−ikxdx/\\e}atio\\slash= 0fork/\\e}atio\\slash= 0, β0=/integraldisplay\nTb(x)dx= 0.\nFor any time T >2πand any (y0,ξ0)∈L2(T)2decomposed as in (1.11), if\n/summationdisplay\nk/\\e}atio\\slash=0|βk|−1(|k|6|ck|+|k|4|dk|)<∞andc0=d0= 0, (1.14)\nthen there exists a control h∈L2(0,T)such that the solution of (1.12)-(1.13)satisfiesy(.,T) =\nyt(.,T) = 0.\nBy Lemma 2.3(see below) there exist simple functions bsuch that |βk|decreases like 1 /|k|3,\nso that ( 1.14) holds for ( y0,ξ0)∈Hs+2(T)×Hs(T) withs>15/2.\nThe second problem we consider is\nytt−yxx−ytxx=b(x−t)h(x,t), x∈T,t>0, (1.15)\ny(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T, (1.16)\nwherethe control function his here allowed to dependalso on x. For that internal controllability\nproblem, the following result will be established.\nTheorem 1.2. Letb=1ωwithωa nonempty open subset of T. Then for any time T >2π\nand any (y0,ξ0)∈Hs+2(T)×Hs(T)withs>15/2there exists a control h∈L2(T×(0,T))such\nthat the solution of (1.15)-(1.16)satisfiesy(.,T) =yt(.,T) = 0.\nWe now turn our attention to some internal controls acting on a single moving point. The\nfirst problem we consider reads\nytt−yxx−ytxx=h(t)δt, x∈T, t>0, (1.17)\ny(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T, (1.18)4 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nwhereδx0represents the Dirac measure at x=x0. We can as well replace δtbydδt\ndxin (1.17),\nwhich yields another control problem:\nytt−yxx−ytxx=h(t)dδt\ndx, x∈T, t>0, (1.19)\ny(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T. (1.20)\nThen we will obtain the following results.\nTheorem 1.3. For any time T >2πand any (y0,ξ0)∈Hs+2(T)×Hs(T)withs>9/2, there\nexists a control h∈L2(0,T)such that the solution of (1.17)-(1.18)satisfiesy(T,.)−[y(T,.)] =\nyt(T,.) = 0, where[f] = (2π)−1/integraltext2π\n0f(x)dxis the mean value of f.\nTheorem 1.4. For any time T >2πand any (y0,ξ0)∈Hs+2(T)×Hs(T)withs >7/2and\nsuch that/integraltext\nTy0(x)dx=/integraltext\nTξ0(x)dx= 0, there exists a control h∈L2(0,T)such that the solution\nof(1.19)-(1.20)satisfiesy(T,.) =yt(T,.) = 0.\nThe paper is organized as follows. Section 2is devoted to the proofs of the above theorems:\nin subsection 2.1we investigate the wellposedness and the spectrum of ( 1.9) forc=−1; in sub-\nsection2.2the null controllability of ( 1.12)-(1.13), (1.17)-(1.18) and (1.19)-(1.20) are formulated\nas moment problems; Theorem 1.1is proved in subsection 2.4thanks to a suitable biorthogonal\nfamily which is shown to exist in Proposition 2.2; Theorem 1.2is deduced from Theorem 1.1\nin subsection 2.5; finally, the proofs of Theorems 1.3and1.4, that are almost identical to the\nproof of Theorem 1.1, are sketched in subsection 2.6. The rather long proof of Proposition 2.2is\npostponed to Section 3. It combines different results of complex analysis about enti re functions\nof exponential type, sine-type functions, atomization of m easures, and Paley-Wiener theorem.\n2.Proof of the main results\n2.1.Spectral decomposition. The free evolution equation associated with ( 1.9) reads\nvtt−2vxt−vtxx+vxxx= 0. (2.1)\nLetvbe as in ( 2.1), and letw=vt. Then (2.1) may be written as\n/parenleftbigg\nv\nw/parenrightbigg\nt=A/parenleftbigg\nv\nw/parenrightbigg\n:=/parenleftbigg\nw\n2wx+wxx−vxxx/parenrightbigg\n. (2.2)\nThe eigenvalues of Aare obtained by solving the system\n/braceleftbigg\nw=λv,\n2λvx+λvxx−vxxx=λ2v.(2.3)\nExpanding vas a Fourier series v=/summationtext\nk∈Zvkeikx, we see that ( 2.3) is satisfied provided that for\neachk∈Z\n(λ2+(k2−2ik)λ−ik3)vk= 0. (2.4)\nForvk/\\e}atio\\slash= 0, the only solution of ( 2.4) reads\nλ=λ±\nk=−(k2−2ik)±√\nk4−4k2\n2· (2.5)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 5\n−180 −160 −140 −120 −100 −80 −60 −40 −20 0−10 −5 0510 \nΛk+\nΛk−\nΛ2\nFigure 1. Spectrum of ( 2.1) splits into a hyperbolic part (Λ+\nkin blue), a para-\nbolic part (Λ−\nkin green) and a finite dimensional part (Λ 2in red).\nNote that\nλ±\n0= 0, λ±\n2=−2+2i, λ±\n−2=−2−2i\nwhile\nλ+\nk/\\e}atio\\slash=λ−\nlfork,l∈Z\\{0,±2}withk/\\e}atio\\slash=l.\nFor|k| ≥3,λ±\nk=−k2±k2(1−2k−2+O(k−4))\n2+ik. Hence\nλ+\nk=−1+ik+O(k−2) as|k| → ∞, (2.6)\nλ−\nk=−k2+1+ik+O(k−2) as|k| → ∞. (2.7)\nThe spectrum Λ = {λ±\nk;k∈Z}may be split into Λ = Λ+∪Λ−∪Λ2where\nΛ+=/braceleftbig\nλ+\nk;k∈Z\\{0,±2}/bracerightbig\n,\nΛ−=/braceleftbig\nλ−\nk;k∈Z\\{0,±2}/bracerightbig\n,\nΛ2={0,−2±2i}\ndenote the hyperbolic part, the parabolic part, and the set of double eigenvalues, respectively. It\nis displayed on Figure 1. (See also [ 13] for a system whose spectrum may also be decomposed\ninto a hyperbolic part and a parabolic part.)\nAn eigenvector associated with the eigenvalue λ±\nk,k∈Z, is/parenleftbigg\neikx\nλ±\nkeikx/parenrightbigg\n, and the correspond-\ning exponential solution of ( 2.1) reads\nv±\nk(x,t) =eλ±\nkteikx.\nFork∈ {0,±2}, we denote λk=λ+\nk=λ−\nk,vk(x,t) =eλkteikx, and introduce\n˜vk(x,t) :=teλkteikx.6 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nThen we easily check that ˜ vksolves (2.1) and\n/parenleftbigg\n˜vk\n˜vkt/parenrightbigg\n(x,0) =/parenleftbigg0\neikx/parenrightbigg\n.\nAny solution of ( 2.1) may be expressed in terms of the v±\nk’s, thevk’s, and the ˜ vk’s. Introduce\nfirst the Hilbert space H=H1(T)×L2(T) endowed with the scalar product\n/a\\}bracketle{t(v1,w1),(v2,w2)/a\\}bracketri}htH=/integraldisplay\nT[(v1v2+v′\n1v′\n2)+w1w2]dx.\nPick any/parenleftbigg\nv0\nw0/parenrightbigg\n=/parenleftbigg/summationtext\nk∈Zckeikx\n/summationtext\nk∈Zdkeikx/parenrightbigg\n∈H. (2.8)\nFork∈Z\\{0,±2}, we write\n/parenleftbigg\nckeikx\ndkeikx/parenrightbigg\n=a+\nk/parenleftbigg\neikx\nλ+\nkeikx/parenrightbigg\n+a−\nk/parenleftbigg\neikx\nλ−\nkeikx/parenrightbigg\n(2.9)\nwith\na+\nk=dk−λ−\nkck\nλ+\nk−λ−\nk, (2.10)\na−\nk=dk−λ+\nkck\nλ−\nk−λ+\nk· (2.11)\nFork∈ {0,±2}, we write\n/parenleftbigg\nckeikx\ndkeikx/parenrightbigg\n=ak/parenleftbigg\neikx\nλkeikx/parenrightbigg\n+˜ak/parenleftbigg0\neikx/parenrightbigg\n(2.12)\nwith\nak=ck,˜ak=dk−λkck. (2.13)\nIt follows that the solution ( v,w) of\n/parenleftbigg\nv\nw/parenrightbigg\nt=A/parenleftbigg\nv\nw/parenrightbigg\n,/parenleftbigg\nv\nw/parenrightbigg\n(0) =/parenleftbigg\nv0\nw0/parenrightbigg\n(2.14)\nmay be decomposed as\n/parenleftbigg\nv(x,t)\nw(x,t)/parenrightbigg\n=/summationdisplay\nk∈Z\\{0,±2}{a+\nkeλ+\nkt/parenleftbigg\neikx\nλ+\nkeikx/parenrightbigg\n+a−\nkeλ−\nkt/parenleftbigg\neikx\nλ−\nkeikx/parenrightbigg\n}\n+/summationdisplay\nk∈{0,±2}{akeλkt/parenleftbigg\neikx\nλkeikx/parenrightbigg\n+˜akeλkt/parenleftbigg\nteikx\n(1+λkt)eikx/parenrightbigg\n}. (2.15)\nProposition 2.1. Assume that (v0,w0)∈Hs+1(T)×Hs(T)for somes≥0. Then the solution\n(v,w)of(2.14)satisfies (v,w)∈C([0,+∞);Hs+1(T)×Hs(T)).CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 7\nProof.Assume first that ( v0,w0)∈C∞(T)×C∞(T). Decompose ( v0,w0) as in (2.8), and leta±\nk\nfork∈Z\\ {0,±2}, andak,˜akfork∈ {0,±2}, be as in ( 2.10)-(2.11) and (2.13), respectively.\nThen, from the classical Fourier definition of Sobolev space s, we have that\n||(v0,w0)||Hs+1(T)×Hs(T)∼/parenleftigg/summationdisplay\nk∈Z(|k|2+1)s/parenleftbig\n(|k|2+1)|ck|2+|dk|2/parenrightbig/parenrightigg1\n2\n∼\n/summationdisplay\nk∈Z\\{0,±2}|k|2s(k2|a+\nk|2+k4|a−\nk|2)+/summationdisplay\nk∈{0,±2}(|ak|2+|˜ak|2)\n1\n2\n.\nFor the last equivalence of norms, we used ( 2.6)-(2.7) and (2.9)-(2.11). Since\n|eλ+\nkt|+|eλ−\nkt| ≤Cfor|k|>2, t≥0,\nwe infer that\n/summationdisplay\nk∈Z\\{0,±2}|k|2s/parenleftig\nk2|a+\nkeλ+\nkt|2+k4|a−\nkeλ−\nkt|2/parenrightig\n≤C/summationdisplay\nk∈Z\\{0,±2}|k|2s/parenleftbig\nk2|a+\nk|2+k4|a−\nk|2/parenrightbig\n<∞,\nhence\n||(v,w)||L∞(R+, Hs+1(T)×Hs(T))≤C||(v0,w0)||Hs+1(T)×Hs(T)· (2.16)\nThe result follows from ( 2.15) and (2.16) by a density argument. /square\n2.2.Reduction to moment problems.\n2.2.1.Internal control. We investigate the following control problem\nvtt−2vxt−vtxx+vxxx=b(x)h(t), (2.17)\nwhereb∈L2(T), suppb⊂ω⊂Tandh∈L2(0,T). The adjoint equation to ( 2.17) reads\nϕtt−2ϕxt+ϕtxx−ϕxxx= 0. (2.18)\nNote thatϕ(x,t) =v(2π−x,T−t) is a solution of ( 2.18) ifvis a solution of ( 2.17) forh≡0.\nPick any (smooth enough) solutions vof (2.17) andϕof (2.18), respectively. Multiplying each\nterm in ( 2.17) byϕand integrating by parts, we obtain\n/integraldisplay\nT[vtϕ+v(−ϕt+2ϕx−ϕxx)]/vextendsingle/vextendsingle/vextendsingle/vextendsingleT\n0dx=/integraldisplayT\n0/integraldisplay\nThbϕdxdt. (2.19)\nPick firstϕ(x,t) =eλ±\n−k(T−t)eikx=eλ±\nk(T−t)eikxfork∈Z. Then (2.19) may be written\n/a\\}bracketle{tvt(T),eikx/a\\}bracketri}ht+(λ±\nk−2ik+k2)/a\\}bracketle{tv(T),eikx/a\\}bracketri}ht−eλ±\nkTγ±\nk\n=/integraldisplayT\n0h(t)eλ±\nk(T−t)dt/integraldisplay\nTb(x)e−ikxdx, (2.20)\nwhere/a\\}bracketle{t.,./a\\}bracketri}htstands for the duality pairing /a\\}bracketle{t.,./a\\}bracketri}htD′(T),D(T), and\nγ±\nk=/a\\}bracketle{tvt(0),eikx/a\\}bracketri}ht+(λ±\nk−2ik+k2)/a\\}bracketle{tv(0),eikx/a\\}bracketri}ht.8 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nIf we now pick ϕ(x,t) = (T−t)eλk(T−t)eikxfork∈ {0,±2}, then (2.19) yields\n/a\\}bracketle{tv(T),eikx/a\\}bracketri}ht−/braceleftig\nTeλkT/a\\}bracketle{tvt(0),eikx/a\\}bracketri}ht+[1+T(λk−2ik+k2)]eλkT/a\\}bracketle{tv(0),eikx/a\\}bracketri}ht/bracerightig\n=/integraldisplayT\n0(T−t)h(t)eλk(T−t)dt/integraldisplay\nTb(x)e−ikxdx k∈ {0,±2}.(2.21)\nSetβk=/integraltext\nTb(x)e−ikxdxfork∈Z. The control problem can be reduced to a moment problem.\nAssume that there exists some function h∈L2(0,T) such that\nβk/integraldisplayT\n0eλ±\nk(T−t)h(t)dt=−eλ±\nkTγ±\nk∀k∈Z, (2.22)\nβk/integraldisplayT\n0(T−t)eλk(T−t)h(t)dt\n=−TeλkT/a\\}bracketle{tvt(0),eikx/a\\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\\}bracketle{tv(0),eikx/a\\}bracketri}ht ∀k∈ {0,±2}.(2.23)\nThen it follows from ( 2.20)-(2.23) that\n/a\\}bracketle{tvt(T),eikx/a\\}bracketri}ht+(λ±\nk−2ik+k2)/a\\}bracketle{tv(T),eikx/a\\}bracketri}ht= 0∀k∈Z, (2.24)\n/a\\}bracketle{tv(T),eikx/a\\}bracketri}ht= 0∀k∈ {0,±2}. (2.25)\nSinceλ+\nk/\\e}atio\\slash=λ−\nkfork∈Z\\{0,±2}, this yields\nv(T) =vt(T) = 0. (2.26)\n2.2.2.Point control. Let us consider first the control problem\nvtt−2vxt−vtxx+vxxx=h(t)dδ0\ndx· (2.27)\nThen the right hand side of ( 2.19) is changed into/integraltextT\n0h(t)/a\\}bracketle{tdδ0\ndx,ϕ/a\\}bracketri}htdt. Forϕ(x,t) =eλ±\nk(T−t)eikx,\nwe have\n/a\\}bracketle{tdδ0\ndx,ϕ/a\\}bracketri}ht=−/a\\}bracketle{tδ0,∂ϕ\n∂x/a\\}bracketri}ht=ikeλ±\nk(T−t)\nhence the right hand sides of ( 2.20) and (2.21) are changed into/integraltextT\n0(ik)eλ±\nk(T−t)h(t)dtand/integraltextT\n0(ik)(T−t)eλk(T−t)h(t)dt, respectively. Let βk=ikfork∈Z. Note that β0= 0 and\nthat (2.20)-(2.21) fork= 0 read\n/a\\}bracketle{tvt(T),1/a\\}bracketri}ht−/a\\}bracketle{tvt(0),1/a\\}bracketri}ht= 0, (2.28)\n/a\\}bracketle{tv(T),1/a\\}bracketri}ht −T/a\\}bracketle{tvt(0),1/a\\}bracketri}ht −/a\\}bracketle{tv(0),1/a\\}bracketri}ht= 0. (2.29)\nThus, the mean values of vandvtcannot be controlled. Let us formulate the moment problem\nto be solved. Assume that\n/a\\}bracketle{tv(0),1/a\\}bracketri}ht=/a\\}bracketle{tvt(0),1/a\\}bracketri}ht= 0, (2.30)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 9\nand that there exists some h∈L2(0,T) such that\nik/integraldisplayT\n0eλ±\nk(T−t)h(t)dt=−eλ±\nkTγ±\nk∀k∈Z\\{0}, (2.31)\nik/integraldisplayT\n0(T−t)eλk(T−t)h(t)dt\n=−TeλkT/a\\}bracketle{tvt(0),eikx/a\\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\\}bracketle{tv(0),eikx/a\\}bracketri}ht ∀k∈ {±2}.(2.32)\nThen we infer from ( 2.20)-(2.21) (with the new r.h.s.) and ( 2.28)-(2.32) that\nv(T) =vt(T) = 0.\nFinally, let us consider the control problem\nvtt−2vxt−vtxx+vxxx=h(t)δ0· (2.33)\nThen the computations above are valid with the new values of βkgiven by\nβk=/a\\}bracketle{tδ0,eikx/a\\}bracketri}ht= 1, k∈Z.\nIt will be clear from the proof of Theorem 1.1that/a\\}bracketle{tvt(T),1/a\\}bracketri}htcan be controlled, while /a\\}bracketle{tv(T),1/a\\}bracketri}ht\ncannot. To establish Theorem 1.3, we shall have to find a control function h∈L2(0,T) such\nthat\n/integraldisplayT\n0eλ±\nk(T−t)h(t)dt=−eλ±\nkTγ±\nk∀k∈Z, (2.34)\n/integraldisplayT\n0(T−t)eλk(T−t)h(t)dt\n=−TeλkT/a\\}bracketle{tvt(0),eikx/a\\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\\}bracketle{tv(0),eikx/a\\}bracketri}ht ∀k∈ {±2}.(2.35)\n2.3.A Biorthogonal family. To solve the moments problems in the previous section, we nee d\nto construct a biorthogonal family to the functions eλ±\nkt,k∈Z, andteλkt,k∈ {±2}. More\nprecisely, we shall prove the following10 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nProposition 2.2. There exists a family {ψ±\nk}k∈Z\\{0,±2}∪{ψk}k∈{0,±2}∪{˜ψk}k∈{±2}of functions\ninL2(−T/2,T/2)such that\n/integraldisplayT/2\n−T/2ψ±\nk(t)eλ±\nltdt=δl\nkδ−\n+k,l∈Z\\{0,±2}, (2.36)\n/integraldisplayT/2\n−T/2ψ±\nk(t)eλltdt=/integraldisplayT/2\n−T/2ψ±\nk(t)teλptdt= 0k∈Z\\{0,±2}, l∈ {0,±2}, p∈ {±2},(2.37)\n/integraldisplayT/2\n−T/2ψl(t)eλ±\nktdt=/integraldisplayT/2\n−T/2˜ψp(t)eλ±\nktdt= 0l∈ {0,±2}, k∈Z\\{0,±2}, p∈ {±2},(2.38)\n/integraldisplayT/2\n−T/2ψl(t)eλktdt=δk\nl,/integraldisplayT/2\n−T/2ψl(t)teλptdt= 0l,k∈ {0,±2}, p∈ {±2}, (2.39)\n/integraldisplayT/2\n−T/2˜ψp(t)eλktdt= 0,/integraldisplayT/2\n−T/2˜ψp(t)teλqtdt=δq\npp,q∈ {±2}, k∈ {0,±2}, (2.40)\n||ψ+\nk||L2(−T/2,T/2)≤C|k|4k∈Z\\{0,±2}, (2.41)\n||ψ−\nk||L2(−T/2,T/2)≤C|k|2e−T\n2k2+2√\n2π|k|k∈Z\\{0,±2}, (2.42)\nwhereCdenotes some positive constant.\nIn Proposition 2.2,δl\nkandδ−\n+denote Kronecker symbols ( δl\nk= 1 ifk=l, 0 otherwise, while\nδ−\n+= 1 if we have the same signs in the l.h.s of ( 2.36), 0 otherwise). The proof of Proposition\n2.2is postponed to Section 3. We assume Proposition 2.2true for the time being and proceed\nto the proofs of the main results of the paper.\n2.4.Proof of Theorem 1.1.Pick any pair ( y0,ξ0)∈L2(T)2fulfilling ( 1.14). From ( 1.10) with\nc=−1, we have that v(0) =y0,vt(0) =dy0\ndx+ξ0, so that\nγ±\nk=/a\\}bracketle{tdy0\ndx+ξ0,eikx/a\\}bracketri}ht+(λ±\nk−2ik+k2)/a\\}bracketle{ty0,eikx/a\\}bracketri}ht,\n=/a\\}bracketle{tξ0,eikx/a\\}bracketri}ht+(λ±\nk−ik+k2)/a\\}bracketle{ty0,eikx/a\\}bracketri}ht, k∈Z.\nLet\nγk=γ±\nkfork∈ {0,±2}.\nThe result will be proved if we can construct a control functi onh∈L2(0,T) fulfilling ( 2.22)-\n(2.23). Let us introduce the numbers\nα±\nk=−β−1\nkeλ±\nkT\n2γ±\nk, k∈Z\\{0,±2},\nαk=−β−1\nkeλkT\n2γk, k∈ {±2},\n˜αk=−β−1\nk/parenleftbiggT\n2eλkT\n2γk+eλkT\n2/a\\}bracketle{ty0,eikx/a\\}bracketri}ht/parenrightbigg\n, k∈ {±2},\nand\nψ(t) =/summationdisplay\nk∈Z\\{0,±2}α+\nkψ+\nk(t)+/summationdisplay\nk∈Z\\{0,±2}α−\nkψ−\nk(t)+/summationdisplay\nk∈{±2}[αkψk(t)+ ˜αk˜ψk(t)].CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 11\nFinally leth(t) =ψ(T\n2−t). Note that h∈L2(0,T) with\n||h||L2(0,T)=||ψ||L2(−T\n2,T\n2)\n≤C\n/summationdisplay\nk∈{±2}(|dk|+|ck|)+/summationdisplay\nk∈Z\\{0,±2}|βk|−1(|dk|+|k|2|ck|)|k|4\n+/summationdisplay\nk∈Z\\{0,±2}|βk|−1(|dk|+|ck|)|k|2e−T|k|2+2√\n2π|k|\n\n<∞,\nby (1.14). Then it follows from ( 1.14) and (2.36)-(2.40) that fork∈Z\\{0,±2}\nβk/integraldisplayT\n0eλ±\nk(T−t)h(t)dt=βkeλ±\nkT\n2/integraldisplayT/2\n−T/2eλ±\nkτψ(τ)dτ=βkeλ±\nkT\n2α±\nk=−eλ±\nkTγ±\nk.\nand also that\nβk/integraldisplayT\n0eλk(T−t)h(t)dt=−eλkTγkfork∈ {0,±2},\nβk/integraldisplayT\n0(T−t)eλk(T−t)h(t)dt=−TeλkTγk−eλkT/a\\}bracketle{ty0,eikx/a\\}bracketri}htfork∈ {0,±2},\nas desired. /square\n2.5.Proof of Theorem 1.2.Setǫ= (T−2π)/2,v(x,t) =y(x+t,t) andξ(x,t) =yt(x,t). We\nfirst steer to 0 the components of vandvtalong the mode associated to the double eigenvalue\nλ0= 0. Denote γ(t) =/integraltext\nTv(x,t)dxandη(t) =/integraltext\nTvt(x,t)dx. According to ( 1.11),γ(0) = 2πc0,\nη(0) = 2πd0and\ndγ\ndt=η,dη\ndt=/integraldisplay\nω˜h(x,t)dx.\nTake aC∞scalar function ̟(t) on [0,ǫ] with̟(0) = 1 and ̟(ǫ) = 0 and such that the support\nofd̟/dtlies inside [0 ,ǫ]. Consider another C∞function of x,¯b(x) with support inside ωand\nsuch that/integraltext\nω¯b(x)dx= 1. Then the C∞control\n˜h(x,t) =¯b(x)¯h(t) with¯h(t) =d2\ndt2((c0+d0t)̟(t))\nsteers (γ,η) from (c0,d0) at timet= 0 to (0,0) at timet=ǫ. Its support lies inside [0 ,ǫ]. Since\nγ(ǫ) =/integraltext\nTy(x,ǫ)dxandη(ǫ) =/integraltext\nTξ(x,ǫ)dx, we can assume that c0=d0= 0 up to a time shift\nofǫ.\nSinceωis open and nonempty, it contains a small interval [ a,a+ 2σπ] whereσ >0 is a\nquadratic irrational ; i.e., an irrational number which is a root of a quadratic equ ation with\nintegral coefficients. Set for t∈[ǫ,T]\nh(x,t) =/parenleftbig\n1[a,a+σπ](x−t)−1[a+σπ,a+2σπ](x−t)/parenrightbig/tildewideh(t)12 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nwhere/tildewidehdenotes a control input independent of x. Thenb(x−t)h(x,t) =/tildewideb(x−t)/tildewideh(t) where\n/tildewideb(x) =1[a,a+σπ](x)−1[a+σπ,a+2σπ](x)\nsatisfies/integraltext\nT/tildewideb(x)dx= 0. Moreover there exists by Lemma 2.3(see below) a number C >0 such\nthat for all k∈Z∗\n/tildewideβk=/integraldisplay\nT/tildewideb(x)e−ikxdx≥C\n|k|3.\nAccording to Theorem 1.1we can find/tildewideh∈L2(ǫ,T) steeringy(.,ǫ) andξ(.,ǫ) toy(.,T) =\nξ(.,T) = 0 as soon as\n/summationdisplay\nk/\\e}atio\\slash=0k6|/tildewideck|+k4|/tildewidedk|\n|/tildewideβk|<∞,\nwith\ny(x,ǫ) =/summationdisplay\nk∈Z/tildewideckeikx, ξ(x,ǫ) =/summationdisplay\nk∈Z/tildewidedkeikx.\nLetWdenote the space of the couples (ˆ y,ˆξ)∈L2(T)2such that ||(ˆy,ˆξ)||W:=|ˆc0|+|ˆd0|+/summationtext\nk/\\e}atio\\slash=0(|k|9|ˆck|+|k|7|ˆdk|)<∞, where ˆy(x) =/summationtext\nk∈Zˆckeikxandˆξ(x) =/summationtext\nk∈Zˆdkeikx. Clearly,W\nendowed with the norm ||·||W, is a Banach space. Standard estimations based on the spectr al\ndecomposition used to prove Proposition 2.1show that if the initial value ( y0,ξ0) lies inW,\nthen the solution of ( 1.12)-(1.13) (withh≡0) remains in W. Therefore, since/summationtext\nk/\\e}atio\\slash=0(|k|9|ck|+\n|k|7|dk|)<∞and since the control is C∞with respect to x∈Tandt∈[0,ǫ], we also have/summationtext\nk/\\e}atio\\slash=0|k|9(|/tildewideck|+|k|7|/tildewidedk|)<∞(see e.g. [ 5,20]). Since (y0,ξ0)∈Hs+2(T)×Hs(T) withs>15/2,\nwe have by Cauchy-Schwarz inequality for ς= 2s−15>0 that\n/summationdisplay\nk/\\e}atio\\slash=0(|k|9|ck|+|k|7|dk|)≤2(/summationdisplay\nk/\\e}atio\\slash=0|k|−1−ς)1\n2(/summationdisplay\nk/\\e}atio\\slash=0|k|19+ς|ck|2+|k|15+ς|dk|2)1\n2<∞./square\nLemma 2.3. Letσ∈(0,1)be a quadratic irrational, and let ˜b,˜βkbe defined as above. Then\n˜β0= 0and there exists C >0such that for all k∈Z∗,|˜βk| ≥C\n|k|3.\nProof.Being a quadratic irrational, σis approximable by rational numbers to order 2 and to\nno higher order [ 8, Theorem 188]); i.e., there exists C0>0 such that for any integers pandq,\nq/\\e}atio\\slash= 0,/vextendsingle/vextendsingle/vextendsingleσ−p\nq/vextendsingle/vextendsingle/vextendsingle≥C0\nq2. On the other hand, |˜βk|=4\n|k|sin2(π\n2kσ) fork/\\e}atio\\slash= 0. Pick any k/\\e}atio\\slash= 0, take\np∈Zsuch that 0 ≤π\n2kσ−pπ < πand use the elementary inequality sin2θ≥4θ2\nπ2valid for\nθ∈[−π\n2,π\n2]. Then two cases occur.\n(i) If 0≤π\n2kσ−pπ≤π\n2, then\nsin2(π\n2kσ) = sin2(π\n2kσ−pπ)≥4\nπ2(π\n2kσ−pπ)2=k2/parenleftig\nσ−2p\nk/parenrightig2\n≥C2\n0\nk2;\n(ii) If−π\n2≤π\n2kσ−(p+1)π≤0, then\nsin2(π\n2kσ−(p+1)π)≥4\nπ2(π\n2kσ−(p+1)π)2=k2/parenleftig\nσ−2(p+1)\nk/parenrightig2\n≥C2\n0\nk2.\nThe lemma follows with C= 4C2\n0. /squareCONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 13\n2.6.Proofs of Theorem 1.3and Theorem 1.4.The proofs are the same as for Theorem\n1.1, with the obvious estimate\n/summationdisplay\n|k|>2|k|p(|k|2|ck|+|dk|)≤Cε\n/summationdisplay\n|k|>2{|k|2p+5+ε|ck|2+|k|2p+1+ε|dk|2\n1\n2\nforp∈ {3,4},ε>0. /square\n3.Proof of Proposition 2.2\nThis section is devoted to the proof of Proposition 2.2. The method of proof is inspired from\nthe one in [ 6,7,18]. We first introduce an entire function vanishing precisely at theiλ±\nk’s,\nnamely the canonical product\nP(z) =z(1−z\niλ2)(1−z\niλ−2)/productdisplay\nk∈Z\\{0,±2}(1−z\niλ+\nk)/productdisplay\nk∈Z\\{0,±2}(1−z\niλ−\nk)· (3.1)\nNext, following [ 1,7], we construct a multiplier mwhich is an entire function that does not\nvanish at the λ±\nk’s, such that P(z)m(z) is bounded for zreal whileP(z)m(z) has (at most) a\npolynomial growth in zas|z| → ∞on each line Im z=const.Next, fork∈Z\\ {0,±2}we\nconstruct a function I±\nkfromP(z) andm(z) and we define ψ±\nkas the inverse Fourier transform\nofI±\nk. The other ψk’s are constructed in a quite similar way. The fact that ψ±\nkis compactly\nsupported in time is a consequence of Paley-Wiener theorem.\n3.1.Functions of type sine. To estimate carefully P(z), we use the theory of functions of\ntype sine (see e.g. [ 16, pp. 163–168] and [ 26, pp. 171–179]).\nDefinition 3.1. An entire function f(z)of exponential type πis said to be of type sine if\n(i)The zerosµkoff(z)are separated; i.e., there exists η>0such that\n|µk−µl| ≥η k/\\e}atio\\slash=l;\n(ii)There exist positive constants A,BandHsuch that\nAeπ|y|≤ |f(x+iy)| ≤Beπ|y|∀x∈R,∀y∈Rwith|y| ≥H. (3.2)\nSome of the most important properties of an entire function o f type sine are gathered in the\nfollowing\nProposition 3.2. (see[16, Remark and Lemma 2 p. 164] ,[26, Lemma 2 p. 172] ) Letf(z)be\nan entire function of type sine, and let {µk}k∈Jbe the sequence of its zeros, where J⊂Z. Then\n(1)For anyε>0, there exist some constants Cε,C′\nε>0such that\nCεeπ|Imz|≤ |f(z)| ≤C′\nεeπ|Imz|if dist{z,{µk}}>ε.\n(2)There exist some constants C1,C2such that\n0<C1<|f′(µk)|<C2<∞ ∀k∈J.\nFinally, we shall need the following result.14 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nTheorem 3.3. (see[16, Corollary p. 168 and Theorem 2 p. 157] Letµk=k+dkfork∈Z,\nwithµ0= 0,µk/\\e}atio\\slash= 0fork/\\e}atio\\slash= 0, and(dk)k∈Zbounded, and let\nf(z) =z/productdisplay\nk∈Z\\{0}(1−z\nµk) = lim\nK→∞z/productdisplay\nk∈{−K,...,K}\\{0}(1−z\nµk)·\nThenfis a function of type sine if, and only if, the following three properties are satisfied:\n(1) inf k/\\e}atio\\slash=l|µk−µl|>0;\n(2)There exists some constant M >0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈Z(dk+τ−dk)k\nk2+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤M∀τ∈Z;\n(3) limsup\ny→+∞log|f(iy)|\ny=π,limsup\ny→−∞log|f(−iy)|\n|y|=π.\nCorollary 3.4. Assume that µk=k+dk, whered0= 0anddk=d+O(k−1)as|k| → ∞for\nsome constant d∈C, and thatµk/\\e}atio\\slash=µlfork/\\e}atio\\slash=l. Thenf(z) =z/producttext\nk∈Z\\{0}(1−z\nµk)is an entire\nfunction of type sine.\nProof.We check that the conditions (1), (2) and (3) in Theorem 3.3are fulfilled.\n(1) Fromµk−µl=k−l+O(k−1,l−1) and the fact that µk−µl/\\e}atio\\slash= 0 fork/\\e}atio\\slash=l, we infer that (1)\nholds.\n(2) Let us write dk=d+ekwithek=O(k−1). Then for all τ∈Z\n/parenleftigg/summationdisplay\nk∈Z(dk+τ−dk)2/parenrightigg1\n2\n≤2/parenleftigg/summationdisplay\nk∈Z|ek|2/parenrightigg1\n2\n<∞.\nTherefore, for any τ∈Z, by Cauchy-Schwarz inequality\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈Z(dk+τ−dk)k\nk2+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftigg/summationdisplay\nk∈Z(dk+τ−dk)2/parenrightigg1\n2/parenleftigg/summationdisplay\nk∈Z(k\nk2+1)2/parenrightigg1\n2\n≤2/parenleftigg/summationdisplay\nk∈Z|ek|2/parenrightigg1\n2/parenleftigg/summationdisplay\nk∈Z(k\nk2+1)2/parenrightigg1\n2\n=:M <∞.\n(3) We first notice that\nf(z) =z∞/productdisplay\nk=1(1−z\nµk)(1−z\nµ−k)·\nLetz=iy, withy∈R. Then\n(1−z\nµk)(1−z\nµ−k) = 1−y2+i(µk+µ−k)y\nµkµ−k\nwith\nµkµ−k=−k2+O(k), µk+µ−k= 2d+O(k−1).CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 15\nIt follows that for any given ε∈(0,1), there exist k0∈N∗and some numbers C1,C2>0 such\nthat\n1+(1−ε)y2−C1|y|\n|µkµ−k|≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1−z\nµk)(1−z\nµ−k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+y2+C2|y|\n|µkµ−k|(3.3)\nfory∈Randk≥k0>0. Let\nn(r) := #{k∈N∗;|µkµ−k| ≤r}.\nSince|µkµ−k| ∼k2ask→ ∞andµk/\\e}atio\\slash= 0 fork/\\e}atio\\slash= 0, we obtain that\n√r−C3≤n(r)≤√r+C3forr>0, (3.4)\nn(r) = 0 for 0 <r<r 0, (3.5)\nfor some constants C3>0,r0>0. It follows that\nlimsup\n|y|→∞log|f(iy)|\n|y|≤limsup\n|y|→+∞|y|−1∞/summationdisplay\nk=1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1−iy\nµk)(1−iy\nµ−k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤limsup\n|y|→∞|y|−1∞/summationdisplay\nk=1log(1+y2+C2|y|\n|µkµ−k|),\nwhere we used the fact that\nlim\n|y|→∞|y|−1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−iy\nµ±k/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0 for 1 ≤k≤k0.\nOn the other hand, setting ρ=y2+C2|y| ≥0, we have that\n∞/summationdisplay\nk=1log(1+ρ\n|µkµ−k|) =/integraldisplay∞\n0log(1+ρ\nt)dn(t)\n=ρ/integraldisplay∞\n0n(t)\nt(t+ρ)dt\n=/integraldisplay∞\n0n(ρs)\ns(s+1)ds\n≤√ρ/integraldisplay∞\n0ds√s(s+1)+C3/integraldisplay∞\nr0/ρds\ns(s+1)\n≤ |y|/radicalbig\n1+C2|y|−1π+C3log/parenleftbig\n1+r−1\n0(y2+C2|y|)/parenrightbig\n.\nThus\nlimsup\n|y|→∞log|f(iy)|\n|y|≤π.\nUsing again ( 3.3), we obtain by the same computations that\nlimsup\ny→+∞log|f(iy)|\ny≥π,and limsup\ny→−∞log|f(iy)|\n|y|≥π.\nThe proof of (3) is completed. /square16 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nIn what follows, arg zdenotes the principal argument of any complex number z∈C\\R−;\ni.e., argz∈(−π,π), and\nlogz= log|z|+iargz,√z=/radicalbig\n|z|eiargz\n2.\nWe introduce, for k∈Z\\{0},\nµk= sgn(k)/radicalig\n−λ−\nk=k/radicaligg\n1+√\n1−4k−2\n2−ik−1=:k+dk, k∈Z\nwith\ndk=−i\n2+O(k−1).\nandµ0= 0. Let\nP1(z) =z/productdisplay\nk∈Z\\{0}(1+z\niλ+\nk), (3.6)\nP2(z) =z/productdisplay\nk∈Z\\{0}(1+z\niλ−\nk), (3.7)\nP3(z) =z2/productdisplay\nk∈Z\\{0}(1+z2\nλ−\nk), (3.8)\nandP4(z) =z/productdisplay\nk∈Z\\{0}(1−z\nµk). (3.9)\nIt follows from ( 2.7) that the convergence in ( 3.7) is uniform in zon each compact set of C, so\nthatP2is an entire function. Note also that\nP2(z) =iP3(e−iπ\n4√z), (3.10)\nP3(z) =−P4(z)P4(−z), (3.11)\nP(z) =P1(−z)P2(−z)\nz(1−z\niλ2)(1−z\niλ−2)· (3.12)\nApplying Corollary 3.4toP1, noticing that\n−iλ+\nk=k+i+O(k−2)\nwithλ+\nk/\\e}atio\\slash=λ−\nlfork/\\e}atio\\slash=l, andλ+\n0= 0, we infer that P1(z) is an entire function of sine type. Thus,\nfor givenε>0 there are some positive constants C4,C5,C6such that\nC4eπ|y|≤ |P1(x+iy)| ≤C5eπ|y|,dist (x+iy,{−iλ+\nk})>ε (3.13)\n|P′\n1(−iλ+\nk)| ≥C6, k∈Z. (3.14)\nNext, applying Corollary 3.4toP4, noticing that\nµk=k−i\n2+O(k−1)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 17\nwithµk/\\e}atio\\slash=µlifk/\\e}atio\\slash=landµ0= 0, we infer that P4(z) is also an entire function of sine type. In\nparticular, it is of exponential type π\n|P4(z)| ≤Ceπ|z|, z∈C. (3.15)\nTherefore, we have for any ε>0 and for some positive constants C7,C8,C9\nC7eπ|y|≤ |P4(x+iy)| ≤C8eπ|y|,dist (x+iy,{µk})>ε (3.16)\n|P′\n4(µk)| ≥C9, k∈Z. (3.17)\nIn particular, P3is an entire function of exponential type 2 πwith\nC2\n7e2π|y|≤ |P3(x+iy)| ≤C2\n8e2π|y|dist (±(x+iy),{µk})>ε. (3.18)\nCombined to ( 3.10), this yields\n|P2(z)| ≤Ce2π√\n|z|z∈C. (3.19)\nSubstituting e−iπ\n4√ztox+iyin (3.18) yields\nC2\n7exp(2π|Im(e−iπ\n4√z)|)≤ |P2(z)| ≤C2\n8exp(2π|Im(e−iπ\n4√z)|) dist(±e−iπ\n4√z,{µk})>ε.\n(3.20)\nFrom (3.20) (applied for xlarge enough) and the continuity of P2onC, we obtain that\n|P2(x)| ≤Ce√\n2π√\n|x|. (3.21)\nWe are now in a position to give bounds for the canonical produ ctPin (3.1).\nProposition 3.5. The canonical product Pin(3.1)is an entire function of exponential type at\nmostπ. Moreover, we have for some constant C >0\n|P(x)| ≤C(1+|x|)−3e√\n2π√\n|x|, x∈R, (3.22)\n|P′(iλ+\nk)| ≥C−1|k|−3e√\n2π√\n|k|k∈Z\\{0,±2}, (3.23)\n|P′(iλ−\nk)| ≥C−1|k|−7eπk2k∈Z\\{0,±2}. (3.24)\nProof.Note first that dist( R,{−iλ+\nk;k/\\e}atio\\slash= 0})>0 from ( 2.5). Since (1 +is\nz)P1(z) is also an\nentire function of sine type for s≫1, with dist( R,{−iλ+\nk;k/\\e}atio\\slash= 0} ∪ {is})>0, we infer from\nProposition 3.2that for some constant C >0\n|P1(x)| ≤C∀x∈R.\nCombined to ( 3.12) and (3.21), this yields ( 3.22). Let us turn to ( 3.23). Note first that for\nk∈Z\\{0,±2}\nP′(iλ+\nk) =P′\n1(−iλ+\nk)P2(−iλ+\nk)\n(−iλ+\nk)(1−λ+\nk\nλ2)(1−λ+\nk\nλ−2)· (3.25)\nClearly, for some δ>0,|λ+\nk−λ−\nl|>δfor allk∈Z\\{0,±2},l∈Z, and\n|Im (e−iπ\n4/radicalig\n−iλ+\nk)|=|Im/parenleftbig1−i√\n2/radicalbig\nk+i+O(k−2)/parenrightbig\n|=/radicalbigg\n|k|\n2+O(|k|−1\n2).18 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nWith (3.20), this gives\n|P2(−iλ+\nk)| ≥Ce√\n2π√\n|k|. (3.26)\nIt follows then from ( 3.14), (3.25), and (3.26) that\n|P′(iλ+\nk)| ≥Ce√\n2π√\n|k|\n|k|3\nfor some constant C >0 independent of k∈Z\\{0,±2}. On the other hand\nP′(iλ−\nk) =P′\n2(−iλ−\nk)P1(−iλ−\nk)\n(−iλ−\nk)(1−λ−\nk\nλ2)(1−λ−\nk\nλ−2)· (3.27)\nBy (2.7) and (3.13), we have that\n|P1(−iλ−\nk)| ≥Ceπk2, k∈Z\\{0,±2}·\nFrom (3.10)-(3.11), we have that\nP′\n2(z) =eiπ\n4\n2√z/bracketleftig\nP′\n4(e−iπ\n4√z)P4(−e−iπ\n4√z)−P4(e−iπ\n4√z)P′\n4(−e−iπ\n4√z)/bracketrightig\n.\nForz=−iλ−\nk,e−iπ\n4√z=/radicalig\n−λ−\nk= sgn(k)µk, hence\nP′\n2(−iλ−\nk) =1\n2µkP′\n4(µk)P4(−µk).\nSince|µk+µl|>δ>0 fork∈Z\\{0}andl∈Z, we have from ( 3.16) that|P4(−µk)| ≥cwhile,\nby (3.17),|P′\n4(µk)|>c>0. It follows that for some constant C >0\n|P′\n2(−iλ−\nk)| ≥C\n|k|∀k∈Z\\{0}.\nTherefore,\n|P′(iλ−\nk)| ≥Ceπk2\n|k|7, k∈Z\\{0}.\n/square\nWe seek for an entire function m(the so-called multiplier ) such that\n|m(x)| ≤C(1+|x|)e−√\n2π√\n|x|, x∈R,\n|m(iλ+\nk)| ≥C−1|k|−3e−√\n2π√\n|k|, k∈Z\\{0},\n|m(iλ−\nk)| ≥C−1eaπk2−2√\n2π√\n|k|, k∈Z\\{0}.\nWe shall use the same multiplier as in [ 7], providing additional estimates required to evaluate it\nat the points iλ−\nkfork∈Z. Let\ns(t) =at−b√\nt, t> 0 (3.28)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 19\nwhere the constants a>0 andb>0 will be chosen later. Note that sis increasing for t>/parenleftbigb\n2a/parenrightbig2\nand thats(B) = 0 where B= (b/a)2. Let\nν(t) =/braceleftbigg\n0t≤B,\ns(t)t≥B.(3.29)\nIntroduce first\ng(z) =/integraldisplay∞\n0log(1−z2\nt2)dν(t) =/integraldisplay∞\nBlog(1−z2\nt2)ds(t)z∈C\\R,(3.30)\nU(z) =/integraldisplay∞\n0log|1−z2\nt2|dν(t) =/integraldisplay∞\nBlog|1−z2\nt2|ds(t)z∈C. (3.31)\nNote thatgis holomorphic on C\\RandUis continuous on C, withU(z) = Reg(z). Next we\natomize the measure µin the above integrals, setting\n˜g(z) =/integraldisplay∞\n0log(1−z2\nt2)d[ν(t)]z∈C\\R, (3.32)\n˜U(z) =/integraldisplay∞\n0log|1−z2\nt2|d[ν(t)]z∈C, (3.33)\nwhere [x] denotes the integral part of x. Again, ˜gis holomorphic on C\\Rand˜Uis continuous\nonCwith˜U(z) = Reg(z). Actually, exp˜ gis an entire function. Indeed, if {τk}k≥0denotes the\nsequence of discontinuity points for t/ma√sto→[ν(t)], thenτk∼k/aask→ ∞and\n˜g(z) =/summationdisplay\nk≥0log(1−z2\nτ2\nk), z∈C\\R· (3.34)\nTherefore,\ne˜g(z)=/productdisplay\nk≥0(1−z2\nτ2\nk), (3.35)\nthe product being uniformly convergent on any compact set in C. We shall pick later m(z) =\nexp(˜g(z−i)) witha=T\n2π−1 andb=√\n2. The strategy, which goes back to [ 1], consists in\nestimating carefully U, and nextU−˜U. Let forx>0\nw(x) =−π√x+xlog/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+1\nx−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle−√xlog/vextendsingle/vextendsingle/vextendsingle/vextendsingle√x+1√x−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle+2√xarctan(√x).(3.36)\nNote thatw∈L∞(R+), for lim x→∞w(x) =−2 andw(0+) = 0.\nLemma 3.6. [7]It holds\nU(x)+bπ/radicalbig\n|x|=−aBw(|x|)∀x∈R (3.37)\nOur first aim is to extend that estimate to the whole domain C.\nLemma 3.7. There exists some positive constant C=C(a,b)such that\n−C−bπ(1+1√\n2)/radicalbig\n|y| ≤U(z)+bπ/radicalbig\n|x|−aπ|y| ≤C, z =x+iy∈C.(3.38)20 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nProof.We follow the same approach as in [ 7]. We first use the following identity from [ 7, (36)]\n(note thatUis even)\nU(z) =|Imz|(πa+1\nπ/integraldisplay∞\n−∞U(t)\n|z−t|2dt). (3.39)\nTo derive ( 3.38), it remains to estimate the integral term in ( 3.39) forz=x+iy∈C. We may\nassume without loss of generality that y>0. From Lemma 3.6, we can write\nU(t) =−bπ/radicalbig\n|t|−aBw(|t|)\nwherew∈L∞(R+). Then, with t=ys,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingley\nπ/integraldisplay∞\n−∞aBw(|t|)\n(x−t)2+y2dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ ||w||L∞(R+)aBy\nπ/integraldisplay∞\n−∞ds\ny((x\ny−s)2+1)\n=aB||w||L∞(R+)=:C. (3.40)\nOn the other hand, still with t=ys, and using explicit computations in [ 7] of some integral\nterms,\ny\nπ/integraldisplay∞\n−∞(−bπ)/radicalbig\n|t|\n(x−t)2+y2dt=−b√y/integraldisplay∞\n−∞/radicalbig\n|s|\n(x\ny−s)2+1ds\n=−b√y\nπ/radicalbigg\n2/radicalig\n1+x2\ny2−2x\ny+π/radicalbigg\n2/radicalig\n1+x2\ny2+2x\ny\n\n=−bπ√\n2(/radicalig/radicalbig\nx2+y2−x+/radicalig/radicalbig\nx2+y2+x).\nRoutine computations give\n/radicalbig\n2|x| ≤/radicalig/radicalbig\nx2+y2−x+/radicalig/radicalbig\nx2+y2+x≤/radicalbig\n2|x|+(√\n2+1)√y∀x∈R,∀y>0.\nTherefore\n−bπ/radicalbig\n|x|−bπ(1+1√\n2)√y≤y\nπ/integraldisplay∞\n−∞(−bπ)/radicalbig\n|t|\n(x−t)2+y2dt≤ −bπ/radicalbig\n|x|.\nCombined to ( 3.39) and (3.40), this yields ( 3.38). /square\nIn order to obtain estimates for ˜U(z), we need to give bounds from above and below for\n˜U(z)−U(z) =/integraldisplay∞\n0log|1−z2\nt2|d([ν](t)−ν(t)).\nWe need the following lemma, which is inspired from [ 11, Vol. 2, Lemma p. 162]\nLemma 3.8. Letν:R+→R+be nondecreasing and null on (0,B). Then for z=x+iywith\ny/\\e}atio\\slash= 0, we have\n−log+|x|\n|y|−log+x2+y2\nB2−log2≤I=/integraldisplay∞\n0log|1−z2\nt2|d([ν](t)−ν(t))≤log+|x|\n|y|·(3.41)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 21\nProof.The proof of the upper bound is the same as in [ 11]. It is sketched here just for the sake\nof completeness. Pick any z=x+iywithy/\\e}atio\\slash= 0. Integrate by part in Ito get\nI=/integraldisplay∞\n0(ν(t)−[ν(t)])∂\n∂tlog|1−z2\nt2|dt.\nLetζ=z2\nt2. If Rez2≤0 (i.e. if |x| ≤ |y|), then the distance |1−ζ|is decreasing w.r.t. t\n(t∈(0,+∞)), so that I≤0. If Rez2>0, then|1−ζ|decreases to the minimal value|Imz2|\n|z2|\ntaken att=t∗:=|z|2\n|x|, and then it increases. Since 0 ≤ν(t)−[ν(t)]≤1, we have that\nI≤/integraldisplay∞\nt∗∂\n∂tlog|1−z2\nt2|dt= log|z2|\n|Imz2|= log(|x|\n2|y|+|y|\n2|x|)≤log|x|\n|y|·\nLet us pass to the lower bound. If Re z2≤0,\nI≥/integraldisplay∞\nB∂\n∂tlog|1−z2\nt2|dt=−log|1−z2\nB2|·\nAssume now that Re z2>0. Ift∗=|z2|\n|x|≤B,I≥0. Ift∗>B, then\nI≥/integraldisplayt∗\nB∂\n∂tlog|1−z2\nt2|dt=−log|z2|\n|Imz2|−log|1−z2\nB2|·\nNote that\nlog|1−z2\nB2| ≤log(1+|z\nB|2)≤log+x2+y2\nB2+log2.\nTherefore\nI≥ −log+|x|\n|y|−log+x2+y2\nB2−log2.\n/square\nGathering together Lemma 3.7and Lemma 3.8, we obtain the\nProposition 3.9. There exists some positive constant C=C(a,b)such that for any complex\nnumberz=x+iywithy/\\e}atio\\slash= 0,\n−C−bπ(1+1√\n2)/radicalbig\n|y|−log+|x|\n|y|−log+(x2+y2\nB2)−log2≤˜U(z)+bπ/radicalbig\n|x|−aπ|y| ≤C+log+|x|\n|y|.\n(3.42)\nPick now\na=T\n2π−1>0, b=√\n2,andm(z) = exp˜g(z−i) (3.43)\nNote that |m(z)|= exp˜U(z−i). The needed estimates for the multiplier mare collected in the\nfollowing22 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nProposition 3.10. mis an entire function on Cof exponential type at most aπ. Furthermore,\nthe following estimates hold for some constant C >0:\n|m(x)| ≤C(1+|x|)e−√\n2π√\n|x|, x∈R (3.44)\n|m(iλ+\nk)| ≥C−1|k|−3e−√\n2π√\n|k|, k∈Z\\{0} (3.45)\n|m(iλ−\nk)| ≥C−1eaπk2−2√\n2π|k|, k∈Z\\{0}. (3.46)\nProof.(3.44) follows at once from ( 3.42) (withy=−1). We infer from ( 2.5) that fork∈Z\\{0}\nIm (iλ±\nk)≤ −1\n2. (3.47)\nIt follows then from ( 2.6) and (3.42) that\n|m(iλ+\nk)|= exp˜U(−k−2i(1+O(k−2)))≥C|k|−3e−√\n2π√\n|k|(k/\\e}atio\\slash= 0).\nFinally, from ( 2.7) and (3.42), we infer that\n|m(iλ−\nk)|= exp˜U(−k−i(k2+O(k−2)))\n≥Cexp(−√\n2π/radicalbig\n|k|+aπk2−(√\n2+1)π|k|−4log|k|)\n≥Cexp(aπk2−2√\n2π|k|).\n/square\nWe are in a position to define the functions in the biorthogona l family. Pick first any k∈\nZ\\{0,±2}, and set\nI±\nk(z) =P(z)\nP′(iλ±\nk)(z−iλ±\nk)·m(z)\nm(iλ±\nk)·(1−z\niλ2)(1−z\niλ−2)\n(1−λ±\nk\nλ2)(1−λ±\nk\nλ−2)·\nClearly,I±\nkis an entire function of exponential type at most π(1+a) =T/2. Furthermore, we\nhave that\nI±\nk(iλ±\nl) =δl\nkδ−\n+∀l∈Z, (3.48)\nwhereδ−\n+is 1 if the two signs in the l.h.s. are the same, and 0 otherwise . Moreover,\n(I±\nk)′(iλ±2) = 0. (3.49)\nOn the other hand, by ( 2.6), (3.22), (3.23), (3.44) and (3.45), we have that\n|I+\nk(x)| ≤C|k|4\n|x−iλ+\nk|≤C|k|4\n1+|k+x|·\nThusI+\nk∈L2(R) with\n||I+\nk||L2(R)≤C|k|4. (3.50)\nFinally, by ( 2.7), (3.22), (3.24), (3.44) , and (3.46), we have that\n|I−\nk(x)| ≤C|k|3\n|x−iλ−\nk|e−(a+1)πk2+2√\n2π|k|≤C|k|3\n|x+k|+k2e−T\n2k2+2√\n2π|k|.CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 23\nThus\n||I−\nk||L2(R)≤C|k|2e−T\n2k2+2√\n2π|k|. (3.51)\nIt remains to introduce the functions I0(z),I2(z),I−2(z),˜I2(z), and˜I−2(z). We set\nI0(z) =P(z)\nP′(0)z·m(z)\nm(0)·(1−z\niλ2)(1−z\niλ−2),\n˜I2(z) =−iP(z)\nP′(iλ2)·m(z)\nm(iλ2)·1−z\niλ−2\n1−λ2\nλ−2,\n˜I−2(z) =−iP(z)\nP′(iλ−2)·m(z)\nm(iλ−2)·1−z\niλ2\n1−λ−2\nλ2,\nK2(z) =i˜I2(z)\nz−iλ2, I2(z) =K2(z)−iK′\n2(iλ2)˜I2(z),\nK−2(z) =i˜I−2(z)\nz−iλ−2, I−2(z) =K−2(z)−iK′\n2(iλ−2)˜I−2(z).\nThen we have that\nI0(0) = 1, I0(iλ±\nk) = 0k∈Z\\{0}, I′\n0(iλ±2) = 0, (3.52)\n˜I2(iλ±\nk) = 0k∈Z,˜I′\n2(iλ2) =−i,˜I′\n2(iλ−2) = 0, (3.53)\n˜I−2(iλ±\nk) = 0k∈Z,˜I′\n−2(iλ−2) =−i,˜I′\n−2(iλ2) = 0, (3.54)\nI2(iλ±\nk) = 0k∈Z\\{2}, I2(iλ2) = 1, I′\n2(iλ±2) = 0, (3.55)\nI−2(iλ±\nk) = 0k∈Z\\{−2}, I−2(iλ−2) = 1, I′\n−2(iλ±2) = 0. (3.56)\nMoreover,I0,˜I2,˜I−2, I2, andI−2are entire functions of exponential type at most π(1+a) and\nthey belong all to L2(R).\nLetψ±\nk,ψk, and˜ψkdenote the inverse Fourier transform of I±\nk,Ik, and˜Ikfork∈Z\\{0,±2},\nk∈ {0,±2}andk∈ {±2}, respectively. Then, by Paley-Wiener theorem, the functio nsψ±\nk,ψk\nand˜ψkbelong toL2(R), and are supported in [ −T/2,T/2]. On the other hand, if I(z) =ˆψ(z) =/integraltext∞\n−∞ψ(t)e−itzdtwithψ∈L2(R),suppψ⊂[−T/2,T/2], then\n/integraldisplayT\n2\n−T\n2ψ(t)eλtdt=I(iλ) and −i/integraldisplayT\n2\n−T\n2tψ(t)eλtdt=I′(iλ).\nThus (2.36)-(2.40) follow from ( 3.48)-(3.49) and (3.52)-(3.56), while ( 2.41)-(2.42) follow from\n(3.50)-(3.51). The proof of Proposition 2.2is complete.\n4.Concluding remark\nIn this paper, the equation ytt−yxx−ytxx=b(x−u(t))h(t) is proved to be null controllable\non the torus (i.e. with periodic boundary conditions) when t he support of the scalar control\nh(t) moves at a constant velocity c(u(t) =ct). What happens for a domain with boundary?24 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON\nMore precisely, we may wonder under which assumptions on the initial conditions, the control\ntimeT, the support of the controller band its pulsations ωthe null controllability of the system\nytt−yxx−ytxx=b(x−cos(ωt))h(t), x∈(−1,1), t∈(0,T),\ny(−1,t) =y(1,t) = 0, t∈(0,T)\nholds.\nAcknowledgements\nLR was partially supported by the Agence Nationale de la Rech erche, Project CISIFS, grant\nANR-09-BLAN-0213-02.\nReferences\n[1] A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support , Acta Math. 107(1962)\n291–309.\n[2] C. Castro, Exact controllability of the 1-d wave equation from a moving interior point , preprint.\n[3] C.Castro, E.Zuazua, Unique continuation and control for the heat equation from a lower dimensional manifold ,\nSIAM J. Cont. Optim., 42(4), (2005) 1400–1434.\n[4] C. Castro, E. 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Khapalov, Mobile point controls versus locally distributed ones for t he controllability of the semilinear\nparabolic equation , SIAM J. Cont. Optim., 40(1), (2001) 231–252.\n[11] P. Koosis, The Logarithmic Integral, vol. I,II, Cambri dge Stud. Adv. Math., vol. 12, Cambridge University\nPress, Cambridge, 1988; Cambridge Stud. Adv. Math., vol. 21 , Cambridge University Press, Cambridge, 1992.\n[12] I. Lasiecka, R. Triggiani, Exact null controllability of structurally damped and ther mo-elastic parabolic models.\nAtti. Accad. Naz Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lince i (9), Mat. Appl. 9(1998), 43-69.\n[13] G. Lebeau, E. Zuazua, Null-controllability of a system of linear thermoelastici ty, Arch. Rational Mech. Anal.\n141(1998), no. 4, 297–329.\n[14] G. Leugering, Optimal controllability in viscoelasticity of rate type , Math. Methods Appl. Sci. 8(1986),\n368–386.\n[15] G. Leugering, E. J. P. G. Schmidt, Boundary control of a vibrating plate with internal damping , Mathematical\nMethods in the Applied Sciences, 11(1989), 573–586.\n[16] B. Ya. Levin, Lectures on Entire Functions, Translatio ns of Mathematical Monographs, American Mathe-\nmatical Society, Vol. 150, 1996.\n[17] J.-L. Lions, Pointwise control for distributed systems , in Control and estimation in distributed parameter\nsystems, edited by H. T. Banks, SIAM, 1992.\n[18] W. A. J. Luxemburg, J. Korevaar, Entire functions and M¨ untz-Sz´ asz type approximation , Transactions of the\nAmerican Mathematical Society 157(1971), 23–37.\n[19] S. Micu, On the controllability of the linearized Benjamin-Bona-Ma hony equation , SIAM J. Control Optim.\n39(2001), 1677–1696.CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 25\n[20] A. Pazy, Semigroups of linear operators and applications to partial differential equations , Applied Mathemat-\nical Sciences, vol. 44, Springer-Verlag, 1983.\n[21] M. Pellicer, J. Sol` a-Morales, Analysis of a viscoelastic spring-mass model , J. Math. Anal. Appl. 294(2),\n(2004) 687–698.\n[22] L. Rosier, P. Rouchon, On the controllability of a wave equation with structural da mping, Int. J. Tomogr.\nStat.5(2007), no. W07, 79–84.\n[23] L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equatio n: recent progresses , Jrl\nSyst & Complexity 22(2009), 647–682.\n[24] L. Rosier, B.-Y. Zhang, Unique continuation property and control for the Benjamin- Bona-Mahony equation ,\npreprint.\n[25] D. L. Russell, Mathematical models for the elastic beam and their control- theoretic implications , in H. Brezis,\nM. G. Crandall and F. Kapper (eds), Semigroup Theory and Applications , Longman, New York (1985).\n[26] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, 1980.\n[27] X.Zhang, Exact internal controllability of Maxwell’s equations , Appl.Math. Optim.41 (2000), no. 2, 155–170.\nCentre Automatique et Syst `emes, Mines ParisTech, 60 boulevard Saint-Michel, 75272 Pa ris\nCedex, France\nE-mail address :philippe.martin@mines-paristech.fr\nInstitut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 70239, 54506 Vandœuvre-l `es-Nancy\nCedex, France\nE-mail address :rosier@iecn.u-nancy\nCentre Automatique et Syst `emes, Mines ParisTech, 60 boulevard Saint-Michel, 75272 Pa ris\nCedex, France\nE-mail address :pierre.rouchon@mines-paristech.fr"    },    {        "title": "1201.3553v1.Magnetic_vortex_echoes__application_to_the_study_of_arrays_of_magnetic_nanostructures.pdf",        "content": "arXiv:1201.3553v1  [cond-mat.mes-hall]  17 Jan 2012Magnetic vortex echoes: application to the study of arrays o f magnetic nanostructures\nF. Garcia1, J.P. Sinnecker2, E.R.P. Novais2, and A.P. Guimar˜ aes2∗\n1Laborat´ orio Nacional de Luz S´ ıncrotron, 13083-970, Camp inas, SP, Brazil and\n2Centro Brasileiro de Pesquisas F´ ısicas, 22290-180, Rio de Janeiro, RJ, Brazil\n(Dated: December 13, 2018)\nWe propose theuse ofthe gyrotropic motion of vortexcores in nanomagnets toproduce amagnetic\necho, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g.,\nnanodisks, is magnetized with an in-plane ( xy) field, and after a time τa field pulse inverts the\ncore magnetization; the echo is a peak in Mxyatt= 2τ. Its relaxation times depend on the\ninhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant α.\nIts feasibility is demonstrated using micromagnetic simul ation. To illustrate an application of the\nechoes, we have determined the inhomogeneity and measured t he magnetic interaction in an array\nof nanodisks separated by a distance d, finding a d−ndependence, with n≈4.\nPACS numbers: 75.70.Kw,75.78.Cd,62.23.Eg,76.60.Lz\nThe interest in magnetic vortices and their properties\nand applications has grown steadily in the last years[1–\n4]. Vortices have been observed, for example, in disks\nand ellipses having sub-micron dimensions[5]. More re-\ncently, the question of the intensity of the coupling be-\ntweenneighbordiskswith magneticvortexstructureshas\nattracted an increasing interest[6–9].\nAt thevortexcorethe magnetizationpointsperpendic-\nularlyto the plane: this characterizesits polarity, p= +1\nfor the + zdirection and p=−1 for−z. The direc-\ntion of the moments in the vortex defines the circulation:\nc=−1 for clockwise (CW) direction, and +1 for CCW.\nIf removed from the equilibrium position at the center\nof the nanodisk by, for example, an in-plane field, and\nthen left to relax, a vortex core will perform a gyrotropic\nmotion, with angular frequency ω, given for thin disks[3]\nbyωG≈(20/9)γMsβ(β=h/Ris the aspect ratio)[10].\nWe propose in this paper that, manipulating the dy-\nnamic properties of the vortex in an analogous way as it\nis done in Nuclear Magnetic Resonance (NMR), a new\nphenomenon results, the magnetic vortex echo (MVE),\nsimilar to the spin echo observed in NMR[11]. This new\necho may provide information on fundamental properties\nof arrays of nanodisks, e.g., their inhomogeneity and in-\nteractions. Despite the fact that applications of vortices\nnecessarilyinvolvearrays,mostoftherecentpublications\ndeal with the analysis of individual nanodisks or arrays\nwith a few elements. Therefore, the possibility of char-\nacterizing large arrays is of much interest. In this paper\nwe have examined the motion of vortex cores in an ar-\nray of nanometric disks under the influence of a pulsed\nmagnetic field, using micromagnetic simulation.\nLet us consider an array of nanodisks where the vor-\ntex cores precess with a distribution of angular frequen-\ncies centered on ω0, of width ∆ ω, arising from any type\nof inhomogeneity (see note [10]). We assume that the\nfrequencies vary continuously, and have a Gaussian dis-\ntribution P(ω) with mean square deviation ∆ ω. To sim-\nplify we can assume that the polarization of every vortex\nFIG. 1. (Color online) Diagram showing the formation of\nmagnetic vortex echoes; the disks are described from a refer -\nence frame that turns with the average translation frequenc y\nω0: a) disks with in-plane magnetizations Mialong the same\ndirection (defined by the white arrows); b) after a time τ\nthe disks on the left, center and right have turned with fre-\nquencies, respectively, lower, higher and equal to ω0; c) the\npolarities of the vortex cores are reversed, and the ωiof the\nvortex cores (and of the Mi) change sign, and d) after a sec-\nond interval τthe cores (and Mi) are again aligned, creating\nthe echo.\nis the same: pi= +1. This is not necessary for our argu-\nment, but, if required, the system can be prepared; see\nref. [12] and the references therein.\nSince the direction of rotation of the magnetic vortex\ncores after removal of the in-plane field is defined exclu-\nsively by p, all the cores will turn in the same direction;\nas the vortex core turns, the in-plane magnetization of\nthe nanodot also turns.2\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49\n/s32/s32\n/s116/s32/s40/s110/s115/s41/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s53/s77/s32/s40/s49/s48/s45/s49/s52\n/s65/s47/s109/s41\n/s32/s61/s32/s50/s48/s32/s110/s109/s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s32/s61/s32/s50/s48/s32/s110/s109 /s44/s32 /s61/s32/s49/s48/s32/s110/s115 /s44/s32 /s61/s32/s52/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s49\n/s100/s99/s98/s97/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s51/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48\n/s32\nFIG. 2. (Color online) Micromagnetic simulation of magneti c\nvortex echoes, for 100 nanodisks, with d= infinity, and a)\nσ= 10nm, τ= 30ns, α= 0; b) σ= 20nm, τ= 10ns\nandτ= 40ns (two pulses), and α= 0.001; c)σ= 20nm,\nτ= 20ns, α= 0; d)σ= 10nm, τ= 20ns, α= 0.005. The\ninversion pulses ( Bz=−300mT) are also shown (in red).\n/s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50 /s48/s46/s48/s48/s51 /s48/s46/s48/s48/s52 /s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s32/s32/s49/s47/s84\n/s50/s32/s40/s110/s115/s45/s49\n/s41\n/s68/s97/s109/s112/s105/s110/s103/s32/s99/s111/s110/s115/s116/s97/s110/s116/s32\nFIG. 3. (Color online) Variation of the inverse of the re-\nlaxation times T2(diamonds) obtained by fitting the curves\nof echo intensity versus time interval τtoM0exp(−τ/T2),\nas a function of α, for simulations made with D= 250nm,\nσ= 10nm, separation infinite; the continuous line is a linear\nleast squares fit.\nLet us assume that all the vortex cores have been dis-\nplaced from their equilibrium positions alongthe positive\nxaxis[13], by a field B. The total in-plane magnetiza-\ntion (that is perpendicular to the displacement of the\ncore) will point along the yaxis, therefore forming an\nangleθ0= 0 att= 0. Using the approach employed in\nthe description of magnetic resonance (e.g., see [14, 15]),one can derive the total in-plane magnetization:\nMy(t) =My(0)/integraldisplay∞\n−∞e/bracketleftBig\n−1\n2(ω−ω0)2\n∆ω2/bracketrightBig\n∆ω√\n2πcos(ωt)dω(1)\nan integral that is[16] the Fourier transform of the func-\ntionP(ω); usingT∗\n2= 1/∆ω:\nMy(t) =My(0)exp/parenleftbigg\n−1\n2t2\nT∗2\n2/parenrightbigg\ncos(ω0t) (2)\nThis result shows that the total magnetization tends to\nzero, as the different contributions to My(t) get gradu-\nally out of phase. This decay is analogous to the free\ninduction decay (FID) in NMR; its characteristic time is\nT∗\n2= 1/∆ω.\nAfter a time τ, the angle rotated by each vortex core\nwill beωτ; if att=τwe invert the polarities of the vor-\ntices in the array, using an appropriate pulse, the motion\nof the cores will change direction (i.e., ω→ −ω), and one\nobtains:\nMy(t−τ) =My(0)/integraldisplay∞\n−∞e−1\n2(ω−ω0)2\n∆ω2\n∆ω√\n2πcos[ω(τ−t)]dω(3)\nThe magnetization at a time t > τis then:\nMy(t) =My(0)e[−1\n2(t−2τ)2\nT∗2\n2]e(−t−τ\nT2)cos(ω0t) (4)\nThis means that the magnetization component My(t) in-\ncreases for τ < t < 2τ, reaching a maximum at a time\nt= 2τ: this maximum is the magnetic vortex echo , anal-\nogous to the spin echo observed in magnetic resonance,\nwhich has important applications in NMR, including in\nMagnetic Resonance Imaging (MRI)[14, 15] (Fig. 1). In\nthe case of the NMR spin echo, the maximum arises from\nthe refocusing of the in-plane components of the nuclear\nmagnetization.\nIn Eq. 4 we have included a relaxation term contain-\ning the time constant T2- also occurring in NMR -, to\naccount for a possible decay of the echo amplitude with\ntime; its justification will be given below.\nIn the arrayof nanodisks, there will be in principle two\ncontributionsto the defocusing ofthe magnetization, i.e.,\ntwo mechanisms for the loss of in-plane magnetization\nmemory: 1) the spread in values of βandH(see note\n[10]), producing an angular frequency broadening term\n∆ω, and 2) irreversible processes that are characterized\nby a relaxation time T2: thus 1/T∗\n2= ∆ω+1/T2.\nThe second contribution is the homogeneous term\nwhose inverse, T2, is the magnetic vortex transverse re-\nlaxationtime, analogousto thespin transverserelaxation\ntime (or spin-spin relaxation time) T2in magnetic reso-\nnance. The irreversible processes include a) the interac-\ntion between the disks, which amounts to random mag-\nnetic fields that will increase or decrease ωof a given3\ndisk, producing a frequency spread of width ∆ ω′= 1/T′\n2,\nand b) the loss in magnetization (of rate 1 /Tα) arising\nfrom the energy dissipation related to the Gilbert damp-\ning constant α. Identifying Tαto the NMR longitudinal\nrelaxation time T1, one has [14]: 1 /T2= 1/T′\n2+1/2Tα.\nTherefore the relaxation rate 1 /T∗\n2is given by:\n1\nT∗\n2= ∆ω+1\nT2= ∆ω+1\nT′\n2+1\n2Tα(5)\n1/T∗\n2is therefore the total relaxation rate of the in-plane\nmagnetization, composed of a) ∆ ω, the inhomogeneity\nterm, and b) 1 /T2, the sum ofall the other contributions,\ncontaining1 /T′\n2, duetotheinteractionbetweenthedisks,\nand 1/Tα, the rate of energy decay. The vortex cores\nwill reach the equilibrium position at r= 0 after a time\nt∼Tα, therefore there will be no echo for 2 τ≫Tα.\nThe vortex echo maximum at t= 2τ, from Eq. 4, is\nMy(2τ)∝exp(−τ/T2); one should therefore note that\nthe maximum magnetization recovered at a time 2 τde-\ncreases exponentially with T2, i.e., this maximum is only\naffected by the homogeneous part of the total decay rate\ngiven by Eq. 5. In other words, the vortex echo cancels\nthe loss in Mydue to the inhomogeneity ∆ ω, but it does\nnot cancel the decrease in Mydue to the interaction be-\ntween the nanodisks (the homogeneous relaxation term\n1/T′\n2), or due to the energy dissipation (term 1 /2Tα).\nNote also that if one attempted to estimate the inho-\nmogeneityofan arrayofnanodotsusing anothermethod,\nfor example, measuring the linewidth of a FMR spec-\ntrum, one would have the contribution of this inhomo-\ngeneity together with the other terms that appear in Eq.\n5, arising from interaction between the dots and from\nthe damping. On the other hand, measuring the vor-\ntex echo it would be possible to separate the intrinsic\ninhomogeneity from these contributions, since T2can be\nmeasured separately, independently of the term ∆ ω.T2\ncan be measured by determining the decay of the echo\namplitude for different values of the interval τ.\nThe Fourier transform of either the vortex free induc-\ntiondecayorthetimedependenceoftheecho My(t) gives\nthe distribution of gyrotropic frequencies P(ω).\nFor the experimental study of vortex echoes, the se-\nquence of preparation (at t=0) and inversion fields (at\nt=τ) should of course be repeated periodically, with a\nperiodT≫Tα. As in pulse NMR, this will produce\nechoes on every cycle, improving the S/N ratio of the\nmeasured signals. Also note that the time T∗\n2can be ob-\ntained either from the initial decay (FID, Eq. 2) or from\nthe echo (Eq. 4), but T2can only be obtained from the\nMVE.\nIn order to demonstrate the MVE, we have performed\nmicromagnetic simulations of an assembly of 100 mag-\nnetic nanodisks employing the OOMMF code[17]. The\nsimulated system was a square array of 10 ×10 disks,\nthickness 20nm, with distance dfrom center to center.\nIn order to simulate the inhomogeneity of the system,we have introduced a Gaussian distribution of diame-\nters, centered on 250nm and mean square deviation σ;\nσ= 10nm corresponds to ∆ ω≈1.6×108s−1. The\ndisks were placed at random on the square lattice. The\ninitial state of the disks ( p= +1 and c=−1) was pre-\npared by applying an in-plane field of 25mT; the po-\nlarity was inverted with a Gaussian pulse of amplitude\nBz=−300mT, with width 100ps. The results for the\ncased=∞were simulations made on the disks one at a\ntime, adding the individual magnetic moments µi(t).\nWe have successfully demonstrated the occurrence of\nthe magnetic vortex phenomenon, and have shown its\npotential as a characterization technique. The simula-\ntions have confirmed the occurrence of the echoes at the\nexpected times ( t= 2τ). For different values of σ, the\nT∗\n2time, and consequently the duration of the FID and\nthe width of the echo are modified (Fig. 2a, 2c); increas-\ningαresults in a faster decay of the echo intensity as a\nfunction of time (Fig. 2a, 2d). We have also obtained\nmultiple echoes, by exciting the system with two pulses\n(Fig. 2b)[18].\nFig. 3 shows the dependence of T2onαforσ= 10nm;\nessentially the same result is obtained for σ= 20nm,\nsinceT2does not depend on ∆ ω(Eq. 5). Taking a linear\napproximation, 1 /T2=Aα, and since for d=infinity\nthere is no interaction between the disks, 1 /T2= 1/2Tα,\nand therefore:\n1\nTα= 2Aα (6)\nFrom the least squares fit (Fig. 3), A= 1.6×1010s−1.\nThis relation can be used to determine experimentally α,\nmeasuring T2with vortex echoes, for an array of well-\nseparated disks.\nRecently some workers have analyzed the important\nproblem of the interaction between disks exhibiting mag-\nnetic vortices, obtaining that it varies with a d−ndepen-\ndence: Vogel and co-workers [6], using FMR, obtained\nfor a 4×300 array a dependence of the form d−6, the\nsame found by Sugimoto et al. [8] using a pair of disks\nexcited with rf current. Jung et al. [7] studying a pair of\nnanodisks with time-resolved X-ray spectroscopy, found\nn= 3.91±0.07 and Sukhostavets et al. [9], also for\na pair of disks, in this case studied by micromagnetic\nsimulation, obtained n= 3.2 and 3.7 for the xandy\ninteraction terms, respectively.\nAs a first approximation one can derive the depen-\ndence of the contribution to 1 /T∗\n2related to the distance\nbetween the disks as:\nT∗\n2=B+Cd−n(7)\nFrom our simulations, and using Eq. 7 we found, from\nthe best fit, that this interaction varies as d−n, withn=\n3.9±0.1, in a good agreement with [7] and reasonable\nagreement with Sukhostavets et al.[9].4\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48/s54/s46/s53\n/s32/s32/s84\n/s50/s42/s32/s40/s110/s115/s41\n/s100/s45/s52\n/s32/s40/s49/s48/s45/s49/s49\n/s110/s109/s45/s52\n/s41/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s77 /s97 /s103 /s110 /s101 /s116/s105/s99 /s32/s109 /s111 /s109 /s101 /s110 /s116/s32/s120 /s32/s49 /s48 /s49/s52/s32\n/s40/s65 /s32/s109 /s50\n/s41\n/s84/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 4. (Color online) Variation of the relaxation times\nT∗\n2versusd−4for an array of 10 ×10 magnetic nanodisks\nwith a distribution of diameters centered on D= 250nm\n(σ= 10nm), damping constant α= 0.001 and separation\ndbetween their centers; the continuous line is a linear least\nsquares fit. The inset shows a vortex echo simulation for the\narray, with d= 500nm, τ= 30ns, α= 0.001.\nDetermining 1 /T′\n2has allowed us to obtain the inten-\nsity of the interaction between the disks as a function\nof separation dbetween them. Substituting Eq. 6 and\nEq. 7 in Eq. 5, we can obtain the expression for the\ninteraction as a function of d:\n1\nT′\n2=1\nB+Cd−n−Aα−∆ω≈dn\n|C|−Aα−∆ω; (8)\n(approximation valid for dsmall). In Fig. 4 we show the\nresults of the simulations with σ= 10nm and α= 0.001.\nAssuming n= 4 and making a linear squares fit, we\nobtained B= 6.15×10−9s,C=−4.03×10−35s m4.\nA new phenomenon, the magnetic vortex echo, anal-\nogous to the NMR spin echo, is proposed and demon-\nstrated here through micromagnetic simulation. Appli-\ncationsofthe magneticvortexechoincludesthe measure-\nment of the inhomogeneity, such as, distribution of di-\nmensions, aspect ratios, defects, and perpendicular mag-\nnetic fields and so on, in a planar array of nanodisks or\nellipses; it may be used to study arrays of nanowires or\nnanopillars containing thin layers of magnetic material.\nThese properties cannot be obtained directly, for exam-\nple, from the linewidth of FMR absorption.\nThe MVE is a tool that can be used to evaluate the\ninteractionbetween the elements ofalargearrayofnano-\nmagnets with vortex ground states. It can also be used\nto determine the Gilbert damping constant αin thesesystems.\nThe authors would like to thank G.M.B. Fior for the\ncollaboration; we are also indebted to the Brazilianagen-\ncies CNPq, CAPES, FAPERJ, FAPESP.\n∗Author to whom correspondence should be\naddressed: apguima@cbpf.br\n[1] A. P. Guimar˜ aes, Principles of Nanomagnetism\n(Springer, Berlin, 2009)\n[2] C. L. Chien, F. Q. Zhu, and J.-G. Zhu, Physics Today\n60, 40 (2007)\n[3] K. Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev.\nLett.100, 027203 (2008)\n[4] F. Garcia, H. Westfahl, J. Schoenmaker, E. J. Carvalho,\nA. D. Santos, M. Pojar, A. C. Seabra, R. Belkhou,\nA. Bendounan, E. R. P. Novais, and A. P. Guimar˜ aes,\nAppl. Phys. Lett. 97, 022501 (2010)\n[5] E. R. P. Novais, P. Landeros, A. G. S. Barbosa, M. D.\nMartins, F. Garcia, and A. P. Guimar˜ aes, J. Appl. Phys.\n110, 053917 (2011)\n[6] A. Vogel, A. Drews, T. Kamionka, M. Bolte, and\nG. Meier, Phys. Rev. Lett. 105, 037201 (2010)\n[7] H. Jung, K.-S. Lee, D.-E. Jeong, Y.-S. Choi, Y.-S.\nYu, D.-S. Han, A. Vogel, L. Bocklage, G. Meier, M.-\nY. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 59, 1\n(2011/08/10/online)\n[8] S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Bar-\nman, and Y. Otani, Phys. Rev. Lett. 106, 197203 (2011)\n[9] O.V.Sukhostavets,J. M.Gonzalez, andK.Y.Guslienko,\nAppl. Phys. Express 4, 065003 (2011)\n[10] The sources of inhomogeneity are the spread in radii, in\nthickness or the presence of defects. An external perpen-\ndicular field Haddsacontributionto ω[19],ω=ωG+ωH,\nwithωH=ω0p(H/Hs), where pis the polarity and Hs\nthe field that saturates the nanodisk magnetization. A\ndistribution ∆ His another source of the spread ∆ ω\n[11] E. L. Hahn, Phys. Rev. 80, 580 (1950)\n[12] R. Antos, M. Urbanek, and Y. Otani, J. Phys.: Conf.\nSeries200, 042002 (2010)\n[13] If the disks have different circulations ( c=±1) the cores\nwill be displaced in opposite directions, butthe effect will\nbe the same, since the Miwill all point along the same\ndirection.\n[14] C. P. Slichter, Principles of Magnetic Resonance, 3. ed.\n(Springer, Berlin, 1990)\n[15] A. P. Guimar˜ aes, Magnetism and Magnetic Resonance in\nSolids(John Wiley & Sons, New York, 1998)\n[16] T. Butz, Fourier Transformation for Pedestrians\n(Springer, Berlin, 2006)\n[17] Available from http://math.nist.gov/oommf/\n[18] These echoes, however, are not equivalent to the stimu-\nlated echoes observed in NMR with two 90opulses[11]\n[19] G. de Loubens, A. Riegler, B. Pigeau, F. Lochner,\nF. Boust, K. Y. Guslienko, H. Hurdequint, L. W.\nMolenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberke-\nvich, N.Vukadinovic,andO.Klein,Phys.Rev.Lett. 102,\n177602 (2009)"    },    {        "title": "1202.3450v1.Current_induced_motion_of_a_transverse_magnetic_domain_wall_in_the_presence_of_spin_Hall_effect.pdf",        "content": "Current -induced motion of a transverse magnetic domain \nwall in the presence of  spin Hall effect  \n \nSoo-Man Seo1, Kyoung-Whan Kim2, Jisu Ryu2, Hyun -Woo Lee2,a), and Kyung -Jin Lee1,3,4,b) \n \n1Department of M aterials Science and Engineering, Korea University , Seo ul 136 -701, Korea  \n2PCTP and Department of Physics, Pohang University of Science and Technology , Kyungbuk 790 -\n784, Korea  \n3Center for Nanoscale Science and Technology , National Institute of Standards and Technology, \nGaithersburg, Maryland 20899 -8412, USA   \n4Maryland Nanocenter, University of Maryland, College Park, MD 20742, USA  \n \nWe theoretically  study  the current -induced dynamics of a transverse magnetic domain \nwall in bi-layer nanowire s consisting of a ferromagnet  on top of a nonmagnet having \nstrong spin-orbit coupling . Domain wall  dynamics is characterized  by two threshold \ncurrent densities, \nWB\nthJ  and \nREV\nthJ , where \nWB\nthJ  is a threshold for the chirality switching \nof the domain wall  and \nREV\nthJ  is another threshold for the reversed domain wall motion  \ncaused by spin Hall effect . Domain wall s with a certain chirality may move opposite to \nthe electron -flow direction with high speed in the current range \nWB\nthREV\nth JJ J  for the \nsystem designed to satisfy the conditio ns \nWB REV\nth thJJ  and \n , where \n  is the \nGilbert damping constant and \n  is the  nonadiabaticity of spin torque . Micromagnetic \nsimulation s confirm the validity of analytical results. 1 \n                                                      \na)Electronic mail:  hwl@postech.ac.kr.  \nb)Electronic mail:  kj_lee@korea.ac.kr .    Electric  manipulati on of domain wall s (DW s) in magnetic nanowire s can be  realized by  \nthe spin-transfer torque (STT) due to the coupling  between local  magnetic moment s of the \nDW and spin-polarized current s1,2. Numerous studie s on this subject  have addressed  its \nfundamental physic s3-5, and to explore  its potential in application s such as data storage and \nlogic devices6. Up until now, h owever, most studies have focused on the effect of the spin \ncurrent that i s polarized by a ferromagnet ic layer . \nAnother  way to generate a spin curren t is the spin Hall effect (SHE)7,8. In ferromagnet \n(FM) |nonmagnet (NM) bi -layer system s, an in-plane  charge current density (Jc) passing \nthrough the NM is converted into a perpendic ular spin current density (Js) owing to the SHE . \nThe ratio of Js to Jc is parameterized by spin  Hall angle . This spin current caused by SHE \nexerts  a STT (= SHE -STT)  on the FM and consequently modifies its m agnetization dynamics. \nDuring the last decade , most studies on the SHE  have focused on measuring the spin  Hall \nangle9-14. Recently the magnetization  switching15 and the modulation of propagating spin \nwaves  by SHE -STT were investigated16-18. However, the effect of SHE -STT on current -\ninduced DW dynamics has not been treated . \nIn this Letter, we study DW dynamics including  all current -induced STT s in a nanowire \nconsisting of FM/NM bi-layers  (Fig. 1) , where FM has a n in-plane magnetic anisotropy and \nNM has strong spin-orbit coupling ( SOC ) responsible for  the SHE. A charge current passing \nthrough the FM generates  conventional adiabatic and nonadiabatic STTs19-21, whereas a \ncharge current flowing through the NM experiences SHE and generates SHE -STT on the FM . \nFor the current running in the x axis, t he modified Landau-Lifshitz -Gilbert equation including \nall the STTs  is given by  \n),ˆ ( y cxbxbt tJ SH J J \n\n\nm mmmmm mmm Hmm\neff    \n   (1) where m is the unit vector along the magnetization, α is the Gilbert damping constant, bJ  \n) 2 (S F B eM PJg\n is the magnitude of adiabatic STT, β is the nonadiabatic ity of STT, \nJ SHc\n( 2 )SH N S FJ eM t\n is the magnitude of SHE -STT, θSH is an effective spin Hall angle \nfor the bi -layer system , γ is the gyromagnetic ratio,  g is the Land é g-factor, μB is the Bohr \nmagneton, P is the spin polarization in the FM, e is the electron charge, MS is the saturation \nmagnetization of the FM, and JF (JN) is the current density in the FM (NM). JF and JN are \ndetermined by a simple circuit model; i.e.,  \n) /() (0 N N FF F N F F t t ttJ J    and \n) /() (0 N N FF N N F N t t ttJ J  \n, where J0 is the total current density in the bi -layer \nnanowire , σF (σN) is the conductivity  of the FM (NM), and tF (tN) is the thickness of the FM \n(NM).  We assume that θSH is smaller than 1  as is usually the case experimentally . \nFor a nanowire with an in-plane magnetic anisotropy, a  net effective field is given by  \n2\n22ˆ ,mH x Heff K x d\nSAHmMx  \n                      (2) \nwhere A is the exchange stiffness constant, HK is the easy axis anisotropy field along the x \naxis, and \ndH  is the magnetostatic field  given by \n)() (~r )(3rmrr rH Nd MS d , where \nthe components of the tensor \nN~  are given by \n3 2 2/]/31[ r rx Nxx , \n5/3 rxy Nxy  [22]. \nOther components are defined in a  similar way . For a one-dimensional  DW as shown in Fig. \n1, the spatial profile of the magnetization is described by  \n)sin sin, cos sin, (cos m, \nwhere \n  ) ( sech sin Xx , \n  ) ( tanh cos Xx , \n)(tX  is the  DW position, \n)(t  is \nthe DW tilt angle , and λ is the DW  width . By using the procedure developed by Thiele23, we \nobtain the equation s of motion for the two collective coordinates \nX  and \n  in the rigid DW \nlimit,  \nsin(2 ),2d\nJH Xbtt    \n                       (3) ,eff JXbtt                               (4) \nwhere \n) 2(S d d MK H , \n  sin 1SH eff B , \n/2SH SH N F FB J t PJ  , and Kd is the \nhard-axis anisotropy energy density . From Eqs. (3) and (4), one finds that the effect of SHE -\nSTT on DW d ynamics is captured by replacing β by βeff. Assuming that FM is Permall oy (Py: \nNi80Fe20) and NM is Pt, for the parameters of  tF = 4 nm, tN = 3 nm, σF = σN, θSH = 0.1, β ≈ \n0.01 to 0.03 [ 24], P = 0.7, and λ = 30 nm, we find \nSHB  ≈ 18 to 56, which is not small . \nTherefore, βeff can be much larger than β unless  sin is extremely small. Furthermore, it is \npossible that βeff is even negative  if \n1 sinSHB . \nTo get an insight into the effect of SHE -STT on DW dynamics, we derive several \nanalytical solutions from  Eqs. (3) and (4). It is known that DW dynamics in a nanowire can \nbe classified into two regimes; i.e., below and above the Walker breakdown25. Below the \nWalker breakdown, \n  increases in the initial time stage and then becomes saturat ed to a \ncertain value over time.  In this limit (\n0 t  as \nt ), we obtain  \n,) (22sin\neffd\nJHb\n\n                             (5) \nThreshold adiabatic STT for the Walker breakdown  (\nWB\nJb ) is obtained from the maximum \nvalue of  the right -hand -side of  Eq. ( 5); i.e., \n ) (22sin maxeff dWB\nJ H b    . Note that \nWB\nJb\n is not simply \n2( )d effH      because  \neff  also includes \n . When  BSH = 0, Eq. \n(5) reduces to \n   2dWB\nJ H b , reproducing the previ ous result  [26] in the absence  of \nSHE.  \n For \nWB\nJJbb  (below t he Walker breakdown) and using  the small -angle approximation , \nDW velocity  (vDW) is given by \n,) (1\n\n\n\n\nSHJ dJ\nSH J DWBb HbB b v\n\n                (6) \nwhere  the sign “+” and “” in the parenthesis corresponds  to the initial tilt angle s \n0  \nand \n00 , respectively.  This \n0  dependen ce of  \nDWv  originates from the fact that SHE -\nSTT acts like a damping or an anti -damping term depending on \n0 . When  \n , \nJ DW b v\n so that vDW does not depend  on SHE -STT. However, this condition is hardly \nrealized  in the bi -layer system that we consider since the strong SOC in NM increases the \nintrinsic α of FM  through the spin pumping effect27. When BSH = 0, \nJ DW b v  , \nconsistent with the DW velocity in the absence of  SHE26. Note that in our sign convention, a \nnegative bJ corresponds to the electron -flow in + x direction and a positive \nDWv  corresponds \nto the DW motion along the electron -flow direction.  Therefore, when the term in the \nparenthesis of Eq. ( 6) is negative, the DW moves against the electron -flow direction instead \nof along it. Threshold adiabatic STT for this re versed DW motion (\nREV\nJb ) is given by  \n.REV d\nJ\nSHHbB\n                                     (7) \n   For \nWB\nJJbb\n  (far above the Walker breakdown) , the time-average d values of \nsin  \nand \n2sin  can be set to zero because of the precession of \n . In this limit, \nDWv  is \ndetermined by Eq. (3) and becomes bJ so that the DW moves along the electron -flow \ndirection and its motion does not depend on SHE -STT. \n   Based on the above investigations , there are  two interesting effects of SHE on  current -\ninduced DW dynamics. First, current -induced DW dynamics is determined  by two thre sholds, WB\nJb and \nREV\nJb . Whe n \nWB\nJ JREV\nJ b b b  , the DW can move against the electron -flow \ndirection. Note that the existence of such bJ range implicitly assumes  \nWB\nJREV\nJ b b . When \nthis inequality is not satisfied, the DW always moves along the electron -flow direction.  For \nall cases, \nDWv  can be larger than \nJb  depending on the parameters (see Eq. ( 6)). \nSecond , vDW is asymmet ric against  the initial tilt angle \n0  for a fixed current polarity. A \nsimilar argument is also valid for a fixed \n0  but with varying the current polarity;  i.e., \nDWv  \nis asymmetric with respect to the current polarity  for a fixed \n0 . This behavior follows \nbecause SHE -STT acts like a damping term for one sign of the current but acts like an anti -\ndamping term for the other sign. Therefore, although the condi tion of \nREV WB\nJ J Jb b b  is \nsatisfied, the reversed DW motion is expected to be observed  only for one current polarity.  \nTo verify the analytical results , we perform a one-dimensional micromagnetic simulation \nby numerically solving Eq. (1). We con sider a Py/Pt bi -layer  nano wire of (length × width × \nthickness ) = (2000 n m × 80 nm × 4 nm (Py) and 3 nm (Pt)) (Fig. 1) . Py m aterial parameters \nof MS = 800 kA/m, A = 1.3×10−11 J/m, P = 0.7, α = 0.02, and  β = 0.01  to 0.03 are used . The \ncrystalline anisotropy and the temperature are assumed to be zero. Conductivit ies of both \nlayers are assumed to be the same as σPy = σPt = 6.5 (μΩm)1, and thus J0 = JF = JN. For all \ncases, the in itial DW tilt angle \n0  is set to zero.  \nAnalytical and numerical results are compared in Fig. 2. DW velocity (\nDWv ) and DW tilt \nangle (\nDW ) as a function of the total current density of the bi -layer (J0) for three values of \nθSH (= +0.1, 0.0, 0.1) and β = 0.01  (thus  > ) are shown in Fig. 2(a) and (b) , respectively . \nDWv\n is estimated from  the terminal velocity . Here, we test both positive and negative values \nof θSH since the spin Hall angle can have e ither sign .  Current dependence s of \nDWv  (Fig. 2(a)) and \nDW  (Fig. 2(b)) show close correlation , \nmeaning that the DW tilting plays  a crucial role for the effect of SHE on DW dynamics  as \ndemonstrated analytically . In Fig. 2(a) , the numerical  results (symbols) are in agreement with \nthe results obtained from Eq. (6)  (lines) . For θSH = 0, \nDWv  is linearly proportional to  J0 and \nthe DW always move s along the electron -flow direction . However, for 0.5×1012 ≤ J0 ≤ \n1.0×1012 A/m2 with θSH = 0.1 (1.0×1012 A/m2 ≤ J0 ≤ 0.5×1012 A/m2 with θSH = 0.1 ), \nDWv  \nhas the same polarity as the current . Thus, the DW moves along  the current -flow direction  for \nthese ranges of the current . The threshold for the  reversed DW motion  is consistent with the \nanalytical solution of Eq. (7); i.e., \nREV\nJb  = ±26.6 m/s corresponding to J0 = ±0.52×1012 A/m2. \nThe maximum \nDWv  is obtained at J0 = ±1 .0×1012 A/m2 immediately before the DW \nexperiences Walker breakdown and switches  its chirality . As shown in the Fig. 2  (c), the \nnormalized y-component of the magnetization at the DW center  (my) abruptly change s from \n1\n to \n1  for J0 = \n 1.0×1012 A/m2 and θSH = \n 0.1. This current density is consist ent \nwith the threshold for Walker breakdown (\nWB\nJb ); i.e.,  \nWB\nJb  = ±53 m/s corresponding to J0 = \n±1.04 5×1012 A/m2. At this current density, vDW is enhanced by a factor of 5  compared to the \ncase for θSH = 0.  \nFig. 3 (a) and (b) show  \nDWv  and \nDW  as a function of J0 for three values of θSH (= +0.1, \n0.0, 0.1) and β = 0.0 3 (thus \n ). Similar ly to the cases for β = 0.0 1, \nDWv  is closely \ncorrelated to \nDW  and significantly enhance d near \nWB\nJb . In this case, in contrast to the case \nfor \n , reversed DW motion is not observed . It is because the sign of  the \n()  term \nin Eq. (6) is negative  in this case , and thus the overall sign of \nDWv  corresponds to the DW \nmotion along the electron -flow direction.  We find that the current -induced Oersted field has \nonly a negligible  effect  on \nDWv  (not shown).  Thus, the numerical results confirm the validity of the analytical solutions; the DW moves along the current -flow directio n at the limited \nrange of the current (i.e., \nREV WB\nJ J Jb b b ) when  \n . In addition this reversed DW \nmotion appears only for one current polarity . \nFinally, we remark the effect of  SHE on DW dynamics  in the nanowire with a \nperpendicular anisotropy . It was experimentally reported that the DW moves along the \ncurrent -flow direction with a high \nDWv  (≈ 400 m/s ) in the perpendicularly magnetized \nnanowire consisting of Pt/Co/AlO x [28, 29 ]. We n ote that  this D W dynamics cannot be \nexplained by the SHE only. Considering the materials parameter s in Ref. [ 29] as MS = 1090 \nkA/m, K = 1.2×106 J/m3, A = 1.3×10−11 J/m, α = 0.2, P = 0.7, λ = 5 nm, and assuming θSH = \n0.1 and β = 0.1, we find BSHλ = 18.8 that is comparable  to the value for  the Py/Pt  bi-layer \ntested in this work . For Pt/Co/AlO x, however,  \nREV\nJb  and \nWB\nJb are respectively 1.5 and 3 \nm/s (corresponding to J0 = 0.4×1011 and 0.8×1011 A/m2). These thresholds  are much \nsmaller t han th ose of the Py/Pt bi-layer since Hd of DW in a perpendicular system is smaller \nthan in an in-plane system  (i.e., Hd = 848  mT for the system  of Py/Pt  in this w ork, 33 mT  for \nthe system in Ref. [29])22. Note that the maximum DW velocity moving along  the current -\nflow direction (\nREV\nDWv ) is obtained at \nWB\nJJbb . The \nWB\nJb (= 3 m/s)  in Pt/Co/AlO x system is \ntoo small to allow such a high \nREV\nDWv  (≈ 400 m/s) . Indeed, the numerically obtained \nmaximum  \nREV\nDWv  is 8.2 m/s at J0 = 0.71×1011 A/m2 (bJ = 2.64 m/s ) (not shown) , which is \nmuch smaller than the experimentally obtained value, 400 m/s . More importantly, in the \nPt/Co/AlO x system, the reversed DW motion was observ ed at both current polarities31 \nwhereas the SHE allows the reversed motion at only one current polarity. On the other hand, \nwe theoretically demonstrated that the DW dynamics reported in Ref. [ 28, 29] can be \nexplained by STTs caused by Rashba SOC32. We als o remark that one of us reported the effect of SOC on current -driven DW motion recently33. In Ref. [33], h owever, the effect of \nSOC within FM  was investigated , in contrast to the present work where the effect of SOC in \nNM of the FM /NM bi -layer system is investigated . \nTo conclude, we present the analytical model for current -induced  DW motion  in the \npresence of SHE . We demonstrate  that DW dynamics is significantly affected by the SHE . In \nparticular, for the case of \n , the SHE  enables t he reversed DW motion with high speed \nat one current polarity when the system is designed to satisfy the condition of \nWB\nJREV\nJ b b  \nand the current density is selected to be in the range between the two thresholds. Our result \ndemo nstrates that the engineering of SOC and thus the SHE provides an  important \nopportunity for an efficient operation of spintronic devices.  \nThis work was supported by the NRF ( 2010 -0014109, 2010 -0023798, 2011 -0009278, \n2011 -0028163 , 2011 -0030046 ) and the MKE/KEIT (2009 -F-004-01). K.J.L. acknowledges \nsupport under the Cooperative Research Agreement between the University of Maryland and \nthe National Institute of Standards and Technology Center for Nanoscale Science and \nTechnology, Award 70NANB10H193, through the University of Maryland.  REFERENCE  \n \n[1] J. C. Slonczewski, J. Magn. Mag. Mater. 159, L1 (1996) . \n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).  \n[3] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, \n077205 (2004).  \n[4] M.  Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London) 428, 539 (2004).  \n[5] M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. \nHeyderman, F. Nolting, and U. Rüdiger, Phys. Rev. Lett. 94, 106601 (2005).  \n[6] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).  \n[7] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).  \n[8] S. Zhang, Phys. Rev. Lett. 85, 393 (2000 ). \n[9] S. O. Valenzuela and M. 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Auffret,  B. Rodmacq, A. Schul, S. \nPizzini, J. V ogel, and M. Bonfim, Appl. Phys. Lett. 93, 262504 (2008); ibid 95, 179902 \n(2009).  \n[29] I. M. Miron , T. Moore, H. Szambolics, L. D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. \nPizzini, J. V ogel, M. Bonfim, A. Schul, and  G. Gaudin,  Nat. Mater. 10, 189 (2011).  \n[31] I. M. Miron, private communication.  \n[32] K.-W. Kim, S. -M. Seo, J. Ryu, K. -J. Lee, and H. -W. Lee, arXiv:1111.3422v2.  \n[33] A. Manchon and K. -J. Lee, Appl. Phys. Lett. 99, 022504 (2011); ibid 99, 229905 (2011).  FIGURE  CAPTION  \n \nFIG. 1. (Color online)  Schematic s of FM/NM bi-layer nanowire . (top) Structure. (lower left) \nSpatial profile of DW. The color ed contour  shows x component  of the magnetization  for 2 -D \nmicromagnetics . (lower rig ht) Width -averaged magnetization components .  \n \nFIG. 2. (Color online)  Domain wall velocity  for  > . (a) DW velocity (\nDWv ) as a function of  \nthe total current density of bi -layer ( J0) for three  values of θSH (= +0.1, 0 .0, 0.1) and β = 0.01  \n( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6) . (b) DW \ntilt angle (\nDW ) as a function of J0. Filled  symbols above the chiral switching threshold (| J0| = \n1.1×1012 A/m2) are shifted from their original values by –180° (filled green triangles) and \n+180° (filled red circles) . (c) Normalized y component of the magnetization at the DW center \n(my) as a function of J0.  \n \nFIG. 3. (Color online) Domain wall velocity for  < . (a) DW velocity (\nDWv ) as a function of \nthe total current density of bi -layer ( J0) for three values of θSH (= +0.1, 0.0, 0.1) and β = 0.0 3 \n( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6). (b) DW \ntilt angle (\nDW ) as a function of J0. Filled  symbols represent the cases that the chirality of \nDW switches from its  initial tilt angle \n00 . (c) N ormalized y component of the \nmagnetization at the DW center (my) as a  function of J0.  \n  \n \n \n \n \nFIG. 1. Seo et al.  \n \n \n \n \nFIG. 2. Seo et al. \n \n  \n \n \n \n \nFIG. 3. Seo et al. \n \n \n \n "    },    {        "title": "1203.0495v2.Damping_Antidamping_Effect_on_Comets_Motion.pdf",        "content": "arXiv:1203.0495v2  [astro-ph.EP]  11 Oct 2013Damping-Antidamping Effect on Comets\nMotion\nG.V. L´ opez∗and E. M. Ju´ arez\nDepartamento de F´ ısica de la Universidad de Guadalajara,\nBlvd. Marcelino Garc´ ıa Barrag´ an 1421, esq. Calzada Ol´ ım pica,\n44430 Guadalajara, Jalisco, M´ exico\nPACS: 45.20.D−,45.20.3j, 45.50.Pk, 95.10.Ce, 95.10.Eg, 96.30.Cw,\n03.67.Lx, 03.67.Hr, 03.67.-a, 03.65w\nSeptember, 2013\nAbstract\nWe make an observation about Galilean transformation on a 1- D mass\nvariable systems which leads us to the right way to deal with m ass vari-\nable systems. Then using this observation, we study two-bod ies gravitational\nproblem where the mass of one of the bodies varies and suffers a d amping-\nantidamping effect due to star wind during its motion. For this system, a\nconstant of motion, a Lagrangian and a Hamiltonian are given for the radial\nmotion, and the period of the body is studied using the consta nt of motion of\nthe system. Our theoretical results are applied to Halley’s comet.\n∗gulopez@udgserv.cencar.udg.mx\n11 Introduction\nThere is not doubt that mass variable systems have been relevant s ince the founda-\ntion of the classical mechanics and modern physics (L´ opez et al 20 04). These type of\nsystems have been known as Gylden-Meshcherskii problems (Gylde n 1884; Meshch-\nerskii 1893, 1902; Lovett 1902; Jeans 1924; Berkovich 1981; Be kov 1989; Prieto and\nDocobo 1997), and among these type of systems one could mention : the motion\nof rockets (Sommerfeld 1964), the kinetic theory of dusty plasma (Zagorodny et al\n2000), propagation of electromagnetic waves in a dispersive-nonlin ear media (Seri-\nmaa et al 1986), neutrinos mass oscillations (Bethe 1986; Commins a nd Bucksbaum\n1983), black holes formation (Helhl et al 1998), and comets intera cting with solar\nwind (Daly1989). This last system belong to the so called ”gravitation al two-bodies\nproblem” which is one of the most studied and well known system in clas sical me-\nchanics (Goldstein 1950). In this type of system, one assumes nor mally that the\nmasses of the bodies are fixed and unchanged during the dynamical motion. How-\never,when one is dealing with comets, beside to consider its mass varia tion due to\nthe interaction with the solar wind, one would like to have an estimation of the the\neffect of the solar wind pressure on the comet motion. This pressur e may produces\na dissipative-antidissipative effect on its motion. The dissipation effec t must be felt\nby the comet when this one is approaching to the sun (or star), and the antidissi-\npation effect must be felt by the comet when this one is moving away fr om the sun.\nTo deal with these type of mass variation problem, it has been propo sed that the\nNewton equation must be modified (Sommerfeld 1964; Plastino and Mu zzio 1992)\nsince the system becomes non invariant under change of inertial sy stems (Galileo\ntransformation).\nIn this paper, we will make first an observation about this statemen t which in-\ndicates the such a proposed modification of Newton’s equation has s ome problems\nand rather the use of the original Newton equation is the right appr oach to deal\nwith mass variation systems, which it was used in previous paper (L´ o pez 2007) to\nstudy two-bodies gravitational problem with mass variation in one of them, where\nwe were interested in the difference of the trajectories in the spac es (x,v) and (x,p).\nAs a consequence, there is an indication that mass variation problem s must be dealt\nas non invariant under Galilean transformation. Second, we study t he two-bodies\ngravitational problem taking into consideration the mass variation o f one of them\nand its damping-antidamping effect due to the solar wind. The mass of the other\nbody is assumed big and fixed , and the reference system of motion is chosen just\nin this body. In addition, we will assume that the mass lost is expelled fr om the\n2bodyradially to its motion. Doing this, the three-dimensional two-bo dies problem is\nreduced to a one-dimensional problem. Then, a constant of motion , the Lagrangian,\nand the Hamiltonian are deduced for this one-dimensional problem, w here a radial\ndissipative-antidissipative force proportional to the velocity squa re is chosen. A\nmodel for the mass variation is given, and the damping-antidamping e ffect is stud-\nied on the period of the trajectories, the trajectories themselve s, and the aphelion\ndistance of a comet. We use the parameters associated to comet H alley to illustrate\nthe application of our results.\n2 Mass variation problem and Galileo transfor-\nmation.\nTo simplify our discussion and without losing generality, we will restrict myself to\none degree of freedom,. Newton equation of motion is given by\nd\ndt/parenleftbig\nm(t)v/parenrightbig\n=F(x,v,t), (1)\nwherem(t)vis the quantity of movement, Fis the total external force acting on the\nobject,m(t) andv=dx/dtare its time depending mass and velocity of the body\n(motion of the mass lost is not considered). Galileo transformations to another\ninertial frame ( S′) which is moving with a constant velocity urespect our original\nframeSare defined as\nx′=x−ut (2a)\nt′=t (2b)\nwhichimplies thefollowing relationbetween thevelocity seenintherefe rencesystem\nS,v, and the velocity seen in the reference system S′,v′,\nv′=v−u. (3)\nMultiplying the last term by m(t′) and making the differentiation with respect to\nt′, one gets\nd\ndt′/parenleftbig\nm(t′)v′/parenrightbig\n=F′(x′,v′,t′), (4)\n3whereF′is given by\nF′(x′,v′,t′) =F(x′+ut′,v′+u,t′)−udm(t′)\ndt′. (5)\nTherefore, Eq. 1 and Eq. 4 have the same form but the force is diffe rent since\nin addition to the transformed force term F(x′+ut′,v′+u,t′), one has the term\nudm(t′)/dt′. This non invariant form of the force under Galilean transformation has\nleadto propose(Sommerfeld 1964; Plastino andMuzzio 1992)that N ewtonequation\n(1) to modify Newton’s equation of motion for mass variation object s, to keep the\nprinciple of invariance of equation under Galilean transformations, o f the form\nm(t)dv\ndt=F(x,v,t)+wdm(t)\ndt, (6)\nwherewis the relative velocity of the escaping mass with respect the center of mass\nof the object. When one does a Galilean transformation on this equa tion, one gets\nm(t′)dv′\ndt′=F′(x′,v′,t′), (7)\nwhereF′is given by\nF′(x′,v′,t′) =F(x′+ut′,v′+u,t′)+wdm(t′)\ndt′, (8)\nwhich has the same form as Eq. 6. However, assume for the moment thatw=\nconstant andF= 0. So, from Eq. 6, it follows that\nv(t) =v0+ln/parenleftbiggm(t)\nm0/parenrightbiggw\n, (9)\nwherem0=m(0). In this way, if we have a mass variation of the for m(t) =m0e−αt\n(for example), one would have a velocity behavior like\nv(t) =v0−wαt, (10)\nwhich is not acceptable since one can have v >0,v= 0 and v <0 depending on\nthe value wαt. Even more, since for F= 0, the equation resulting in the reference\nsystemS′is the same, i.e. in S′one gets the same type of solution,\nv′(t′) =v0+ln/parenleftbiggm(t′)\nm0/parenrightbigg\n(11)\n4which is independent on the relative motion of the reference frames , and this must\nnot be possible due to relation (3).\nIn addition, it worths to mention that special theory of relativity ca n be seen as\nthe motion of mass variation problem, where the mass depends on th e velocity of\nthe particle of the form m(v) =m0(1−v2/c2)−1/2, withcbeing the speed of light.\nThis system is obviously not invariant under Galilean transformation, and given\nthe force, Newton’s equation motion is always kept in the same form t o solve a\nrelativistic problem, d/parenleftbig\nm(v)v/parenrightbig\n/dt=F(x,v,t), (C. Møller 1952, L´ opez et al 2004).\n3 Mass variation and equations of motion.\nHaving explained and clarify the problem of mass variation (Spivak 201 0), Newton’s\nequations of motion for two bodies interacting gravitationally, seen from arbitrary\ninertial reference system, and with radial dissipative-antidissipat ive force acting in\none of them are given by\nd\ndt/parenleftbigg\nm1dr1\ndt/parenrightbigg\n=−Gm1m2\n|r1−r2|3(r1−r2) (12a)\nand\nd\ndt/parenleftbigg\nm2dr2\ndt/parenrightbigg\n=−Gm1m2\n|r2−r1|3(r2−r1)−γ\n|r1−r2|/bracketleftbiggd|r1−r2|\ndt/bracketrightbigg2\n(r2−r1),(12b)\nwherem1andm2are the masses of the two bodies, r1= (x1,y1,z1) and\nr2= (x2,y2,z2) are their vectors positions from the reference system, Gis the\ngravitational constant ( G= 6.67×10−11m3/Kg s2),γis the nonnegative constant\nparameter of the dissipative-antidissipative force, and\n|r1−r2|=|r2−r1|=/radicalbig\n(x2−x1)2+(y2−y1)2+(z2−z1)2\nis the Euclidean distance between the two bodies. Note that if γ >0 and\nd|r1−r2|/dt>0one has dissipation since the force acts against the motion of the\nbody, and for d|r1−r2|/dt<0one has anti-dissipation since the force pushes the\nbody. Ifγ <0 this scheme is reversed and corresponds to our actual situation with\nthe comet mass lost.\nIt will be assumed the mass m1of the first body is constant and that the mass\n5m2of the second body varies. Now, It is clear that the usual relative, r, and center\nof mass,R, coordinates defined as r=r2−r1andR= (m1r1+m2r2)/(m1+m2) are\nnot so good to describe the dynamics of this system. However, one can consider the\ncase form1≫m2(which is the case star-comet), and consider to put our referenc e\nsystem just on the first body ( r1=˜0). In this case, Eq. (12a) and Eq. (12b) are\nreduced to the equation\nm2d2r\ndt2=−Gm1m2\nr3r−˙m2˙r−γ/bracketleftbiggdr\ndt/bracketrightbigg2\nˆr, (13)\nwhere one has made the definition r=r2= (x,y,z),ris its magnitude, r=/radicalbig\nx2+y2+z2, andˆr=r/ris the unitary radial vector. Using spherical coordinates\n(r,θ,ϕ),\nx=rsinθcosϕ , y=rsinθsinϕ , z=rcosθ , (14)\none obtains the following coupled equations\nm2(¨r−r˙θ2−r˙ϕ2sin2θ) =−Gm1m2\nr2−˙m2˙r−γ˙r2, (15a)\nm2(2˙r˙θ+r¨θ−r˙ϕ2sinθcosθ) =−˙m2r˙θ , (15b)\nand\nm2(2˙r˙ϕsinθ+r¨ϕsinθ+2r˙ϕ˙θcosθ) =−˙m2r˙ϕsinθ . (15c)\nTaking ˙ϕ= 0 as solution of this last equation, the resulting equations are\nm2(¨r−r˙θ2) =−Gm1m2\nr2−˙m2˙r−γ˙r2, (16a)\nand\nm2(2˙r˙θ+r¨θ)+ ˙m2r˙θ= 0. (16b)\nFrom this last expression, one gets the following constant of motion (usual angular\nmomentum of the system)\nlθ=m2r2˙θ , (17)\nand with this constant of motion substituted in Eq. (16a), one obta ins the following\none-dimensional equation of motion for the radial part\nd2r\ndt2=−Gm1\nr2−˙m2\nm2/parenleftbiggdr\ndt/parenrightbigg\n−γ\nm2˙r2+l2\nθ\nm2\n2r3. (18)\n6Now, let us assume that m2is a function of the distance between the first and the\nsecond body, m2=m2(r). Therefore, it follows that\n˙m2=m′\n2˙r , (19)\nwherem′\n2is defined as m′\n2=dm2/dr. Thus, Eq. (18) is written as\nd2r\ndt2=−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2/parenleftbiggdr\ndt/parenrightbigg2\n, (20)\nwhich, in turns, can be written as the following autonomous dynamica l system\ndr\ndt=v;dv\ndt=−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2v2. (21)\nNote from this equation that m′\n2is always a non-positive function of rsince it\nrepresents the mass lost rate. On the other hand, γis a negative parameter in our\ncase.\n4 Constant of Motion, Lagrangian and Hamilto-\nnian\nA constant of motion for the dynamical system (21) is a function K=K(r,v) which\nsatisfies the partial differential equation (L´ opez 1999)\nv∂K\n∂r+/bracketleftbigg−Gm1\nr2+l2\nθ\nm2\n2r3−m′\n2+γ\nm2v2/bracketrightbigg∂K\n∂v= 0. (22)\nThe general solution of this equation is given by (John 1974)\nK(x,v) =F(c(r,v)), (23)\nwhereFis an arbitrary function of the characteristic curve c(r,v) which has the\nfollowing expression\nc(r,v) =m2\n2(r)e2γλ(r)v2+/integraldisplay/parenleftbigg2Gm1\nr2−2l2\nθ\nm2\n2r3/parenrightbigg\nm2\n2(r)e2γλ(r)dr, (24)\nand the function λ(r) has been defined as\nλ(r) =/integraldisplaydr\nm2(r). (25)\n7During a cycle of oscillation, the function m2(r) can be different for the comet\napproaching the sun and for the comet moving away from the sun. L et us denote\nm2+(r) for the first case and m2−(r) for the second case. Therefore, one has two\ncases to consider in Eq. (23) which will denoted by ( ±). Now, if mo\n2±denotes the\nmass at aphelium (+) or perielium (-) of the comet, F(c) =c±/2mo\n2±represents the\nfunctionality in Eq. (23) such that for m2constant and γequal zero, this constant\nof motion is the usual gravitational energy. Thus, the constant o f motion can be\nchosen as K±=c(r,v)/2m0\n2±, that is,\nK±=m2\n2±(r)\n2mo\n2±e2γλ±(r)v2+V±\neff(r), (26a)\nwhere the effective potential Veffhas been defined as\nV±\neff(r) =Gm1\nmo\n2±/integraldisplaym2\n2±(r)e2γλ±(r)dr\nr2−l2\nθ\nmo\n2±/integraldisplaye2γλ±(r)dr\nr3(26b)\nThis effective potential has an extreme at the point r∗defined by the relation\nr∗m2\n2(r∗) =l2\nθ\nGm1(27)\nwhich is independent on the parameter γand depends on the behavior of m2(r).\nThis extreme point is a minimum of the effective potential since one has\n/parenleftBigg\nd2V±\neff\ndr2/parenrightBigg\nr=r∗>0. (28)\nUsing the known expression (Kobussen 1979; Leubner 1981; L´ op ez 1996) for the\nLagrangian in terms of the constant of motion,\nL(r,v) =v/integraldisplayK(r,v)dv\nv2, (29)\nthe Lagrangian, generalized linear momentum and the Hamiltonian are given by\nL±=m2\n2±(r)\n2mo\n2±e2γλ±(r)v2−V±\neff(r), (30)\np=m2\n2±(r)v\nmo\n2±e2γλ±(r), (31)\n8and\nH±=mo\n2±p2\n2m2\n2±(r)e−2γλ±(r)+V±\neff(r). (32)\nThe trajectories in the space ( x,v) are determined by the constant of motion (26a).\nGiven the initial condition ( ro,vo), the constant of motion has the specific value\nK±\no=m2\n2±(ro)\n2mo\n2±e2γλ±(ro)v2\no+V±\neff(ro), (33)\nand the trajectory in the space ( r,v) is given by\nv=±/radicalBigg\n2mo\n2±\nm2\n2±(r)e−γλ±(r)/bracketleftbigg\nK±\no−V±\neff(r)/bracketrightbigg1/2\n. (34)\nNote that one needs to specify ˙θoalso to determine Eq. (17). In addition, one\nnormallywants toknowthetrajectoryintherealspace, thatis, t heacknowledgment\nofr=r(θ). Since one has that v=dr/dt= (dr/dθ)˙θand Eqs. (17) and (34), it\nfollows that\nθ(r) =θo+l2\nθ/radicalbig2mo\n2±/integraldisplayr\nrom2±(r)eγλ±(r)dr\nr2/radicalBig\nK±o−V±\neff(r). (35)\nThe half-time period (going from aphelion to perihelion (+), or backwa rd (-)) can\nbe deduced from Eq. (34) as\nT±\n1/2=1/radicalbig2mo\n2±/integraldisplayr2\nr1m2±(r)eγλ±(r)dr/radicalBig\nK±o−V±\neff(r), (36)\nwherer1andr2are the two return points resulting from the solution of the following\nequation\nV±\neff(ri) =K±\noi= 1,2. (37)\nOntheother hand, thetrajectory inthespace ( r,p) isdetermine by theHamiltonian\n(32), and given the same initial conditions, the initial poandH±\noare obtained from\nEqs. (32) and (31). Thus, this trajectory is given by\np=±/radicalBigg\n2m2\n2±(r)\nmo\n2±eγλ±(r)/bracketleftbigg\nH±\no−V±\neff(r)/bracketrightbigg1/2\n. (38)\n9It is clear just by looking the expressions (34) and (38) that the tr ajectories in the\nspaces (r,v) and (r,p) must be different due to complicated relation (31) between v\nandp(L´ opez 2007).\n5 Mass-Variable Model and Results\nAs a possible application, consider that a comet looses material as a r esult of the\ninteraction with star wind in the following way (for one cycle of oscillatio n)\nm2±(r) =\n\nm2−(r2(i−1))/parenleftbigg\n1−e−αr/parenrightbigg\nincoming (+)v <0\nm2+(r2i−1)−b/parenleftbigg\n1−e−α(r−r2i−1)/parenrightbigg\noutgoing (−)v >0(39)\nwhere the parameters b >0 andα >0 can be chosen to math the mass loss rate in\nthe incoming and outgoing cases. The index ”i” represent the ith-se mi-cycle, being\nr2(i−1)andr2i−1the aphelion( ra) and perihelion( rp) points ( rois given by the initial\nconditions, and one has that m2−(ro) =mo). For this case, the functions λ+(r) and\nλ−(r) are given by\nλ+(r) =1\nαmaln/parenleftbigg\neαr−1/parenrightbigg\n, (40a)\nand\nλ−(r) =−1\nα(b−mp)/bracketleftBigg\nαr+ln/parenleftbig\nmp−b(1−e−α(r−rp))/parenrightbig/bracketrightBigg\n. (40b)\nwhere we have defined ma=m2(ra) andmp=m2(rp). Using the Taylor expansion,\none gets\ne2γλ+(r)=e2γr/ma/bracketleftbigg\n1−2γ\nαmae−αr+1\n22γ\nαma/parenleftbigg2γ\nαma−1/parenrightbigg\ne−2αr+.../bracketrightbigg\n,(41a)\nand\ne2γλ−(r)=e−2γr\n(b−mp)\n(mp−b)2γ\nα(mp−b)/bracketleftbigg\n1+2γ\nα(mp−b)e−α(r−rp)\nmp−b\n+1\n22γ\nα(mp−b)/parenleftbigg2γ\nα(mp−b)−1/parenrightbigge−2α(r−rp)\n(mp−b)2+.../bracketrightbigg\n.(41b)\n10The effective potential for the incoming comet can be written as\nV+\neff(r) =/bracketleftbigg\n−Gm1ma\nr+l2\nθ\n2ma1\nr2/bracketrightbigg\ne2γr/ma+W1(γ,α,r), (42a)\nand for the outgoing comet as\nV−\neff(r) =/bracketleftbigg\n−Gm1ma\nr+l2\nθ\n2ma1\nr2/bracketrightbigge2γr\n(mp−b)\n(mp−b)2γ\nα(mp−b)+W2(γ,α,r),(42b)\nwhereW1andW2are given in the appendix A.\nWe will use the data corresponding to the sun mass (1 .9891×1030Kg) and the\nHalley comet ( Cevolani et al 1987, Brandy 1982, Jewitt 2002)\nmc≈2.3×1014Kg, r p≈0.6au, r a≈35au, l θ≈10.83×1029Kg·m2/s,(34)\nwith a mass lost of about δm≈2.8×1011Kgper cycle of oscillation. Although,\nthe behavior of Halley comet seem to be chaotic (Chirikov and Veches lavov 1989),\nbut we will neglect this fine detail here. Now, the parameters αand ”b” appearing\non the mass lost model, Eq. (39), are determined by the chosen mas s lost of the\ncomet during the approaching to the sun and during the moving away from the sun\n(we have assumed the same mass lost in each half of the cycle of oscilla tion of the\ncomet around the sun). Using Eq. (42a) and Eq. (42b) in the expre ssion (34), the\ntrajectories can be calculated in the spaces ( r,v) . Fig. 1 shows these trajectories\nusingδm= 2×1010Kg(orδm/m= 0.0087%) for γ= 0 and (continuos line), and\nforγ=−3Kg/m(dashed line), starting both cases from the same aphelion dis-\ntance. As one can see on the minimum, dissipation causes to reduce a little bit the\nvelocity of the comet , and the antidissipation increases the comet v elocity, reaching\na further away aphelion point. Also, when only mass lost is considered (γ= 0) the\ncomet returns to aphelion point a little further away from the initial o ne during\nthe cycle of oscillation. Something related with this effect is the chang e of period\nas a function of mass lost ( γ= 0). This can be see on Fig. 2, where the period is\ncalculated starting always from the same aphelion point ( ra). Note that with a mass\nlost of the order 2 .8×1011Kg(Halley comet), which correspond to δm/m=.12%,\nthe comet is well within 75 years period. The variation of the ratio of t he change\nof aphelion distance as a function of mass lost ( γ= 0) is shown on Fig.3. On Fig.\n4, the mass lost rate is kept fixed to δm/m= 0.0087%, and the variation of the\nperiod of the comet is calculated as a function of the dissipative-ant idissipative pa-\nrameterγ <0 (using |γ|for convenience). As one can see, antidissipation always\n11wins to dissipation, bringing about the increasing of the period as a fu nction of\nthis parameter. The reason seems to be that the antidissipation ac ts on the comet\nwhen this ones is lighter than when dissipation was acting (dissipation a cts when\nthe comet approaches to the sun, meanwhile antidissipation acts wh en the comet\ngoes away from the sun). Since the period of Halley comets has not c hanged much\nduring many turns, we can assume that the parameter γmust vary in the interval\n(−0.01,0]Kg/m. Finally, Fig. 5 shows the variation, during a cycle of oscillation, of\nthe ratio of the new aphelion ( r′\na) to old aphelion ( ra) as a function of the parameter\nγ.\n6 Conclusions and comments\nWe have shown that the proposed modified Newton equation for mas s variation\nsystems has some problems. Therefore, we have considered that it is better to keep\nNewton’s equations of motion for mass variable systems to have a co nsistent ap-\nproach to these problems. Having this in mind, the Lagrangian, Hamilt onian and\na constant of motion of the gravitational attraction of two bodies were given when\none of the bodies has variable mass and the dissipative-antidissipativ e effect of the\nsolar wind is considered. By choosing the reference system in the ma ssive body,\nthe system of equations is reduce to 1-D problem. Then, the const ant of motion,\nLagrangian and Hamiltonian were obtained consistently. A model for comet-mass-\nvariation was given, and with this model, a study was made of the varia tion of\nthe period of one cycle of oscillation of the comet when there are mas s variation\nand dissipation-antidissipation. When mass variation is only considere d, the comet\ntrajectory is moving away from the sun, the mass lost is reduced as the comet is\nfartheraway (according toour model), andtheperiodofoscillation s becomes bigger.\nWhen dissipation-antidissipation is added, this former effect become s higher as the\nparameter γbecomes higher.\n127 Appendix A\nExpression for W1andW2:\nW1=Gm2\n2−\nmo\n2+/braceleftBigg\n−p(p−1)e(−4+p)αr\n2r+αpEi(αpr)−2αp(p−1)Ei/parenleftbig\n(−4+p)αr/parenrightbig\n+αp2(p−1)\n2Ei/parenleftbig\n(−4+p)αr/parenrightbig\n+p(p−1)\nr/bracketleftbig\ne(p−3)αr+3α(1−p)rEi/parenleftbig\n(p−3)αr/parenrightbig/bracketrightbig\n+p(p+3)\n2/bracketleftbigg\n−e(p−2)αr\nr+α(p−2)Ei/parenleftbig\n(p−2)αr/parenrightbig/bracketrightbigg\n+p+2\nr/bracketleftbig\ne(p−2)αr+α(p−1)rEi/parenleftbig\n(p−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n+l2\nθ\n2m2\n2+r2/braceleftBigg\np(p−1)\n2e(p−2)αr−pe(p−1)αr−αp(p−1)e(p−2)αr+αp(p−1)\n2epαr\n+α2p(p−1)r\n2e(p−2)αr+pαre(p−1)αr−p2αre(p−1)αr−p2α2r2Ei/parenleftbig\npαr/parenrightbig\n−α2(p−2)2p(p−1)r2\n2Ei/parenleftbig\n(p−2)αr/parenrightbig\n+pα2r2Ei/parenleftbig\n(p−1)αr/parenrightbig\n−2α2p2r2Ei/parenleftbig\n(p−1)αr/parenrightbig\n+p3α2r2Ei/parenleftbig\n(p−1)αr/parenrightbig/bracerightBigg\n(A1)\nwheremais the mass of the body at the aphelion, and we have made the definitio ns\np=2γ\nαma(A2)\nand the function Eiis the exponential integral,\nEi(z) =/integraldisplay∞\n−ze−t\ntdt (A3)\n13W2=Gm2\n2−\nmo\n2+/braceleftBigg\ne(q−2)αr\nr/bracketleftbigg\n1+q(q−1)\n2(mp+αq)e2qαr+2q\nmp+αqeqαr/bracketrightbigg\n+qαEi/parenleftbig\nqαr/parenrightbig\n−q(q−1)e2qαr\n(mp+αq)2r/bracketleftbig\ne(q−3)αr−α(q−3)rEi/parenleftbig\n(q−3)αr/parenrightbig/bracketrightbig\n+qeqαr\n(mp+αq)r/bracketleftbig\ne(q−3)αr−α(q−3)rEi/parenleftbig\n(q−3)αr/parenrightbig/bracketrightbig\n−2αEi/parenleftbig\n(q−2)αr/parenrightbig\n+αqEi/parenleftbig\n(q−2)αr/parenrightbig\n−q(q−1)αe2qαr\n(mp+αq)2Ei/parenleftbig\n(q−2)αr/parenrightbig\n+q2(q−1)αe2qαr\n2(mp+αq)2Ei/parenleftbig\n(q−2)αr/parenrightbig\n−4αeqαr\nmp+αqEi/parenleftbig\n(q−2)αr/parenrightbig\n+2q2αeqαr\n(mp+αq)rEi/parenleftbig\n(q−2)αr/parenrightbig\n+2\nr/bracketleftbig\ne(q−1)αr−(q−1)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig\n+qeqαr\n(mp+αq)r/bracketleftbig\ne(q−1)αr−(q−1)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n+l2\nθ\n2m2\n2+(mp+αq)q/braceleftBigg\n−qαeqαr\nr+q2α2Ei/parenleftbig\nqαr/parenrightbig\n+q(q−1)e(3q−2)αr\n2(mp+αq)2r2/bracketleftbig\n−1+2αr−qαr+(2−q)2α2r2e(2−q)αrEi/parenleftbig\n(q−2)αr/parenrightbig/bracketrightbig\n−qe(2q−1)αr\n(mp+αq)r2/bracketleftbig\n−1+αr+qαr+(q−1)2α2r2e(1−q)αrEi/parenleftbig\n(q−1)αr/parenrightbig/bracketrightbig/bracerightBigg\n(A4)\nwherempis the mass of the body at the perihelion, and we have made the definit ion\nq=2γ\nα(mp−b)(A5)\n14References\nBekov A.A., 1989, Astron. Zh., 66, 135\nBerkovich L.M., 1981, Celestial Mechanics, 24 ,407\nBethe H.A., 1986, Phys. Rev. Lett., 56, 1305\nBrandy J.L., 1982, J. Brit. Astron. Assoc., 92, no. 5, 209\nCevolani G., Bortolotti G. and Hajduk A., 1987, IL Nuo. Cim. C, 10, n o.5,\n587\nChirikov B.V. and Vecheslavov V.V., 1989, Astron. Astrophys., 221, 146\nCommins E.D. and Bucksbaum P.H., 1983, Weak Interactions of Lepto ns and\nQuarks, Cambridge University Press\nDaly P.W., 1989, Astron. Astrophys., 226, 318\nGoldstein H., 1950, Classical Mechanics, Addison-Wesley, M.A.\nGylden, H., 1884, Astron. Nachr., 109, no. 2593,1\nHelhl F.W., Kiefer C. and Metzler R.J.K., 1998, Black Holes: Theory and\nObservation, Springer-Verlag\nJeans J.H., 1924, MNRAS, 85, no. 1, 2\nJewitt D.C., 2002, Astron. Jour., 123, 1039\nJohn F., 1974, Partial Differential Equations, Springer-Verlag, Ne w York\nKobussen J.A., 1979, Acta Phys. Austr. 51,193\nLeubner C., 1981, Phys. Lett. A, 86, 2\nL´ opez G., Barrera L.A., Garibo Y., Hern´ andez H., Salazar J.C., 2004, Int.\nJour. Theo. Phys., 43, no. 10, 1\nL´ opez G., 1999, Partial Differential Equations of First Order and T heir\nApplications to Physics, World Scientific\nL´ opez G., 1996, Ann. of Phys., 251, no. 2 , 372\nL´ opez G, 2007, Int. Jour. Theo. Phys., 46, no. 4, 806\nLovett E.O., 1902, Astron. Nachr., 158, no. 3790, 337\nMeshcherskii I.V., 1893, Astron. Nachr., 132, no. 3153, 93\nMeshcherskii I.V., 1902, Astron. Nachr., 159, no. 3807, 229\nMøller C., 1952 Theory of Relativity , Oxford University Press.\nPlastino A.R. and Muzzio J.C., 1992, Cel. Mech. Dyn. Ast. , 53, 227\nPrieto C. and Docobo J.A., 1997, Astron. Astrophys., 318, 657\nSerimaa O.T., Javanainen J., and Varr´ o S., 1986, Phys. Rev. A, 33, 2 913\nSpivak M.,2010, Physics for Mathematicians, Mechanics I Publish or Pe rish\nInc., chapter 3\nSommerfeld A., 1964, Lectures on Theoretical Physics, Vol. I, Aca demic Press\n15Zagorodny A.G., Schram P.P.J.M., and Trigger S.A., 2000, Phys. Rev. Le tt.,\n84, 3594\n161/Multiply10122/Multiply10123/Multiply10124/Multiply10125/Multiply10126/Multiply1012r/LParen1m/RParen1\n/Minus20000/Minus100001000020000v/LParen3m\ns/RParen3\nFigure 1: Trajectories in the ( r,v) space with δm/m= 0.009.0 0.1 0.2 0.3 0.4 0.5 0.6\nδm/m (%)100150200250300350400450500T (years)\nFigure 2: Period of the comet as a function of the mass lost ratio.\n170 0.1 0.2 0.3 0.4 0.5 0.6\nδm/m (%)012345δ ra/ra\nFigure 3: Ratio of aphelion distance change as a function of the mass lost rate.0 1 2 3 4 5 6 7\n| γ |708090100110120130140T (years)δm=2x1010\nKg(δm/m=0.0087%)\nFigure 4: Period of the comet as a function of the parameter γ.\n180 1 2 3 4 5 6 7\n| γ |11.21.41.61.82ra'/raδm/m=0.0087%\nFigure 5: Ratio of the aphelion increasing as a function of the parame terγ.\n19"    },    {        "title": "1203.0607v1.Scaling_of_intrinsic_Gilbert_damping_with_spin_orbital_coupling_strength.pdf",        "content": "arXiv:1203.0607v1  [cond-mat.mtrl-sci]  3 Mar 2012Scaling of intrinsic Gilbert damping with spin-orbital cou pling strength\nP. He1,4, X. Ma2, J. W. Zhang4, H. B. Zhao2,3, G. L¨ upke2, Z. Shi4, and S. M. 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. The experimental results\ndeepen the understanding the mechanism of α0in\nmagnetic metallic materials and provide a new clue to\nexplore ideal ferromagnets with reasonably low α0and\nhigh PMA as storage media for the next generation\nmicrowave-assisted magnetic recording.\nAcknowledgements This work was supported by the\nMSTC under grant No. 2009CB929201, (US) DOE\ngrant No. DE-FG02-04ER46127, NSFC under Grant\nNos. 60908005, 51171129 and 10974032, and Shanghai\nPuJiang Program (10PJ004).\n1Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar,\nand Th. Rasing, Nature (London) 418, 509 (2002).\n2H.W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas,\nand J. Miltat, Phys. Rev. Lett. 90, 017204 (2003).\n3S. I. Kiselev, J. C. Sankey, I.N. Krivorotov, N.C. Emley, R.\nJ. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n(London) 425, 380 (2003).\n4S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S.\nE. Russek, and J. A. Katine, Nature (London) 437, 389\n(2005)\n5E. 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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": "1203.2094v1.Collective_Light_Emission_of_a_Finite_Size_Atomic_Chain.pdf",        "content": "arXiv:1203.2094v1  [quant-ph]  9 Mar 2012Collective Light Emission of a Finite Size Atomic Chain\nHashem Zoubi\nInstitut f ¨ur Theoretische Physik, Universit ¨at Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austr ia\n(Dated: 09 March, 2012)\nRadiative properties of collective electronic states in a o ne dimensional atomic chain are investi-\ngated. Radiative corrections are included with emphasize p ut on the effect of the chain size through\nthe dependence on both the number of atoms and the lattice con stant. The damping rates of collec-\ntive states are calculated in considering radiative effects for different values of the lattice constant\nrelative to the atomic transition wave length. Especially t he symmetric state damping rate as a\nfunction of the number of the atoms is derived. The emission p attern off a finite linear chain is also\npresented. The results can be adopted for anychain of active material, e.g., a chain of semiconductor\nquantum dots or organic molecules on a linear matrix.\nPACS numbers: 37.10.Jk, 42.50.-p, 71.35.-y\nI. INTRODUCTION\nOptical lattice ultracold atoms continue to be of in-\nterest for more and more researches of different branches\nof physics [1]. Big attention is given for the realization\nof different condensed matter models that provide a test\nsystem for achieving a deep understanding of fundamen-\ntal physics and answering open questions in the subject\n[2], beside their applications for quantum information\nprocessing [3]. In general, the main objective is to con-\nsider optical lattice ultracold atoms as artificial crystals\nwith a wide range of controllable parameters.\nOptical lattices form of counter propagating laser\nbeams to get standing wavesin which ground state ultra-\ncold atoms are loaded [4, 5]. The atoms experience opti-\ncal lattice potential with lattice constant of half wave\nlength of the laser. Low dimensional lattices can be\nachieved with different geometric structures and symme-\ntries [6]. In conventional solid crystals the lattice con-\nstant and the symmetry of the lattice is fixed through\nthe different chemical bonds that responsible for the for-\nmation of the crystal. The advantage of optical lattices\nis due to the controllability of the lattice constant and\nsymmetry through controlling the external laser field [1].\nCollective states of electronicexcitations play a central\nrule in solidcrystalsand molecularclusters and they usu-\nally termed excitons [7, 8]. They induced by electrostatic\ninteractions among the lattice atoms or molecules, where\nan electronic excitation can be delocalized in the crystal\nthrough energy transfer. Collective states can dominate\nthe electrical and optical properties of the material, and\nespeciallytheystronglyaffecttheexcitationlifetimes and\ngive rise to dark and superradiant states. In such mate-\nrial the lattice constant is few angstroms which is much\nsmaller than the electronic transition wavelength, and\nhence one can use electrostatic interactions, e.g. reso-\nnance dipole-dipole interactions, and to neglect radiative\ncorrections altogether.\nElectronicexcitationsin opticallatticeultracoldatoms\nare of big importance, e.g., for optical lattice clocks [9],\nand for optical lattice Rydberg atoms [10]. In our previ-\nous work we introduced excitons for optical lattice ultra-cold atoms in one and two dimensional set-ups [11, 12].\nWe concentratedmainly in the Mott insulatorphasewith\none and two atomsper lattice site. We treated both large\nand finite atomic chains [13–15], and we calculated the\ndamping rate of excitons into free space and their emis-\nsion pattern [16–18]. In all of our previous researches we\nexploited electrostatic interactions for the formation of\ncollective states, mainly resonance dipole-dipole interac-\ntions. But for typical optical lattices the lattice constant\nis few thousands of angstroms, which can be of the order\nof the electronic transition wavelength, and hence radia-\ntive corrections can be significant.\nIn the present paper we investigate a one dimensional\nfinite chain of atoms where the lattice constant can take\nany value relative to the atomic transition wavelength.\nFinite atomic chains have been realized recently in a\nnumber of optical lattice experiments [19, 20]. We em-\nphasize the influence of radiative corrections on the for-\nmation of collective sates and their damping rates, where\nweexploit generalcollectivestateswith emphasizeon the\nmost symmetric one. We derive the condition for the\nvalidity of applying electrostatic interactions, which we\nusedinourpreviouswork. Few studiestreatedthecollec-\ntive effect on the optical properties of finite atomic chain\nof several atoms [21], but extensive study done for two\natoms in the radiative regime [22], and in which we com-\npare our results. We extract how the damping rate de-\npends on the chain size, namely on the number of atoms\nin the lattice. Furthermore, we calculate the emission\npattern off a finite atomic chain.\nThe paper is organized as follows: in section 2 we\npresent a finite one dimensional atomic chain and dis-\ncuss the energy transfer parameter due to dipole-dipole\ninteractionsin the radiative regime. Then in section 3 we\ncalculate the damping rates for different collective states\nand several chain sizes. The emission pattern for col-\nlective states is calculated in section 4. The summary\nappears in section 5.2\nII. FINITE ONE-DIMENSIONAL ATOMIC\nCHAIN\nWe consider a finite one dimensional atomic lattice,\nwhere the number of atoms is Nwith lattice constant\na, as seen in figure (1). The atoms are considered to\nbe two-level systems with electronic transition energy\nEA=/planckover2pi1ωA. An electronic excitation can delocalize in\nthe lattice by transferring among the atoms. The elec-\ntronic excitation Hamiltonian is given by\nHex=/summationdisplay\nn/planckover2pi1ωAB†\nnBn+/summationdisplay\nnm/planckover2pi1JnmB†\nnBm,(1)\nwhereB†\nnandBnare the creation and annihilation oper-\nators of an electronic excitation at atom n. For a single\nexcitation the operators can be assumed to obey boson\ncommutation relations.\n...a µ\n1 2 N                               \n                           ϕ\nFIG. 1: A finite lattice of Natoms. The lattice constant is a,\nand the transition dipole µmakes an angle ϕwith the lattice\ndirection.\nThe energy transfer among two atoms, nandm, is a\nfunction of the interatomic distance and given by [23]\nJ(qARnm) =3\n4ΓA/braceleftBigg/bracketleftBigg\nsin(qARnm)\n(qARnm)2+cos(qARnm)\n(qARnm)3/bracketrightBigg\n×/parenleftbig\n1−3cos2ϕ/parenrightbig\n−cos(qARnm)\nqARnm/parenleftbig\n1−cos2ϕ/parenrightbig/bracerightbigg\n, (2)\nwhere the distance between the two atoms is Rnm=\n|n−m|a, andµis the magnitude of the electronic excita-\ntion transition dipole, which makes an angle ϕwith the\nlattice direction, see figure (1). qAis the atomic transi-\ntion wave number given by EA=/planckover2pi1cqA. Here Γ Ais the\nsingle excited atom damping rate\nΓA=ω3\nAµ2\n3πǫ0/planckover2pi1c3. (3)\nIn the limit of λA> a, whereλAis the atomic transition\nwave length defined by EA=hc/λA, we can consider\nonly energy transfer among nearest neighbor atoms with\nJ(qAa) where we take Rnm=a.\nIn figure (2) we plot J(qAa)/ΓAas a function of\nqAafor two different polarization directions. Note that\nfor typical optical lattice we have EA= 1eV, with\nλA≈12405˚A, andqA≈4×10−4˚A−1. Fora= 1000˚A\nwe getqAa≈0.5, anda/λA≈0.08. For ϕ= 0◦we\nobtainJ(0.5)/ΓA≈ −13.4, and for ϕ= 90◦we get13579−2−1012J(qAa)/ΓA\nqAa\nFIG. 2: The scaled interaction J(qAa)/ΓAvs.qAa. The full\nline is for ϕ= 0◦, and the dashed line for ϕ= 90◦.\nJ(0.5)/ΓA≈5.4. For large qAathe coupling tend to zero\nwith oscillations, and the atoms are almost independent.\nInthe limit λA≫a, orqAa≪1, wecanneglectthera-\ndiativeterms(aswedidinourpreviousworks[11–18]), to\nget the electrostatic resonance dipole-dipole interaction\nJ≈3\n4ΓA\n(qAa)3/parenleftbig\n1−3cos2ϕ/parenrightbig\n. (4)\nUsing the previous numbers, ϕ= 0◦yieldsJ(0.5)/ΓA≈\n−12, and ϕ= 90◦yieldsJ(0.5)/ΓA≈6, which are\nslightly different from the above exact results. For\nsmallerqAawe get much better agreement. In figure\n(3) we plot equations (2) and (4) for ϕ= 0◦, and in\nfigure (4) for ϕ= 90◦. The results justify the use of\nelectrostatic dipole-dipole interactions for optical lattice\nultracold atoms when qAa <1.\n13579−2−1.5−1−0.50J(qAa)/ΓA\nqAa\nFIG. 3: The scaled interaction J(qAa)/ΓAvs.qAaforϕ=\n0◦. The full line is for equation (2), and the dashed line for\nequation (4).3\n13579−0.200.20.40.60.8J(qAa)/ΓA\nqAa\nFIG. 4: The scaled interaction J(qAa)/ΓAvs.qAaforϕ=\n90◦. The full line is for equation (2), and the dashed line for\nequation (4).\nIII. COLLECTIVE EXCITATION DAMPING\nRATE\nWe start in presenting the free space radiation field\nand its coupling to a finite atomic chain. The free space\nradiation field Hamiltonian is\nHrad=/summationdisplay\nqλEph(q)a†\nqλaqλ, (5)\nwherea†\nqλandaqλare the creation and annihilation op-\nerators of a photon with wave vector qand polarization\nλ, respectively. The photon energy is Eph(q) =/planckover2pi1cq. The\nelectric field operator is\nˆE(r) =i/summationdisplay\nqλ/radicalbigg\n/planckover2pi1cq\n2ǫ0V/braceleftBig\naqλeqλeiq·r−a†\nqλe∗\nqλe−iq·r/bracerightBig\n,\n(6)\nwhereeqλis the photon polarization unit vector, and V\nis the normalization volume.\nThe atomic transition dipole operator is\nˆµ=µN/summationdisplay\nn=1/parenleftbig\nBn+B†\nn/parenrightbig\n. (7)\nThematter-field couplingis givenformallybythe electric\ndipole interaction HI=−ˆµ·ˆE. In the rotating wave\napproximation and for linear polarization, we get\nHI=−i/summationdisplay\nqλ,n/radicalbigg\n/planckover2pi1cq\n2ǫ0V(µ·eqλ)\n×/braceleftBig\naqλB†\nneiqzna−a†\nqλBne−iqzna/bracerightBig\n.(8)\nIn the following we treat a single electronic excitation in\nthe atomic chain. We start in treating the most symmet-\nric collective state and then the general collective state.A. Symmetric Collective Excitation\nWe consider a single excitation in the system with the\nsymmetric collective state\n|i/an}bracketri}hts=1√\nN/summationdisplay\ni|g1,···,ei,···,gN/an}bracketri}ht.(9)\nThis state is an eigenstate ofthe Hamiltonian in the limit\nofqAa >1 withJ/ΓA<1, where the atoms are almost\nindependent. The other limit of qAa <1 treated by us\nin other work [11–18].\nWe calculate the damping rate of such collective state\nthrough the emission of a photon into free space and the\ndamping into the final ground state\n|f/an}bracketri}ht=|g1,···,gN/an}bracketri}ht. (10)\nWe apply the Fermi goldenrule to calculatethe collective\nsymmetric state damping rate\nΓs=2π\n/planckover2pi1/summationdisplay\nqλ|/an}bracketle{tf|HI|i/an}bracketri}ht|2δ(EA−Eph),(11)\nwhich in the present case reads\nΓs=/summationdisplay\nqλπcq\nǫ0VN(µ·eqλ)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nn=1e−iqzna/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδ(EA−Eph).\n(12)\nThe summation over the photon polarization yields\n/summationdisplay\nλ(µ·eqλ)2=µ2−(q·µ)2\nq2. (13)\nThe summation over qcan be converted into the integral\n/summationdisplay\nq→V\n(2π)3/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθ/integraldisplay∞\n0q2dq.(14)\nWe use\nq=q(sinθcosφ,sinθsinφ,cosθ),(15)\nand the transition dipole is taken to be\nµ=µ(sinϕ,0,cosϕ). (16)\nThe integration over φ, and the change of the variable\ny=qAa cosθ, gives\nΓs=µ2q2\nA\n8πǫ0/planckover2pi1aN/integraldisplay+qAa\n−qAady/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nn=1e−iny/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n×/bracketleftbigg/parenleftbig\n1+cos2ϕ/parenrightbig\n−y2\n(qAa)2/parenleftbig\n3cos2ϕ−1/parenrightbig/bracketrightbigg\n.(17)\nUsing the relation\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nn=1e−iny/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=N+N/summationdisplay\nn<m=12cos[(n−m)y],(18)4\nwe reach, after the integration over y, the result\nΓs= ΓA/braceleftBigg\n1+2\nNN/summationdisplay\nn<m=1F[qAa(n−m)]/bracerightBigg\n,(19)\nwhere\nF(x) =3\n2/braceleftbiggsinx\nx/parenleftbig\n1−cos2ϕ/parenrightbig\n+/bracketleftbiggcosx\nx2−sinx\nx3/bracketrightbigg/parenleftbig\n1−3cos2ϕ/parenrightbig/bracerightbigg\n.(20)\nIn figure (5) we plot the function F(x), for two different\npolarization directions. Using the previous numbers, for\nϕ= 0◦we getF(0.5) = 0.9752, and for ϕ= 90◦we get\n0.9507, which justifies the use of F= 1 for optical lattice\nultracold atoms in our previous works [11–18].\n0 5 10 15−0.500.51F(x)\nx\nFIG. 5: The function F(x) vs.x. The full line is for ϕ= 0◦,\nand the dashed line for ϕ= 90◦.\nFor different number of atoms we get\nΓs(1) = Γ A,\nΓs(2) = Γ A{1+F(qAa)},\nΓs(3) = Γ A/braceleftbigg\n1+2\n3[2F(qAa)+F(2qAa)]/bracerightbigg\n,\n··· (21)\nThe symmetric damping rate can be written in the form\nΓs(N) = ΓA/braceleftBigg\n1+2N−1/summationdisplay\nn=1(N−n)\nNF(qAan)/bracerightBigg\n.(22)\nIn figure (6) we plot Γ /ΓAasa function of qAaforN= 5,\nand for the polarizations ϕ= 0◦andϕ= 90◦.\nLets consider F(qAan) to represent a bond between\ntwo atoms that separated by a distance ( an), then in the\nabove summation the function F(qAan) is multiplied by\nthe number of bonds of this length which is ( N−n). In\nthe limit of qAa≪1 we get F≃1, and then Γ s(N)≈051015202530012345Γ/ΓA\nqAa\nFIG. 6: The symmetric scaled damping Γ /ΓAvs.qAa, for\nN= 5. The full line is for ϕ= 0◦, and the dashed line for\nϕ= 90◦.\nNΓA. In the limit of qAa≫1 we get F≃0, and then\nΓs(N)≈ΓAwith oscillations.\nNow we emphasize the dependence of the symmetric\nstate damping rate as a function of the number of atoms\nN. We plot the scaled damping rate Γ s/ΓAas a function\nofNfor different values of qAa. In figures (7 −9) we\nplot for qAa= 0.001,qAa= 0.1, andqAa= 1, in the\ntwo cases of ϕ= 0◦andϕ= 90◦. The damping rate of\nthe symmetric state grows linearly with the number of\natoms for small N, and approach a finite value for large\nN. ForqAa≥1 the damping rate approach the finite\nvalue faster than for qAa≪1. In figure (10) we plot\nΓs/ΓAas a function of φforN= 100 at qAa= 0.1.\nSignificant difference appears between the damping rates\nforϕ= 0◦andϕ= 90◦.\n0 500 100002004006008001000Γs/ΓA\nN  \nFIG. 7: The symmetric scaled damping rate Γ /ΓAvs.N, for\nqAa= 0.001. The full line is for ϕ= 0◦, and the dashed line\nforϕ= 90◦.5\n0 50 10001020304050Γs/ΓA\nN  \nFIG. 8: The symmetric scaled damping rate Γ /ΓAvs.N, for\nqAa= 0.1. The full line is for ϕ= 0◦, and the dashed line for\nϕ= 90◦.\n0 50 10012345Γs/ΓA\nN  \nFIG. 9: The symmetric scaled damping rate Γ /ΓAvs.N, for\nqAa= 1. The full line is for ϕ= 0◦, and the dashed line for\nϕ= 90◦.\nB. General Collective Excitation\nHere we consider the case of a single excitation but for\na general collective state, which is given by\n|i/an}bracketri}ht=1√\nN/summationdisplay\niCi|g1,···,ei,···,gN/an}bracketri}ht,(23)\nwhereCi=±1, and/summationtext\niC2\ni=N. Equation (17) reads\nΓ =µ2q2\nA\n8πǫ0/planckover2pi1aN/integraldisplay+qAa\n−qAady/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nn=1Cne−iny/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n×/bracketleftbigg/parenleftbig\n1+cos2ϕ/parenrightbig\n−y2\n(qAa)2/parenleftbig\n3cos2ϕ−1/parenrightbig/bracketrightbigg\n.(24)0 1 2 3202530354045Γs/ΓA\nφ\nFIG. 10: The symmetric scaled damping rate Γ /ΓAvs.ϕ, for\nN= 100 at qAa= 0.1.\nWe use\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nn=1Cne−iny/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=N+N/summationdisplay\nn/negationslash=m=1CnCme−i(n−m)y,(25)\nwhereCnCm=±1. After integration over y, we get\nΓ(N) = ΓA/braceleftBigg\n1+2\nNN/summationdisplay\nn<m=1CnCmF[qAa(n−m)]/bracerightBigg\n.\n(26)\nHere we present the results for two examples. For N=\n2 we have the symmetric state\n|i/an}bracketri}hts=|e1,g2/an}bracketri}ht+|g1,e2/an}bracketri}ht√\n2, (27)\nwith the damping rate\nΓs= ΓA{1+F(qAa)}, (28)\nand the antisymmetric state\n|i/an}bracketri}hta=|e1,g2/an}bracketri}ht−|g1,e2/an}bracketri}ht√\n2, (29)\nwith the damping rate\nΓa= ΓA{1−F(qAa)}. (30)\nThe results for N= 2 agree with the known results [22].\nIn figure (11) we plot Γ /ΓAfor the symmetric state of\nN= 2 as a function of qAa, for the polarizations ϕ= 0◦\nandϕ= 90◦. In figure (12) the plot is for the antisym-\nmetric state.\nForN= 3, for the symmetric state\n|i/an}bracketri}hts=|e1,g2,g3/an}bracketri}ht+|g1,e2,g3/an}bracketri}ht+|g1,g2,e3/an}bracketri}ht√\n3,(31)6\n0510152025300.511.52Γ/ΓA\nqAa\nFIG. 11: The scaled damping Γ /ΓAvs.qAa, forN= 2 with\nthe symmetric state. The full line is for ϕ= 0◦, and the\ndashed line for ϕ= 90◦.\n05101520253000.511.5Γ/ΓA\nqAa\nFIG. 12: The scaled damping Γ /ΓAvs.qAa, forN= 2 with\nthe antisymmetric state. The full line is for ϕ= 0◦, and the\ndashed line for ϕ= 90◦.\nwe get\nΓs= ΓA/braceleftbigg\n1+2\n3[2F(qAa)+F(2qAa)]/bracerightbigg\n.(32)\nFor the antisymmetric state\n|i/an}bracketri}hta=|e1,g2,g3/an}bracketri}ht−|g1,e2,g3/an}bracketri}ht+|g1,g2,e3/an}bracketri}ht√\n3,(33)\nwe get\nΓa= ΓA/braceleftbigg\n1−2\n3[2F(qAa)−F(2qAa)]/bracerightbigg\n.(34)\nIn the limit of qAa≪1 we have F≃1, then for the sym-\nmetric state we get Γ ≈3ΓA, and for the antisymmetric\none we get Γ ≈ΓA/3. In the limit of qAa≫1 we haveF≃0, then for the symmetric and antisymmetric states\nwe get Γ ≈ΓA.\nIn figures (13) we plot Γ /ΓAfor the symmetric state of\nN= 3 as a function of qAa, for the polarizations ϕ= 0◦\nandϕ= 90◦. In figure (14) the plot is for the antisym-\nmetric state.\n0510152025300.511.522.53Γ/ΓA\nqAa\nFIG. 13: The scaled dampingΓ /ΓAvs.qAa, of the symmetric\nstate for N= 3. The full line is for ϕ= 0◦, and the dashed\nline forϕ= 90◦.\n05101520253000.511.52Γ/ΓA\nqAa\nFIG. 14: The scaled damping Γ /ΓAvs.qAa, of the antisym-\nmetric state for N= 3. The full line is for ϕ= 0◦, and the\ndashed line for ϕ= 90◦.\nIV. COLLECTIVE EXCITATION EMISSION\nPATTERN\nHere we calculate the emission pattern of a collec-\ntive state in a chain of Natoms separated by a dis-\ntancea. The transition dipole of each atom is µ=\nµ(sinϕ,0,cosϕ), at positions Rn= (0,0,Rn). For sim-\nplicitytheobservationpointistakentobeat r= (x,0,0),7\nas seen in figure (15). We concentrate here in the limit of\nqAa >1 where the atoms can be treated independently.\nThe other limit of qAa <1 investigated by us in previous\nwork[18]. The positive electric field operatorof the atom\n(n), in the far zone field where x≫λA, is given by [24]\nˆE(+)\nn(r,t) =µq2\nA\n4πǫ0sinφn\n|r−Rn|B/parenleftbigg\nt−|r−Rn|\nc/parenrightbigg\nˆen,(35)\nwhereφnis the angle between µandr−Rn, and the\nunit vector ˆenis defined by\nˆen=ˆnn׈y,ˆnn=r−Rn\n|r−Rn|. (36)\nWe have\nr−Rn= (x,0,−Rn),|r−Rn|2=x2+R2\nn,(37)\nand\nφn=π−ϕ−αn,tanαn=x/Rn,(38)\nwith\nˆnn=(x,0,−Rn)/radicalbig\nx2+R2n,ˆen=(Rn,0,x)/radicalbig\nx2+R2n.(39)\nµ\nϕa\n... ...nx\nz\nyrr−R\nRαn en n\nφnnnn\nFIG. 15: The observation point at ralong the xaxis, and the\nlattice is along the zaxis. The angle between the transition\ndipoleµandr−Rnisφn. The electric field direction off\natom (n) isˆen.\nFor the atomic transition operators we use the expec-\ntation values\n/an}bracketle{tBi(t−ti)/an}bracketri}ht=/an}bracketle{tBi(0)/an}bracketri}hte−iωA(t−ti)e−ΓA(t−ti)/2,\n/an}bracketle{tB†\ni(t−ti)Bi(t−ti)/an}bracketri}ht=/an}bracketle{tB†\ni(0)Bi(0)/an}bracketri}hte−ΓA(t−ti),(40)\nand\n/an}bracketle{tB†\ni(t−ti)Bj(t−tj)/an}bracketri}ht=/an}bracketle{tB†\ni(0)Bj(0)/an}bracketri}hte−ΓA[t−(ti+tj)/2]\n×e−iωA(ti−tj), (41)\nwhereti=|r−Ri|/c. In the limit qAa >1 the single\nexcitation collective states decay with the single excited\natom damping rate Γ A.The total electric field at the observation point is\nˆE(+)(r,t) =/summationdisplay\niˆE(+)\ni(r,t), (42)\nand the intensity is\nI(r,t) =1\n2ǫ0c/an}bracketle{tˆE(−)(r,t)·ˆE(+)(r,t)/an}bracketri}ht.(43)\nExplicitly we can write\nI(r,t) =/summationdisplay\niIi(r,t)+/summationdisplay\ni/negationslash=jGij(r,t),(44)\nwhere the i-th intensity is\nIi(r,t) =1\n2ǫ0c/an}bracketle{tˆE(−)\ni(r,t)ˆE(+)\ni(r,t)/an}bracketri}ht,(45)\nand the correlation function is\nGij(r,t) =1\n2ǫ0c/an}bracketle{tˆE(−)\ni(r,t)·ˆE(+)\nj(r,t)/an}bracketri}ht.(46)\nWe get\nIi(r,t) =µ2ω4\nA\n32π2ǫ0c3sin2φi\n|r−Ri|2/an}bracketle{tB†\ni(0)Bi(0)/an}bracketri}hte−ΓA(t−ti),\n(47)\nand\nGij(r,t) =µ2ω4\nA\n32π2ǫ0c3e−ΓA[t−(ti+tj)/2]e−iωA(ti−tj)\n× /an}bracketle{tB†\ni(0)Bj(0)/an}bracketri}htsinφi\n|r−Ri|sinφj\n|r−Rj|(ˆni·ˆnj).\n(48)\nA. Two-Atoms Chain\nWepresenttheresultsforthesimplecaseoftwoatoms.\nOne atom is located at the origin R1= (0,0,0), and the\nsecond at R2= (0,0,a). The observation point is at\nr= (x,0,0), where r−R1= (x,0,0), andr−R2=\n(x,0,−a), with|r−R1|=x, and|r−R2|=√\nx2+a2.\nWe have the angles φ1=π\n2−ϕ, andφ2=π−ϕ−α,\nwhere tan α=x/a. We get the times t1=x/c, andt2=√\nx2+a2/c. Also we have the unit vectors ˆe1= (0,0,1),\nandˆe2=(a,0,x)√\nx2+a2, thenˆn1= (1,0,0), andˆn2=(x,0,−a)√\nx2+a2,\nhence (ˆn1·ˆn2) =x√\nx2+a2. We obtain\nI1(r,t) =µ2ω4\nA\n32π2ǫ0c3sin2φ1\nx2/an}bracketle{tB†\n1(0)B1(0)/an}bracketri}hte−ΓA(t−x\nc),\nI2(r,t) =µ2ω4\nA\n32π2ǫ0c3sin2φ2\nx2+a2/an}bracketle{tB†\n2(0)B2(0)/an}bracketri}ht\n×e−ΓA/parenleftbigg\nt−√\nx2+a2\nc/parenrightbigg\n, (49)8\nand\nG12(r,t) =µ2ω4\nA\n32π2ǫ0c3sinφ1sinφ2\nx2+a2/an}bracketle{tB†\n1(0)B2(0)/an}bracketri}ht\n×e−ΓA/bracketleftbigg\nt−/parenleftbigg\nx+√\nx2+a2\n2c/parenrightbigg/bracketrightbigg\ne−iωA/parenleftbigg\nx−√\nx2+a2\nc/parenrightbigg\n,\nG21(r,t) =µ2ω4\nA\n32π2ǫ0c3sinφ1sinφ2\nx2+a2/an}bracketle{tB†\n2(0)B1(0)/an}bracketri}ht\n×e−ΓA/bracketleftbigg\nt−/parenleftbigg\nx+√\nx2+a2\n2c/parenrightbigg/bracketrightbigg\neiωA/parenleftbigg\nx−√\nx2+a2\nc/parenrightbigg\n.(50)\nNow we consider the two initial states of symmetric and\nantisymmetric collective states.\nFor the symmetric collective state\n|i/an}bracketri}ht=|e1,g2/an}bracketri}ht+|g1,e2/an}bracketri}ht√\n2, (51)\nwe have\n/an}bracketle{tB†\n1(0)B1(0)/an}bracketri}ht=/an}bracketle{tB†\n2(0)B2(0)/an}bracketri}ht\n=/an}bracketle{tB†\n1(0)B2(0)/an}bracketri}ht=/an}bracketle{tB†\n2(0)B1(0)/an}bracketri}ht=1\n2,(52)\nthen we get\nI(r,t) =I0(x)\n4/braceleftbigg\nsin2φ1e−ΓA(t−x\nc)+x2sin2φ2\nx2+a2\n×e−ΓA/parenleftbigg\nt−√\nx2+a2\nc/parenrightbigg\n+x2sinφ1sinφ2\nx2+a22cos/bracketleftBigg\nωA/parenleftBigg\nx−√\nx2+a2\nc/parenrightBigg/bracketrightBigg\n×e−ΓA/bracketleftbigg\nt−/parenleftbigg\nx+√\nx2+a2\n2c/parenrightbigg/bracketrightbigg/bracerightBigg\n, (53)\nwhere we defined the intensity\nI0(x) =µ2ω4\nA\n16π2ǫ0c3x2. (54)\nIn figures (16 −18) we plot the relative intensity\nI(r,t)/I0(x) as a function of afor the angles ϕ= 0◦,\nϕ= 45◦andϕ= 90◦, at the observation point x= 106˚A\nat the moment t= 2x/c. We use EA= 1eV,µ= 1e˚A\nand ΓA= 108Hz.\nFor small athe relative intensity is maximum for the\npolarization angle ϕ= 0◦and decreases for largerangles.\nIt is half for ϕ= 45◦, and becomes zero for ϕ= 90◦. The\nmaximum of the relative intensity moves into larger a\nwith increasing the angle ϕ. The relative intensity oscil-\nlates in changing aand tend to a finite value for large\na. Interesting case is for ϕ= 90◦, where the intensity\nis zero for small aand increases with increasing atill it\nreach a maximum at a= 106˚A(for the given numbers),\nand decreases back towards a finite value for larger a.In the limit of x≫a, where√\nx2+a2∼x+a2\n2x, and\nasφ≈π\n2−ϕ, we can write\nI(r,t)≃µ2ω4\nA\n64π2ǫ0c3x2e−ΓA(t−x\nc)cos2ϕ\n×/braceleftbigg\n1+eΓAa2\n2cx+2cos/parenleftbigg\nωAa2\n2cx/parenrightbigg\neΓAa2\n4cx/bracerightbigg\n.(55)\n0 1 2 3\nx 10600.20.40.60.81I/I0\na [Angs.]\nFIG. 16: The symmetric state scaled intensity I(r,t)/I0(x)\nvs.a, forϕ= 0◦atx= 106˚Aandt= 2x/c.\n0 1 2 3\nx 10600.20.40.60.8I/I0\na [Angs.]\nFIG. 17: The symmetric state scaled intensity I(r,t)/I0(x)\nvs.a, forϕ= 45◦atx= 106˚Aandt= 2x/c.\nFor the antisymmetric collective state\n|i/an}bracketri}ht=|e1,g2/an}bracketri}ht−|g1,e2/an}bracketri}ht√\n2, (56)\nwe have\n/an}bracketle{tB†\n1(0)B1(0)/an}bracketri}ht=/an}bracketle{tB†\n2(0)B2(0)/an}bracketri}ht=1\n2,\n/an}bracketle{tB†\n1(0)B2(0)/an}bracketri}ht=/an}bracketle{tB†\n2(0)B1(0)/an}bracketri}ht=−1\n2,(57)9\n012345\nx 10600.020.040.060.08I/I0\na [Angs.]\nFIG. 18: The symmetric or antisymmetric state scaled in-\ntensityI(r,t)/I0(x) vs.a, forϕ= 90◦atx= 106˚Aand\nt= 2x/c.\nthen we can write\nI(r,t) =I0(x)\n4/braceleftbigg\nsin2φ1e−ΓA(t−x\nc)+x2sin2φ2\nx2+a2\n×e−ΓA/parenleftbigg\nt−√\nx2+a2\nc/parenrightbigg\n−x2sinφ1sinφ2\nx2+a22cos/bracketleftBigg\nωA/parenleftBigg\nx−√\nx2+a2\nc/parenrightBigg/bracketrightBigg\n×e−ΓA/bracketleftbigg\nt−/parenleftbigg\nx+√\nx2+a2\n2c/parenrightbigg/bracketrightbigg/bracerightBigg\n. (58)\nIn figures (19 −20) we plot the relative intensity\nI(r,t)/I0(x) as a function of afor the angles ϕ= 0◦and\nϕ= 45◦. The case of ϕ= 90◦is the same as in figure\n(23). As before, the observation point is at x= 106˚Aat\nthe moment t= 2x/c, with the other previous numbers.\nTheresultsaresimilartothesymmetriconesexceptfrom\nthe case of small awhere the relative intensity tends to\nzero as expected.\nIn the limit of x≫a, asφ≈π\n2−ϕ, we can write\nI(r,t)≃µ2ω4\nA\n64π2ǫ0c3x2e−ΓA(t−x\nc)cos2ϕ\n×/braceleftbigg\n1+eΓAa2\n2cx−2cos/parenleftbigg\nωAa2\n2cx/parenrightbigg\neΓAa2\n4cx/bracerightbigg\n.(59)\nV. SUMMARY\nIn the present paper we investigated optical properties\nof a one dimensional atomic chain, in which the lattice\nconstant can range from a few angstroms up to thou-\nsands of angstroms. Namely, the lattice constant can\nchange from being smaller than the atomic transition0 1 2 3\nx 10600.20.40.60.81I/I0\na [Angs.]\nFIG. 19: The antisymmetric state scaled intensity\nI(r,t)/I0(x) vs.a, forϕ= 0◦atx= 106˚Aandt= 2x/c.\n0 1 2 3\nx 10600.20.40.60.8I/I0\na [Angs.]\nFIG. 20: The antisymmetric state scaled intensity\nI(r,t)/I0(x) vs.a, forϕ= 45◦atx= 106˚Aandt= 2x/c.\nwavelength up to much larger one. In the limit of lattice\nconstant smaller than the atomic transition wavelength\nthe electrostatic interactions are applicable, which found\nuseful for most of the typical experiments on optical lat-\ntice ultracold atoms. In our previous workwe limited the\ndiscussion to electrostatic interactions, where we consid-\neredonly resonancedipole-dipole interactions, and which\nis justified in the present work. For small lattice constant\nthe electrostatic interactions are responsible for the for-\nmation of excitons, where we did extensive study in this\nregimewithemphasizeontheexcitonlifetimes. Forlarge\nlattice constant the inclusion of radiative corrections are\nnecessary, which is the main issue in the present paper.\nFor large lattice constant the radiative corrections are\nincluded, and in this regime we found that the coupling\nparameter for the energy transfer among even the near-\nest neighbor atom sites is smaller than a single excited\natom damping rate. Hence, the energy transfer is not10\nfavorable, and we treated the atoms as independently\nsetting on the lattice sites. Then, we calculated the\ndampingratesofdifferentcollectiveelectronicexcitations\nin including the radiative corrections by considering the\neffect of the existence of all the other atom sites, de-\nspite their large distances from the excited atom. Big\nattention we gave for the most symmetric state, where\nwe emphasized the dependence of its damping rate on\nthe number of atoms for different lattice constant. We\nfound the symmetric damping rate to behave linearly at\nsmall atom numbers and saturate at large numbers. The\ndamping rate of symmetric and antisymmetric collective\nstates tend to that of a single excited atom with oscilla-\ntions due to the radiative effect through the exchange of\nvirtual photons. The differences between damping rates\nof collective states appear for small lattice constant, in\nwhich the symmetric states have superradiant damping\nrate, that is Ntime the single excited atom rate. Here,\npart of the antisymmetric states become dark with zero\ndamping rate, and other part is metastable with a frac-\ntion of the single excited atom damping rate. Moreover\nwe calculated the emission pattern off a chain of atomswith a large lattice constant in which the atoms can be\nconsidered independently. The emission intensities off\ntwo atoms with symmetric and antisymmetric states are\npresented as a function of the interatomic distance.\nTheresultsofthepresentpaperareillustratedinterms\nof optical lattice ultracold atoms, but they are general\nand can be adopted for any chain of optically active ma-\nterial. For example, chains of semiconductor quantum\ndots fit exactly in the regime of large lattice constant,\nwhere radiative corrections are unavoidable, and the life\ntimesoftheircollectivestatescanbetreatedaccordingto\nthe present paper. Other system that exploits the regime\nofthe present paper is a lattice oflarge organicmolecules\nsitting on a matrix with a given large lattice constant,\nthe collective damping rate and emission pattern are ex-\npected to behave according to our present results.\nThe author acknowledge very fruitful discussions with\nHelmut Ritsch. The work was supported by the Austrian\nScience Funds (FWF) via the project (P21101), and by\nthe DARPA QuASAR program.\n[1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n[2] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A.\nSen De, and U. Sen, Adv. in Phys. 56, 243 (2007).\n[3] D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The\nphysics of Quantum Information , (Springer, NY, 2000).\n[4] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and\nI. Bloch, Nature415, 39 (2002).\n[5] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and\nP. Zoller, Phys. Rev. Lett. 81, 3108 (1998).\n[6] I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys.\nRev. Lett. 98, 080404 (2007).\n[7] S. Davydov, Theory of Molecular Excitons , (Plenum,\nNew York, 1971).\n[8] V. M. Agranovich, Excitations in Organic Solids , (Ox-\nford, UK, 2009).\n[9] M. Takamoto, F-L Hong, R Higashi, and H Katori, Na-\nture435, 321 (2005).\n[10] M. Viteau, M. G. Bason, J. Radogostowicz, N. Malossi,\nD. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev.\nLett.107, 060402 (2011).\n[11] H. Zoubi, and H. Ritsch, Phys. Rev. A 76, 013817 (2007).\n[12] H. Zoubi, and H. Ritsch, Europhys. Lett. 82, 14001(2008).\n[13] H. Zoubi, and H. Ritsch, Europhys. Lett. 87, 23001\n(2009).\n[14] H. Zoubi, and H.Ritsch, New J. Phys. 12, 103014 (2010).\n[15] H. Zoubi, and H. Ritsch, J. Phys. B 44, 205303 (2011).\n[16] H. Zoubi, and H. Ritsch, Europhys. Lett. 90, 23001\n(2010).\n[17] H. Zoubi, and H. Ritsch, Phys. Rev. A 83, 063831 (2011).\n[18] H. Zoubi, and H. Ritsch, arXiv:1106.4923.\n[19] E. Vetsch, D. Reitz, G. Sague, R. Schmidt, S. T.\nDawkins, and A. Rauschenbeutel, Phys. Rev. Lett. 104,\n203603 (2010).\n[20] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau,\nP. Schauss, T. Fukuhara, I. Bloch, and S. Kuhr, Nature\n471, 319 (2011).\n[21] C. J. Mewton, and Z. Ficek, J. Phys. B 40, S181 (2007).\n[22] Z. Ficek and R. Tanas, Physics Reports 372, 369 (2002).\n[23] D. P. Craig and T. Thirunamachandran, Molecular\nQuantum Electrodynamics , (Academic Press, London,\n1984).\n[24] R. Loudon, The Quantum Theory of Light , 3Ed. (Oxford\nUniversity Press, 2000)."    },    {        "title": "1203.4735v1.Approximate_rogue_wave_solutions_of_the_forced_and_damped_Nonlinear_Schrödinger_equation_for_water_waves.pdf",        "content": "Approximate rogue wave solutions of the forced and\ndamped Nonlinear Schr odinger equation for water waves\nMiguel Onorato and Davide Proment\nDipartimento di Fisica, Universit\u0012 a degli Studi di Torino, Via Pietro Giuria 1, 10125\nTorino, Italy, EU\nINFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy, EU\nAbstract\nWe consider the e\u000bect of the wind and the dissipation on the nonlinear\nstages of the modulational instability. By applying a suitable transforma-\ntion, we map the forced/damped Nonlinear Schr odinger (NLS) equation into\nthe standard NLS with constant coe\u000ecients. The transformation is valid\nas long asj\u0000tj\u001c 1, with \u0000 the growth/damping rate of the waves due to\nthe wind/dissipation. Approximate rogue wave solutions of the equation are\npresented and discussed. The results shed some lights on the e\u000bects of wind\nand dissipation on the formation of rogue waves.\nKeywords: rogue waves, water waves, breathers\n1. Introduction\nModulational instability, also known as the Benjamin-Feir instability in\nthe water wave community, has been discovered in the late sixties indepen-\ndently by Benjamin and Feir [1] and Zakharov [2] (see [3] for an historical\nreview on the subject and possible applications). It describes the exponen-\ntial growth of an initially sinusoidal long wave perturbation of a plane wave\nsolution of the one dimensional water wave problem. For water waves the\ncondition of instability in in\fnite water depth is that 2p\n2a0k0N > 1, where\na0is the amplitude of the plane wave and k0is the corresponding wave\nnumber;N=k0=\u0001Kis the number of waves under the perturbation of\nwavenumber \u0001 K. The modulaitonal instability is frequently studied within\nthe Nonlinear Schr odinger (NLS) equation that describes weakly nonlinear\nand dispersive waves in the narrow band approximation. In this context,\nPreprint submitted to Physics Letters A March 22, 2012arXiv:1203.4735v1  [nlin.CD]  21 Mar 2012the nonlinear stages of the modulational instability are described by exact\nsolutions of the NLS, known as Akhmediev breathers [4, 5]. Other exact\nNLS solutions which describe the focussing of an initially non-small pertur-\nbation have been derived in [6, 7]. Such solutions have been considered as\nprototypes of rogue waves [8, 9].\nWithin the one dimensional NLS equation, the modulational instability is\nwell understood. What is probably less clear is the modulation of waves and\nthe formation of rogue waves in forced (by wind) or damped (by dissipation)\nconditions. In this regard in the past there has been a number of experimental\nworks, [10, 11, 12], which did not gave a clear picture on the e\u000bect of the wind\non the modulaitonal instability. A careful discussion of the discrepancy of the\nresults presented in the above papers can be found in [12]. According to their\ndiscussion the role of the wind is twofold: i) the wind changes the growth\nrate of the instability; ii) the natural selection of the sideband frequency is\naltered with respect to the no wind conditions.\nConcerning damping e\u000bects, it has been showed in [13] that any amount\nof dissipation stabilizes the modulational instability, questioning the role of\nthe modulational instability in the formation of rogue waves, [14]. More\nrecently, the role of dissipation and wind in the modulational instability has\nbeen considered together within the NLS equation, [15] (then con\frmed by\nfully nonlinear simulations, [16]). The authors performed a linear stability\nanalysis and numerical simulations and found that, in the presence of wind,\nyoung waves are more sensitive to modulational instability than old waves.\nThe just mentioned numerical results (except the one in [16]) are all based\non the following forced and damped Nonlinear Schr odinger equation:\ni@A\n@t\u0000\u000b@2A\n@x2\u0000\fjAj2A=i\u0000A: (1)\nAis the wave envelope, \u000band\fare two coe\u000ecients that depend on the\nwavenumber, k0, of the carrier wave. The right-hand side is responsible for\nthe forcing, \u0000 >0, and/or dissipation, \u0000 <0. The two e\u000bects are additive\nso that \u0000 is in general the sum of forcing coe\u000ecient plus a damping one.\nThe wind forcing depends on the ratio between air and water density and\nthe dissipation on the water viscosity, therefore the absolute value of \u0000 is\nalways a small quantity. Finding analytical solutions of equation (1) is not\nan obvious task. In the present paper we take advantage of the smallness of \u0000\nand, after a suitable transformation, we are able to \fnd breather solutions of\n2the forced-damped NLS equation. In the following sections we \frst describe\nthe transformation and then present the rogue wave analytical solutions.\n2. Reduction of the forced/damped NLS to the standard NLS\nWe considered the NLS equation discussed in [15]\ni\u0012@A\n@t+cg@A\n@x\u0013\n\u00001\n8!0\nk2\n0@2A\n@x2\u00001\n2!0k2\n0jAj2A=i\u0000A (2)\nwith\n\u0000 =1\n2g\u00142\u001aa\n\u001aw\r!0\u0012u\u0003\nc\u00132\n\u00002\u0017k2\n0 (3)\nhere\u0014is the Von Karman constant and u\u0003is the friction velocity, gis the\ngravity acceleration, \u001aaand\u001aware the air and water density, respectively;\n\ris a coe\u000ecient to be determined from the solution of Rayleigh equation\nassociated to the stability of the wind wave problem (see also [17] for a\njusti\fcation of the wind forcing term); cis the phase velocity, \u0017is the water\nkinematic viscosity. In [15] the equation is written in a nondimensional form\nand the coe\u000ecient K= \u0000=!0is introduced). The surface elevation is related\nto the envelope as follows:\n\u0011(x;t) =1\n2\u0012\nA(x;t) exp[i(k0x\u0000!0t)] +c:c\u0013\n: (4)\nNote that we use a di\u000berent de\fnition of the surface elevation from the one\nin [15] where the 1/2 factor is not included (the consequence is that the\ncoe\u000ecient in the nonlinear term in equation (2) di\u000bers by a factor of 4 from\nthe one in equation (3.1) in [15]). If \u000fis the small parameter in the derivation\nof the NLS, then it is assumed that the right-hand side term in (2) is of the\norder of\u000f2as the nonlinear and the dispersive term.\nWe consider the following new variable:\nB(x;t) =A(x;t)e\u0000\u0000t(5)\nand by selecting a coordinate system moving with the group velocity we get:\ni@B\n@t\u0000\u000b@2B\n@x2\u0000\fexp2\u0000tjBj2B= 0 (6)\n3were\u000band\fare the coe\u000ecients of the dispersive and nonlinear term, re-\nspectively. Written in the above form the e\u000bect of the forcing/damping term\nenters as a factor in front of nonlinear term and has the role of enhanc-\ning/decreasing the nonlinearity of the system as the wave evolve in time.\nRecalling that \u0000 is usually small, we Taylor expand the exponential and\nre-write the equation as follows:\ni@B\n@t\u0000\u000b@2B\n@x2\u0000\fp(t)jBj2B= 0 (7)\nwithp(t) = 1=(1\u00002\u0000t). Let's introduce the following change of coordinates:\n\u001f(x;t) =p(t)x; \u001c (t) =p(t)t (8)\nand scale the wave envelope function Bas follows\n (\u001f;\u001c) =B(x;t)p\np(t)exp\u0014\n\u0000i\u0012\u0000p(t)x2\n2\u000b\u0013\u0015\n: (9)\nAfter the transformation, the equation (6) results in:\ni@ \n@\u001c\u0000\u000b@2 \n@\u001f2\u0000\fj j2 = 0 (10)\ni.e., the NLS equation with constant coe\u000ecients. We have transformed the\nforced/damped NLS equation into the standard NLS equation whose solu-\ntions can be studied analytically. From a physical point of view the trans-\nformation (and consequently the validity of the solutions) is valid as long\nas 2j\u0000tj\u001c 1 (the transformation is singular for 2 j\u0000tj= 1). We underline\nthat the transformation of the forced/damped NLS equation to the standard\none has been possible only for 1 =p(t) equal to a linear function in t. For\nother functional dependences, the transformation does not seem to be possi-\nble. Our result is consistent with ones reported in [18, 19] where analytical\nsolutions of the variable coe\u000ecient NLS equation are described.\n3. Rogue wave solutions\nIn the following we will present three analytical solutions corresponding\nto the Peregrine, the Akhmediev and the Kuznetsov-Ma breathers for the\nstandard NLS.\n4The Peregrine solution also known as rational solution, has been origi-\nnally proposed in [5]. It has the peculiarity of being not periodic in time and\nin space: it is a wave that \\appears out of nowhere and disappears without\ntrace\" [20, 21]; its maximum amplitude reaches three times the amplitude of\nthe unperturbed waves. For the above reasons it has been considered as spe-\ncial prototype of freak wave, [21]. The Peregrine solution has been recently\nreproduced experimentally in wave tank laboratories [22] and in optical \fbers\n[23]. Below we present an exact analytical solution of equation (7) which is\nthe analogous of the Peregrine solutions but for the forced/damped case:\nB(x;t) =B0G(x;t)\u00124(1\u0000i2\fB2\n0p(t)t)\n\u000b+\u000b(2\fB2\n0p(t)t)2+ 2\fB2\n0(p(t)x)2\u00001\u0013\n(11)\nwith\nG(x;t) =p\np(t) exp\u0014\ni\u0012\u0000p(t)x2\n2\u000b\u0000\fB2\n0p(t)t\u0013\u0015\n: (12)\nIn \fgure 1 we show an example of such solution for steepness 0.1 and forcing\ncoe\u000ecientK= 0:0004 (the same value has been used in [15]). The axis are\nnormalized by the wave period, the wavelength and the initial wave amplitude\nB0. The e\u000bect of the wind/dissipation is to increase/reduced the amplitude\nof the plane wave. As in the case of the standard NLS, the wave appears\nonly once in time and space.\nThe Akhmediev solution [4] describes the modulational instability in its\nnonlinear regime; it is periodic in space. It is characterized by an ampli\f-\ncation factor which ranges from 1 to 3 (this last value corresponds to the\nPeregrine solution). In the presence of a forcing or damping, the breather\nhas the following analytical form:\nB(x;t) =B0G(x;t) p\n2~\u00172cosh[\u001bp(t)t]\u0000ip\n2~\u001bsinh[\u001bp(t)t)]p\n2 cosh[\u001bp(t)t]\u0000p\n2\u0000~\u00172cos[\u0017p(t)x]\u00001!\n(13)\nand\n\u0017=k0\nN;~\u0017=\u0017\nB0r\u000b\n\f;~\u001b= ~\u0017p\n2\u0000~\u00172; \u001b =\fB2\n0~\u001b: (14)\nThe function G(x;t) is reported (12). It should be noted that the function\nis periodic in space with a period that changes in time. In \fgure 2 we show\nan example of such solution for steepness 0.1, N= 5 and forcing coe\u000ecient\nofK= 0:0004.\n5Figure 1: The Peregrine solution of the forced NLS equation.\nFigure 2: The Akhmediev solution of the forced NLS equation.\n6Figure 3: The Kuznetsov-Ma solution of the forced NLS equation.\nThe Kuznetsov-Ma solution [6] is periodic in time and decrease exponen-\ntially in space. While for the Akhmediev breather the large time (positive or\nnegative) limit is a plane wave plus a small perturbation, the modulation for\nthe Ma breather is never small. The solution for the forced/damped equation\nis here reported:\nB(x;t) =B0G(x;t) \n\u0000p\n2~\u00162cos[\u001ap(t)t] +ip\n2~\u001asin[\u001ap(t)t)]p\n2 cos[\u001ap(t)t]\u0000p\n2 + ~\u00162cosh[\u0017p(t)x]\u00001!\n(15)\nwith\n\u0016=B0~\u0016r\n\f\n\u000b;~\u001a= ~\u0016p\n2 + ~\u00162; \u001a =\fB2\n0~\u001a: (16)\n~\u0016is a parameter related to the ampli\fcation factor. In \fgure 3 we show an\nexample of such solution for steepness 0.1, ~ \u0016=p\n2 and forcing coe\u000ecient of\nK= 0:0004. The periodicity (appearance of maxima) changes in time and\nincrease in the presence of forcing and decrease for the damping case.\n4. Conclusion\nIn the present Letter we have considered the problem of generation of\nrogue waves in the presence of wind forcing or dissipation. Our work is\nbased on the one dimensional forced/damped NLS equation.\n7Under the assumption of 2 j\u0000tj\u001c 1, where \u0000 is the forcing (\u0000 >0) or\ndamping (\u0000 <0) term, we have shown how the equation can be mapped in\nthe standard NLS equation with constant coe\u000ecients. In this framework, we\nhave found explicit analytical breather solutions.\nAs mentioned the e\u000bect of wind/dissipation is to increase/reduce in time\nthe coe\u000ecient in front of the nonlinear term. This has an impact on the\nmodulational instability; in particular, an initially stable (unstable) wave\npacket could be destabilized (stabilized) by the wind (dissipation). Similar\nresults have been obtained for the interaction of waves and current (see [24]).\nThe present results should be tested in wind waves tank facilities.\nAcknowledgments The E.U. project EXTREME SEAS (SCP8-GA-\n2009-234175) is acknowledged. M.O. thanks Dr. GiuliNico for discussions\nand ONR (grant N000141010991) for support. We are thankful to J. Dudley\nfor pointing us out reference [18].\nReferences\n[1] T. B. Benjamin, J. E. Feir, The disintegration of wave trains on deep\nwater. Part I. Theory, J. Fluid Mech. 27 (1967) 417{430.\n[2] V. Zakharov, Stability of period waves of \fnite amplitude on surface of\na deep \ruid, J. Appl. Mech. Tech. Phys. 9 (1968) 190{194.\n[3] V. Zakharov, L. Ostrovsky, Modulation instability: the beginning, Phys-\nica D: Nonlinear Phenomena 238 (5) (2009) 540{548.\n[4] N. Akhmediev, V. Eleonskii, N. Kulagin, Exact \frst-order solutions\nof the nonlinear Schr odinger equation, Theoretical and Mathematical\nPhysics 72 (2) (1987) 809{818.\n[5] D. Peregrine, Water waves, nonlinear schr odinger equations and their\nsolutions, J. Austral. Math. Soc. Ser. B 25 (1) (1983) 16{43.\n[6] Y. Ma, The perturbed plane-wave solutions of the cubic Schr odinger\nequation, Studies in Applied Mathematics 60 (1979) 43{58.\n[7] E. Kuznetsov, Solitons in a parametrically unstable plasma, in:\nAkademiia Nauk SSSR Doklady, Vol. 236, 1977, pp. 575{577.\n[8] K. B. Dysthe, K. Trulsen, Note on breather type solutions of the nls as\nmodels for freak-waves, Physica Scripta T82 (1999) 48{52.\n8[9] A. Osborne, M. Onorato, M. Serio, The nonlinear dynamics of rogue\nwaves and holes in deep{water gravity wave train, Phys. Lett. A 275\n(2000) 386{393.\n[10] L. Bliven, N. Huang, S. Long, Experimental study of the in\ruence of\nwind on benjamin-feir sideband instability, J. Fluid Mech 162 (1986)\n237{260.\n[11] T. Hara, C. Mei, Frequency downshift in narrowbanded surface waves\nunder the in\ruence of wind, Journal of Fluid Mechanics 230 (1) (1991)\n429{477.\n[12] T. Waseda, M. Tulin, Experimental study of the stability of deep-water\nwave trains including wind e\u000bects, Journal of Fluid Mechanics 401 (1)\n(1999) 55{84.\n[13] H. Segur, D. Henderson, J. Carter, J. Hammack, C. Li, D. Phei\u000b,\nK. Socha, Stabilizing the benjamin-feir instability, Journal of Fluid Me-\nchanics 539 (2005) 229{272.\n[14] H. Segur, D. Henderson, J. Hammack, Can the benjamin-feir instability\nspawn a rogue wave?, in: Proceedings of the Aha Huliko'a Hawaiian\nWinter Workshop, University of Hawaii, Retrieved August, Vol. 5, 2008.\n[15] C. Kharif, R. Kraenkel, M. Manna, R. Thomas, The modulational in-\nstability in deep water under the action of wind and dissipation, Journal\nof Fluid Mechanics 664 (1) (2010) 138{149.\n[16] C. Kharif, J. Touboul, Under which conditions the benjamin-feir insta-\nbility may spawn an extreme wave event: A fully nonlinear approach,\nThe European Physical Journal-Special Topics 185 (1) (2010) 159{168.\n[17] S. Leblanc, Ampli\fcation of nonlinear surface waves by wind, Physics\nof Fluids 19 (2007) 101705.\n[18] Q. Tian, Q. Yang, C. Dai, J. Zhang, Controllable optical rogue waves:\nRecurrence, annihilation and sustainment, Optics Communications 284\n(2011) 2222{2225.\n[19] J. Zhang, C. Dai, Q. Yang, J. Zhu, Variable-coe\u000ecient f-expansion\nmethod and its application to nonlinear schr odinger equation, Optics\ncommunications 252 (4-6) (2005) 408{421.\n9[20] N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere\nand disappear without a trace, Physics Letters A 373 (6) (2009) 675{678.\n[21] V. Shrira, V. Geogjaev, What makes the Peregrine soliton so special as\na prototype of freak waves?, Journal of Engineering Mathematics 67 (1)\n(2010) 11{22.\n[22] A. Chabchoub, N. Ho\u000bmann, N. Akhmediev, Rogue wave observation\nin a water wave tank, Physical Review Letters 106 (20) (2011) 204502.\n[23] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhme-\ndiev, J. Dudley, The Peregrine soliton in nonlinear \fbre optics, Nature\nPhysics 6 (10) (2010) 790{795.\n[24] M. Onorato, D. Proment, A. To\u000boli, Triggering rogue waves in op-\nposing currents, Phys. Rev. Lett. 107 (2011) 184502. doi:10.1103/\nPhysRevLett.107.184502 .\nURL http://link.aps.org/doi/10.1103/PhysRevLett.107.184502\n10"    },    {        "title": "1204.5051v1.Rotating_skyrmion_lattices_by_spin_torques_and_field_or_temperature_gradients.pdf",        "content": "arXiv:1204.5051v1  [cond-mat.str-el]  23 Apr 2012Rotating skyrmion lattices by spin torques and field or tempe rature gradients\nKarin Everschor,1Markus Garst,1Benedikt Binz,1Florian Jonietz,2\nSebastian M¨ uhlbauer,3Christian Pfleiderer,2and Achim Rosch1\n1Institute of Theoretical Physics, University of Cologne, D -50937 Cologne, Germany\n2Physik-Department E21, Technische Universit¨ at M¨ unchen , D-85748 Garching, Germany\n3Forschungsneutronenquelle Heinz Maier Leibnitz (FRM II),\nTechnische Universit¨ at M¨ unchen, D-85748 Garching, Germ any\n(Dated: May 28, 2022)\nChiral magnets like MnSi form lattices of skyrmions, i.e. ma gnetic whirls, which react sensitively\nto small electric currents jabove a critical current density jc. The interplay of these currents with\ntiny gradients of either the magnetic field or the temperatur e can induce a rotation of the magnetic\npattern for j > j c. Either a rotation by a finite angle of up to 15◦or – for larger gradients –\na continuous rotation with a finite angular velocity is induc ed. We use Landau-Lifshitz-Gilbert\nequations extended by extra damping terms in combination wi th a phenomenological treatment\nof pinning forces to develop a theory of the relevant rotatio nal torques. Experimental neutron\nscattering data on the angular distribution of skyrmion lat tices suggests that continuously rotating\ndomains are easy to obtain in the presence of remarkably smal l currents and temperature gradients.\nPACS numbers:\nI. INTRODUCTION: SPINTORQUES AND\nSKYRMION LATTICES\nManipulating magnetic structures by electric current\nis one of the main topics in the field of spintronics. By\nstrong current pulses one can, for example, switch mag-\nnetic domains in multilayer devices1,2, induce microwave\noscillations in nanomagnets3or move ferromagnetic do-\nmain walls4,5. The latter effect may be used to develop\nnew types of non-volatile memory devices6. It is there-\nfore a question of high interest to study the coupling\nmechanisms of currents to magnetic structures7,8.\nHere, the recent discovery9,10ofthe so-called skyrmion\nlattice in chiral magnets like MnSi provides a new oppor-\ntunity for studying the manipulation of magnetism by\nelectric currents both experimentally and theoretically.\nThe skyrmions in MnSi form a lattice of magnetic whirls,\nsimilar to the superfluid whirls forming the vortex lattice\nin type-II superconductors. While in ordinary ferromag-\nnets, currents couple only to the canted spin configura-\ntions at domain walls, the peculiar magnetic structure\nof the skyrmion lattice allows for an efficient bulkcou-\npling. Furthermore, the smooth magnetic structure of\nthe skyrmion lattice decouples efficiently from the un-\nderlying atomic lattice and from impurities. As a conse-\nquence, itwasobserved10that the criticalcurrentdensity\nneeded to affect the magnetic structure was more than\nfive orders of magnitude smaller than in typical spin-\ntorque experiments.\nThese low current densities open opportunities for new\ntypes of experiments to study quantitatively the physics\nof spin transfer torques. Due to the much lower cur-\nrent densities it is now possible to perform spintorque\nexperiments in bulk materials and thus avoid the surface\neffects that dominate in nanoscopic samples. Moreover\nfor smaller currents the effects of heating and Oersted\nmagnetic fields created by the current are suppressed.\nFigure 1: Schematic plot of the forces on a skyrmion lattice\nperpendicular and parallel to the current flowing in vertica l\ndirection. For a static, non-moving skyrmion lattice the re d\nhorizontal arrows correspond to the Magnus force and the\ngreen vertical arrows to dissipative forces. In the presenc e of\na temperature or field gradient, these forces change smoothl y\nacross a domain, thereby inducing rotational torques which\ndepend sensitively on the relative orientation of current a nd\ngradient (and on the direction in which the skyrmion lattice\nmoves). Small black arrows: local orientation of the magnet i-\nzation projected into the plane perpendicular to the magnet ic\nfieldB. Ineachunitcell themagnetization windsoncearound\nthe unit sphere.\nIn this paper we suggest experiments and develop a\ntheory with the goal to exploit the rotational motion in-\nsteadofjust translationalmotionto investigatethe inter-\nplay of electric currents and moving magnetic structures.\nOurtheoryisdirectlymotivatedbyrecentexperiments10,\nwhere a change of orientation of the skyrmion lattice as a\nfunction ofthe applied electric currentwas observedwith\nneutron scattering. In Ref. [10] we have shown that the2\nrotation arises from the interplay of a tiny thermal gradi-\nent parallel to the current and the Magnus forces arising\nfromthespintorquecouplingofcurrentandskyrmionlat-\ntice. For example, the rotation angle could be reversed\nby reversing either the current direction or the direction\nof the thermal gradient.\nThe basic idea underlying the theoretical analysis of\nour paper is sketched in Fig. 1. In the presence of an\nelectric current several forces act on the skyrmion lat-\ntice. First, dissipative forces try to drag the skyrmion\nlattice parallel to the (spin-) current. Second, the inter-\nplay of dissipationless spin-currents circulating around\neach skyrmion and the spin-currents induced by the elec-\ntric current lead to a Magnus force oriented perpendic-\nular to the current for a static skyrmion lattice (for the\nrealistic case of moving skyrmions the situation is more\ncomplicated). In the presence of any gradient across the\nsystem (e.g. a temperature or field gradient), indicated\nby the color gradient, these forces will vary in strength\nacross a skyrmion domain.\nAs in the experiment, we assume that the gradients\nare tiny: on the length scale set by the skyrmion distance\nthe gradients have negligible effects. However, multiply-\ning the tiny gradient with a large length, i.e., the size of\na domain of the skyrmion lattice (which can be11several\nhundred µm), one obtainsa sizablevariationofthe forces\nacross the domain. These inhomogeneous forces can give\nrisetorotationaltorques. Whetherthetorquearisesfrom\nthe Magnus forcesor the dissipative forces depends, how-\never, on the relative orientation of current and gradient\nand also on the direction in which the skyrmion lattice\ndrifts. Fig. 1 givesa simple example: if, forexample, cur-\nrent and gradient are parallel to each other (right panel)\ntheforcesperpendiculartothecurrentdirection(redhor-\nizontal arrows) give rise to rotational torques while the\nparallel forces do not contribute. The situation is re-\nversed when current and gradient are perpendicular (left\npanel).\nWe therefore suggest to use the rotation of magnetic\nstructures as a function of the relative orientation of cur-\nrent and further gradients as a tool to explore the cou-\npling of magnetism and currents. We will show that the\nresulting rotations depend very sensitively both on the\nrelative size of the various forces affecting the skyrmion\ndynamics and on how these forces depend on the induced\ngradients. While we apply our theory here to skyrmion\nlattices, our theoretical approaches can also be used for\nother complex magnetic textures and our results should\nalso have ramifications for other setups12,13. Quantita-\ntively, we will only study the role of gradients induced\nby changes in temperature or magnetic field but other\noptions are also possible. For example, macroscopic vari-\nations of the cross section of a sample will lead to gradi-\nents in the current density. Also changes in the chemical\ncomposition or strain in the sample can induce gradients.\nIt is also essential to investigate the effect of pin-\nningofthe magneticstructurebyinhomogeneitiesarising\nfrom crystalline imperfections. Inhomogeneities distortthe perfect skyrmion lattice and lead to forces prohibit-\ning (up to a very small creep) the motion of the mag-\nnetic structure as long as the current is below a criti-\ncal value, j < jc. Also for j/greaterorsimilarjc, inhomogeneities in-\nduce an effective, velocity dependent frictional force on\nthe moving skyrmion lattice connected to local, time-\ndependent distortions of the skyrmion lattice. Pinning\nhas widely been studied both experimentally and the-\noretically for charge density waves and vortex lattices\nin superconductors14–17. As the dynamics of skyrmions\ndiffers qualitatively (and quantitatively) from these two\ncases it is not clear which of these results can be trans-\nferred to skyrmion lattices. Due to the non-linear depen-\ndence of the pinning forces on the velocity, they can not\nbe described by a simple damping term. Within this pa-\nper we will not try to develop a theory ofpinning but will\ninstead use a simple phenomenologicalansatz to describe\nand discuss pinning effects.\nRotational torques can also arise in the absence of the\ntypes of gradients discussed above. In Ref. [18] we have\nstudied the role of distortions of the skyrmion lattice by\ntheunderlyingatomiclatticeextendingthemethodsused\nbyThiele19torotationaltorques(thismethodwillalsobe\nused below). Such distortions indeed induce small rota-\ntional torques in a macroscopicallyhomogeneous system,\ni.e. withoutanyexternalgradients. Similarly,alsodistor-\ntions induced by disorder can induce rotational torques\nwithout external gradients as has been discussed in the\nseminal paper by Hauger and Schmid14. But all these\neffects are very small and have notbeen observed in the\nexperimental setup of Ref. [10] as no rotation has been\nobserved in the absence of gradients. Therefore they will\nbe neglected in the following.\nIn the following we will first describe briefly the rele-\nvant Ginzburg-Landau model and the Landau-Lifshitz-\nGilbert equation used to model the dynamics of the\nskyrmions. Here we include a novel damping term α′\nrecently introduced in Refs. [20,21] (we also add the cor-\nresponding β′term). We then derive effective equations\nfor the translational and rotational mode where pinning\nphysics is taken into account by an extra phenomeno-\nlogical term. This allows to develop predictions both for\nstaticrotationsbyafiniteangleandcontinuousrotations.\nIn the light of our results we interpret experimental re-\nsults on the angular distribution of skyrmion lattices in\nthe presence of currents and gradients.\nII. SETUP\nA. Ginzburg-Landau model\nThe starting point of our analysis is the standard\nGinzburg-Landau model of a chiral magnet in the pres-\nence of a Dzyaloshinskii-Moriya interaction22,23. After a\nrescaling of the length r, the local magnetization M(r)\nand the magnetic field Bthe free energy functional re-3\nduces to9\nF=γF/integraldisplay\nd3r/bracketleftbig\n(1+t)M2+(∇M)2\n+2M·(∇×M)+M4−B·M/bracketrightbig\n,(1)\nHeret∝T−TMF\ncparametrizesthe distance to the mean-\nfield phase transition at B= 0 from a phase with helical\nmagnetic order ( t <0) to a paramagnetic phase ( t >\n0)22,23. In the presence of weak disorder t(and strictly\nspeaking also the prefactors of all other terms) fluctuates\nslightly as a function of r.\nThe skyrmion lattice (stabilized by thermal fluctua-\ntions) exists for a small temperature and field range9.\nIt is translationally invariant parallel to Band shows a\ncharacteristic winding of the magnetization in the plane\nperpendicular to B, see Fig. 1.\nB. Landau-Lifshitz-Gilbert equation\nTo describe the dynamics of the orientation ˆΩ(r,t) =\nM(r,t)/|M(r,t)|of the magnetization M(r,t) in the\npresence of spin-transfer torques due to electric cur-\nrentsweuse the standardLandau-Lifshitz-Gilbert(LLG)\nequation,7,8,24extended by a new dissipative term20,21\n(∂t+vs∇)ˆΩ=−ˆΩ×Heff+αˆΩ×/parenleftBig\n∂t+β\nαvs∇/parenrightBig\nˆΩ\n−α′/bracketleftBig\nˆΩ·/parenleftbig\n∂iˆΩ×(∂t+β′\nα′vs∇)ˆΩ/parenrightbig/bracketrightBig\n∂iˆΩ.(2)\nHerevsis an effective spin velocity parallel to the spin\ncurrent density. More precisely, for smooth magnetic\nstructures with constant amplitude of the magnetization\nit is given by the ratio of the spin current25and the size\nof the local magnetization, |M|. In a good metal (for ex-\nample, MnSi) vsis expected to be parallel to the applied\nelectric current and to depend only weakly on temper-\nature and field. The magnetization precesses in the ef-\nfective magnetic field Heff≈ −1\nMδF\nδˆΩ. Strictly speaking\nEq. (2) is only valid for a constant amplitude of the mag-\nnetization, |M|= const. Since |M|varies only weakly9\nin the skyrmion phase, we use as a further approxima-\ntionHeff≈ −1\nMδF\nδM∂M\n∂ˆΩwhereMis the average local\nmagnetization, M2=∝angb∇acketleftM2∝angb∇acket∇ight.\nThe last two terms in Eq. (2) describe dissipation.\nαis called the Gilbert damping and βparametrizes\nthe dissipative spin transfer torque. The new damping\nterm proportional to α′was introduced (for β′= 0) in\nRefs. [20,21]. It arises from the ohmic damping of elec-\ntrons coupled by Berry phases to the spin texture as can\nbe seen by rewriting Eq. (2) in the form\n−δF\nδˆΩ=MˆΩ×(∂t+vs∇)ˆΩ+αM/parenleftBig\n∂t+β\nαvs∇/parenrightBig\nˆΩ\n+MˆΩ×α′/bracketleftBig\nEe\ni+β′\nα′(vs×Be)i/bracketrightBig\n∂iˆΩ.(3)whereEe\ni=ˆΩ·(∂iˆΩ×∂tˆΩ) can be interpreted as the\nemergent electric field and Be\ni=1\n2ǫijkˆΩ·(∂jˆΩ×∂kˆΩ) as\nthe emergent magnetic field26,27. These fields describe\nthe forces on the electrons arising from Berry phases\nwhich they pick up when their spin adiabatically follows\nˆΩ(r,t). They couple to the spin rather to the charge:\nelectrons with magnetic moment parallel (antiparallel)\ntoˆΩcarry the ”emergent charge” −1/2 (+1/2), respec-\ntively. For vs= 0 the change of the free energy density\nis given by\n∂tF=δF\nδˆΩ∂tˆΩ=−αM(∂tˆΩ)2−α′M(Ee)2.(4)\nwhich shows that the last term describes the dissi-\npated power ∝(Ee)2arising from the emergent electric\nfield.α′Mis therefore approximately given by the spin-\nconductivity σs.\nWe have also added a new β′–term. The presence of\nsuch a term becomes evident if one considers the special\ncaseofaGalileaninvariantsystem. In thiscase, allforces\nhave to cancel when the magnetic structure is comoving\nwiththe conductionelectrons, ˆΩ(r,t) =ˆΩ(r−vst). This\nis only possible for α=βandα′=β′. Solids are not\nGalilean invariant and therefore β′is different from α′\nbut one can, nevertheless, expect that the two quantities\nare of similar order of magnitude.\nWhich of the damping terms will dominate? As\npointed out in Refs. [20,21], the naive argument, that the\nα′terms are suppressed compared to the αterms as they\ncontain two more derivatives, is not correct. The dis-\ntance of skyrmions is9proportional to 1 /λSO, whereλSO\nparametrizes the strength of spin-orbit coupling. While\ntheα′term has two more gradients compared to the α\nterm, the contribution arising from α′is, nevertheless,\nof the same order in powers of λSO, if we assume that\nαarises only from spin-orbit coupling, α∝λ2\nSO, while\nα′∝λ0\nSO(ohmic damping (see above) does not require\nspin-orbit effects). As furthermore αis proportional to a\nscattering rate while α′is proportional to a conductivity\nand therefore the scattering time20,21,α′andβ′might\nbe the dominating damping terms in good metals.\nIII. DYNAMICS OF SKYRMIONS\nOur goal is to describe both the drift and the rotation\nofthe skyrmion lattice in the limit of small currentdensi-\nties and small magnetic or thermal gradients. We there-\nforeassumethat vsissmallcomparedtoallcharacteristic\nvelocity scales of the skyrmion lattice (e.g. Tc−Tmulti-\nplied with the skyrmion distance). The gradients should\nbe so small that the total change across a domain of ra-\ndiusrdremains small, rd∇λ≪λwhereλisBorTc−T\nfor magnetic or thermal gradients, respectively. In this\nlimit, both the drift velocities vd/lessorsimilarvsand the angular\nvelocity ∂tφ∝vs·∇λcharacterizing rotational motion\nremain small. Below we will show, that even ∂tφrd, the4\nvelocity at the boundary of the domain remains small in\nthe considered limit.\nWe can therefore neglect macroscopic deformations of\nthe magnetic structure and consider the following ansatz\nˆΩ(r,t) =Rφ(t)·ˆΩ0/parenleftBig\nR−1\nφ(t)·(r−vdt)/parenrightBig\n(5)\nHereˆΩ0(r) describes the static skyrmion lattice, Rφis\na matrix describing a rotation by the angle φaround the\ndirection of the skyrmion lines (i.e. around the field di-\nrection when anisotropies are neglected, which will be\nassumed in the following) and vdtdescribes the loca-\ntion of the center of the skyrmion domain. This ansatz\ndescribes a magnetic domain which rotates around its\ncenter, while the center is moving with the velocity vd.\nWhen the torque forces are too weak to induce a steady-\nstate rotation, such that ∂tφ= 0, we will study rotations\nby the finite angle φas in the experiment of Ref. [9].\nA. Drift of domains\nTo obtain an equation for the drift velocity vdwe fol-\nlow Thiele19and project Eq. (3) onto the translational\nmode by multiplying Eq. (3) with ∂iˆΩand integrating\nover a unit cell (UC). We thereby obtain to order ( ∇λ)0\n(where no rotations occur) an equation for the force per\n2d magnetic unit cell (and per length)28\nG×(vs−vd)+D(˜βvs−˜αvd)+Fpin= 0 (6)\nGi=/integraldisplay\nUCd2rMBe\ni=GˆBi,G= 4πMW\nDij=/integraldisplay\nUCd2rM∂iˆΩ∂jˆΩ=DPij\nD′=/integraldisplay\nUCd2rM(Be)2\n˜α=α+α′D′/Dand˜β=β+β′D′/D\nHere the first term describes the Magnus force which is\nproportionalto the topologicalwinding number Wwhich\nis for the skyrmion lattice exactly given by W=−1.\nGis called the gyromagnetic coupling vector following\nThiele19. The second term are the dissipative forces with\nthe projector Pinto the plane perpendicular to B,P=\n(1−ˆB·ˆBT).\nBesides the forces discussed above, also pinning forces,\ndescribed by the last term in Eq. (6), have to be con-\nsidered. Formally, they are encoded in spatial fluctua-\ntions of δF/δˆΩin Eq. (3). The Thiele approach, used\nabove, which considers only a global shift (or a global\nrotation18, see below) of the magnetic structure does not\ncapture these pinning effects as for a perfectly rigid mag-\nnetic structure, random pinning forces average to zero,\nsuch that no net effect remains in the limit of a large\ndomain. To describe pinning, it is necessary14,15to take\ninto account that the magnetic structure adjusts locallyto the pinning forces, a complicated problem for which\npresently no full solution exists17,29and which is far be-\nyond the scope of the present paper. Instead, we use a\nphenomenological ansatz and write for a finite drift ve-\nlocityvd\nFpin=−4πMvpinf(vd/vpin)ˆvd (7)\nto describe a net pinning force, which is oriented op-\nposite to the direction of motion. Its strength, which\ndepends both on the number (and nature) of defects re-\nsponsible for pinnning and the elastic properties of the\nskyrmion lattice, is parametrized by the ‘pinning veloc-\nity’vpin. The function f(x) withf(x→0) = 1 and\nf(x→ ∞) =xνparametrizes the non-linear dependence\nof the pinning force on the velocity. Presently, it is not\nclear to what extent f(x) depends on microscopic details\nand also the exponent νis not known. For large driving\nvelocities, however, pinning becomes less and less impor-\ntant (ν <1)16,17,29. If the driving forces are smaller\nthan the force 4 πMvpin, needed to depin the lattice, vd\nvanishes and the pinning forces cancel exactly the driv-\ning forces. Note that we do not consider creep, i.e. a\ntinymotiondrivenbythermal(orquantum)fluctuations,\nwhich occurs even in the pinning regime17. If the dissi-\npative forces can be neglected, it is in principle possible\nto obtain f(x) from a measurement of the velocity of the\nskyrmion lattice27.\nIn the limit vs≫vpin, whereFpincan be neglected,\nwe solve Eq. (6) for vs⊥Bto obtain\nvd=˜β\n˜αvs+˜α−˜β\n˜α3(D/G)2+ ˜α/parenleftBig\nvs+ ˜αD\nGˆB×vs/parenrightBig\n(8)\nwith ˜α=α+α′D′/Dand˜β=β+β′D′/D.\nB. Rotational torques\nBy symmetry, a small uniform current cannot induce\nany rotational torques on a skyrmion lattice with per-\nfect sixfold rotation symmetry and therefore all effects\narise from gradients. To derive an equation for the ro-\ntational torques which determine the rotations around\ntheBaxis, we follow18a similar procedure as used for\nthe translations by multiplying (3) by the generator of\nrotations applied to ˆΩ\n∂φˆΩ=ˆB׈Ω−(ˆB(∆r×∇))ˆΩ (9)\nwith∆r=r−vdtandintegratingover r. Thisprocedure\nleads to several types of contributions.\nFor the first type of contribution, we observe that the\nsecond term in Eq. (9), linear in ∆ r, is much larger than\nthe first one which we can therefore neglect whenever\nthe second term contributes. The second term induces\ntorques of the form r×fwhere the force fiis obtained\nby multiplying ∇iˆΩwith the terms of Eq. (3). In the5\npresence of gradients of the parameter λwe obtain\n/integraldisplay\nˆB·[r×f(λ(r))]≈/integraldisplay/parenleftBig\nˆB·[r×∂λf]/parenrightBig\n(r·∇λ)\n≈A\n4πˆB·[∇λ×∂λ/integraldisplay\nf] (10)\nwhereAis the area of the domain. Here it is essential to\ntake the derivative with respect to λforfixedvdreflect-\ning that due to the rigidity of the skyrmion crystal vdis\nconstant across the domain. As the sum of all relevant\nforces vanishes [Eq. (6)],/summationtext\nifi(λ,vd) = 0, one obtains\nd\ndλ/summationtext\nifi= 0 while∂\n∂λ/summationtext\nifi/vextendsingle/vextendsingle\nvdis finite. In Eq. (10)\nwe have implicitely assumed a symmetrically shaped do-\nmain, where integrals odd in rvanish. In general, there\nwill also be a shape dependent torque Tshapearising even\nin the absence of a gradient. As its sign is random, it\ncan easily be distinguished from the other torques (and\nappears to be relatively small in the MnSi experiments9).\nMore difficult is the question what happens at the inter-\nface of different domains or when a domain comes close\nto the surface ofthe sample. Nominally surfaceforcesare\nsuppressed by a factor proportional to 1 /√\nAcompared\nto the bulk terms considered above but the relevant pref-\nactors are difficult to estimate. We will neglect in the\nfollowing formulas both extra surface forces and shape\ndependent torques.\nA different contribution arises from the time deriva-\ntives∂tˆΩ=∂tφ∂φˆΩ−(vd∇)ˆΩin Eq. (3). The contri-\nbution proportional to vdis of the form discussed above.\nThe term proportionalto ∂tφleads to extra torquesinde-\npendent of ∇λ. By combiningthe linearterm in ∆ rfrom\n∂φˆΩwith the second term of Eq. (9) we obtain for exam-\nple the contribution α∂tφ/integraltext\nM[(ˆB[∆r×∇])Ω]2which is\nalso linear in A. Physically this term describes the fric-\ntional torque which is linear in the angular velocity ∂tφ.\nThe frictional torque per volume is proportional to Abe-\ncause the velocity and therefore the frictional forces grow\nlinearly with the distance from the center of the rotating\ndomain.\nFinally, a contribution exists which is independent of\nthe gradients ∇λ, the angular velocity ∂tφand ofvs.\nThis contribution describesthat in the absence ofanyex-\nternal perturbation the skyrmion lattice has a preferred\norientation relative to the atomic lattice. Such terms ex-\npressthat angularmomentum canbe transferreddirectly\nfrom the skyrmion lattice to the underlying atomic lat-\nticemediatedbyspin-orbitcouplingandsmallanisotropy\nterms (not included in Eq. (1)). These terms have been\ndiscussed in detail in Ref. [18]. This torque per unit cell\nTL=−/integraldisplay\nUCd2rδF\nδˆΩ(ˆGrotˆΩ) =−∂FUC\n∂φ≈ −χsin(6φ)\n(11)\ncan be expressed by the change of free energy per unit\ncell,FUC, upon rotation by the angle φ, where φ= 0\nreflects the equilibrium position and sin6 φreflects the\nsixfold symmetry of the skyrmion lattice. As has been\ndiscussed in Ref. [10], the absolute value of χin materials-1.5-1-0.500.511.5\nλ  Vs-15-10-5051015 φ  φ \n-1.5-1-0.500.511.50\n-1.5-1-0.500.511.5\n ω γ ω γ ∆.\nFigure 2: Rotation angle φ(in units of 1◦) and angular ve-\nlocity ¯ω(times the prefactor γ) as a function of ∇λ·Vsde-\ntermined from Eq. (13).\nlike MnSi is tiny as it arises only to high order in spin-\norbitcoupling and, in contrastto allother terms, it is not\nlinear in the size of the domain. Nevertheless, we have\nto consider this term, as it is the leading contribution\narising to zeroth order in ∇λandvs.\nBalancing all torques (per unit cell) we obtain as our\ncentral result\n0 =TL+TG+Tpin+TD (12)\nTG=A\n4π∇λ·/bracketleftbigg∂(Gvs)\n∂λ−∂G\n∂λvd/bracketrightbigg\nTpin=A\n4π∇λ·[ˆB׈vd]∂Fpin\n∂λ, Fpin≡ |Fpin|\nTD=−A˜αD\n2π∂tφ\n−A\n4π∇λ·/bracketleftbigg\nˆB×/parenleftBig∂(D˜βvs)\n∂λ−∂(D˜α)\n∂λvd/parenrightBig/bracketrightbigg\nThe direction of the torques, which depends on the rel-\native orientation of velocities and currents, is for vd= 0\n(and∂tφ= 0) fully consistent with the simple picture\nshown in Fig. 1: the dissipative torques TDarise when\ngradient and current are perpendicular to each other\nwhile the reactive torque TGarising from the Magnus\nforce is activated for a parallel alignment of gradients\nand currents. For finite vd, however, this simple intu-\nitive picture cannot be used especially as some of the\ntorques tend to cancel when vdapproaches vs.\nC. Rotation angle and angular velocity\nEq. (12) can be rewritten in the compact form\nsin6φ=−γ∂tφ+∇λ·Vs (13)\nwhereγ=A˜αD\n2πχand the vector Vs=Vs[vs] can be ob-\ntainedbyfirstsolvingEq.(6)toobtain vdasafunctionof6\nvs. This function is inserted into Eq. (12) which, finally,\nis devided by −χ. The function Vs[vs] withVs[0] = 0 is\nproportional to the area Aof the domain and encodes all\ninformation how the current couples to small gradients\nand includes contributions from Magnus forces, dissipa-\ntive forces and pinning.\n1. Dependence on size of gradients\nQualitatively, three different regimes have to be dis-\ntinguished. For j < jc, when pinning forces cancel all\nreactive and dissipative forces, there is neither a motion\nnor a rotation of the skyrmion lattice, Vs= 0,φ= 0,\nwithin our approximation. Note, however, that it is\nwell known from the physics of charge density waves or\nvortices17that even below jca slow creep motion is pos-\nsible. Whether during this creep also rotations are possi-\nble is unclear, but the rather sharp onset of the rotation\nin the experiments of Ref. 10, see Fig. 8, seems to con-\ntradict a scenario of pronounced rotations during creep.\nForj > jc, the domains move and Vswill generally be\nfinite. In this case, one can control the size and direction\nof rotations by the size of ∇λas shown in Fig. 2. For\n|∇λ·Vs|<1, one obtains a solution where ∂tφ= 0 but\nthe gradients induce a rotation by a finite angle\nφ=1\n6arcsin∇λ·Vs, (14)\nwhich grows upon increasing ∇λfrom zero until it\nreachesthe maximalpossiblevalue π/12 = 15◦(rotations\nby an average angle of 10◦have already been observed10,\nsee Fig. 8). For |∇λ·Vs|>1 the domain rotates (see\nFig. 2) with the (average) angular velocity\n¯ω=/radicalbig\n(∇λ·Vs)2−1\nγ(15)\nand Eq. (13) is solved by\nφ(t) =1\n3arctan/bracketleftBigg\n1+γ¯ωtan(3¯ωt)/radicalbig\n1+γ2¯ω2/bracketrightBigg\n.(16)\ndisplayed in the inset of Fig. 3. As both γandVsare\nlinear in the area Aof the domain, ¯ ω≈(∇λ·Vs)/γbe-\ncomesindependent ofthe domainsizefor A→ ∞. Inthis\nlimit, the domain rotates continuously, φ= ¯ωt. Close to\nthe threshold, ∇λ·Vs= 1, however, the rotation be-\ncomes very slow close to an angle of 15◦(plus multiples\nof 60◦).\nA way to detect the rotation of the magnetization is\nto exploit the emergent electric field Eewhich obtains\na contribution proportional to ∝∂tφand can be mea-\nsured in a Hall experiment27. In Fig. 3 we therefore\nshow the modulus of the Fourier components, |cn|=\n|/integraltext\nei6¯ωnt∂tφdt|of∂tφas a function of ∇λ·Vs. At the\nthreshold, all Fourier components are of equal weight\nwhile for large gradients the rotation gets more uniform.0 1 2 3 4 5\nλ Vs0 010 1020 2030 3040 4050 5060 60cnc0c1c2c3c4c5\n0 0.1 0.2 0.3 0.4 0.5\nωt/2π060120180φ1.01\n1.3\n10∆.\nFigure 3: Inset: Rotation angle (in units of 1◦) as a function\nof time for three values of ∇λVs>1, see Eq. (16). For\ntorques close to the value where rotations sets in, the rotat ion\nis strongly anharmonic. This can also be seen by considering\nthe Fourier coefficients cn=|/integraltext2π/6¯ω\n0∂tφein6¯ωtdt|shown in\nthe main panel as a function of ∇λVs.\nFor fixed ¯ ωthe velocities at the boundary of the do-\nmain,vb= ¯ωrd, grow linearly with the radius of the do-\nmainrd. As we have assumed that the gradients across\nthe sample and therefore also across a single domain are\nsmall,rd∇λ≪λ, the velocities nevertheless remain\nsmall,vb≪ |Vs|λ/γ/lessorsimilarvs/˜α. While our estimate does\nnot rule out that vbcan become somewhat larger than vs\norvd, we expect that the typical situation is that the ve-\nlocityvbarising from the rotation remains smaller than\nthe overall drift velocity of the domain vd. This estimate\nalso implies that violent phenomena like the breakup of\ndomains due to the rotation will probably not occur.\n2. Domain size dependence and angular distribution\nIn a real system, there will always be a distribution\nof domain sizes A. BothVsandγare linear in Aand\ntherefore both the rotation angle (14) and the angular\nvelocity (15) will in general depend on the domain size\nand therefore on the distribution of domains.\nOnly in the limit |∇λ·Vs| ≫1, the dependence on A\ncancels in Eq. (15) and all domains rotate approximately\nwiththesameangularvelocity. For |∇λ·Vs|/lessorsimilar1onewill\nin general obtain a distribution of rotation angles which\ncan be calculated from the distribution of domain sizes\nPd(A). For the static domains only angles up to 15◦are\npossible with\nPs\nφ=/integraldisplayAc\n0dAPd(A)δ(φ−arcsin(A/Ac)\n6)\n= 6Accos(6φ)Pd(Acsin(6φ)) for 0 ≤φ≤π\n12(17)7\n0 10 20 30 40 50 60\n φ 02468PφA0/Ac=0.25\nA0/Ac=0.5\nA0/Ac=1.0\nA0/Ac=2.0\nA0/Ac=4.0\nFigure 4: Angular distribution Pφof the rotation angle of the\nskyrmion lattice for various values of A0/Ac∝ ∇λ(see text).\nHere we assumed a distribution of domain sizes of the form\nPd(A) =e−A/A0A\nA2\n0. While static domains contribute only for\n0≤φ≤15◦one obtains a smooth angular distribution when\none takes the rotating domains with A > A cinto account.\nwhereAc=A/(∇λ·Vs) is the size of a ‘critical’ domain\nwhich just starts to rotate continuously.\nThe continuously rotating domains also have a non-\ntrivial angular distribution as their rotation will be\nslowed down when the counterforces are strongest, i.e.,\nforφ= 15◦, see inset of Fig. 3. The angular distribution,\nPr\nφ, of the rotating domains is calculated from distribu-\ntion ofdomainsizes, Pd(A), andthe angulardistribution,\npr\nφ(A), of a single domain\nPr\nφ=/integraldisplay∞\nAcdAPd(A)pr\nφ(A)\npr\nφ(A) =1\nT/integraldisplayT\n0δ(φ−φ(t))dt=1\nT∂tφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nφ(t)=φ\n=3\nπ/radicalbig\nA2−A2c\nA−Acsin6φ(18)\nwhereT= 2π/(6¯ω). While both Ps\nφandPr\nφare non-\nanalytic at φ= 15◦, the total distribution, Pφ=Ps\nφ+Pr\nφ\nissmoothfor φ >0andnormalizedto1,/integraltext2π/6\n0Pφdφ= 1.\nIn Fig. 4 we show Pφassuming the domain distribution\nPd(A) =e−A/A0A\nA2\n0for various values of A0/Ac.\nIn elastic neutron scattering, the skyrmion phase is\nobservedby six Braggspots forming a regularhexagonin\naplaneperpendiculartothemagneticfield. Arotationof\nthe skyrmion domain results in a rotation of these Bragg\nspots. Therefore the angular distribution Pφof rotation\nangles is directly observable (see Sec. IIID below) by\nmeasuring the scattering intensity as a function of angle.\nBy comparing angulardistributions for different strength\nof the current or gradient, one can – at least in principle\n– obtain not only Acasa function of ∇λorjbut alsothedistribution of domain sizes. The latter can be extracted\nmost easily in the regime where most of the domains\ndo not rotate continuously by plotting Pφ/cos6φas a\nfunction of sin6 φusing Eq. (17).\n3. Dependence on strength of current\nWhile the behavior of φand ¯ωas a function of ∇λis\nrather universal and independent of microscopic details,\nits dependence on the strength of the current for fixed\n∇λis much more complex. As discussed above, Vs= 0\nforj < jc. Directly at jc, when the domain starts to\nmove with vd≈0,Vsjumps to the finite value\nVs|vs=vpin=−A\n4πχ/bracketleftbigg/parenleftbigg\n−∂Gvs\n∂λ+Gvs\nFpin∂Fpin\n∂λ/parenrightbigg\n+ˆB×/parenleftBigg\n∂D˜βvs\n∂λ−D˜βvs\nFpin∂Fpin\n∂λ/parenrightBigg/bracketrightBigg\n.(19)\nNote that the jump is independent of αandα′as well as\nof their gradients, as the skyrmions are not moving di-\nrectlyatthe depinningtransition(seeFig.6). Depending\non the direction and size of ∇λ, the jump of Vseither\nleads to a jump of the rotation angle for |∇λ·Vs|<1 or\nimmediately to a continuous rotation for |∇λ·Vs|>1.\nUpon increasing the current, ∇λ·Vscan either in-\ncrease, decrease or even change its sign depending on\n(i) the direction of ∇λand (ii) on the question which\nof the forces changes most strongly when varying λ(i.e.,\ntemperature or magnetic field).\nMotivatedbyexistingexperimentaldata(discussedbe-\nlow in Sec. IIID) we study the case of a temperature gra-\ndient,λ=t, based on the following assumptions. First,\nwe assume that all damping constants are temperature\nindependent (this assumption is relaxed later). Second,\nwe need also a theory for the temperature dependence of\nthe pinning force. Here we use the experimental obser-\nvation27that the critical current is almost temperature\nindependent at least for a certain range of temperatures.\nWithin our theory, Eqs. (6) and (7), this implies that\nall temperature dependence of Fpin(i.e., the dependence\non the parameter tin Eq. (1)) arises from the tempera-\nture dependence of the magnetization Mwhich we cal-\nculate from the Ginzburg-Landau theory (1). From the\nGinzburg-Landautheory, we obtain alsothe temperature\ndependence of the other parameters, see Fig. 7.\nIn Fig. 5 we show a typical result (for temperature-\nindependent dissipation constants) for the rotation angle\nand angular velocity of a skyrmion domain as a function\nofvsin the presence of a temperature gradient. For a\ntemperature gradientperpendicular to the current(lower\npanel of Fig. 5), the rotation angle increases after the\ninitial jump. For the gradient parallel to the current,\nhowever, we obtain that the rotation angle dropsafter\nthe initial jump (upper panel). For larger values of vs\nthe angle rises again until it reaches its maximal value of8\n-30 -20 -10 0 10 20 30vs-15-10-5051015φφ\n-2-1012\nω  γω γ\nvs\n∆t\n-10 -5 0 510vs-15-10-5051015φφ\n-2-1012\nω  γω γ\nt    vs\n∆\nFigure 5: Rotation angle φ(in units of 1◦) and angular veloc-\nity,γ¯ω, as a function of vsfor a temperature gradient parallel\n(∇t∝ba∇dblvs,∇t= (−0.1,0,0), upper panel) and perpendicular\n(∇t⊥vs,∇t= (0,−0.05,0), lower panel) to the current\n(α= 0.2,β= 0.45,α′= 0.01,β′= 0.2,A/χ= 200,t=\n−1,B= (0,0,1/√\n2),vpin= 1,f= 1). For both geometries\none observes a jump of φatvs≈vpinfrom zero to a finite\nrotation angle. After the initial jump the rotation angle in -\ncreases for the perpendicular configuration (panel b) while for\nthe parallel arrangement first a drop and then an increase up\nto the maximal angle of 15◦occurs. For larger vsa contin-\nuous rotation characterized by the angular velocity ¯ ωsets in\nfor both configurations. For the calculation we assumed that\nthe damping parameters and vpinare independent of t.\n15◦. This qualitative shape of the curve appears to be\nrather independent of the precise values of the various\nparameters ifwe assume that all damping parameters\nare temperature independent.\nIn Fig. 6 we plot the rotation angle for small cur-\nrent densities taking an extra effect into account which is\npresent in the experiments described in Ref. [10]: as the\ntemperature gradients are induced by the currents, they\ngrowquadraticallywith vs. This doesnot giverisetoany\nqualitative changes. The thin blue curve Fig. 6 thereby\nreflects the same physics as the corresponding curve in-1 0 1vs-15 -15-10 -10-5 -50 05 510 1015 15φφ (   α = 0.035   t)\nφ (   α = 0 )\n∆ ∆∆∆t      vs\n-1 0 1vs-15 -15-10 -10-5 -50 05 510 1015 15φφ (   α = 0.035   t)\nφ (   α = 0 )\nt     vs\n∆\n∆∆\n∆\nFigure 6: Rotation angle φ(in units of 1◦) as a function of vs\nfor a temperature gradient parallel ( ∇t∝ba∇dblvs, upper panel) and\nperpendicular ( ∇t⊥vs, lower panel) to the current. The pa-\nrameters are the same as in Fig. 5 with two exceptions. First,\nwe have taken into account that in the experiments of Ref. 10\nthe temperature gradient grow with the square of the applied\ncurrent, ∇t= (−0.1v2\ns,0,0) and∇t= (0,−0.05v2\ns,0)), for\ncurrent parallel an perpendicular to vs, respectively. For the\nthin blue curve we assumed (as in Fig. 5) that the damping\nconstants are independent of twhile for the thick green curve\na weak temperature dependence of the damping constant α,\n∇α= 0.035∇t, was assumed. This parameter has been cho-\nsen to reflect the experimental observation, see Fig. 8. For\neven stronger currents (not measured experimentally and no t\nshown in the figure) the size of the torque drops again and a\nfinite rotation angle is obtained for 1 .57/lessorsimilarvs/lessorsimilar2.53 in the\nparallel configuration with the temperature dependent damp -\ning constant.\nFig. 5 (note the different scale on the xaxis). The thick\ngreen curve of Fig. 6 shows that one can, however, ob-\ntain qualitative different results (an increase rather than\na reduction of the rotation angle after the initial jump\nforTgradients parallel to the current, upper panel) by\nincluding a small temperature dependence of the Gilbert\ndamping α. As we will discuss in Sec. IIID, this can9\n-1.1 -1 -0.9 -0.8t468101214 -G,  D ,  D’-G\nD\nD'\nFigure 7: Change of G,DandD′defined in Eq. (6) with\ntemperature t. The applied magnetic field is h/√\n2(0,0,1).\nDashed lines are for h= 0.9 and continuous lines for h= 1.1.\nreproduce qualitatively the experimentally observed be-\nhavior.\n4. Dependence on orientation of gradients\nFig. 5 shows that the rotational torques on the sys-\ntem depend strongly on the relative orientation of gradi-\nent and current. More importantly, one probes different\nphysical mechanism for gradients parallel or perpendicu-\nlartothecurrent. Thiseffectwasalreadydiscussedinthe\nintroduction, see Fig. 1, where, however, only the simple\ncase of a static domain without pinning was described.\nIn reality, the situation is more complex. All directional\ninformation is encoded in the function Vs(vs) which can\nbe obtained by first solving Eq. (6) to obtain vdand then\ncomparing Eqs. (12) and (13). Unfortunately, a rather\nlarge number of unknown parameters (most importantly,\nthe pinning forces and their dependence on λ) enters the\ndescription. Therefore we will discuss in the following\nonly a few limiting cases.\nA drastically simplified picture occurs in regimes when\nonly two forces dominate in Eq. (6). For example, close\nto the pinning transition, the Magnus force is of the\nsame order as the pinning force while the two dissipa-\ntive forces are typically much smaller. In this case one\ncan use Eq. (6) to show that ˆvdbecomes proportional\ntoˆB×(vs−vd). Thus, for an λ-independent vs, both\nthe reactive rotational coupling vector and the rotational\npinning vector become proportional to ∇λ·(vs−vd)\n(here we neglect a possible λ-dependence of vs). There-\nfore theratioof the component of Vsparallel ( Vs/bardbl) and\nperpendicular ( Vs⊥) tovsdepends only on the directionin which the skyrmion lattice drifts.\nVs/bardbl\nVs⊥≈(vs−vd)/bardbl\n(vs−vd)⊥=−v⊥\nd\nv/bardbl\nd(20)\nThe ratioVs/bardbl\nVs⊥can be obtained experimentally by mea-\nsuring the rotation angle or the angular velocity for ∇λ\nparallel and perpendicular to the current, from which\none can obtain directlyVs/bardbl\nVs⊥using Eqs. (14) and (15).\nFor small angles, arcsin x≈x, for example, one obtains\nVs/bardbl\nVs⊥directly from the ratio of the two rotation angles. A\ndifferent, but probably more precise way to determine\nthis ratio is to find experimentally the “magic angle”\nφmof gradient vs. current, where all rotations vanish,\n∇λ·Vs= 0. In this case one obtains\nVs/bardbl\nVs⊥=1\ntanφm(21)\nThis should allow for a quantitative determination of\nv⊥\nd/v/bardbl\nd. Asv/bardbl\ndcan be measured independently us-\ning emergent electric fields generated by the motion of\nskyrmions27, one can obtain the complete information\non the drift motion by combining both experiments. It\nis also instructive to compare skyrmions and vortices in\na superconductor. Vortices and skyrmions follow essen-\ntially the same equation of motions, Eq. (6). The rele-\nvant parameters (and therefore also the pinning physics)\nare, however, rather different. For vortices in conven-\ntional superconductors17,29the dissipation is very large\nDα≫ G. Therefore, vortices drift – up to small correc-\ntions – predominantly perpendicular to the current while\nfor magnetic skyrmions we expect that at least not too\nclose to the depinning transition, the motion is domi-\nnantly parallel to the current.\nIn the limit where the pinning forces can be neglected,\ni.e.,vs≫vpin, to linear order in ˜βand ˜αthe vector Vs\nis given by\nVs=−A\n4πχ/parenleftBig\nˆB×vs/parenrightBig/parenleftBigg\n(˜β−˜α)∂G\n∂λD\nG+∂D(˜β−˜α)\n∂λ/parenrightBigg\n(22)\n=−A\n4πχ/parenleftBig\nˆB×vs/parenrightBig1\nG∂\n∂λ/parenleftBig\nDG(˜β−˜α)/parenrightBig\n(23)\nHere we also neglected a possible λ-dependence of vs. In\nthis limit the rotation can be induced primarily by gra-\ndients perpendicular to vsreflecting that the motion of\nskyrmions is mainly parallel to the current, see Eq. (20)\nand Eq. (8). This is also consistent with the behavior\nshown in Fig. 5 where we used a two-times smaller gradi-\nentfortheperpendicularconfigurationandobtainednev-\nertheless an onset of the rotational motion for values of\nvsmuch smaller than in the parallel configuration. Note\nthat in a Galilean invariant system, ˜ α=˜β, no torques\ncan be expected.10\nD. Experimental situation\nOur study is directly motivated by recent neutron\nscattering experiments in the skyrmion lattice phase of\nMnSi10. In the presence of a sufficiently large current,\na rotation of the magnetic diffraction pattern by a finite\nangle was observed when simultaneously a temperature\ngradient was present (only temperature gradients paral-\nlel to the current have been studied). The rotation angle\ncould be reversed by reversing either the direction of the\ncurrent, the direction of the magnetic field or the direc-\ntion of the temperature gradient. This clearly showed\nthat rotational torques in the experiment were driven by\nthe interplay of gradients and currents as studied in this\npaper.\nIn Fig. 8a we reproduce Fig. 3 (A) of Ref. [10], which\nshows the average rotation angle (defined as the maxi-\nmum of the azimuthal distribution of the scattering in-\ntensity) as a function of current density. Above a critical\ncurrent, j > jc, the rotation sets in. The rotation angle\ninitially increases abruptly, followed by a slower increase\nfor larger current densities. When comparing these re-\nsults with our theory one has to take into account that\nthe temperaturegradientin theexperiment wasnotinde-\npendent of the strength of the applied electrical current\ndensity as it originated in the resistive heating in the\nsample. Therefore the temperature gradient was grow-\ning with j2(i.e. the heating rate due to the electric cur-\nrent). This was taken into account in Fig. 6 as discussed\nabove. For a full quantitative comparison of theory and\nexperiment, it would be desirable to have data, where\nthe applied current as well as both the strength and the\ndirection of the gradients are changed independently. As\nsuch data is presently not available, we restrict ourselves\nto a few more qualitative observations.\nIn our theory we expect a jump of the rotation angle\natjc, which depends on the domain size. This appears\nto be consistent with the steep increase of the rotation\nangle as observed experimentally at jc, especially when\ntaking into account the experimental results are subject\nto a distribution of domain sizes.\nInterestingly, the experimentally observed increase of\nthe rotation angle after its initial jump is apparently\nnotconsistent with the predictions from the extended\nLandau-Lifshitz-Gilbert equation shown in Eq. (2) ifwe\nassumeα,α′,β,β′are independent of temperature. As\nshown in Fig. 6, we can, however, describe the experi-\nmentally observed behavior if we assume a weak temper-\nature dependence of the Gilbert damping.\nAn important question concerns, whether the existing\nexperiments already include evidence of some larger do-\nmainsthatrotatecontinuously. Fig.8ashowsthatforthe\nlargest currents averagerotation angles of up to 10◦have\nbeen obtained. As this is rather close to the maximally\npossiblevalueof15◦forstaticdomains, thissuggeststhat\ncontinuously rotating domains are either already present\nin the system or may be reached by using slightly larger\ncurrents or temperature gradients.We have therefore investigated the angular distribu-\ntion of the scattering pattern using the same set of ex-\nperimental data analyzed in Ref. [10] (technical details\nof the experimental setup are reported in this paper). In\nFig. 8b weshowthe azimuthal intensity distribution with\nand without applied current. Already for zero current\na substantial broadening of the intensity distribution is\nobserved. The origin of this broadening are demagneti-\nzation effects which lead to small variations of the orien-\ntation of the local magnetic fields in the sample tracked\nclosely by the skyrmions. It has been shown11that this\neffect can be avoided in thin samples when illuminating\nonly the central part of this sample. For the existing\ndata this implies that a quantitative analysis of Pφis not\npossible. We observed that the measured experimental\ndistribution of angles extents up just to 15◦. Therefore,\nfrom the present data we can neither claim nor exclude\nthat continuously rotating domains already exist for this\nset of data but slightly larger current densities or gradi-\nents should be sufficient to create those.\nIV. CONCLUSIONS\nThemagneticskyrmionlattices,firstobservedinMnSi,\nhave by now been observed in a wide range of cubic,\nchiral materials including insulators30,31, doped semicon-\nductors32andgoodmetals9,33. Thisisexpectedfromthe-\nory: in any material with B20 symmetry, which would\nbe ferromagnetic in the absence of spin-orbit coupling,\nweak Dzyaloshinskii Moriya interaction induce skyrmion\nlattices in a small magnetic field. While in bulk they are\nonly stabilized in a small temperature window by ther-\nmal fluctuations close to the critical temperature, they\nare much more stable in thin films34,35.\nFrom the viewpoint of spintronics, such skyrmions\nare ideal model systems to investigate the coupling of\nelectric-, thermal- or spin currents to magnetic textures:\n(i) the coupling by Berry phases to the quantized wind-\ningnumber providesa universalmechanism tocreateeffi-\ncientlyMagnusforces,(ii)skyrmionlatticecanbemanip-\nulated by extremely small forces induced by ultrasmall\ncurrents10,27, (iii) the small currents imply that also new\ntypes of experiments (e.g., neutron scattering on bulk\nsamples) are possible.\nWe think that the investigation of the rotational dy-\nnamics of skyrmion domains provides a very useful\nmethod to learn in more detail which forces affect the dy-\nnamics of the magnetic texture. As we have shown, the\nrotational torques can be controlled by both the strength\nand the direction of field- or temperature gradients in\ncombination with electric currents. They react very sen-\nsitively not only on the relative strength of the various\nforces but also on how the forces depend on temperature\nand field.\nWhile some aspects of the theory, e.g. the dependence\non the strength of the gradients, can be worked out in\ndetail, many other questions remain open. An important11\nFigure 8: a) Average rotation angle ∆ φ(in units of 1◦) of the\nskyrmion lattice in MnSi measured by neutron scattering in\nthe presence of an electric current and a temperature gradie nt\nparallel to the current. The figure is taken from Ref. [10]\nwhere further details on the experimental setup can be found .\nb) Angular distribution Pφof the intensity normalized to 1\nfor currents of strength j= 0 (black diamonds) and j≈\n−2.07·106A/m2forT= 27.4K (red circles). The lines are\nGaussian fits servingas aguide to theeye. The distribution o f\nangles extentsup the maximally possible rotation angle of 1 5◦\nwhich suggests that some of the larger domains are rotating\nwith finite angular velocity for this parameter range.question is, for example, to identify the leading damping\nmechanisms and their dependence on temperature and\nfield. Also an understanding of the interplay of pinning\nphysics, damping and the motion of magnetic textures\nis required to control spin torque effects. Here future\nrotation experiments are expected to give valuable in-\nformation. Furthermore, it will be interesting to study\nthe pinning physics in detail and to learn to what ex-\ntent skyrmions and vortices in superconductors behave\ndifferently.\nOnewaytoobservetherotationoftheskyrmionlattice\nis to investigate the angular distribution of the neutron\nscattering pattern as discussed in Sec. IIIC. This does,\nhowever, only provide indirect evidence on the expected\ncontinuous rotation of the skyrmion lattice. Therefore\nit would be interesting to observe the continuous rota-\ntion more directly. For example, one can use that time-\ndependent Berry phases arising from moving skyrmions\ninduce “emergent” electrical fields which can be directly\nmeasured27in a Hall experiment. Here it would be in-\nteresting to observe higher harmonics in the signal which\nare expected to appear close to the threshold where con-\ntinuous rotations set in, see Fig. 3.\nIn future, it might also be interesting to use instead of\nelectrical current other methods, e.g. pure spin currents\northermalcurrents, tomanipulateskyrmionlattices(e.g.\nin insulators). We expect that also in such systems the\ninvestigationofrotationalmotiondrivenbygradientswill\ngive useful insight in the control of magnetism beyond\nthermal equilibrium.\nAcknowledgments\nWe gratefully acknowledgediscussionswith M. Halder,\nM. 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In 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": "1205.1199v2.Fractional_wave_equation_and_damped_waves.pdf",        "content": "Fractional wave equation and damped waves\nYuri Luchko\nDepartment of Mathematics, Physics, and Chemistry\nBeuth Technical University of Applied Sciences Berlin\nLuxemburger Str. 10, 13353 Berlin, Germany\ne-mail: luchko@beuth-hochschule.de\nAbstract\nIn this paper, a fractional generalization of the wave equation that\ndescribes propagation of damped waves is considered. In contrast to\nthe fractional di\u000busion-wave equation, the fractional wave equation\ncontains fractional derivatives of the same order \u000b;1\u0014\u000b\u00142 both\nin space and in time. We show that this feature is a decisive factor\nfor inheriting some crucial characteristics of the wave equation like a\nconstant propagation velocity of both the maximum of its fundamental\nsolution and its gravity and mass centers. Moreover, the \frst, the\nsecond, and the Smith centrovelocities of the damped waves described\nby the fractional wave equation are constant and depend just on the\nequation order \u000b. The fundamental solution of the fractional wave\nequation is determined and shown to be a spatial probability density\nfunction evolving in time that possesses \fnite moments up to the order\n\u000b. To illustrate analytical \fndings, results of numerical calculations\nand numerous plots are presented.\nMSC 2010 : 26A33, 35C05, 35E05, 35L05, 45K05, 60E99\nKey Words : Caputo fractional derivative, Riesz fractional derivative, frac-\ntional wave equation, propagation velocity, gravity center, mass center, cen-\ntrovelocity, maximum of fundamental solution, damped waves\n1 Introduction\nDuring the last few decades, fractional order di\u000berential equations have been\nsuccessfully employed for modeling of many di\u000berent processes and systems,\nsee e.g. [27] for di\u000berent applications of derivatives and integrals of fractional\norder in physics, chemistry, engineering, astrophysics, etc. and [4] for appli-\ncations of fractional di\u000berential equations in classical mechanics, quantum\nmechanics, nuclear physics, hadron spectroscopy, and quantum \feld theory.\nFor other interesting models in form of fractional di\u000berential equations we\nrefer the reader to [3], [5], [9]-[12], [15]-[17], [20] to mention only few of many\nrecent publications.\n1arXiv:1205.1199v2  [math-ph]  15 May 2012Among many other applications, models for anomalous transport pro-\ncesses in form of time- and/or space-fractional advection-di\u000busion-wave\nequations enjoyed a particular attention and have been considered by a\nnumber of researches since 1980's. In particular, this kind of phenomena is\nknown to occur in viscoelastic media that combine characteristics of solid-\nlike materials that exhibit waves propagation and \ruid-like materials that\nsupport di\u000busion processes (see e.g. the recent book [17]). It is impor-\ntant to note that anomalous transport models are usually \frst formulated\nin stochastic form in terms of the so called continuous time random walk\nprocesses. The time- and/or space-fractional di\u000berential equations are then\nderived from the stochastic models for a special choice of the probability\ndensity functions with the in\fnite \frst or/and second moments (see e.g.\n[10], [12], or [20]).\nIt is well known that whereas di\u000busion equation describes a process,\nwhere a disturbance of the initial conditions spreads in\fnitely fast, the\npropagation velocity of the disturbance is constant for the wave equa-\ntion. In a certain sense, the time-fractional di\u000busion-wave equation of order\n\u000b;1< \u000b < 2 interpolates between these two di\u000berent behaviors: its re-\nsponse to a localized disturbance spreads in\fnitely fast, but the maximum\nof its fundamental solution disperses with a \fnite velocity v(t;\u000b) that is\ndetermined by the formula (see e.g. [6] or [14])\nv(t;\u000b) =C\u000bt\u000b\n2\u00001: (1)\nFor\u000b= 1 (di\u000busion), the propagation velocity is equal to zero because of\nC1= 0, for\u000b= 2 (wave propagation) it remains constant and is equal to\nC2= 1, whereas for all intermediate values of \u000bthe propagation velocity\nof the maximum point depends on time tand is a decreasing function that\nvaries from +1at timet= 0+ to zero as t!+1. This fact makes it\ndi\u000ecult to interpret solutions to the fractional di\u000busion-wave equation as\nwaves in case 1 <\u000b< 2.\nIn this paper, a fractional wave equation that contains fractional deriva-\ntives of the same order \u000b;1\u0014\u000b\u00142 both in space and in time is considered.\nThe fractional derivative in time is interpreted in the Caputo sense whereas\nthe space-fractional derivative is taken in form of the inverse operator to\nthe Riesz potential (Riesz fractional derivative). It turns out that this fea-\nture of the fractional wave equation is a decisive factor for inheriting some\ncrucial characteristics of the wave equation. In particular, we show that six\ndi\u000berent velocities of the damped waves that are described by the funda-\nmental solution of the fractional wave equation (propagation velocity of its\n2maximum, its gravity and mass centers, the \frst, the second, and the Smith\ncentrovelocities) are constant and depend just on the equation order \u000b.\nFrom the mathematical viewpoint the fractional wave equation we deal\nwith in this paper has been considered in all probability for the \frst time in\n[7], where an explicit formula for the fundamental solution of this equation\nwas derived. In [18], a space-time fractional di\u000busion-wave equation with\nthe Riesz-Feller derivative of order \u000b2(0;2] and skewness \u0012has been inves-\ntigated in detail. A particular case of this equation called neutral-fractional\ndi\u000busion equation that for \u0012= 0 corresponds to our fractional wave equa-\ntion has been shortly mentioned in [18]. Still, to the best knowledge of the\nauthor, both in-depth mathematical treatment and physical interpretation\nof the fractional wave equation seem to be not yet given in the literature.\nThe rest of the paper is organized as follows. In the 2nd section, basic\nde\fnitions, problem formulation, and some analytical results for an initial-\nvalue problem for a model one-dimensional fractional wave equation are\npresented. The fundamental solution G\u000bfor this problem is derived in terms\nof elementary functions for all values of \u000b;1\u0014\u000b < 2. Moreover, G\u000bis\ninterpreted as a spatial probability density function evolving in time whose\nmoments up to order \u000bare \fnite. For the fundamental solution G\u000bboth its\nmaximum location and its maximum value are determined in closed form.\nRemarkably, the product of the maximum location and the maximum value\nofG\u000bis time-independent and just a function of \u000b. Finally we show that\nboth the maximum and the gravity and mass centers of the fundamental\nsolutionG\u000bpropagate with constant velocities like in the case of the wave\nequation but in contrast to the wave equation ( \u000b= 2) these velocities are\ndi\u000berent each to other for a \fxed value of \u000b;1<\u000b< 2. Moreover, the \frst,\nthe second, and the Smith centrovelocities of the damped waves described by\nthe fractional wave equation are shown to be constants that depend just on\nthe equation order \u000b. To illustrate analytical \fndings, results of numerical\ncalculations, numerous plots, their physical interpretation and discussion\nare presented in 3rd section. The last section contains some conclusions and\nopen problems for further research.\n32 Analysis of the fractional wave equation\n2.1 Problem formulation\nIn this paper, we deal with the model one-dimensional fractional wave equa-\ntion\nD\u000b\ntu=@\u000bu\n@jxj\u000b; x2IR; t2IR+;1\u0014\u000b\u00142: (2)\nIn (2),u=u(x;t) is a real \feld variable,@\u000b\n@jxj\u000bis the Riesz space-fractional\nderivative of order \u000bthat is de\fned below, and D\u000b\ntis the Caputo time-\nfractional derivative of order \u000b:\n(D\u000bf)(t) := (In\u0000\u000bf(n))(t); n\u00001<\u000b\u0014n; n2IN (3)\nI\u000b; \u000b\u00150 being the Riemann-Liouville fractional integral\n(I\u000bf)(t) :=(1\n\u0000(\u000b)Rt\n0(t\u0000\u001c)\u000b\u00001f(\u001c)d\u001c; \u000b> 0;\nf(t); \u000b= 0;\nand \u0000 the Euler gamma function. For \u000b=n; n2IN, the Caputo fractional\nderivative coincides with the standard derivative of order n.\nAll quantities in (2) are supposed to be dimensionless, so that the coef-\n\fcient by the Riesz space-fractional derivative can be taken to be equal to\none without loss of generality.\nFor the equation (2), the initial-value problem\nu(x;0) ='(x);@u\n@t(x;0) = 0; x2IR (4)\nis considered. In this paper, we are mostly interested in behavior and prop-\nerties of the fundamental solution (Green function) G\u000bof the equation (2),\ni.e. its solution with the initial condition '(x) =\u000e(x),\u000ebeing the Dirac\ndelta function.\nFor a su\u000eciently well-behaved function fthe Riesz space-fractional\nderivative of order \u000b;0<\u000b\u00142 is de\fned as a pseudo-di\u000berential operator\nwith the symbol\u0000j\u0014j\u000b([23]):\n(Fd\u000b\ndjxj\u000bf)(\u0014) :=\u0000j\u0014j\u000b^f(\u0014); (5)\nFbeing the Fourier transform of a function f. For 0< \u000b < 2,\u000b6= 1, (5)\ncan be written in the form ([24])\nd\u000b\ndjxj\u000bf(x) =\u00001\n2\u0000(\u0000\u000b) cos(\u000b\u0019)Z1\n0f(x+\u0018)\u00002f(\u0018) +f(x\u0000\u0018)\n\u0018\u000b+1d\u0018: (6)\n4For\u000b= 1, the relation (5) can be interpreted in terms of the Hilbert trans-\nform\nd1\ndjxj1f(x) =\u00001\n\u0019d\ndxZ+1\n\u00001f(\u0018)\nx\u0000\u0018d\u0018;\nwhere the integral is understood in the sense of the Cauchy principal value.\nIn particular, the equation (2) with \u000b= 1 that we call modi\fed advection\nequation is written in the form\n@u\n@t=\u00001\n\u0019d\ndxZ+1\n\u00001f(\u0018)\nx\u0000\u0018d\u0018 (7)\nthat is of course di\u000berent from the standard advection equation.\nFor\u000b= 2, equation (2) is reduced to the one-dimensional wave equation.\nIn what follows, we focus on the case 1 \u0014\u000b < 2 because the case \u000b= 2\n(wave equation) is well studied in the literature.\n2.2 Fundamental solution of the fractional wave equation\nWe start our analysis by applying the Fourier transform with respect to\nthe space variable xto the equation (2) with 1 < \u000b < 2 and to the initial\nconditions (4) with '(x) =\u000e(x). Using de\fnition of the Riesz fractional\nderivative, for the Fourier transform ^G\u000bwe get the initial-value problem\n(^G(\u0014;0) = 1;\n@^G\n@t(\u0014;0) = 0(8)\nfor the fractional di\u000berential equation\n(D\u000b^G\u000b)(t) +j\u0014j\u000b^G\u000b(\u0014;t) = 0: (9)\nThe unique solution of (8), (9) is given by the expression (see e.g. [13])\n^G\u000b(\u0014;t) =E\u000b(\u0000j\u0014j\u000bt\u000b) (10)\nin terms of the Mittag-Le\u000fer function\nE\u000b(z) =1X\nk=0zk\n\u0000(1 +\u000bk); \u000b> 0: (11)\nThe well known formula (see e.g. [21])\nE\u000b(\u0000x) =\u0000mX\nk=1(\u0000x)\u0000k\n\u0000(1\u0000\u000bk)+O(jxj\u00001\u0000m); m2IN; x!+1\n5for asymptotics of the Mittag-Le\u000fer function that is valid for 0 < \u000b < 2\nand the formula (10) show that ^G\u000bbelongs toL1(IR) with respect to \u0014for\n1< \u000b < 2. Therefore we can apply the inverse Fourier transform and get\nthe representation\nG\u000b(x;t) =1\n2\u0019Z+1\n\u00001e\u0000i\u0014xE\f(\u0000j\u0014j\u000bt\f)d\u0014; x2IR; t> 0 (12)\nfor the Green function G\u000b. The last formula shows that the fundamental\nsolutionG\u000bis an even function in x\nG\u000b(\u0000x;t) =G\u000b(x;t); x2IR; t> 0\nand (12) can be rewritten as the cos-Fourier transform:\nG\u000b(x;t) =1\n\u0019Z1\n0cos(\u0014x)E\f(\u0000\u0014\u000bt\f)d\u0014; x2IR; t> 0: (13)\nRemarkably, the fundamental solution G\u000bcan be represented in terms of\nelementary functions for every \u000b;1<\u000b< 2. Indeed, for x= 0 the integral\nin the right-hand side of (13) is reduced to the Mellin integral transform of\nthe Mittag-Le\u000fer function at the point s=1\n\u000b(for de\fnition and proper-\nties of the Mellin integral transform see e.g. [19]). It converges under the\nconditions\u000b>1 and its value is given by the formula ([19])\n1\n\u0019Z1\n0E\u000b(\u0000\u0014\u000bt\u000b)d\u0014=1\n\u0019\u000btZ1\n0E\u000b(\u0000u)u1\n\u000b\u00001du (14)\n=1\n\u0019\u000bt\u0000(1\n\u000b)\u0000(1\u00001\n\u000b)\n\u0000(1\u0000\u000b1\n\u000b)= 0; t> 0\nbecause the gamma function has a pole at the point z= 0: 1=\u0000(0) = 0.\nSinceG\u000bis an even function, we consider the integral in the right-hand\nside of (13) just in the case x=jxj>0 and recognize that this integral\nis the Mellin convolution of the functions g(\u0018) =E\f(\u0000\u0018\u000bt\u000b) andf(\u0018) =\n1\n\u0019x\u0018cos(1=\u0018) in point 1 =x. Using the known Mellin integral transforms of\nthe cos-function and the Mittag-Le\u000fer function as well as some elementary\nproperties of the Mellin integral transform ([19]) we get the formulas:\ng\u0003(s) =Z1\n0g(\u001c)\u001cs\u00001d\u001c=1\n\u000bts\u0000\u0000s\n\u000b\u0001\n\u0000\u0000\n1\u0000s\n\u000b\u0001\n\u0000(1\u0000s);0<<(s)<\u000b;\nf\u0003(s) =1p\u0019x2s\u0000\u00001\n2\u0000s\n2\u0001\n\u0000\u0000s\n2\u0001;0<<(s)<1:\n6These formulas together with the convolution rule and the inverse Mellin\nintegral transform lead to the representation\nG\u000b(x;t) =1p\u0019\u000bx1\n2\u0019iZ\r+i1\n\r\u0000i1\u0000\u0000s\n\u000b\u0001\n\u0000\u0000\n1\u0000s\n\u000b\u0001\n\u0000(1\u0000s)\u0000\u00001\n2\u0000s\n2\u0001\n2s\u0000\u0000s\n2\u0001\u0012t\nx\u0013\u0000s\nds;0<\r < 1\n(15)\nof the fundamental solution G\u000bin terms of the general Fox H-function (see,\nfor example, [19], [24], [25]). The representation (15) can be simpli\fed to\nthe form\nG\u000b(x;t) =1\n\u000bx1\n2\u0019iZ\r+i1\n\r\u0000i1\u0000(s\n\u000b)\u0000(1\u0000s\n\u000b)\n\u0000(1\u0000s\n2)\u0000(s\n2)\u0012t\nx\u0013\u0000s\nds;0<\r <\u000b (16)\nand then to the form\nG\u000b(x;t) =1\n\u000bx1\n2\u0019iZ\r+i1\n\r\u0000i1sin(\u0019s=2)\nsin(\u0019s=\u000b )\u0012t\nx\u0013\u0000s\nds;0<\r <\u000b (17)\nby using the duplication and re\rection formulas for the gamma function.\nA useful representation\nG\u000b(x;t) =1\nxL\u000b(t=x); x=jxj>0; t> 0 (18)\nof the fundamental solution G\u000bin terms of an auxiliary function L\u000bthat\ndepends on the quotient t=xcan be obtained from (16) or (17). Moreover,\nbecause (17) is in form of an inverse Mellin transform we deduce from (17)\nthe Mellin transform of the function L\u000bas follows:\nL\u0003\n\u000b(s) =Z1\n0L\u000b(\u001c)\u001cs\u00001d\u001c=1\n\u000bsin(\u0019s=2)\nsin(\u0019s=\u000b ): (19)\nIt follows from (17) that (19) holds true at least for 0 <<(s)<\u000bbut as we\nsee later, in fact (19) is valid even for \u0000\u000b<<(s)<\u000b.\nNow let us represent the special case (16) of the H-function in form of\nsome convergent series that can be summated in explicit form in terms of\nsome elementary functions. General theory of the Mellin-Barnes integrals\npresented e.g. in [19] says that the integral in (16) is convergent under the\ncondition 0 < \u000b < 2. For 0< t < x , the contour of integration in the\nintegral (16) can be transformed to the loop L\u00001starting and ending at\nin\fnity and encircling all poles sk=\u0000\u000bk; k = 0;1;2;::: of the function\n\u0000(s=\u000b). Taking into account the relation\nress=\u0000k\u0000(s) =(\u00001)k\nk!; k= 0;1;2;:::;\n7the residue theorem provides us with the desired series representation:\nG\u000b(x;t) =1\n\u000bx1X\nk=0\u000b(\u00001)k\nk!\u0000(1 +k)\n\u0000\u0000\n\u0000\u000b\n2k\u0001\n\u0000\u0000\n1\u0000\u000b\n2k\u0001\u0012t\nx\u0013\u000bk\n(20)\nthat can be transformed to the form\nG\u000b(x;t) =\u00001\n\u0019x1X\nk=1sin(\u000b\u0019k= 2)\u0012\n\u0000t\u000b\nx\u000b\u0013k\n(21)\nby using the re\rection formula for the gamma function.\nNow we use the summation formula\n1X\nk=1rksin(ka) == 1X\nk=1rkeika!\n==\u0012reia\n1\u0000reia\u0013\n=rsina\n1\u00002rcosa+r2(22)\nthat is valid for a2IR;jrj<1 to summate the series in (21) and obtain the\nnice representation\nG\u000b(x;t) =1\n\u0019x\u000b\u00001t\u000bsin(\u0019\u000b=2)\nt2\u000b+ 2x\u000bt\u000bcos(\u0019\u000b=2) +x2\u000b(23)\nfor the Green function G\u000bthat is valid for 0 <t<x .\nIn the case 0 < x < t we can transform the contour of integration in\n(16) to the loop L+1encircling all poles sk=\u000b(1 +k); k= 0;1;2;:::\nof the function \u0000\u0000\n1\u0000s\n\u000b\u0001\n. Applying the residue theorem we arrive at the\nrepresentation\nG\u000b(x;t) =1\n\u000bx1X\nk=0\u000b(\u00001)k\nk!\u0000(1 +k)\n\u0000\u0000\u000b\n2(k+ 1)\u0001\n\u0000\u0000\n1\u0000\u000b\n2(k+ 1)\u0001\u0010x\nt\u0011\u000b(k+1)\n(24)\nthat can be transformed to the form\nG\u000b(x;t) =\u00001\n\u0019x1X\nk=1sin(\u000b\u0019k= 2)\u0012\n\u0000x\u000b\nt\u000b\u0013k\n(25)\nby using the re\rection formula for the gamma function. The formula (22)\napplied to the series from the right-hand side of (25) again leads to the\nrepresentation (23), this time for 0 <x<t . Finally, validity of the formula\n(23) for 0<x=tfollows from the principle of analytic continuation for the\nMellin-Barnes integrals. Thus the fundamental solution G\u000bfor the fractional\nwave equation is given by the formula\nG\u000b(x;t) =1\n\u0019jxj\u000b\u00001t\u000bsin(\u0019\u000b=2)\nt2\u000b+ 2jxj\u000bt\u000bcos(\u0019\u000b=2) +jxj2\u000b; t> 0; x2IR (26)\nfor 1<\u000b< 2.\n82.3 Fundamental solution as a pdf\nWe begin by a remark that the formula (26) is valid for \u000b= 1 (modi\fed\nadvection equation (7)), too, that can be proved by direct calculations. In\nthis case we get the well known Cauchy kernel\nG1(x;t) =1\n\u0019t\nt2+x2(27)\nthat is a spatial probability density function evolving in time.\nFor\u000b= 2 (wave equation), the Green function G2is known to be given\nby the formula\nG2(x;t) =1\n2(\u000e(x\u0000t) +\u000e(x+t)): (28)\nThe representation (26) thus leads to an interesting relation\nlim\n\u000b!2\u00000jxj\u000b\u00001t\u000bsin(\u0019\u000b=2)\nt2\u000b+ 2jxj\u000bt\u000bcos(\u0019\u000b=2) +jxj2\u000b=\u0019\n2(\u000e(x\u0000t)+\u000e(x+t)); t> 0; x2IR\nfor the Dirac \u000e-function.\nFor 1< \u000b < 2, the Green function (26) is a spatial probability density\nfunction evolving in time, too. Indeed, the function (26) is evidently non-\nnegative for all t > 0. Furthermore, for all t > 0 and 1< \u000b < 2 the\nintegral\nZ1\n\u00001G\u000b(x;t)dx=2\n\u0019\u000bZ+1\n0sin(\u0019\u000b=2)\n1 + 2ucos(\u0019\u000b=2) +u2du (29)\nis identically equal to 1 that can be checked by direct calculations. Thus G\u000b\ngiven by (26) is a spatial probability density function evolving in time that\ncan be considered to be a fractional generalization of the Cauchy kernel (27)\nfor the case of an arbitrary index \u000b;1\u0014\u000b<2.\nNow let us study the properties of the fundamental solution (26) as a\npdf. Because G\u000bis an even function we can restrict our attention to the\ncasex\u00150 and consider the function\nG+\n\u000b(x;t) =1\n\u0019x\u000b\u00001t\u000bsin(\u0019\u000b=2)\nt2\u000b+ 2x\u000bt\u000bcos(\u0019\u000b=2) +x2\u000b; x\u00150; t> 0;1<\u000b< 2:\nIt is easy to see that G+\n\u000bbehaves like a power function in xboth atx= 0\nand atx= +1for a \fxedt>0:\nG+\n\u000b(x;t)\u0019(\nx\u000b\u00001; x!0;\nx\u0000\u000b\u00001; x!+1:(30)\n9This means that the pdf G\u000bpossesses all \fnite moments up to the order \u000b.\nIn particular, the mean value of G\u000b(its \frst moment) exists for all \u000b > 1\n(we note that the Cauchy kernel does not possess a mean value). Let us\nnow evaluate the moments of the one-sided fractional Cauchy kernel G+\n\u000bfor\na \fxedt>0. To do this, we refer to the representation (18) of G+\n\u000bin terms\nof the auxiliary function L\u000bthat can be now represented in the form\nL\u000b(\u001c) =1\n\u0019\u001c\u000bsin(\u0019\u000b=2)\n\u001c2\u000b+ 2\u001c\u000bcos(\u0019\u000b=2) + 1; \u001c > 0;1<\u000b< 2: (31)\nTaking into account this formula, the function1\nj\u001cjL\u000b(j\u001cj) can be interpreted\nas a fractional Cauchy pdf of order \u000b.\nThe moment of the order \f;j\fj<\u000bofG+\n\u000bcan be represented in terms\nof the Mellin integral transform of L\u000bthat is known (see the formula (19))\nand thus evaluated:\nZ1\n0G+\n\u000b(x;t)x\fdx=t\fZ1\n0L\u000b(\u001c)\u001c\u0000\f\u00001d\u001c=t\f\n\u000bsin(\u0019\f=2)\nsin(\u0019\f=\u000b ): (32)\nIn particular, we get the formula\nZ1\n0G+\n\u000b(x;t)dx=1\n2(33)\nthat is in accordance with (29) because G\u000bis an even function.\nWe mention also important formula\nZ1\n0G+\n\u000b(x;t)xdx =t\n\u000bsin(\u0019=\u000b);1<\u000b\u00142 (34)\nfor the \frst moment of the one-sided fractional Cauchy kernel G+\n\u000b.\n2.4 Extrema points, gravity and mass centers of G\u000b, and\nlocation of its energy\nNow we derive some important analytical properties of the fractional Cauchy\nkernel (26). First we remark that G\u000b(0;t) = 0 andG\u000b(x;t)>0 forx6= 0,\nso thatx= 0 is a minimum point for the fundamental solution G\u000bfor any\nt >0. Because G\u000bis an even function we again consider its restriction to\nx\u00150, i.e. the function G+\n\u000b.\nTo determine locations of maxima of G+\n\u000bfor \fxed values of tand\u000bwe\nsolve the equation\n@G+\n\u000b\n@x(x;t) = 0\n10that turns out to be equivalent to the quadratic equation\n(\u000b+ 1)\u0012x\u000b\nt\u000b\u00132\n+ 2 cos(\u0019\u000b=2)\u0012x\u000b\nt\u000b\u0013\n\u0000(\u000b\u00001) = 0\nwith solutions given by\nx\u000b\nt\u000b=\u0000cos(\u0019\u000b=2)\u0006p\n\u000b2\u0000sin2(\u0019\u000b=2)\n\u000b+ 1:\nSince we are interested in nonnegative solutions the only candidate for this\nrole is the point\nx\u000b\nt\u000b=c\u000b; c\u000b:=\u0000cos(\u0019\u000b=2) +p\n\u000b2\u0000sin2(\u0019\u000b=2)\n\u000b+ 1: (35)\nBecause@G+\n\u000b\n@x(x;t) is positive forx\u000b\nt\u000b< c\u000band negative forx\u000b\nt\u000b> c\u000bwe\nconclude that the point\nx?\n\u000b(t) =vp(\u000b)t; vp(\u000b) := (c\u000b)1\n\u000b (36)\nwithc\u000bgiven by (35) is the only maximum point of the fractional Cauchy\nkernelG\u000bforx\u00150. Of course, the point \u0000x?\n\u000b(t)<0 is another maximum\npoint ofG\u000bbecauseG\u000bis an even function.\nTo determine the maximum value of the function G\u000bthat coincides with\nthe maximum value of G+\n\u000band is denoted by G?\n\u000b(t) we substitute the point\nx=x?\n\u000b(t) given by (36) into the function G+\n\u000band get\nG?\n\u000b(t) =1\n\u0019vp(\u000b)tc\u000bsin(\u0019\u000b=2)\n1 + 2c\u000bcos(\u0019\u000b=2) +c2\u000b; (37)\nwherevp(\u000b) andc\u000bare de\fned as in the formulas (35) and (36).\nIt follows from the formulas (36) and (37) that for a \fxed value of \u000b;1<\n\u000b < 2 the product p\u000bof the maximum value G?\n\u000b(t) and the maximum\nlocationx?\n\u000b(t) is time-independent:\np\u000b=G?\n\u000b(t)\u0001x?\n\u000b(t) =1\n\u0019c\u000bsin(\u0019\u000b=2)\n1 + 2c\u000bcos(\u0019\u000b=2) +c2\u000b: (38)\nThis means that the maximum point ( x?\n\u000b(t); G?\n\u000b(t)) of the fundamental so-\nlutionG\u000bwith a \fxed value of \u000b;1<\u000b< 2 moves in time along a hyperbola\nthat is completely determined by the value of \u000b(see the formula (38)). It\n11is interesting to note that the product p\u000bcoincides with the value of the\nfundamental solution G\u000bat the point (1 ;vp(\u000b)):\np\u000b=1\n\u0019c\u000bsin(\u0019\u000b=2)\n1 + 2c\u000bcos(\u0019\u000b=2) +c2\u000b=G\u000b(1;vp(\u000b)):\nThis property can be probably used to give a physical interpretation of the\nformula (38).\nNow we calculate the location of the gravity center xg\n\u000b(t) of the funda-\nmental solution G\u000bthat is de\fned by the formula (we recall that G\u000bis an\neven function)\nxg\n\u000b(t) =R1\n0xG\u000b(x;t)dxR1\n0G\u000b(x;t)dx: (39)\nUsing the formulas (33) and (34) we get the following result:\nxg\n\u000b(t) =2t\n\u000bsin(\u0019=\u000b): (40)\nThe \"mass\"-center xm\n\u000b(t) ofG\u000bis determined by the formula ([8])\nxm\n\u000b(t) =R1\n0xG2\n\u000b(x;t)dxR1\n0G2\u000b(x;t)dx: (41)\nSubstituting (26) into (41) and transforming the obtained integrals we get\nthe representation\nxm\n\u000b(t) =vm(\u000b)t; vm(\u000b) =R1\n0\u001c\u00001L2\n\u000b(\u001c)d\u001cR1\n0\u001c\u00002L2\u000b(\u001c)d\u001c; (42)\nwhere the function L\u000bis de\fned by (31). Because the Mellin transform of\nL\u000bis known (see the formula (19)), the integrals\nZ1\n0\u001c\fL2\n\u000b(\u001c)d\u001c;\u00002\u000b\u00001<\f < 2\u000b\u00001 (43)\ncan be interpreted as Mellin convolutions of the functions \u001c\f+1L\u000b(\u001c) and\nL\u000b(1=\u001c) in pointx= 1 and thus expressed in terms of an H-function with\nparameters depending on \u000band\fand with the argument x= 1. Because\nthere are no routines for numerical evaluation of the H-function available,\nwe prefer to stay by the representation of vm(\u000b) given in (42) and not to\ntransform it to a quotient of two H-functions. Remarkably, there exists an\n12explicit formula for the integrals (43) in the case \u000b= 1 (modi\fed advection\nequation), namely ([22])\nZ1\n0\u001c\fL2\n1(\u001c)d\u001c=1 +\f\n4\u00191\ncos(\u0019\f=2);\u00003<\f < 1: (44)\nThe \"mass\"-center xm\n1ofG1can be then represented by the simple formula\nxm\n1(t) =2\n\u0019t: (45)\nFinally we mention that \"location of energy\" of the damped wave that\nis represented by the fundamental solution G\u000bis given by the formula ([2])\ntc\n\u000b(x) =R1\n0tG2\n\u000b(x;t)dtR1\n0G2\u000b(x;t)dt(46)\nand can be represented in the form\ntc\n\u000b(x) =x\nvc(\u000b); vc(\u000b) =R1\n0L2\n\u000b(\u001c)d\u001cR1\n0\u001cL2\u000b(\u001c)d\u001c; (47)\nwhere the function L\u000bis de\fned by (31). Because the integralR1\n0\u001cL2\n\u000b(\u001c)d\u001c\ndiverges for \u000b= 1,vc(\u000b) tends to 0 as \u000b!1.\n2.5 The velocities of the damped waves\nIt is well known (see e.g. [1], [2], [8], or [26]) that several di\u000berent de\fnitions\nof the wave velocities and in particular of light velocity can be introduced.\nFor the damped waves that are described by the fractional wave equation\n(2) we calculate propagation velocity of the maximum of its fundamental so-\nlutionG\u000bthat can be interpreted as the phase velocity, propagation velocity\nof the gravity center of G\u000b, the velocity of its \"mass\"-center or the pulse\nvelocity, and three di\u000berent kinds of its centrovelocity. It turns out that all\nthese velocities are constant in time and depend just on the order \u000bof the\nfractional wave equation. Whereas four out of six velocities are di\u000berent\neach to other, the \fst centrovelocity coincides with the Smith centrovelocity\nand the the second centrovelocity is the same as the pulse velocity.\nWe start with the phase velocity and determine it using the formula\n(36) that leads to the result that the maximum location of the fundamental\nsolutionG\u000bpropagates with a constant velocity vp(\u000b) that is given by the\nexpression\nvp(\u000b) :=dx?\n\u000b(t)\ndt= \n\u0000cos(\u0019\u000b=2) +p\n\u000b2\u0000sin2(\u0019\u000b=2)\n\u000b+ 1!1\n\u000b\n:(48)\n13For\u000b= 1 (modi\fed advection equation (7)), the propagation velocity of\nthe maximum of G\u000bis equal to zero (the maximum point stays at x= 0)\nwhereas for \u000b= 2 (wave equation) the maximum point propagates with the\nconstant velocity 1.\nTo determine the propagation velocity vg(\u000b) of the gravity center of G\u000b\nwe employ the formula (40) and get the following result:\nvg(\u000b) :=dxg\n\u000b(t)\ndt=2\n\u000bsin(\u0019=\u000b): (49)\nvg(\u000b) is thus time-independent and determined by the order \u000bof the frac-\ntional wave equation. Evidently, vg(2) = 1 and vg(\u000b)!+1as\u000b!1 + 0.\nThe velocity vm(\u000b) of the \"mass\"-center of G\u000bor its pulse velocity ([8])\nis obtained from the formula (42) and is equal to\nvm(\u000b) :=dxm\n\u000b(t)\ndt=R1\n0\u001c\u00001L2\n\u000b(\u001c)d\u001cR1\n0\u001c\u00002L2\u000b(\u001c)d\u001c; (50)\nwhere the function L\u000bis de\fned by (31). For \u000b= 1, the pulse velocity is\nequal to2\n\u0019\u00190:64 (see the formula (45)).\nFollowing [2] we de\fne the second centrovelocity v2(\u000b) as the mean pulse\nvelocity computed from 0 to time t. It follows from (42) and (50) that for the\ndamped wave that is described by the fundamental solution of the fractional\nwave equation the second centrovelocity is equal to its pulse velocity vm(\u000b):\nv2(\u000b) :=xm\n\u000b(t)\nt=vm(\u000b) =R1\n0\u001c\u00001L2\n\u000b(\u001c)d\u001cR1\n0\u001c\u00002L2\u000b(\u001c)d\u001c: (51)\nThe Smith centrovelocity vc(\u000b) ([26]) of the damped waves describes the\nmotion of the \frst moment of their energy distribution and can be evaluated\nin explicit form using the formula (47):\nvc(\u000b) :=\u0012dtc\n\u000b(x)\ndx\u0013\u00001\n=R1\n0L2\n\u000b(\u001c)d\u001cR1\n0\u001cL2\u000b(\u001c)d\u001c; (52)\nwhere the function L\u000bis de\fned by (31). Because the integralR1\n0\u001cL2\n\u000b(\u001c)d\u001c\ndiverges for \u000b= 1, the Smith centrovelocity tends to 0 as \u000b!1.\nFinally, we calculate the \frst centrovelocity v1(\u000b) that is de\fned as the\nmean centrovelocity from 0 to x([2]). It follows from (47) and (52) that for\nthe damped wave G\u000bthe \frst centrovelocity is equal to the Smith centrov-\nelocityvc(\u000b):\nv1(\u000b) :=x\ntc\u000b(x)=vc(\u000b) =R1\n0L2\n\u000b(\u001c)d\u001cR1\n0\u001cL2\u000b(\u001c)d\u001c: (53)\n14−0.5−0.4−0.3−0.2−0.1 00.10.20.30.40.50246810\nxG1.01t=0.1\nt=0.3t=0.2\nStudent Version of MATLAB\n−0.5−0.4−0.3−0.2−0.100.10.20.30.40.50246810\nxG1.1t=0.1\nt=0.3t=0.2\nStudent Version of MATLAB\n−0.5−0.4−0.3−0.2−0.1 00.10.20.30.40.502468101214\nxG1.5t=0.1\nt=0.2\nt=0.3\nStudent Version of MATLAB\n−0.5−0.4−0.3−0.2−0.1 00.10.20.30.40.5010203040506070\nxG1.9t=0.1\nt=0.2\nt=0.3\nStudent Version of MATLABFigure 1: Fundamental solution G\u000b: Plots for values of \u000b= 1:01 (1st line,\nleft), 1:1 (1st line, right), 1 :5 (2nd line left), and 1 :9 (2nd line, right) for\n\u00000:5\u0014x\u00140:5 andt= 0:1;0:2;0:3\nAs we have seen, all velocities introduced above are constant in time\nand depend just on the order \u000bof the fractional wave equation. The phase\nvelocity, the velocity of the gravity center of G\u000b, the pulse velocity, and the\nSmith centrovelocity are di\u000berent each to other whereas the \fst centroveloc-\nity coincides with the Smith centrovelocity and the second centrovelocity is\nthe same as the pulse velocity. For the physical interpretation and meaning\nof the velocities that were determined above we refer to e.g. [1], [2], or [8].\n3 Discussion of the obtained results and plots\nTo start with, let us consider evolution of the fundamental solution G\u000b\nin time for some characteristic values of \u000b. In Fig. 1 plots of G\u000bfor\n\u000b= 1:01;1:1;1:5, and 1:9 are presented. As we can see, in all cases maxi-\nmum location is moved linearly in time according to the formula (36) whereas\nthe maximum value decreases according to the formula (37). The behavior\nofG\u000bcan be thus interpreted as propagation of damped waves whose am-\nplitude decreases with time. This phenomena can be very clearly recognized\non the 3D plot presented in Fig. 2. Of course, because of the nonlocal char-\nacter of the fractional derivatives in the fractional wave equation solutions\nto this equation show some properties of di\u000busion processes, too. In par-\n15Student Version of MATLAB\n11.21.41.61.8201234\nαmα\nStudent Version of MATLABFigure 2: Plots of G\u000bfor\u000b= 1:5,\u00000:5\u0014x\u00140:5, and 0< t\u00140:3 (left)\nand of the maximum value m\u000bof the function G\u000bat the time instant t= 1\n(right)\nticular, the fundamental solution G\u000bis positive for all x6= 0 at any small\ntime instance t>0 that means that a disturbance of the initial conditions\nspreads in\fnitely fast and the equation (2) is non relativistic like the clas-\nsical di\u000busion equation. But in contrast to the di\u000busion equation, both the\nmaximum of the fundamental solution G\u000b, its gravity and mass centers and\nlocation of its energy propagate with the \fnite constant velocities like in the\ncase of the fundamental solution of the wave equation.\nPlots of the propagation velocity vpof the maximum of the fundamen-\ntal solution G\u000b(phase velocity), the velocity vgof its gravity center, its\npulse velocity vmand its centrovelocity vcare presented in Fig. 3. As ex-\npected,vp=vc= 0 andvm\u00190:64 for\u000b= 1 (modi\fed advection equation)\nand all velocities smoothly approach the value 1 as \u000b!2 (wave equa-\ntion). For 1 < \u000b < 2,vp; vm;andvcmonotonously increase whereas vg\nmonotonously decreases. It is interesting to note that for all velocities the\npropertydv(\u000b)\nd\u000b(2\u00000) = 0 holds true, i.e. in a small neighborhood of the point\n\u000b= 2 the velocities of G\u000bare nearly the same as in the case of the fundamen-\ntal solution of the wave equation. The velocity vgof the gravity center of G\u000b\ntends to +1for\u000b!1+0 andt>0 (modi\fed advection equation) because\nthe \frst moment of the Cauchy kernel (27) does not exist. It is interesting\nto note that for all \u000b;1< \u000b < 2 the velocities vp; vg; vm; vcare di\u000berent\neach to other and ful\fll the inequalities vc(\u000b)< vp(\u000b)< vm(\u000b)< vg(\u000b).\nFor\u000b= 2, all velocities are equal to 1.\nIn Fig. 2 (right), a plot of the maximum value of the fundamental solu-\ntionG\u000bthat is given by the formula (37) is presented for t= 1. Surprisingly,\n161.21.41.61.82.0Alpha1234V\nV_gV_p\n1.21.41.61.82.0Alpha0.20.40.60.81.0V\nV_mV_cFigure 3: Plots of the phase velocity vp(\u000b) and the gravity center velocity\nvg(\u000b) (left) and of the pulse velocity vm(\u000b) and the centrovelocity vc(\u000b)\n(right)\nthe function m\u000b=G?\n\u000b(1) is not monotone. Numerical calculations show that\nit has a minimum that is located at the point \u000b\u00191:13. At\u000b= 1,G\u000b(x;1)\nhas a (local) maximum with the value1\n\u0019\u00190:32. ThenG\u000b(x;1) monotoni-\ncally decreases to mmin\u00190:28 at\u000bmin\u00191:13 (minimum) and then starts\nto monotonically increase and tends to + 1when\u000b!2\u00000 that is in ac-\ncordance with behavior of the fundamental solution (1\n2(\u000e(x\u0000t) +\u000e(x+t))\nof the wave equation ( \u000b= 2). We note that the location x\u0003\n\u000b(t) of the maxi-\nmum value G\u0003\n\u000b(t) ofG\u000bis given by the formula (36). For t= 1, location of\nthe maximum value coincides with the phase velocity vpthat is presented\nin Fig. 3.\nFinally, some plots of the product p\u000bof the maximum value G?\n\u000b(t) and\nthe maximum location x?\n\u000b(t) given by the formula (38) are presented in Fig.\n4. The function p\u000b=p\u000b(\u000b) is monotone increasing for 1 \u0014\u000b<2 and varies\nfrom 0 at the point \u000b= 1 (the location of the maximum of the Cauchy kernel\n(27) is always at the point x= 0) to +1at the point \u000b= 2 (the maximum\nvalue of the the Green function (1\n2(\u000e(x\u0000t)+\u000e(x+t)) of the wave equation is\n+1). In this sense, the product p\u000bcan be considered to be a characteristic\nproperty of the damped waves that are described by the fractional wave\nequation (2). As we can see on the left plot, the product p\u000bvaries between\n0 and 10 for 1\u0014\u000b\u00141:98 and changes very slowly for 1 :01\u0014\u000b\u00141:9. For\n\u000b!1+0 (see the right plot of Fig. 4) and \u000b!2\u00000,p\u000bgoes to 0 and +1,\nrespectively, very fast. For a damped wave that is supposed to be described\nby the fractional wave equation (2) with an unknown exponent \u000b, the value\nof the product p\u000bcan be measured and used to recover \u000beither from the\nformula (38) or graphically from Fig. 4.\n1711.21.41.61.820246810\nαpα\nStudent Version of MATLAB\n11.011.021.031.041.0500.010.020.030.040.050.060.07\nαpα\nStudent Version of MATLABFigure 4: Plots of the product p\u000bof the maximum value G?\n\u000b(t) and the\nmaximum location x?\n\u000b(t) on the interval 1 \u0014\u000b<2 (left) and on the interval\n1\u0014\u000b\u00141:05 (right)\n4 Conclusions and open questions\nIn this paper, a model one-dimensional fractional wave equation with the\nfractional derivatives of order \u000b;1\u0014\u000b\u00142 both in space and in time is\nintroduced and considered in detail. The fractional wave equation inherits\nsome crucial characteristics of the wave equation like constant propagation\nvelocities of the maximum of its Green function, its gravity and mass centers,\nand its energy location. Because the maximum value of the fundamental\nsolutionG\u000b(wave amplitude) decreases with time whereas its location moves\nwith a constant velocity, solutions to the fractional wave equation can be\ninterpreted as damped waves. Moreover, G\u000bthat turns out to be expressed\nin terms of elementary functions for all values of \u000b;1\u0014\u000b < 2 can be\ninterpreted as a spatial probability density function evolving in time whose\nmoments up to order \u000bare \fnite. For the fundamental solution G\u000b, both its\nmaximum location and its maximum value are determined in closed form.\nRemarkably, the product of the maximum location and the maximum value\nofG\u000bis time-independent and just a function of \u000b.\nAmong problems for further research we mention two- and three-\ndimensional fractional wave equations with di\u000berent initial or/and boundary\nconditions. Of course, it would be interesting to consider fractional wave\nequations with fractional derivatives de\fned in di\u000berent ways. We note here\nthat in [18] a space-time fractional di\u000busion-wave equation with the Caputo\nderivative of order \f2(0;2] in time and the Riesz-Feller derivative of order\n18\u000b2(0;2] and skewness \u0012in space has been investigated in detail. A par-\nticular case of this equation called neutral-fractional di\u000busion equation that\nfor\u0012= 0 corresponds to our fractional di\u000busion equation has been shortly\nmentioned in [18].\nAnother interesting and important problem for further research would be\ndetermination of other velocities like the group velocity or the ratio-of-units\nvelocity (see e.g. [1] or [8]) for the damped waves described by the fractional\nwave equations and comparison them each to other at least for the linear\nequations with the constant coe\u000ecients. Finally, fractional wave equations\nwith non-constant coe\u000ecients as well as qualitative behavior of solutions of\nnon-linear fractional wave equations would be worth to consider.\nAcknowledgment: The author is thankful to Prof. Francesco Mainardi\nfor useful and stimulating discussions regarding the subject of the paper\nduring author's visit to the University of Bologna in December 2011.\nReferences\n[1] S. C. Bloch, Eighth velocity of light. Am. J. Phys. 45(1977), 538-549.\n[2] J. M. Carcione, D. Gei, and S. Treitel, The velocity of energy through\na dissipative medium. Geophysics 75(2010), T37-T47.\n[3] J.L.A. Dubbeldam, A. Milchev, V.G. Rostiashvili, and T.A. Vilgis,\nPolymer translocation through a nanopore: A showcase of anomalous\ndi\u000busion. Physical Review E 76(2007), 010801 (R).\n[4] R. Herrmann, Fractional Calculus: An Introduction for Physicists .\nWorld Scienti\fc, Singapore (2011).\n[5] A. Freed, K. Diethelm, and Yu. Luchko, Fractional-order viscoelasticity\n(FOV): Constitutive development using the fractional calculus . NASA's\nGlenn Research Center, Ohio (2002).\n[6] Y. Fujita, Integrodi\u000berential equation which interpolates the heat equa-\ntion and the wave equation, I, II. Osaka J. Math. 27(1990), 309{321,\n797{804.\n[7] R. Goren\ro, A. Iskenderov, and Yu. Luchko, Mapping between solutions\nof fractional di\u000busion-wave equations. Fract. Calc. Appl. Anal. 3(2000),\n75-86.\n19[8] I. Gurwich, On the pulse velocity in absorbing and nonlinear media and\nparallels with the quantum mechanics. Progress In Electromagnetics\nResearch 33(2001), 69-96.\n[9] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics . World\nScienti\fc, Singapore (2000).\n[10] R. Klages, G. Radons, and I.M. Sokolov (Eds.), Anomalous Transport:\nFoundations and Applications , Wiley-VCH, Weinheim (2008).\n[11] Yu. Luchko and A. Punzi, Modeling anomalous heat transport in\ngeothermal reservoirs via fractional di\u000busion equations. International\nJournal on Geomathematics 1, No 2 (2011), 257-276.\n[12] Yu. Luchko, Anomalous di\u000busion models and their analysis. Forum der\nBerliner mathematischen Gesellschaft 19(2011), 53-85.\n[13] Yu. Luchko, Operational method in fractional calculus. Fract. Calc.\nAppl. Anal. 2(1999), 463-489.\n[14] Yu. Luchko, F. Mainardi, and Yu. Povstenko, Propagation speed of the\nmaximum of the fundamental solution to the fractional di\u000busion-wave\nequation, E-print: arXiv:1201.5313v1 [math-ph].\n[15] R.L. Magin, Fractional Calculus in Bioengineering: Part1, Part 2 and\nPart 3. Critical Reviews in Biomedical Engineering 32(2004), 1-104,\n105-193, 195-377.\n[16] F. Mainardi, Fractional relaxation-oscillation and fractional di\u000busion-\nwave phenomena. Chaos, Solitons and Fractals 7(1996), 1461-1477.\n[17] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity .\nImperial College Press, London (2010).\n[18] F. Mainardi, Yu. Luchko, and G. Pagnini, The fundamental solution\nof the space-time fractional di\u000busion equation. Fract. Calc. Appl. Anal.\n4(2001), 153-192. [E-print http://arxiv.org/abs/cond-mat/0702419].\n[19] O.I. Marichev, Handbook of Integral Transforms of Higher Transcen-\ndental Functions, Theory and Algorithmic Tables . Chichester, Ellis Hor-\nwood (1983).\n[20] R. Metzler and J. Klafter, The restaurant at the end of the random\nwalk: Recent developments in the description of anomalous transport\nby fractional dynamics. J. Phys. A. Math. Gen. 37(2004), R161-R208.\n20[21] I. Podlubny, Fractional Di\u000berential Equations . Academic Press, San\nDiego (1999).\n[22] A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Se-\nries, Vol. 1: Elementary Functions . Gordon and Breach, New York\n(1986).\n[23] A. Saichev, G. Zaslavsky, Fractional kinetic equations: solutions and\napplications. Chaos 7(1997), 753-764.\n[24] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and\nDerivatives: Theory and Applications . Gordon and Breach, New York\n(1993).\n[25] W.R. Schneider, W. Wyss, Fractional di\u000busion and wave equations. J.\nMath. Phys. 30(1989), 134-144.\n[26] R.L. Smith, The velocities of light. American Journal of Physics\n38(1970), 978-984.\n[27] V.V. Uchaikin, Method of fractional derivatives . Artishok, Ul'janovsk\n(2008) (in Russian).\n21"    },    {        "title": "1205.2436v1.On_radiative_damping_in_plasma_based_accelerators.pdf",        "content": "arXiv:1205.2436v1  [physics.plasm-ph]  11 May 2012On radiative damping in plasma-based accelerators\nI. Yu. Kostyukov,∗E. N. Nerush, and A. G. Litvak\nInstitute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia\nRadiative damping in plasma-based electron accelerators i s analyzed. The electron dynamics\nunder combined influence of the constant accelerating force and the classical radiation reaction\nforce is studied. It is shown that electron acceleration can not be limited by radiation reaction.\nIf initially the accelerating force was stronger than the ra diation reaction force then the electron\nacceleration is unlimited. Otherwise the electron is decel erated by radiative damping up to a certain\ninstant of time and then accelerated without limits. Regard less of the initial conditions the infinite-\ntime asymptotic behavior of an electron is governed by self- similar solution providing unlimited\nacceleration. The relative energy spread induced by the rad iative damping decreases with time in\nthe infinite-time limit.\nPACS numbers: 41.75.Jv,52.38.Kd,52.40.Mj\nThe plasma-based methods of electron acceleration\ndemonstrate an impressive progress in the last ten years.\nThe quasimonoenergetic electron bunches are generated\nin laser-plasma acceleration experiments [1]. The elec-\ntron energy in laser wakefield acceleration experiments\nexceeds1GeVforcm-scaleaccelerationlength[2]anden-\nergydoublingof42GeVelectronsinameter-scaleplasma\nwakefield accelerator is demonstrated [3]. Recently the\nphysics of linear colliders based on laser-plasma acceler-\nators have been discussed [4, 5].\nThe accelerating structure in the plasma-based meth-\nods is a plasma wave generated behind the driver which\ncan be the laser pulse or the electron bunch. There is\na number of effects which limit the energy gain in the\nplasma-based accelerators [6]. One of the main limita-\ntions comes from the dephasing. The velocity of the rela-\ntivistic electrons becomesslightly higherthan the plasma\nwave phase velocity, which is determined by the driver\nvelocity. The accelerated electrons slowly outrun the\nplasmawaveandleavetheacceleratingphase. Thisprob-\nlemcanbepartiallysolvedbytheuseofproperlongitudi-\nnal gradient of plasma density [7, 8]. Another limitation\nis caused by the driver depletion as the driver energy\nconverts into the energy of the plasma wave. The driver\nevolution during acceleration(e. g. laserpulse diffraction\norelectronbunch expansion)alsoimposes certainrestric-\ntions on the electron energy gain. In the case of laser-\nplasma accelerators the laser pulse can be guided over\nlong distances in the preformed plasma density channel\n[9] or with relativistic optical guiding when diffraction is\ncompensated by relativistic self-focusing [10]. In general,\nin order to accelerate electrons far beyond the energy\nlimited by these effects the multistage schemes can be\nused.\nThe electron accelerationin the plasma wave is accom-\npanied with the transverse betatron oscillations caused\nby the action of the focusing force on the electron from\nthe plasma wakefield. The accelerating force and the\nfocusing force acting on the relativistic electron near the\ndriveraxiscan be approximatedasfollows Facc=fmcωpandF⊥≃ −mκ2ω2\npr, respectively, where ris the trans-\nverse displacement of the electron from the driver axis, f\nandκare the numerical factor and the focusing constant,\nrespectively, determined by the parameters of the driver\nand the plasma, ωp=/parenleftbig\n4πe2n/m/parenrightbig1/2is the plasma fre-\nquency,nisthe densityofthe backgroundplasma, mand\ne=−|e|are the electron mass and the electron charge,\nrespectively [6]. For example, if the driver is the lin-\nearly polarized Gaussian laser pulse with resonant pulse\nduration then f= 0.35a2\n0≃0.7 andκ2≃0.11, where\na0=eEL/(mcωL) = 21/2is chosen, ELis the laser field\namplitude, ωLis the laser frequency [5]. The period of\nthe betatron oscillations is ωβ=ωpκγ−1/2, whereγis\nthe relativistic gamma-factor of the electron.\nThe electrons undergoing betatron oscillations emit\nsynchrotron radiation [11, 12]. The radiated power can\nbe estimated as follows Prad≃2reγ2F2\n⊥/(3mc), where\nre=e2/(mc2)≃3·10−13cm is the classical elec-\ntron radius, cis the speed of light. Since the power is\nproportional to the square of the electron energy, the\nradiation losses can stop electron acceleration at some\nthreshold value of the electron energy. The threshold\nenergy can be estimated by balancing the accelerating\nforce and the radiation reaction force, Frrf≃Prad/c, so\nthatγ2\nth≃f/(ǫκ4R2\nβ), where Rβ=kpris the normal-\nized amplitude of betatron oscillations, ǫ= 2reωp/(3c)\nandkp=ωp/c. The threshold energy is ∼100GeV for\nf= 0.7,n= 1019cm−3andRβ= 1 and κ2= 0.11.\nTherefore the radiative damping may be a serious limi-\ntation of electron acceleration.\nThe electron acceleration in plasma with the radiation\nreactioneffecthasbeenstudiedtheoretically[4,5,13,14].\nTheradiationreactionhasbeentreatedasaperturbation\n[13]. The first-order radiative correction to the energy\ngain of the accelerated electron bunch and the energy\nspread induced by radiation emission have been derived\nfor the constant accelerating force. The dependence of\nthe electron energy on time has been calculated in the\nplasma channel without the accelerating force and with\nthe radiation reaction force [14]. Here we study the elec-2\ntron acceleration treating the radiation damping unper-\nturbatively and analyzing the infinite-time limit.\nWe start from the relativisticequation for electron mo-\ntion in an electromagnetic field with the radiative reac-\ntion force in Landau-Lifshits form [15]\nγdui\ndt=cre\neFikuk+2r2\ne\n3mcFi\nrad, (1)\nwhereFi\nrad=Fi\n1+Fi\n2+Fi\n3,Fi\n1= (e/re)/parenleftbig\n∂Fik/∂xl/parenrightbig\nukul,\nFi\n2=−FilFkluk,Fi\n3=/parenleftbig\nFklul/parenrightbig/parenleftbig\nFkmum/parenrightbig\nui,Fikis the\nelectromagnetic field tensor, ukis the 4-velocity of the\nelectron. The first term in Eq. (1) corresponds to the\nLorentz force and the last term corresponds to the radia-\ntion reaction force. We assume that the ultrarelativistic\nelectrons ( γ≫1) are accelerated along x-axis by the\nforceFacc≫F⊥v⊥/cand undergo betatron oscillations\ndriven by the focusing force F⊥≃ −mκ2ω2\npy. Under\nour assumptions, F3≫F1, F2and the focusing forces\nmake a major contribution to the energy losses through\nradiation. It is convenient to introduce new variables\nP= (py/mc)ǫ1/2f1/2,Y=ykpf3/2ǫ1/2,T=ωptκ2/f,\nG=γκ2f−2. Then Eq. (1) can be reduced to the form\ndP\ndT=−Y−Y2PG, (2)\ndY\ndT=P\nG, (3)\ndG\ndT= 1−Y2G2, (4)\nThe obtainedequationsdescribethe betatronoscillations\nwith the radiative damping. The first term on the right-\nhand side of Eq. (4) describes the action of the accelerat-\ning force, while the second term describes the radiative\ndamping.\nWhen the number of the betatron oscillations is large,\nwe can use the averaging method [16]. To do this let us\nintroduce a new variable, S, so that 2 S=|U|2=Y2+\nP2/G=R2\nβf3ǫ≃2/angbracketleftbig\nY2/angbracketrightbig\nandUexp/parenleftbig\ni/integraltext\nG−1/2dT/parenrightbig\n=\nY−iG−1/2P.After averaging over the fast time related\nto the betatron oscillations the averaged equations are\ndS\ndT=−1\n2S\nG−1\n4GS2, (5)\ndG\ndT= 1−SG2. (6)\nAsG >0 andS >0 thendS/dt < 0 and the amplitude\nof the betatron oscillations always decreases with time.\nThis means that for arbitrary electron energy the beta-\ntron oscillation amplitude will be small enough at certain\ninstance of time to be radiation reaction force less than\nthe accelerating force.\nAt the absence of the accelerating force ( f= 0), it\nfollows from Eqs. (5) and (6) that SG−1/4= const and\nγ=γ0/parenleftBig\n1+5ǫR2\nβ,0γ0ωpt/16/parenrightBig−4/5\n, which is in agreement0 1 2\nS012345678910\nG\nFIG. 1: The phase portrait of the system governed byEqs. (5)\nand (6).\nwith the solution calculated in Ref. [14], where Rβ,0=\nRβ(t= 0). At the absence of the radiation reaction (the\nlast terms in RHS of Eqs. (5) and (6) are absent) we get\nG=G0+T,√\nGS= const. The radiation reaction effect\ncan be treated as a perturbation. To the first orderin the\nradiation reaction force the normalized electron energy\nisG=G0+T−(2/5)/bracketleftBig\n1−(G0+T)5/2/bracketrightBig\n, which is in\nagreement with the result obtained in Ref. [13].\nThe system of Eqs. (5) and (6) has integral of motion\nI=1−3SG2/2\nS9/4(SG2)3/4= const. (7)\nThe electron trajectoriesin the phase space S−Gare the\nintegral lines determined by Eq. (7). The phase portrait\nof the system governed by Eqs. (5) and (6) is shown in\nFig. 1. It is seen from Fig. 1 that if initially the acceler-\nating force is stronger than the radiation reaction force\n(SG2<1) then the electron energy monotonically in-\ncreases with time. Otherwise the electron energy decays\nup to the time instance when Facc=Frrf(that corre-\nsponds to SG2= 1) and then it monotonically increases\nwith time. It is also seen from Fig. 1 that all electron tra-\njectories merge in the the limit t→ ∞so thatG→ ∞\nandS→0. It follows from Eq. (7) that S= 2G−2/3 in\nthis limit. We will call the electron acceleration in this\nlimit as an asymptotic acceleration regime (AAR).\nWe verify our analytical results by numerical simula-\ntions. The exact equation(1) and the averagedequations\nof motions (5) and (6) are integrated numerically for test\nelectrons for f= 0.1 andn= 1015cm−3. For simplicity,3\n \n0 2·107 107 2880 2960 \nIn \nωpt 0 2·107 107 1 1.5·106 \nγ \nωpt a) \nb) \nFIG. 2: Thedependenceofa) γandb)Inonωptcalculated by\nsolving of the exact Eq. (1) (black solid lines) and by solvin g\nof the approximate Eqs. (5)-(6) (red dashed lines) for f= 0.1,\nκ2= 0.5,n= 1015cm−3and for initial conditions γ0= 2000,\nRβ,0= 0.8,py,0= 0.\nwe consider the structure of the transverse electromag-\nnetic field similar to the bubble regime: κ2= 0.5 and\nE⊥≈H⊥. The dependence of the normalized integral of\nmotionIn=I−1(ǫκf2)−3, andγonωptfor initial con-\nditionγ0= 2000 and Rβ,0= 0.8,py,0= 0 is shown on\nFig. 2. It is seen from Fig. 2 that the solution of the ex-\nact equations and that of the approximate averaged are\nin a good agreement. Moreover, the integral Iis almost\nconstant for the exact equations (1) (see Fig. 2c).\nWe can introduce new variables g=G/Gtr,τ=T/Ttr\nands= (S/Str)(G/Gtr)−1/4, whereGtr=Ttr=S−2\ntr=\nI2/9. Then Eqs. (5), (6) and (7) are reduced to the form\nwhich does not depend on any parameters. Therefore\nthe characteristic time of transition to AAR is ∼Ttr.\nThe solution of the equations can be written in term\nof hypergeometric function, 2F1(a,b;c;z), [18] as fol-\nlowsϕ(s)−ϕ(s0) =τ,ϕ(s) = 24/9/parenleftbig\n3+2s2/parenrightbig5/9s−4/9−\n213/935/9s14/92F1/parenleftbig\n7/9,4/9;16/9;−2s2/3/parenrightbig\n, where s0=\ns(τ= 0). The asymptotic expansions of function ϕ(s)\nareϕ(s)≈3(3s/2)−4/9fors≪1,ϕ(s)≈δ+s−4/3\nfors≫1, where δ≈1.85. Thus in the limit τ≫1\ns∼τ−9/4≪1 andg∼τ≫1.\nTo derive the asymptotic solution the initial condi-tion should be applied. We assume that S0G2\n0≪1 (so\nthats0≪1 andI≃S−3G−3/2) which is typical for\nthe initial parameters of the electron beam. For exam-\nple, this condition is fulfilled for the initial parameters\nγ0mc2<0.1 TeV,n <1018cm−3,Rβ,0= 1,f= 0.7,\nκ2= 0.11. Making of use the asymptotic expansion for\ns≪1 ands0≫1 we have (9 /4)s−4/9≈τ+δ. There-\nfore the normalized electron energy and the square of the\nnormalized betatron amplitude are in the limit T≫Ttr\nG=δ\n3Gtr+1\n3T, S=2\n3G−2. (8)\nWe can conclude that in AAR Frrf= 2Facc/3so that\nthe electron energy increases linearly with time while the\nbetatron amplitude is reversely proportional to the time.\nThe averaged equations of motions (5) and (6) are in-\ntegrated numerically for the test electrons with the same\nparameters as for Fig. 2 for three values of the initial\nbetatron amplitude Rβ,0= 0.8,0.2,0.1. It is seen from\nFig. 3 that the asymptotic solution (8) is in a good agree-\nment with the result of numerical integration.\nThe radiation damping rate varies for the electrons\nwith different betatron oscillation amplitudes. This\ncauses the energy spread in the electron bunch acceler-\nated in the plasma wave. We assume that the ampli-\ntude of the betatron oscillations of the electrons in the\naccelerated bunch is uniformly distributed in the range\nRmin< Rβ,0< RmaxandRmax≫Rmin. We also again\nassume that S0G2\n0≪1. Then the normalized mean en-\nergy and the normalized square of the relative energy\nspread are in AAR\n/angbracketleftG/angbracketright ≃2\nR2maxRmax/integraldisplay\nRminGRβ,0dRβ,0≃Gmaxδ+T\n3,(9)\nσ2\nG=/angbracketleftbig\nG2/angbracketrightbig\n−/angbracketleftG/angbracketright2≃G2\nmaxδ2\n3/parenleftbiggRmax\nRmin/parenrightbigg2/3\n,(10)\nwhereGmax=Gtr(Rβ,0=Rmax). It follows from\nEqs. (9) and (10) that the relative energy spread,\nσG//angbracketleftG/angbracketright, decreases with time in AAR.\nEqs. (2)-(4) are derived under conditions that F⊥gives\nthe main contribution to the radiative damping and\nF3≫F1, F2. However F⊥goes to zero in the limit\nt→ ∞. Therefore we should check: should the acceler-\nating force and terms F1,F2be taken into account in the\nradiation reaction force in this limit? First it is signif-\nicant that the radiation reaction force remains constant\nin AAR because F⊥∼Rβ→0 andγ→ ∞fort→ ∞\nin such way that R2\nβγ2= const. Making of use Eq. (8)\nand relation vy∼ωβywe getF2/F3∼fǫ≪1 and\nF1/F3∼(3/4)κ2fγ−1/2ǫ1/2≪1, where we assume that\nκ∼f∼1. The contribution from the accelerating force\n(or from Ex) toF3is of the order F2/F3≪1. Therefore\nourmodeldefinedbyEqs.(5)and(6)isvalidinAAR.For4\nγ \n ωp t 16·107\n1.2·109 03\n2\n1\nFIG. 3: Thedependenceof γonωptinAAR:analytic solution\n(red dashed lines) and numerical solution (black solid line ) for\nRβ,0= 0.8 (lines 1), Rβ,0= 0.2 (lines 2) and Rβ,0= 0.1 (lines\n3). The other parameters are the same as in Fig. 2.\nhigh energy electrons quantum electrodynamics (QED)\neffects can be important. The energy of the photon emit-\nted by the accelerated electron can be so high that the\nquantum recoil becomes strong. The photon emission\ncan be treated in classical approach if QED parame-\nterχ=/bracketleftBig\n(mcγE+p×H)2−(p·E)2/bracketrightBig1/2\n/(mcEcr)≃\nγF⊥/(eEcr) is much less than unity, where Ecr=\nm2c3/(e/planckover2pi1)≈1.32×1016V/cm is the QED critical\nfield [17]. χcan be estimated in AAR as follows χ≈/bracketleftbig\n(2f/α)/parenleftbig\n/planckover2pi1ωp/mc2/parenrightbig/bracketrightbig1/2≪1, where α=e2//planckover2pi1c≈1/137\nis the fine structure constant. Therefore the classical ap-\nproachfor the radiationreactionforce isvalid in the limit\nt→ ∞because, like for the corrections to the radiation\nreaction force, the growth of γinχis compensated by\ndecreasing of F⊥.\nThe distance passed by the electron before reaching\nAAR is kpltr≃/parenleftbig\nf/κ2/parenrightbig\nTtr≃1.6/parenleftBig\nǫ2γ0R4\nβ,0fκ8/parenrightBig−1/3\n.\nFor the initial parameters n= 1018cm−3,Rβ,0= 1,\nγ0= 2·103,f= 0.7,κ2= 0.11 the electron comes into\nAAR after passing 7800 laser-driven acceleration stages\nwith total distance ltr≃73 m, achieving the energy\nγmc2≃5 TeV and Rβ≃0.008, where the stage dis-\ntance is chosen to be equal to the half dephasing length\n[5] and the distance between the acceleration stages is\nneglected. For the rarefied plasma n= 1015cm−3, AAR\nis achievedin 78 stageswith ltr≃23km,γmc2≃48TeV\nandRβ≃0.005. AAR may be achieved within one accel-\neration stage in the proton-driven acceleration schemes\nbecause of very large dephasing length [19].\nIn conclusions, we have shown that the electron accel-\nerationisnotlimitedbytheradiativedampinginplasma-\nbased accelerators. Even if the radiation reaction force\nis stronger than the accelerating force at the beginning,\nthen acceleration eventually succeeds deceleration with\ntime. The damping of the betatron oscillations leads tothe transition to the self-similar asymptotic acceleration\nregime in the infinite-time limit when the radiation re-\naction force becomes equal to two thirds of the acceler-\nating force. The relative energy spread induced by the\nradiative damping in the accelerated electron bunch de-\ncreases with time in this regime. This opens possibility\nto use high density plasma at the late stages of multi-\nstage plasma-based accelerators despite the fact that the\nradiative damping is enhanced as density increases. The\nhigh density plasma can be favorable because it provides\nhigh accelerating gradient and, thus, reduces the length\nof the acceleration stages. The obtained results can be\nalso applied to any other accelerating systems with the\nlinear focusing forces.\nThisworkwassupportedinpartsbytheRussianFoun-\ndation for Basic Research, the Ministry of Science and\nEducationoftheRussianFederation, theRussianFederal\nProgram “Scientific and scientific-pedagogical personnel\nof innovative Russia”.\n∗Electronic address: kost@appl.sci-nnov.ru\n[1] S. P.D.Mangles et al.,Nature(London) 431, 535(2004);\nC. G. R. Geddes et al., ibid. 431, 538 (2004); J. Faure et\nal., ibid. 431, 541 (2004).\n[2] W. P. Leemans et al.,Nat. Phys. 2, 696 (2006).\n[3] I. Blumenfeld et al., Nature 445, 741 (2007).\n[4] C. B. Schroeder et al.,Phys. Rev. ST Accel. Beams 13,\n101301 (2010).\n[5] K. Nakajima et al.,Phys. Rev. ST Accel. Beams 14,\n091301 (2011).\n[6] E. Esarey et al.Rev. Mod. Phys. 81, 1229 (2009).\n[7] T. Katsouleas, Phys. Rev. A 33, 2056 (1986).\n[8] A.PukhovandI.Kostyukov,Phys.Rev.E 77, 025401(R)\n(2008).\n[9] E. Esarey, J. Krall, and P. Sprangle, Phys. Rev. Lett. 72,\n2887 (1994).\n[10] L. A. Abramyan et al.,Sov. Phys. JETP 75, 978 (1992).\n[11] E. Esarey et al.,Phys. Rev. E 65, 056505 (2002).\n[12] I. Kostyukov, S. Kiselev and A. Pukhov, Phys. Plas-\nmas.10, 4818 (2003).\n[13] P. Michel et al.,Phys. Rev. E 74, 026501 (2006).\n[14] I. Yu.Kostyukov, E. N. NerushandA. M. Pukhov, JETP\n103, 800 (2006).\n[15] L. D. Landau and E. M. Lifshitz, Course of Theoretical\nPhysics, Vol. 2: The Classical Theory of Fields, 7th ed.\n(Nauka, Moscow, 1988; Pergamon, Oxford, 1975).\n[16] N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic\nMethods in the Theory of Nonlinear Oscillations, 4th ed.\n(Nauka, Moscow, 1974; Gordon and Breach, New York,\n1962), p. 217.\n[17] V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii ,\nQuantum Electrodynamics (Pergamon Press, New York,\n1982).\n[18]Handbook of Mathematical Functions , edited by\nM. Abramowitz and I. A. Stegun (Dover, New York,\n1972).\n[19] A. Caldwell et al.,Nat. Phys, 5, 363 (2009);"    },    {        "title": "1206.4819v2.Fast_domain_wall_propagation_in_uniaxial_nanowires_with_transverse_fields.pdf",        "content": "arXiv:1206.4819v2  [cond-mat.mtrl-sci]  18 Aug 2013Fast domain wall propagation in uniaxial nanowires with tra nsverse fields\nArseni Goussev1,2, Ross G. Lund3, JM Robbins3, Valeriy Slastikov3, Charles Sonnenberg3\n1Department of Mathematics and Information Sciences,\nNorthumbria University, Newcastle Upon Tyne, NE1 8ST, UK\n2Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Straße 38, D-01187 Dresden, Germany\n3School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom\n(Dated: May 14, 2018)\nUnder a magnetic field along its axis, domain wall motion in a u niaxial nanowire is much slower\nthan in the fully anisotropic case, typically by several ord ers of magnitude (the square of the di-\nmensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse\nto the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the\nWalker breakdown field, domain walls in a uniaxial wire propa gate faster than in a fully anisotropic\nwire (without transverse field). Beyond this critical field s trength, precessional motion sets in, and\nthe mean velocity decreases. Our results are based on leadin g-order analytic calculations of the\nvelocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.\nPACS numbers: 75.75.-c, 75.78.Fg\nIntroduction\nThe dynamics of magnetic domain walls in ferromag-\nnetic nanowiresunder external magnetic fields [1–12] and\nspin-polarised currents [12–20] is a central problem in\nmicromagnetics and spintronics, both as a basic physical\nphenomenonaswellasacornerstoneofmagneticmemory\nand logic technology [3, 16–18]. From the point of view\nof applications, it is desirable to maximise the domain\nwall velocity in order to optimise switching and response\ntimes.\nPartly because of fabrication techniques, attention has\nbeen focused on nanowires with large cross-sectional as-\npect ratio, typically of rectangular cross-section. In this\ncase, even if the bulk material is isotropic (e.g., permal-\nloy), the domain geometry induces a fully anisotropic\nmagnetic permeability tensor, with easy axis along the\nwire and hard axis along its shortest dimension [21, 22].\nNanowires with uniaxial permeability, characteristic of\nmoresymmetricalcross-sectionalgeometries(e.g., square\nor circular), have been less studied [23–25]. Here we in-\nvestigate domain wall (DW) motion in uniaxial wires in\nthe presence of transverse fields. We show that the DW\nvelocityinuniaxialwiresdependsstronglyonthelongitu-\ndinal applied field H1, increasing with H1up to a certain\ncritical field and thereafter falling off as precessional mo-\ntion sets in. We employ a systematic asymptotic expan-\nsion scheme, which differs from alternative approaches\nbased on approximate dynamics for the DW centre and\norientation; a detailed account of this scheme, also in-\ncluding anisotropy and current-induced torques, will be\ngiven separately [29] .\nWe employ a continuum description of the magnetisa-\ntion. For a thin nanowire, this is provided by the one-\ndimensionalLandau-Lifshitz-Gilbert(LLG)equation[22,26–28], which we write in the non-dimensionalised form\n˙M=γM×H−αM×(M×H). (1)\nHereM(x,t) is a unit-vector field specifying the orien-\ntation of the magnetisation, which we shall also write\nin polar form M= (cosΘ ,sinΘcosΦ ,sinΘsinΦ). The\neffective magnetic field, H(m), is given by\nH=Am′′+K1m1ˆx−K2m2ˆy+Ha.(2)\nHereAis the exchange constant, K1is the easy-axis\nanisotropy, K2>0 is the hard-axis anisotropy, Hais\nthe applied magnetic field (taken to be constant), γis\nthe gyromagnetic ration, and αis the Gilbert damping\nparameter. For convenience we choose units for length,\ntime and energy so that A=K1=γ= 1.Domains cor-\nrespond to locally uniform configurations in which Mis\naligned along one of the local minima, denoted m+and\nm−, of the potential energy\nU(m) =−1\n2(m2\n1−K2m2\n2)−m·Ha.(3)\nTwo distinct domains separated by a DW are described\nby the boundary conditions M(±∞,t) =m±.\nFor purely longitudinal fields Ha=H1ˆxand forH1\nbelowtheWalkerbreakdownfield HW=αK2/2,theDW\npropagates as a travelling wave [1], the so-called Walker\nsolution Θ( x,t) =θW(x−VWt), Φ(x,t) =φW, whereθW\nandφWare given by\nθW(ξ) = 2tan−1(e−ξ/γ),sin2φW=H1/HW.(4)\nThe width of the DW, γ, is given by γ= (1 +\nK2cos2φW)−1/2, and the velocity is given by\nVW=−γ(α+1/α)H1. (5)\nForH1> HW, the DW undergoes non-uniform preces-\nsion and translation, with mean velocity decreasing with2\nH1[1, 5, 6, 9]. The effects of additional transverse fields\nhave been examined recently [7, 11].\nIf the cross-sectional geometry is sufficiently symmet-\nrical (e.g., square or circular), the permeability tensor\nbecomes uniaxial, so that K2= 0 [21, 22]. The dynamics\nin this case is strikingly different. The LLG equation has\nan exact solution, Θ( x,t) =θ0(x−VPt), Φ(x,t) =−H1t,\nin which the DW propagates with velocity\nVP=−αH1 (6)\nand precesses about the easy axis with angular velocity\n−H1[23, 24]. The precessing solution persists for all H1\n– there is no breakdown field – but becomes unstable for\nH1/greaterorsimilar1/2 [25].\nForH1< HW, the ratio VW/VPisgiven by γ(α−2+1).\nFor typical values of α(0.01 – 0.1), the uniaxial velocity\nVPis less than the fully anisotropic velocity VWby sev-\neral orders of magnitude. As we show below, applying a\ntransverse field H2>0 to a uniaxial wire dramatically\nchanges its response to an applied longitudinal field H1.\nThe transverse field, analogous to hard-axis anisotropy,\ninhibits precession and facilitates fast DW propagation.\nForH1lessthanan H2-dependentcriticalfield H1c, given\nin the linear regime by (29) below, there appears a trav-\nelling wave, while for H1> H1c, there appears an oscil-\nlating solution, as in the Walker case. The DW velocity\nof travelling wave exceeds that of oscillating solution.\nVelocity of travelling wave\nWe first obtain a general identity, of independent in-\nterest, which relates the velocity of a travelling wave\nM(x,t) =m(x−Vt) (assuming one exists) to the change\nin potential energy across the profile (for zero transverse\nfield, this coincides with results of [1] and [10]). Noting\nthat˙M=−Vm′, we take the squareof (1) and integrate\nover the length of the wire to obtain\nV2||m′||2= (1+α2)||m×H||2. (7)\nHere we use the notation\n||u||2=/angbracketleftu,u/angbracketright,/angbracketleftu,v/angbracketright=/integraldisplay∞\n−∞u·vdx(8)\nfor theL2-norm and inner product of vector fields (anal-\nogous notation for scalar fields is used below). Next, we\ntake the inner product of (1) with Hto obtain\nV/angbracketleftm′,H/angbracketright=−α||m×H||2. (9)\nNoting that m′·H=/parenleftbig1\n2m′·m′−U(m)/parenrightbig′, we combine\n(7) and (9) to obtain\nV=1\n2(α+1/α)||m′||−2(U(m−)−U(m+)).(10)The identity (10) has a simple physical interpretation;\nthe velocity is proportional to the potential energy dif-\nference across the wire, and inversely proportional to the\nexchange energy of the profile.\nFrom now on, we consider the uniaxial case K2= 0\nandappliedfieldwithlongitudinalandtransversecompo-\nnentsH1,H2>0 (by symmetry, we can assume H3= 0)\nwith|Ha|<1. An immediate consequence of (10) is\nthat, in the uniaxial case, the velocity must vanish as\nH1goes to zero. For when H1= 0, the local minima\nm±are related by reflection through the 23-plane, and\nU(m+) =U(m−).\nSmall transverse field\nIn order to understand travelling wave and oscillating\nsolutions as well as the transition between them, we first\ncarry out an asymptotic analysis in which both H1and\nH2are regarded as small, writing H1=ǫh1,H2=ǫh2\nand rescaling time as τ=ǫt(a systematic treatment in-\ncludingcurrent-inducedtorqueswillbegivenin[29]). We\nseek a solution of the LLG equation (1) of the following\nasymptotic form:\nΘ(x,t) =θ0(x,τ)+ǫθ1(x,τ)+..., (11)\nΦ(x,t) =φ0(x,τ)+ǫφ1(x,τ)+... (12)\nIt is straightforward to check that the boundary condi-\ntions, namely that mapproach distinct minima of Uas\nx→ ±∞, imply that\nm(±∞,τ) = (±1,ǫh2,0)+O(ǫ2).(13)\nThe leading-order equations for Θ and Φ become\nθ0,xx−1\n2(1+φ2\n0,x)sin2θ0= 0, (14)\n/parenleftbig\nsin2θ0φ0,x/parenrightbig\nx= 0. (15)\nThe only physical (finite-energy) solutions of (14) and\n(15) consistent with the boundary conditions (13) are of\nthe form\nφ0(x,τ) =φ0(τ) (16)\nθ0(x,τ) = 2arctanexp( −(x−x∗(τ))),(17)\nwhereφ0andx∗respectivelydescribetheDWorientation\nand centre, and are functions of τalone. It is convenient\nto introduce a travelling coordinate ξ=x−x∗(τ) and\nrewrite the ansatz (11)–(12) as\nΘ(x,t) =θ0(ξ,τ)+ǫθ1(ξ,τ)+..., (18)\nΦ(x,t) =φ0(ξ,τ)+ǫφ1(ξ,τ)+... (19)\nTo obtain equations for φ0(τ) andx∗(τ) we must pro-\nceed to the next order. It is convenient to introduce new3\nvariables at order ǫwhich, in light of the boundary con-\nditions (13), vanish at x=±∞, as follows:\nΘ1:=θ1−h2cosφ0cosθ0, (20)\nu:=φ1sinθ0+h2sinφ0. (21)\nThese satisfy the linear inhomogeneous equations\nLΘ1=f, (22)\nLu=g. (23)\nHereLis the self-adjoint Schr¨ odinger operator given by\nL=−∂2\n∂ξ2+W(ξ), (24)\nwhere\nW=θ′′′\n0\nθ′\n0= 1−2sech2ξ, (25)\nandf(ξ,τ) andg(ξ,τ) are given by\nf= (1+α2)−1sinθ0(−α˙x∗−˙φ0)−h1sinθ0,\ng= (1+α2)−1sinθ0(˙x∗−α˙φ0)+2h2sin2θ0sinφ0.\n(26)\nTheDWposition x∗andorientation φ0aredetermined\nfrom the solvability conditions for (22) – (23). According\nto the Fredholm alternative, given a self-adjoint opera-\ntorLonL2(R), a necessary condition for the equation\nLΘ1=fto have a solution Θ 1is thatfbe orthogonal to\nthe kernel of L. If this is the case, a sufficient condition is\nthatthespectrumof Lisisolatedawayfrom0. From(24)\nand (25) it is clear that θ′\n0belongs to the kernel of L, and\nsince the eigenvalues of a one-dimensional Schr¨ odinger\noperator are nondegenerate, it follows that θ′\n0spans the\nkernel of L. Moreover, since W(ξ)→1 asξ→ ±∞, it\nfollowsthat the spectrum of Lis discrete near0. (In fact,\nWis a special case of the exactly solvable P¨ oschl-Teller\npotential, but we won’t make use of this fact.) Requiring\nfandgin (22) and (23) to be orthogonal to θ′\n0and not-\ning that /angbracketleftθ′\n0,θ′\n0/angbracketright= 2,/angbracketleftθ′\n0,sinθ0/angbracketright=−2,/angbracketleftθ′\n0,1/angbracketright=−π, and\n/angbracketleftθ′\n0,cosθ0/angbracketright= 0, we obtain the following system of ODEs\nforφ0andx∗:\n˙φ0=−h1−απ\n2h2sinφ0, (27)\n˙x∗=−αh1+π\n2h2sinφ0. (28)\nTravelling wave solutions appear provided (27) has\nfixed points; this occurs for h1below a critical field h1,c\ngiven by\nh1,c=απh2\n2, (29)The velocity and orientation of the travelling wave are\ngiven by\n˙x∗=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1, (30)\nsinφ0=−h1\nh1,c. (31)\nThere are two possible solutions for φ0∈[0,2π), only\none of which is stable. Oscillating solutions appear for\nh1> h1c, and are given by\nh1tan1\n2φ0=−h1,c−/radicalBig\nh2\n1−h2\n1,ctan/parenleftBig\n1\n2/radicalBig\nh2\n1−h2\n1,cτ/parenrightBig\n(32)\nwith the period T= 2π//radicalBig\nh2\n1−h2\n1,c. The mean preces-\nsional and translational velocities are obtained by aver-\naging over a period, with result\n/angbracketleftBig\n˙φ0/angbracketrightBig\n=−sgn(h1)/radicalBig\nh2\n1−h2\n1,c, (33)\n/angbracketleft˙x∗/angbracketright=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1+1\nαsgn(h1)/radicalBig\nh2\n1−h2\n1,c.(34)\nNote that for h1=h1,c, (34) coincides with the travelling\nwave velocity (30), whereas for h1≫h1,c, (34) reduces\nto the velocity of the precessing solution given by (6).\nThe behaviour is similar in many respects to the\nWalker case (i.e., K2/negationslash= 0 and H2= 0). Here, the trans-\nverse field rather than hard-axis anisotropy serves to ar-\nrest the precession of the DW (provided the longitudinal\nfield is not too strong). There are differences as well;\nin the transverse-field case there is just one stable trav-\nelling wave, whereas in the Walker case there are two.\nAlso, in the transverse-field case the asymptotic value of\nthe magnetisation has a transverse component, whereas\nin the Walker case it has none.\nModerate transverse field\nWe can extend the travelling wave analysis to the\nregime where H2is no longer regarded as small. We\ncontinue to regard H1as small, writing H1=ǫh1and\nV=ǫv, and expand the travelling wave ansatz Θ( x,t) =\nθ(x−Vt), Φ(x,t) =φ(x−Vt) to first order in ǫ, writing\nθ=θ0+ǫθ1,φ=φ0+ǫφ1. Substituting into the LLG\nequation, we obtain the O(ǫ0) equations\nθ′\n0= (H2−sinθ0), φ0= 0, (35)\nwith boundary conditions sin θ0±=H2,θ0+> π/2 and\nθ0−< π/2. Thus, for H2=O(ǫ0), azimuthal symmetry\nisbrokenatleadingorder,andthestaticprofileisparallel\nto the transversefield (the alternativesolution with φ0=\nπis unstable). The solution of (35) is given by\ntanθ0\n2=κ\nH2tanh/bracketleftbigg\ntanh−1/parenleftbiggH2−1\nκ/parenrightbigg\n−κ\n2ξ/bracketrightbigg\n+1\nH2,\n(36)4\nwhereκ=/radicalbig\n1−H2\n2.\nAt order ǫwe obtain the linear inhomogeneous equa-\ntions\nLθ1=α\n1+α2vθ′\n0−h1sinθ0, (37)\nMφ1=1\n1+α2v(cosθ0)′, (38)\nwhere\nL=−d2\ndξ2+θ′′′\n0\nθ′\n0, M=−d\ndξsin2θ0d\ndξ+H2sinθ0.(39)\nHereθ0is given by (36), and θ1,φ1are requiredto vanish\nasξ→ ±∞. As above, the Fredholm alternative implies\nthat the right-hand side of (37) must be orthogonal to θ′\n0\nin order for a solution to exist. Calculation yields\nV=−/parenleftbigg\nα+1\nα/parenrightbigg\n(1−(H2/κ)cos−1H2)−1H1.(40)\nForH2= 0, this coincides with (30); thus, (40) gives\nH2-nonlinear corrections to the velocity. Moreover, it is\nstraightforward to show that (40) is consistent with the\ngeneral identity (10). Finally, one can also show that\nMhas trivial kernel with spectrum bounded away from\nzero, so that (38) is automatically solvable.\nItisinterestingtocomparetheDWvelocitywithtrans-\nverse field to the Walker case. From (5) and (40),\nVW/V=γ(1−(H2/κ)cos−1H2)<1.(41)\nThus, to leading orderin H1, the DWvelocity in a uniax-\nial wire with transverse field exceeds the Walker velocity.\nNumerical results below establish that this continues to\nhold asH1approaches the critical field H1c.\nNumerical results\nToverifyouranalyticalresults, wesolvetheLLGequa-\ntion (1) using a finite-difference scheme on a domain\n−L≤x≤LwhereL= 100 (the DW has width of\norder 1). Neumann boundary conditions, m′= 0, are\nmaintained at the endpoints. The damping parameter α\nis taken to be 0 .1 throughout. As initial condition we\ntake the stationary profile, with θ0given by (36) and\nφ0= 0. After an initial transient period, during which\nthe asymptotic values of matx→ ±Lconverge to m±,\na stable solution emerges, in which the DW propagates\nwith a characteristic mean velocity V. (For convenience,\nwehavetaken H1<0, sothat Vis positive.) In Figure1,\nnumerically computed values of Vare plotted as a func-\ntion of|H1|for three fixed values of the transverse field:\nH2= 0.2,H2= 0.1, and the limiting case H2= 0, where\nthe dynamics is given by the precessing solution. There\nis good quantitative agreement with the analytic results\nfor small transverse fields, (30), for |H1|< H1,c, and(34), for |H1|> H1,c, In Figure 2, the analytic expres-\nsions for the velocity for small and moderate transverse\nfields are compared to numerical results for H2= 0.2 and\n|H1| ≪H1c. The moderate-field expression (40), which\ndepends nonlinearly in H2, gives excellent agreement for\nsmalldrivingfields. Fornonzero H2, the velocityexhibits\na peak at a critical field |H1c|, which depends on H2.\n0 0.1 0.2 0.3 0.400.050.10.150.20.250.30.35\n|H1|V\n  \nH2= 0.2\nH2= 0.1\nH2= 0\nFIG. 1: Average DW velocity Vas a function of the driving\nfield|H1|for three values of the transverse field H2. The an-\nalytic formulas (solid curves) (30), for |H1|< H1,c, and (34),\nfor|H1|> H1,c, are plotted against numerically computed\nvalues (open circles). For H2= 0, the analytic formula is\nexact.\n00.005 0.01 0.015 0.02 0.02500.050.10.150.20.250.3\n|H1|V\n  \nSmall −H2theory\nModerate −H2theory\nNumerics\nFIG. 2: DW velocity Vas a function of the driving field\n|H1|forH2= 0.2. The expressions for small-transverse field\n(30) (red curve) and moderate-transverse field (40) (light b lue\ncurve) are plotted against numerically computed values (op en\ncircles).\nFigure 3 shows the dependence of the critical field |H1,c|5\nonH2, in close agreement with the analytic result (29).\n0 0.05 0.1 0.15 0.200.0050.010.0150.020.0250.030.035\nH2|H1,c|\nFIG. 3: The critical driving field |H1,c|as a function of the\ntransverse field H2. A linear fit (blue curve) through the nu-\nmerically computed data (blue diamonds) is plotted alongsi de\nthe analytical result (29) (red curve).\nFIG. 4: The magnetization distribution, θ(x,t) andφ(x,t),\nfor two values of the driving field: H1=−0.01 in figures\n(a) and (b), and H1=−0.05 in figures (c) and (d). The\ntransverse field is taken as H2= 0.1 throughout.\nAs in the Walker case, the properties of the propagat-\ning solution are qualitatively different for driving fields\n|H1|below and above the critical field. This is confirmed\nin Figure 4, which shows contour plots of the magne-\ntization in the ( x,t)-plane. Figs. 4(a) and 4(b), where\nH1=−0.01, exemplify the case |H1|<|H1c|. The mag-\nnetisation evolves as a fixed profile translating rigidly\nwith velocity V. For|H1|>|H1c|, as exemplified by\nFigs. 4(c) and 4(d), where H1=−0.05. the magnetiza-\ntion profile exhibits a non-uniform precession as it prop-agates along the nanowire, with mean velocity in good\nagreement with (34).\nSummary\nWe haveestablished, both analyticallyin leading-order\nasymptotics and numerically, the existence of travelling\nwaveandoscillatingsolutionsoftheLLGequationinuni-\naxial wires in applied fields with longitudinal and trans-\nverse components. We have obtained analytic expres-\nsions for the velocity, (30) and (40), and for the critical\nlongitudinal field, (29), above which the travelling wave\nsolution ceases to exist. We have also obtained the mean\nprecessional and linear velocities (33) and (34) for oscil-\nlating solutions. The analytic results are confirmed by\nnumerics.\n[1] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[2] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nT. Shinjo, Phys. Rev. Lett. 92077205 (2004).\n[3] D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson,\nD. Petit and R.P. Cowburn, Science 309, 1688 (2005).\n[4] R.P. Cowburn, Nature (London) 448, 544 (2007).\n[5] G.S.D Beach, C. Nistor, C. Knutson, M. Tsoi, and\nJ.L. Erskine, Nature Mater. 4, 741(2005).\n[6] J. Yang, C. Nistor, G.S.D. Beach, and J.L. Erskine,\nPhys. Rev. B 77, 014413 (2008).\n[7] M.T. Bryan, T. Schrefl, D. Atkinson, D.A. Allwood,\nJ. Appl. Phys. 103, 073906 (2008).\n[8] O.A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Baza-\nliy and O. Tchernyshyov, Phys. Rev. Lett. 100127204\n(2008).\n[9] X.R. Wang, P. Yan, J. Lu, Europhys. Lett. 86, 67001\n(2009).\n[10] X.R. Wang, P. Yan , J. Lu, C. He, Ann. Phys. 324, 1815–\n1820 (2009).\n[11] J. Lu and X.R. Wang, J. Appl.Phys. 107, 083915 (2010).\n[12] A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas and\nJ. Ferr´ e, Europhys. Lett. 78, 57007 (2007).\n[13] Z. Li and S. Zhang, Phys. Rev. Lett. 92207203 (2004).\n[14] A. Thiaville, Y. Nakatani,, J. Miltat and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n[15] G.S.D Beach, C. Knutson, C. Nistor, M. Tsoi, and\nJ.L. Erskine, Phys. Rev. Lett. 97057203 (2006)\n[16] S. S. P. Parkin, M. Hayashi and L. Thomas, Science 320,\n190 (2008).\n[17] M. Hayashi, L. Thomas, R. Moriya, C. Rettner and\nS. S. P. Parkin, Science 320, 209 (2008).\n[18] L. Thomas, R. Moriya, C. Rettner, S. and S. P. Parkin,\nScience330, 1810 (2010).\n[19] O.A. Tretiakov and Ar. Abanov,\nPhys. Rev. Lett. 105157201 (2010).\n[20] O.A. Tretiakov, Y. Liu and Ar. Abanov,\nPhys. Rev. Lett. 108247201 (2012).\n[21] V. Slastikov and C. Sonnenberg, IMA J. Appl. Math. 77\nno. 2, 220 (2012)6\n[22] A. Hubert and R. Sch¨ afer, Magnetic Domains: The\nAnalysis of Magnetic Microstructures (Springer, Berlin,\n1998).\n[23] Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104,\n037206 (2010).\n[24] A. Goussev, J.M. Robbins, V. Slastikov, Phys. Rev. Lett .\n104, 147202 (2010).\n[25] Y. Gou, A. Goussev, J. M. Robbins, V. Slastikov, Phys.\nRev. B84, 104445 (2011)[26] L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu-\nnion8, 153 (1935).\n[27] T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMag.40, 3443 (2004).\n[28] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys.\nRep.194, 117 (1990).\n[29] A. Goussev, R. Lund, J.M. Robbins, C. Sonnenberg,\nV. Slastikov, in preparation."    },    {        "title": "1207.0310v1.Establishing_micromagnetic_parameters_of_ferromagnetic_semiconductor__Ga_Mn_As.pdf",        "content": "Establishing micromagnetic parameters of ferromagnetic\nsemiconductor (Ga,Mn)As\nP. N\u0014 emec,1V. Nov\u0013 ak,2N. Tesa\u0014 rov\u0013 a,1E. Rozkotov\u0013 a,1H. Reichlov\u0013 a,2, 1\nD. Butkovi\u0014 cov\u0013 a,1F. Troj\u0013 anek,1K. Olejn\u0013 \u0010k,2P. Mal\u0013 y,1R. P. Campion,3\nB. L. Gallagher,3Jairo Sinova,4, 2and T. Jungwirth2, 3\n1Faculty of Mathematics and Physics, Charles University in Prague,\nKe Karlovu 3, 121 16 Prague 2, Czech Republic\n2Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n3School of Physics and Astronomy, University of Nottingham,\nNottingham NG7 2RD, United Kingdom\n4Department of Physics, Texas A&M University,\nCollege Station, TX 77843-4242, USA\n(Dated: November 20, 2018)\nPACS numbers: 75.50.Pp,75.30.-m,75.70.Ak\n1arXiv:1207.0310v1  [cond-mat.mtrl-sci]  2 Jul 2012(Ga,Mn)As is at the forefront of research exploring the synergy of magnetism\nwith the physics and technology of semiconductors, and has led to discover-\nies of new spin-dependent phenomena and functionalities applicable to a wide\nrange of material systems. Its recognition and utility as an ideal model material\nfor spintronics research has been undermined by the large scatter in reported\nsemiconducting doping trends and micromagnetic parameters. In this paper\nwe establish these basic material characteristics by individually optimizing the\nhighly non-equilibrium synthesis for each Mn-doping level and by simultane-\nously determining all micromagnetic parameters from one set of magneto-optical\npump-and-probe measurements. Our (Ga,Mn)As thin-\flm epilayers, spannig\nthe wide range of accessible dopings, have sharp thermodynamic Curie point\nsingularities typical of uniform magnetic systems. The materials show system-\natic trends of increasing magnetization, carrier density, and Curie tempera-\nture (reaching 188 K) with increasing doping, and monotonous doping depen-\ndence of the Gilbert damping constant of \u00180:1\u00000:01 and the spin sti\u000bness of\n\u00182\u00003 meV nm2. These results render (Ga,Mn)As well controlled degenerate\nsemiconductor with basic magnetic characteristics comparable to common band\nferromagnets.\nUnder equilibrium growth conditions the incorporation of magnetic Mn ions into III-\nAs semiconductor crystals is limited to approximately 0.1%. To circumvent the solubility\nproblem a non-equilibrium, low-temperature molecular-beam-epitaxy (LT-MBE) technique\nwas employed which led to \frst successful growths of (In,Mn)As and (Ga,Mn)As ternary\nalloys with more than 1% Mn and to the discovery of ferromagnetism in these materials.1{6\nThe compounds qualify as ferromagnetic semiconductors to the extent that their magnetic\nproperties can be altered by the usual semiconductor electronics engineering variables, such\nas doping, electric \felds,7{12or light.13{27By exploiting the large spin polarization of car-\nriers and low saturation moment in (Ga,Mn)As and building on the well established het-\nerostructure growth and microfabrication techniques in III-V semiconductors, (Ga,Mn)As\nhas been extensively used for spintronics research of direct and inverse magneto-transport\nphenomena.28{37Besides the more conventional spintronic e\u000bects based on Mott's two-spin-\nchannel model of conduction in ferromagnets, (Ga,Mn)As has become particularly fruitful\nfor exploring the second, more physically intriguing spintronics paradigm based on Dirac's\n2spin-orbit coupling.34,38{46\nThe apparent potential of (Ga,Mn)As to become the test-bed model material for many\nlines of spintronics research has been hindered by the large scatter in reported semiconduct-\ning doping trends and micromagnetic parameters. Our strategy to tackle this problem begins\nfrom the synthesis of a set of (Ga,Mn)As materials spanning a wide range of Mn dopings. Be-\ncause of the highly non-equilibrium nature of the heavily-doped ferromagnetic (Ga,Mn)As,\nthe growth and post-growth annealing procedures have to be individually optimized for each\nMn-doping level in order to obtain \flms which are as close as possible to idealized uniform\n(Ga,Mn)As mixed crystals with the minimal density of compensating and other uninten-\ntional defects. An extensive set of characterization measurements has to accompany the\nsynthesis to guarantee that the materials show systematic doping trends; monitoring the\nthermodynamic Curie point singularities is essential for assuring the uniformity and high\nmagnetic quality of the materials.10,47{49When omitting the above procedures,50extrinsic\nimpurities and sample inhomogeneities can yield non-systematic doping trends and conceal\nthe intrinsic magnetic properties of (Ga,Mn)As.\nThe focus of the work presented in this paper is on the systematic study of the Gilbert\ndamping and spin sti\u000bness constants which, together with magnetic anisotropy \felds, rep-\nresent the basic micromagnetic parameters of a ferromagnet. A more than an order of mag-\nnitude experimental scatter and a lack of any clear trend as a function of Mn-doping can be\nfound in the literature for the Gilbert damping and spin-sti\u000bness constants.51{60(See Supple-\nmentary information for a detailed discussion of previous experimental works.) This re\rects\npartly the issues related to the control and reproducibility of the synthesis of (Ga,Mn)As\nand partly the di\u000eculty with applying common magnetic characterization techniques, such\nas neutron scattering, to the thin-\flm dilute-moment (Ga,Mn)As samples. Hand-in-hand\nwith the optimization of the material synthesis we have developed experimental capabilities\nbased on the magneto-optical (MO) pump-and-probe method which allow us to simultane-\nously determine the magnetic anisotropy, Gilbert damping, and spin sti\u000bness constants from\none consistent set of measured data. Our results are summarized in Fig. 1. The Curie point\nsingularity in the temperature derivative of the resistivity d\u001a=dT measured throughout the\nseries of optimized ferromagnetic (Ga,Mn)As samples with metallic conduction is shown in\nFig. 1a. The data span the nominal doping range from x\u00191.5 to 13% and corresponding\nCurie temperatures from Tc= 29 to 188 K, and illustrate the high quality of all the epilayers\n30.8 1.0 1.20.40.60.81.0\n01 0 0 2 0 00246dU/dT ( 10\u00105:cm/K)\nT (K)× 0.5\n(dU/dT)*\nT*13% 5.2%1.5%\n3.010a\nbc\nd2468 1 0-200-150-1000255075Ki (mT)\nx (%)Ku\nKc\nKout\n1.01.52.02.5\n2468 1 00246810D (10-2)\nx (%)\nD (meV.nm2)\n05 0 0 1 0 0 0 1 5 0 0 2 0 0 09% 5.2% 3.8% MO signal (rel. units)\nTime delay (ps)2.5%\nFig. 1FIG. 1: Micromagnetic parameters of optimized epilayers of ferromagnetic (Ga,Mn)As.\na, Examples of sharp Curie point singularities in the temperature derivative of the resistivity in the\nseries of optimized ferromagnetic (Ga,Mn)As epilayers with metallic conduction; Tcmonotonously\nincreases with increasing nominal Mn doping between 1.5 and 13%. Inset shows d\u001a=dT normalized\nto its peak value with the temperature axis normalized to Tc.b,Examples of oscillatory parts of\nMO signals measured in 18 nm thick (Ga,Mn)As epilayers with the depicted nominal Mn doping\nfor external magnetic \feld \u00160Hext= 400 mT applied along the [010] crystallographic direction;\nthe curves are normalized and vertically o\u000b-set for clarity. c,Dependence of anisotropy constants\non nominal Mn doping. d,Dependence of the Gilbert damping constant \u000band the spin sti\u000bness\nconstantDon nominal Mn doping.\nwithin the series. Examples of the measured magnetization precession signals by the MO\npump-and-probe method are shown in Fig. 1b. From these time-dependent magnetization\nmeasurements we obtained the magnetic anisotropy constants Ki, Gilbert damping constant\n4\u000b, and spin sti\u000bness constant Dwhich are summarized in Figs.1c,d. We now proceed to the\ndetail discussion of our experimental techniques and the discussion of the measured results\nin the context of physics of degenerate semiconductors and band ferromagnets.\nOptimization of the (Ga,Mn)As synthesis. Our (Ga,Mn)As layers were grown at\nthe growth rate of approximately 0.2 monolayers/second. The Mn \rux, and hence the nom-\ninal Mn doping x, was determined by measuring the ratio of the beam equivalent pressures\n(BEP) of Mn and Ga sources before each growth. The Mn content was cross-checked by\nsecondary ion mass spectroscopy (SIMS) and by comparing the growth rates of GaAs and\n(Ga,Mn)As measured by the oscillations of the re\rection high-energy electron di\u000braction\n(RHEED).\nThere are two critical growth parameters of (Ga,Mn)As: the substrate temperature,\nand the As-to-(Ga+Mn) \rux ratio. At the typical temperatures of \u0018200\u000eC neither an\noptical pyrometer nor a radiatively coupled temperature sensor are applicable. Instead,\nwe used the GaAs band-edge spectrometer to measure the substrate temperature and the\npredictive substrate heater control to stabilize the temperature during the growth. For a\ngiven As:(Ga+Mn) ratio the substrate temperature fully determines the growth regime: the\ngrowth proceeds two-dimensionally at low temperatures, and turns irreversibly into the 3D\ngrowth mode when a critical temperature is exceeded. The scatter of the critical substrate\ntemperature for given xand As:(Ga+Mn) ratio is remarkably small, typically less than\n2\u000eC. In excess As \rux the 2D/3D transition occurs at higher temperature. The highest\nquality samples are grown in a narrow window of the 1:1 stoichiometric As:(Ga+Mn) ratio\nand at the substrate temperature approaching as close as possible from below the 2D/3D\ncritical temperature for given x. The As:(Ga+Mn) ratio was adjusted by the As-cell valve,\nand calibrated using the As-controlled RHEED oscillations. In insets of Fig. 2a we show\nexamples of RHEED patterns for the x= 7% nominally doped (Ga,Mn)As material grown\nat stoichiometric 1:1 ratio of As:(Ga+Mn) for substrate temperature of 225 K which is\nabove the 3D/2D boundary and 210 K which is below the boundary. The optimal growth\ntemperature for this doping is 215 K. In the main panel of Fig. 2a we plot the optimal\ngrowth temperature as a function of nominal Mn doping, showing the rapidly decreasing\ngrowth temperature trend.\nThe next important factor determining the quality of the resulting (Ga,Mn)As materials\nare post-growth annealing conditions. In Fig. 2b we show the dependence of the Curie tem-\n5000.51.0U/dT)*50 100 150 200 250246U (10-3:cm)\nT (K)annealing time\n160 °C\n160170180190TC (K)140°C\n180°Cd\nb ea\n02468 1 0 1 2 1 4100150200250300350\nGrowth T (°C)\nx (%)\n50 100 150 200 2500204060M (emu/cm3)\nT (K)50 100 150 200 250-0.50.0(dUT (K)11 014015016015 nm\nAnnealing time (h)160°C\n01 0 2 0 3 070125150175 200 °C\n100 nmTC (K)\nAnnealing time (h)25 nmc f\nFig. 2FIG. 2: Optimization of the (Ga,Mn)As synthesis. a , Optimal growth temperature as a\nfunction of the nominal Mn doping. Insets show examples of RHEED images of the 2D growth\nat 210\u000eC (lower inset) and 3D growth at 225\u000eC (upper inset) of the 7% Mn-doped (Ga,Mn)As.\nb, Dependence of the Curie temperature on the annealing time for three di\u000berent annealing tem-\nperatures in a 15 nm thick (Ga,Mn)As epilayer with 13% nominal Mn doping. c, Dependence of\nthe Curie temperature on the annealing time for the annealing temperature of 200\u000eC in a 100 nm\nthick (Ga,Mn)As epilayer with 13% nominal Mn doping, and in the same epilayer thinned down to\n25 nm by wet etching. d{f, Temperature dependencies of resistivity \u001a,d, temperature derivative\nof the resistivity d\u001a=dT ,e, and remnant magnetization M,f, in a 20 nm thick (Ga,Mn)As epilayer\nwith 13% nominal Mn doping at successive annealing times at the optimal annealing temperature\nof 160\u000eC for this doping.\nperatureTcon the annealing time for three di\u000berent annealing temperatures for the record\nTc= 188 K sample with nominal 13% Mn doping and \flm thickness 15 nm. These curves\n6illustrate the common trend in annealing (at temperatures close to the growth temperature)\nsuggesting the presence of competing mechanisms. One mechanism yields the increase of Tc\nand is ascribed in a number of reports to the removal of charge and moment compensating\ninterstitial Mn impurities (see e.g. the detailed annealing study in Ref. 10). The removal is\nslowed down by the growth of an oxide surface layer during annealing10and an additional\nmechanism can eventually yield reduction of Tcafter su\u000eciently long annealing times, de-\npending on the annealing temperature. The origin of this detrimental mechanism may be in\nMn clustering or in the competition between the non-equilibrium (Ga,Mn)As phase and the\nequilibrium MnAs second phase. Because of the competing mechanisms, the absolutely high-\nest Curie temperature for the given nominal doping is achieved at intermediate annealing\ntemperature and time, as illustrated in Fig. 2b.\nThe remaining critical parameter of the synthesis is the epilayer thickness. For a given\nnominal doping, the highest attainable Tcis reached only in thin \flms, typically thinner\nthan\u001850 nm. In Fig. 2c we illustrate the importance of the \flm thickness for obtaining\nhigh quality (Ga,Mn)As materials. A 100 nm thick \flm is grown with nominal 13% doping\nand, unlike the thin record Tc\flm discussed above, here the maximum Tcachieved by\nannealing is only about 140 K. However, if the same \flm is thinned down (to e.g. 25 nm)\nby wet etching and annealed at the same conditions, the achieved Curie temperatures are\nsigni\fcantly higher.\nAn increase of Tcis not the only parameter followed to ascertain that a sample is of\nhigh quality. A key characterization tool are the thermodynamic Curie point singularities.47\nThis is illustrated in Figs. 2d-f where we compare resistivity and magnetization measured\nat increasing time steps during the optimizing annealing procedure. The development of\nsharply vanishing magnetization M(T) atTcand the onset of the singularity in d\u001a=dT are\nwell correlated with increasing Tcand conductivity within the annealing sequence.\nAfter \fnding the optimal growth and post-growth conditions for each individual nominal\ndoping we obtained a series of samples spanning the wide range of Mn dopings. The samples\ncan be divided into several groups: at nominal dopings below \u00180:1% the (Ga,Mn)As ma-\nterials are paramagnetic, strongly insulating, showing signatures of the activated transport\ncorresponding to valence band { impurity band transitions at intermediate temperatures,\nand valence band { conduction band transitions at high temperatures (see Fig. 3a).61,62For\nhigher nominal dopings, 0 :5.x.1:5%, no clear signatures of activation from the valence\n72468 1 010-2100102V (:-1cm-1)\n1000/T (K-1)0.05%1%2%7%\n~ 100 meV\nEg/20123p (1021cm-3)\n \n200a\nbc\nd120255075100\n Msat(emu/cm3)NMn (1021cm-3)\n02468 1 0 1 2 1 4050100150200TC (K)\nx (%)\nFig. 3be02468 1 0 1 2 1 410-2100102V (:-1cm-1)\nx (%)FIG. 3: Doping trends in the series of optimized (Ga,Mn)As epilayers. a , Temperature\ndependence of the conductivity \u001b(T) of optimized (Ga,Mn)As epilayers with depicted nominal\nMn doping. Dashed lines indicate the activated parts of \u001b(T) of the insulating paramagnetic\n(Ga,Mn)As with 0.05% Mn doping, corresponding to the Mn acceptor level and the band gap,\nrespectively. b-e, Conductivity, b, hole density, c, saturation magnetization and corresponding\nMn moment density, d, and Curie temperature, e, as a function of the nominal Mn doping in the\nseries of optimized (Ga,Mn)As epilayers.\nband to the impurity band are seen in the dc transport, con\frming that the bands start to\noverlap and mix, yet the materials remain insulating.61,62Atx\u00191:5%, the low-temperature\nconductivity of the \flm increases abruptly by several orders of magnitude (see Fig. 3b), and\nthe system turns into a degenerate semiconductor.61,62The onset of ferromagnetism occurs\nalready on the insulating side of the transition at x\u00191% and the Curie temperature then\nsteadily increases with increasing nominal Mn doping up to \u001913%. The hole concentration\npcan be measured by the slope of the Hall curve at high \felds (see Supplementary infor-\nmation) with an error bar due to the multi-band nature estimated to \u001820%.63Within this\n8uncertainty, the overall trend shows increasing pwith increasing doping in the optimized\nmaterials, as shown in Fig. 3c. Similarly, the saturation moment and Tcsteadily increase\nwith increasing nominal doping up to x\u001913%, as shown in Figs. 3d,e. Assuming 4.5 \u0016B\nper Mn atom64we can estimate the density NMnof uncompesated Mn Gamoments from the\nmagnetization data (see left y-axis in Fig. 3d). An important conclusion can be drawn when\ncomparing this estimate with the hole density estimated from the Hall resistance. Since\nthere is no apparent de\fcit of pcompared to NMn, and since the interstitial Mn impurity\ncompensates one local moment but two holes we conclude that interstitial Mn is completely\n(within the experimental scatter) removed in our optimally annealed epilayers. Hence, our\nseries of optimized (Ga,Mn)As materials have reproducible characteristics, showing an over-\nall trend of increasing saturation moment with increasing x, increasing Tc(reaching 188 K),\nand increasing hole density. The materials have no measurable charge or moment compen-\nsation of the substitutional Mn Gaimpurities and have a large degree of uniformity re\rected\nby sharp Curie point singularities.\nDetermination of the micromagnetic parameters. We now proceed to the de-\ntermination of the magnetic anisotropy, Gilbert damping, and spin sti\u000bness constants of\nour (Ga,Mn)As epilayers from the MO time-resolved measurements of the magnetization\nprecession. In the MO pump-and-probe experiments, we used a femtosecond titan sapphire\nlaser that was spectrally tuned to 1.64 eV, i.e., above the band gap of GaAs. The possibility\nto excite and detect precession of ferromagnetic Mn moments in (Ga,Mn)As by this method\nhas been extensively discussed in previous MO studies.16{27All experiments presented be-\nlow were preformed at temperature of approximately 15 K in re\rection geometry. External\nmagnetic \felds up to 550 mT were applied in the [010] and [110] crystallographic directions.\nThe intensity of the pump pulse was \u001830\u000040\u0016Jcm\u00002, with the pump to probe intensity\nratio\u001820 : 1\u000010 : 1. The penetration depth of the laser beam ( \u0018600 nm) safely exceeds\nthe thickness of the studied (Ga,Mn)As epilayers.\nThe anisotropy constants, shown in Fig. 1c, where obtained combining three complemen-\ntary measurements. In the \frst experiment we measured the external magnetic \feld Hext\ndependence of the precession frequency fof the time resolved MO signal. In the studied\n(Ga,Mn)As/GaAs epilayers, the internal magnetic anisotropy \felds are dominated by three\ncomponents. The out-of-plane component Koutis a sum of the thin-\flm shape anisotropy\nand the magnetocrystalline anisotropy due to the compressive growth strain in (Ga,Mn)As.\n9The cubic magnetocrystalline anisotropy Kcre\rects the zinc-blende crystal structure of the\nhost semiconductor. The additional uniaxial anisotropy component along the in-plane di-\nagonalKuis not associated with any measurable macroscopic strain in the epilayer and is\nlikely of extrinsic origin. The precession frequency is given by,\nf=g\u0016B\nhq\u0000\nHextcos('\u0000'H)\u00002Kout+Kc(3 + cos 4')=2 + 2Kusin2('\u0000\u0019=4) + \u0001Hn\u0001\n\u0002p\n(Hextcos('\u0000'H) + 2Kccos 4'\u00002Kusin 2'+ \u0001Hn); (1)\nwheregis the Land\u0013 e g-factor of Mn moments, \u0016Bthe Bohr magneton, 'and'Hare the\nin-plane magnetization and external magnetic \feld angles measured from the [100] crystal\naxis, and \u0001 Hnis the shift of the resonant \feld for the higher index nspin wave modes\nwith respect to the n= 0 uniform precession mode. In order to uniquely determine the\nanisotropy constants, the \feld-dependent precession frequency measurements were comple-\nmented by MO experiments with variable polarization angle of the probe beam. The latter\nmeasurements allow us to precisely determine the angle of the equilibrium easy axis of the\nmagnetization (see Supplementary information).26,27Finally, we con\frmed the consistency\nof the obtained anisotropy constants by performing static measurements of magnetization\nhysteresis loops by the superconducting quantum interference device (SQUID). Results for\nferromagnetic materials from our series of optimized (Ga,Mn)As epilayers are summarized in\nFig. 1c. Note that the values of KoutandKcfor the given Mn-doping are well reproducible in\nmaterials whose synthesis yields the same optimized values of the basic structural, magnetic\nand transport properties. For the Kuconstant, variations in the width of the optimized\nthin (Ga,Mn)As \flms or of other otherwise insigni\fcant changes of the growth or annealing\nconditions may yield sizable changes of Ku. This con\frms the presumed subtle extrinsic\nnature of this magnetic anisotropy component.\nThe sign of Koutimplies that all studied (Ga,Mn)As/GaAs materials are in-plane fer-\nromagnets. The competing magnitudes of KcandKuand the di\u000berent doping trends of\nthese two in-plane magnetic anisotropy constants (see Fig. 1c) are therefore crucial for the\nmicromagnetics of the materials. The biaxial anisotropy Kcdominates at very low dopings\nand the easy axis aligns with the main crystal axis [100] or [010]. At intermediate dopings,\nthe uniaxial anisotropy Kuis still weaker but comparable in magnitude to Kc. In these\nsamples the two equilibrium easy-axes are tilted towards the [1 \u001610] direction and their angle\nis sensitive to small changes of external parameters such as temperature. This allows for\n10exciting the magnetization precession by laser pulses in the pump-and-probe MO experi-\nments. At very high dopings, the uniaxial anisotropy dominates and the system has one\nstrong easy-axis along the [1 \u001610] in-plane diagonal. In the low-doped and high-doped samples\nwith very stable easy-axes aligned with one of the main crystal directions the dynamical MO\nexperiments become unfeasible.\nThe Gilbert damping constant \u000b, shown in Fig. 1d, is obtained by \ftting the measured\ndynamical MO signal to Landau-Lifshitz-Gilbert (LLG) equations using the experimentally\nobtained magnetic anisotropy constants. The high accuracy of the LLG \fts is demonstrated\nin Figs. 4a,b on data measured in a x= 5:2% doped sample. The obtained dependence of \u000b\non the external magnetic \feld applied along the [010] and [110] directions is shown in Fig. 4c.\nAt smaller \felds, \u000bis not constant and shows a strong anisotropy with respect to the \feld\nangle. When plotted as a function of frequency, however, the dependence on the \feld-angle\ndisappears, as shown in Fig. 4d. Analogous results are obtained for the entire series of the\noptimized materials. We can therefore conclude that the apparent anisotropy of \u000bcan in\nour materials be ascribed fully to the \feld-angle dependence via the precession frequency. In\nall our studied materials, the frequency-independent Gilbert damping constant is isotropic\nand can be accurately determined from MO data with precession frequencies f&15 GHz.\nWe point out that in ferromagnetic resonance (FMR) experiments, the measurement fre-\nquency was limited to two values, f= 9 and 35 GHz which even in the optimized (Ga,Mn)As\nmaterials is not su\u000ecient to reliably separate the intrinsic Gilbert damping constant from\nthe inhomogeneous broadening of the FMR line-width. The dynamical MO measurements,\non the other hand, span a large enough range of frequencies and allow us to extract a con-\nsistent set of frequency-independent values of \u000bfor our series of optimized ferromagnetic\n(Ga,Mn)As materials. We \fnd a systematic doping trend across the series in which the\nGilbert constant decreases from \u00180:1 to 0:01 when the nominal Mn doping increases from\n\u00182% to 5% and then remains nearly constant (see Fig. 1d). The magnitudes of \u000band the\ndoping dependence are consistent with Gilbert damping constants in conventional transi-\ntion metal ferromagnets. In metals, \u000btypically increases with increasing resistivity and is\nenhanced in alloys with enhanced spin-orbit coupling.65{67Similarly, in our measurements\nin (Ga,Mn)As, the increase of \u000bcorrelates with a sizable increase of the resistivity in the\nlower Mn-doped samples. Also, the spin-orbit coupling e\u000bects tend to be stronger in the\nlower doped samples with lower \flling of the valence bands and with the carriers closer to\n1166[110]05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-50050MO signal ( µrad)\nTime delay (ps)Hext || [110]\n05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-10010MO signal ( µrad)\nTime delay (ps)Hext || [010]a b\nc d\n05 1 0 1 5 2 0024D (10-2)\nf (GHz)[010], [110]02 0 0 4 0 0 6 0 0024D (10-2)\nµ0Hext (mT)[010]\nFig. 4FIG. 4: Determination of the Gilbert damping constant from MO experiments. a,b,\nOscillatory part of the MO signal (points) measured in a 18 nm thick epilayer with 5.2% nominal Mn\ndoping for external magnetic \feld \u00160Hext= 100 mT applied along the crystallographic directions\n[010] and [110]; lines are \fts by the LLG equation. c,Dependence of the Gilbert damping on\nexternal magnetic \feld applied along the [010] and [110] crystallographic directions. d,Dependence\nof the Gilbert damping on the precession frequency.\nthe metal-insulator transition.68Theory ascribing magnetization relaxation to the kinetic-\nexchange coupling of Mn moments with holes residing in the disordered, exchange-split, and\nspin-orbit-coupled valence band of (Ga,Mn)As yields a comparable range of values of \u000bas\nobserved in our measurements.51\nSimilar to the Gilbert constant, there has been a large scatter60in previous reports of\nexperimental values of the spin-sti\u000bness in (Ga,Mn)As inferred from FMR,53{56magneto-\noptical studies,59and from complementary static magnetization and domain structure\n12measurements.57,58We attribute the lack of a consistent picture obtained from these measure-\nments to sample inhomogeneities and extrinsic defects in the studied (Ga,Mn)As epilayers\nwith thicknesses typically exceeding 100 nm and to experimental data which allowed only\nan indirect extraction of the spin sti\u000bness constant. The MO pump-and-probe technique\nutilized in our work allows in principle for the direct measurement of the spin sti\u000bness,\nhowever, one has to \fnd the rather delicate balance between thin enough epilayers to avoid\nsample inhomogeneity and thick enough \flms allowing to observe the higher-index Kittel\nspin-wave modes69of a uniform thin-\flm ferromagnet. For these modes, the spin-sti\u000bness\nparameterDis directly obtained from the measured resonant \felds,\n\u0001Hn\u0011H0\u0000Hn=Dn2\nL2\u00192\ng\u0016B; (2)\nwhereLis the thickness of the ferromagnetic \flm. The MO pump-and-probe technique\nhas the key advantage here that, unlike FMR, it is not limited to odd index spin wave\nmodes.69The ability to excite and detect the n= 0, 1, and 2 resonances is essential for the\nobservation of the Kittel modes in our optimized (Ga,Mn)As epilayers whose thickness is\nlimited to \u001850 nm.\nIn Fig. 5a we show an example of the time dependent MO signal measured in a 48 nm\nthick optimized epilayer with 7% nominal Mn doping. Three spin wave resonances (SWRs)\nare identi\fed in the sample with frequencies f0,f1, andf2, as shown in Figs. 5b,c. The asso-\nciation of these SWRs with the Kittel modes, described by Eq. (2), is based on experiments\nshown in Figs. 5b-e. In Fig. 5c we plot the dependence of the three detected precession\nfrequencies on the external magnetic \feld applied along the [010] and [110] crystal axes.\nAt saturation \felds, which for the 7% Mn-doped sample are &70 mT, the equilibrium\nmagnetization vector is aligned with Hextand Eq. (1) with '='Hcan be used to \ft the\ndata. We emphasize that all six displayed dependences fn(Hext) forn= 0, 1, and 2, and\n'H= 45\u000eand 90\u000ecan be accurately \ftted by one set of magnetic anisotropy constants. We\ncan therefore use Eq. (2) to convert the measured frequency spacing of individual SWRs\nto \u0001Hn. In Fig. 5d we show that \u0001 Hnin our optimized epilayers is proportional to n2as\nexpected for the Kittel modes in homogeneous \flms.\nThe magnetic homogeneity and the applicability of Eq. (2) in our epilayers is further\ncon\frmed by the following experiments: We prepared three samples by etching the original\n48 nm thick (Ga,Mn)As \flm down to the thicknesses of 39, 29 and 15 nm, respectively.\n1301 0 0 2 0 0 3 0 0051015f2f1f (GHz)\nµ0Hext (mT)f0\nHext || [010]; [110]\n           \n           \n           f0f05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-50050MO signal ( µrad)\nTime delay (ps)48 nm\n200300400'Hn (mT)a c\nb d\n05 1 0 1 5 2 0FFT (arb. units)\nf (GHz)48 nm\n39 nm\n29 nm\n15 nmf1\nf2012340100µ0'n2\nFig. 50.6 0.9 1.280160240µ0'H1 (mT)\n1/L2 (10-3 nm-2)eFIG. 5: Determination of the spin sti\u000bness constant from MO experiments. a, Oscillatory\npart of the MO signal (points) measured in a 48 nm thick epilayer with 7% nominal Mn doping\nfor external magnetic \feld \u00160Hext= 20 mT applied along the [010] crystallographic direction;\nline is a \ft by a sum of three damped harmonic functions. b,Fourier spectra of oscillatory MO\nsignals (points) measured for \u00160Hext= 20 mT applied along the [010] crystallographic direction\nin samples prepared by etching from the 48 nm thick epilayer. The curves are labeled by the \flm\nthicknesses, normalized, and vertically o\u000b-set for clarity; lines are \fts by a sum of Lorentzian peaks.\nc,Dependence of the measured precession frequency (points) on the magnetic \feld for two di\u000berent\norientations of the \feld in the 48 nm thick epilayer; lines are \fts by Eq. (1). d,Dependence of the\nmeasured mode spacing on square of the mode number in the 48 nm thick epilayer. e,Dependence\nof the spacing between the two lowest modes (\u0001 H1) on the \flm thickness. Lines in dandeare\n\fts by Eq. (2) with spin sti\u000bness D= 2:43 meV nm2.\n14As seen in Fig. 5b, the frequency f0is independent of the \flm thickness which con\frms\nthat it corresponds to the uniform precession mode and that the \flm is homogeneous, i.e.,\nthe magnetic anisotropy constants do not vary across the width of the (Ga,Mn)As epilayer.\nThe spacing \u0001 H1shown in Fig. 5e scales as L\u00002and the values of Dextracted from the\nn-dependence of the resonant \feld spacings in the L= 48 nm epilayer (see Fig. 5d) and from\ntheL-dependence of \u0001 H1(see Fig. 5e) give the same D= 2:43\u00060:15 meVnm2. Identical\nvalue of the spin sti\u000bness was also obtained from measurements in an epilayer grown with\nthe same doping and thickness of 18 nm in which we detected the frequencies f0andf1\nand applied Eq. (2). These measurements con\frm the reliability of extracted values of the\nspin sti\u000bness. We note that the SWR frequencies are determined with high accuracy in\nour measurements and that the indicated error bars in Fig. 1d re\rect the uncertainty of\nthe \flm thickness. As shown in Fig. 1d, we observe a consistent, weakly increasing trend\ninDwith increasing doping and values of Dbetween 2 and 3 meVnm2in the studied\nferromagnetic samples with nominal doping 3.8-9%. (Note that apart from the di\u000eculty\nof exciting magnetization precession in the very low and high-doped samples with stable\neasy-axes, the measurements of Dwere unfeasible on the lower doping side of the series\nbecause of the increasing damping and the corresponding inability to detect the higher SWR\nmodes.) Similar to the Gilbert damping constant, our measured spin sti\u000bness constant in the\noptimized (Ga,Mn)As epilayers is comparable to the spin sti\u000bness in conventional transition\nmetal ferromagnets.70\nWe remark, that we tested the inapplicability of the SWR experiments for the direct\ndetermination of the spin sti\u000bness in thick non-uniform materials. In the Supplementary\ninformation we show measurements in \u0018500 nm thick as-grown and annealed samples with\n7% nominal Mn-doping. The Curie temperatures of \u001860 and 90 K can be inferred only\napproximately from smeared out singularities in d\u001a=dT andM(T) and are signi\fcantly\nsmaller than Tcin the thin optimized epilayers with the same nominal doping. The \flms\nare therefore clearly inhomogeneous and contain compensating defects. Because of the large\nthickness of the epilayers we observe up to \fve SWR modes, however, consistent with the\ninhomogeneous structure of the \flms, the corresponding \u0001 Hndo not show the quadratic\nscaling with nof the Kittel modes of Eq. (2).\nIn the experiments discussed above we have established the systematic semiconducting\ndoping trends and basic magnetic characteristics of epilayers which have been optimized to\n15represent as close as possible the intrinsic properties of idealized, uniform and uncompen-\nsated (Ga,Mn)As. Our study supports the overall view of (Ga,Mn)As as a well behaved\nand understood degenerate semiconductor and band ferromagnet and, therefore, an ideal\nmodel system for spintronics research. We conclude in this paragraph by commenting on\nthe implications of systematic studies of optimized (Ga,Mn)As materials in the context of\nthe recurring alternative proposal of an intricate impurity band nature of conduction and\nmagnetism of (Ga,Mn)As.71In the impurity band picture, the Fermi level in materials with\n\u00181021cm\u00003Mn-acceptor densities is assumed to reside in a narrow impurity band detached\nfrom the valence band, i.e., the band structure keeps the form closely reminiscent of a sin-\ngle isolated Mn Gaimpurity level. Previously, the systematic measurements of the infrared\nconductivity on the extensive set of optimized materials49disproved one of the founding\nelements of the impurity band picture which was the red-shift of the mid-infrared peak with\nincreasing doping.72In the systematic measurements in Ref. 49, the mid-infrared peak was\nobserved to blue-shift49,73and experimentalists focusing on the infrared spectroscopy49,73,74\nreached the consensus that the valence and impurity bands are merged in the highly doped\nferromagnetic (Ga,Mn)As materials. The large values of the spin sti\u000bness of the order\nmeVnm2, experimentally determined in the present work, are consistent with model Hamil-\ntonian and ab initio calculations60,75{77which all consider or obtain the band structure of the\nferromagnetic (Ga,Mn)As with merged valence and impurity bands.62On the other hand,\nfor carriers localized in a narrow impurity band the expected spin sti\u000bness would be small\nin a dilute moment system like (Ga,Mn)As, in which the magnetic coupling between remote\nMn moments is mediated by the carriers.78By recognizing that the bands are merged, the\ndistinction between a \"valence\" and \"impurity\" band picture of ferromagnetic (Ga,Mn)As\nbecomes mere semantics with no fundamental physics relevance. 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Nature Materials 4, 195 (2005). arXiv:cond-mat/0503185.\nAcknowledgment\nWe acknowledge theoretical assistence of Pavel Motloch and support from EU ERC Ad-\nvanced Grant No. 268066 and FP7-215368 SemiSpinNet, from the Ministry of Education of\nthe Czech Republic Grants No. LM2011026, from the Grant Agency of the Czech Republic\nGrant No. 202/09/H041 and P204/12/0853, from the Charles University in Prague Grant\nNo. SVV-2012-265306 and 443011, from the Academy of Sciences of the Czech Repub-\nlic Preamium Academiae, and from U.S. grants onr-n000141110780, NSF-MRSEC DMR-\n0820414, NSF-DMR-1105512. .\n22 1 Establishing micromagnetic parameters of ferromagnetic \nsemiconductor (Ga,Mn)As : Supplementary information  \n \nP. Němec,1 V. Novák,2 N. Tesařová,1 E. Rozkotová,1 H. Reichlová2,1, D. Butkovičová1, \nF. Trojánek,1 K. Olejník,2 P. Malý,1 R. P. Campion,3 B. L. Galla gher,3 Jairo Sinova,4,2 and \nT. Jungwirth2,3 \n \n1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3,  \n121 16 Prague 2, Czech Republic  \n2 Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 16253 Praha 6, Czech Republic  \n3 School of  Physics and Astronomy, University of Nottingham, Nottingham NG72RD,  \nUnited Kingdom  \n4 Department of Physics, Texas A&M University, College Station, TX 77843 -4242, USA  \n \nEXPERI MENTS PRECEEDING OUR DETERMINATION OF MIC ROMAGNETIC \nPARAMETERS OF (GA,MN )AS \n Magn etic anisotropy fields, Gilbert damping  constant  and spin -stiffness are the  basic \nparameters of a ferromagnet which determine its  micromagnetic properties. The anisotropy \nfields are as sociated with the energy required to coherently rotate  magnetic moments of the \nentire ferromagnet. They can  be determined in a straightforward way in magnetization  or \nmagneto -transport  measurements from external mag netic fields required to reorient the \nmagnetization of a  ferromagnetic sample, or in magnetization dynamics ex periments from the \nfield-dependent resonant frequencies  [1, 2, 3, 4]. \n Gilbert damping characterizes  dissipative  process es that drive  the magnetization  \nmotion towards an equilibrium  state.  This phenomenon is usually investigated by the \nfrequency -domain -based ferromagnetic resonance (FMR) experiment where  the \nphenomenological Gilbert damping coefficient  is deduced  from the resonance peak \nlinewidth  [3, 4]. The experimentally measured FMR linewidths contain not only the \nfrequency -dependent linewidth due to  the Gilbert damping but also the frequency -\nindependent inhomogeneous linewidth broadening [ 3, 4]. To separate them, it is necessary to \nmeasure the linewidth s at several microwave frequencies  [3, 4]. In FMR t hese frequencies are \ngiven by the resonant -cavity fre quency  that significantly complicates the frequency change .  2 Therefore, the experiments are usually performed at only two different frequencies (typically, \n9 and 35 GHz [ 4]) that makes the corresponding separation of the individual components in \nthe measure d signal rather questionable. Alternatively ,  can be determined from the time-\ndomain based magneto -optical pump -and-probe experiment by fitting the damping of the \nmeasured oscillatory data by Landa u-Lifshitz -Gilbert  equation [ 5, 6]. However, to obtain the \nGilbert damping coefficient  from the measured value of  it is necessary to take into account \na realistic magnetic anisotropy of the investigated samples (see below). Moreover, also the \nfrequency dependence of  has to be measured for a separation of the intrinsic value of the \nGilbert damping coefficient  from the inhomogeneous parts of  . The absence of these two \nrequirements  and the un -optimized magnetic properties of the investigated samples led to  a \nlarge scatter in the deduced values of  for Ga 1-xMn xAs with a different Mn content x: The \nincrease of  from  0.02 to  0.08 for the increase of x from 3.6% to 7.5% was reported in \nRef. 5. On the contrary, in Ref. 6 the values of  from 0.06 to 0.19 – without any apparent \ndoping trend – were observed for x from 2% to 11%.  \nThe spin -stiffness is associated with the exchange energy of non -uniform local \ndirections of the magnetization, in particular with the energy of small wave -vector spin -wave \nexcitations of the ferromagnet. Considering a specific model of the rmodynamic properties of \nthe studied ferromagnet, the spin -stiffness can be indirectly inferred from the measured \ntemperature dependence of magnetization  [7], Curie temperature  [7], or domain wall width  \n[8]. The direct determination of the spin -stiffness f rom magnetization dynamics experiments \nis significantly more challenging than in the case of the magnetic anisotropy fields  [9-13]. The \nlow-energy non -uniform collective excitations of the system can be strongly affected by \ninhomogeneities or surface prope rties of the ferromagnet for which specific models have to be \nassumed in order to extract the spin -stiffness constant from the measured data. An exception \nare the Kittel spin -wave modes of a uniform thin -film ferromagnet for which the spin -stiffness \nparame ter D is directly obtained from the measured resonant f ields (see below) . To date, spin -\nwave resonance measurements of (Ga,Mn)As have been reported on > 100 nm thick epilayers  \n[9-12]. The Kittel modes with Hn ~ n2 were observed only in a 120 nm thick, 8% Mn doped \n(Ga,Mn)As for magnetic fields applied close to the magnetic easy -axis [11]. Measurements of \nthe same sample in other field orientations showed different trends which indicated the \npresence of strong inhomogeneities and surface dependent effects  [11]. A non -Kittel -like \nlinear or sublinear dependence of the resonant fields on the mode index has been reported also \nin the other ferromagnetic resonance measurements of thick (Ga,Mn)As epilayers  [9-12]. In  3 complementary studies of the magnetization dynami cs induced and detected  by magneto -\noptical pump -and-probe measurements, only two resonant frequencies were identi fied [13]. \nBased on the theoretical modeling, they were not ascribed to the Kittel modes but rather to \ncoupled bulk -surface modes which again m ade the extraction of the spin -stiffness constant \ndependent on the considered model of bulk and surface properties of the studied sample  [13]. \nThe extracted values of the spin -stiffness from all available magnetic resonance data in \n(Ga,Mn)As materials, com plemented by values inferred from magnetization and domain \nstudies  [7, 8], are scattered over more than an order of magnitude and show no clear trend as a \nfunction of Mn -doping or other material parameters of the (Ga,Mn)As ferromagnetic \nsemiconductor  [14]. \nIn this Supplementary material we show how we are able to deduce from a single  \nmagneto -optical pump -and-probe experiment all these micromagnetic parameters. In \nparticular, the anisotropy fields can be determined from the dependence of the precession \nfrequ ency on the external magnetic filed combined with the probe -polarization dependence of \nthe precession signal amplitude. The Gilbert damping constant can be deduced from the \nprecession signal dampin g. Finally, the spin stiffness can be obtained from the mut ual spacing \nof the precession modes  which are present in the measured oscillatory magneto -optical signal.  \n  \nSAMPLES  \nThe time -resolved magneto -optical experim ents described  below were performed in a \nlarge set of optimized  (Ga,Mn)As epilayers  whose selected properties are described in detail \nin the main paper.  In Fig. 1 we show results of the Hall effect measurements at 4.2 K. For this \npurpose the samples were lithographically patterned into Hall -bars of 60 m width. It can be \nseen in the figure that the Hall signal is affected by longitudinal magnetoresistance Rxx of the \nsamples, especially at low dopings. Therefore, we extracted p from high field data and by \nfitting the measured transversal resistance Rxy by \n \n Rxy = B/(epd) + k 1Rxx + k 2Rxx2       (1) \n \nwhere d is the sample thickness and k1 and k2 are fitting constants reflecting the anomalous \nHall effect and possible imperfections in the geometry of the Hall bars. We also emphasize \nthat, apart from the commo n experimental scatter and from the corrections due to the non -zero \nmagnetoresistance and due to the anomalous Hall effect, the carrier density can in principle be  4 inferred only approximately from the slope of the Hall curve in a multi -band, spin -orbit \ncoupled exchange -split system such as the (Ga,Mn)As. The error bar due to the multi -band \nnature is estimated to  be 20% [ 15]. Due to these uncertainties we can only make semi -\nquantitative conclusions based on the measured Hall effect hole densities.  \n0 5 10 15 200.800.850.900.951.00\n 1.5%\n 2%\n 3%\n 5.2%\n 13%RXX / RXX(0)\n0Hext (T)1\n234\n \n0 5 10 15 20020406080\n 1.5%     2%\n 3%        5.2%\n 13%RXY()\n0Hext (T)1\n234  \n \nFig. 1. (a) Longitudinal resistances Rxx [normalized to Rxx(0)], and (b) transversal (Hall) resistances  Rxy as a \nfunction of normal magnetic field μ0Hext measured in (Ga,Mn)As epilayer with depicted Mn concentration x; \nsample s temperature 4.2 K.  \n \n For a n evaluation of material parameters from an experimentally measured data (e.g., \nfor an evaluation of the hole densities from the measured transversal resistances  which is \ndescribed above)  it is necessary to know the (Ga,Mn)As epilayer thicknesses. However, \naccurate determination of layer thicknesses is a nontrivial task in case of thin (Ga,Mn)As \nlayers. Some standard techniques (e.g. , X-ray reflectivity  or optical ellipsometry) are \ninapplicable due to the weak contrast between the (Ga,Mn)As layer and the GaAs substrate, \nor unknown optical parameters. The relative accuracy of other common techniques (e.g. , of \nX-ray diffraction) does not exceed 10% because  of the small thickness of the measured layer. \nTherefore, we used a thickness estimation based on the following quantities: (i) the growth \ntime and the growth rate of the GaAs buffer layer measured by the RHEED oscillations \n(typical accuracy of ±3%); (ii) increase in the growth rate by adding the known Mn -flux \nmeasured by the beam -flux monitor relatively to the Ga flux (typical accuracy of ±5% of the \nMn vs. Ga flux ratio); (iii) reduction of thickness by the native oxidation ( -1.5 nm ± 0.5 nm); \n(iv) reducti on of thickness by thermal oxidation ( -1.0 nm ± 0.5 nm). Relative accuracy of \nsteps (i) and (ii) was verified on separate calibration  growths of (Ga,Mn)As on AlAs, where \nan accurate X-ray reflectivity method to measure the (Ga,Mn)As layer thickness could b e \nused. Typical thicknesses of the native and the thermal oxides in steps (iii) and (iv) were  5 determined by XPS. The resulting total accuracy of the (Ga,Mn)As layer  thickness \ndetermination  is thus 3% (relative random error) and 1 nm (systematic error) . \n \nEXPERIMENTAL DETAILS  ABOUT MAGNETO -OPTICAL EXPERIMENTS  \nWe investigated laser -pulse induced dynamics of magnetization by a pump -and-probe \nmagneto -optical (MO) technique.  A schematic diagram of the experimental set -up is shown \nin Fig. 2.  The output of a femt osecond laser is divided into a strong pump pulse and a weak \nprobe pulse that are focused to a same spot on the sample. Laser pulses, with the time width \nof 200 fs and the repetition rate of 82 MHz, were tuned to 1.64 eV, i.e. above the \nsemiconductor band gap, in order to excite magnetization dynamics by photon absorption. \nThe pump pulses were usually circularly polarized (with a helicity controlled by a wave plate) \nand the probe pulses were linearly polarized. The measured magneto -optical signals \ncorrespon d to the probe polarization rotation induced by the pump pulses (see Fig. 2). The \nexperiment was performed close to the normal incidence geometry ( θi = 2° and 8° for pump \nand probe pulses, respectively) with a sample mounted  in a cryostat, which was placed \nbetween the poles of an electromagnet. All the experimental data in this Supplementary \nmaterial were measured at temperature of 15 K , at pump exci tation intensity 30  -40 J.cm-2, \nand they correspond to the helicity -independent part of the measured signal [ 16]. The external \nmagnetic field Hext was applied in the sample plane at an angle H with respect to the [100] \ncrystallographic direction in the s ample plane (see Fig. 2). Prior to all time -resolved \nexperiments, we always prepared the magnetization in a well-defined state by first applying a \nstrong saturating magnetic field at an angle H and then reducing it to the desired magnitude \nof  Hext. \n \nFig. 2.  Schematic diagram of the experimental set -up for a detection of the magnetization precession induced by \nan impact of the circularly polarized femtosecond laser pump pulse  in (Ga,Mn)As . Rotation of the polarization \nplane of reflected linearly polarize d probe pulses is measured as a function of the time delay  ∆t between pump \nand probe pulses. The orientation of magnetization in the sample is described by the polar angle  and azimuthal \nangle . The external magnetic field Hext is applied in the sample plane at an angle H.  6  \nThere are several microscopic mechan isms that can lead to a precession of \nmagnetization due to the impact of pump laser pulse . In particular, very recently we reported \non the precession of magnetization due to optical spin -transfer torque (OSTT)  [16] and optical \nspin-orbital torque (OS OT) [ 17]. However, the most common mechanism, which is \nresponsible for the oscillatory MO signals measured in the majority of (Ga,Mn)As samples at \nlow excitation intensities, is the change of the sample magnetic anisotropy due to the pump -\ninduced temperature inc rease [17] that is schematically shown in Fig. 3. Before an impact of \nthe pump pulse the magnetization points to the easy axis direction [see Fig. 3(a)]. Absorption \nof the laser pulse leads to a photo -injection of electron -hole pairs. The subsequent fast \nnonradiative recombination of photo -injected electrons induces a transient increase of the \nlattice temperature (within tens of picoseconds after the impact of the pump pulse). The laser -\ninduced change of the lattice temperature  then leads to a change of the  easy axis position  [17]. \nAs a result , magnetization starts to follow the easy axis shift by the precessional motion  [see \nFig. 3(b)]. Finally, dissipation of the heat and recombination of the excess holes lead to the \nreturn of the easy axis to the equilibr ium position and the precession of magnetization is \nstopped by the Gilbert damping  [see Fig. 3(c)]. The most important point from the perspective \nof the present paper is that the precession of magnetization induced by the laser pulses is \ndetermined by the magnetic anisotropy of the sample which makes this method an all -optical \nanalog to FMR [ 18]. \n \n \nFig. 3. Schematic illustration of the thermal laser pulse -induced precession of magnetization.  (a) In the \nequilibrium , the magnetization points to the easy axis  direction, which is located in the sample plane at azimuthal \nangle . (b) Impact of a pump pulse induces a transient increase of the lattice temperature that lead s to a change \nof the easy axis position and, consequently, to the precession of magnetization . (c) Dissipation of the heat lead s \nto the return of the easy axis to the equilibrium position. Simultaneously with this, the precession of \nmagnetization is stopped by the Gilbert damping.  \n  7 ANALYTICAL DESCRIPTION OF MAGNE TIZATION DYNAMICS IN  (GA,MN)AS  \nThe dynamics of magnetization is described by the Landau -Lifshitz -Gilbert (LLG) \nequation. We used LLG equation in spherical coordinates where the time evolution of \nmagnetization magnitude Ms and orientation, which is characterized by the polar  and \nazimuthal  angles, is given by:  \n \n0dtdMs\n ,           (2) \n \nsin 12BAM dtd\ns\n,       (3) \n\n  \nsin sin 12BAM dtd\ns\n,       (4) \n \nwhere  is the Gilbert damping coefficient . The gyromagnetic ratio \n/Bg , where g = 2 \nis the Land é g-factor of Mn moments, B the Bohr magneton, and \n is the reduced Planc k \nconstant . Functions\nddFA  and \nddFB  are the derivatives of the energy density \nfunctional F with respect to  and , respectively. We expressed F in a form  [2]: \n \n\n   \n\n\n\n    \n\n\nH H H extu\nout c\nHKK KMF\n  \ncos sin sin cos cos2sin1 sin2cos cos sin2sin41sin2 2 2 2 2 2\n, (5) \n \nwhere Kc, Ku and Kout are constants that characterize the cubic, uniaxial and out -of-plane \nmagnetic anisotropy  fields in (Ga,Mn)As , respectively,  and Hext is the external magnetic field \nwhose ori entation is given by the angles H and H. For a small deviations  and  from the \nequilibrium values 0 and 0, the solution of Eqs. ( 3) and ( 4) can be written in a form  \n \n      tf eA ttkD2cos0\n,       (6) \n      tf eA ttkD2cos0\n,       (7) \n \nwhere the constants A (A) and  θ () describe the initial amplitude and phase of θ (), \nrespectively. The precession frequency f and oscillation damping rate kD are given by   8  \n\n   \n \n   \n    \n\n\n\n\n\n    \n\n\n \n   \n\n\n \n\n\n22\n22\n2sin3124cos53812 cos2sin2 4cos2 cos4sin2 4cos322 cos2sin2 4cos2 cos4sin224cos32 cos\n21\n      \n  \n\n\nu\nc out H extu c H extuc\nout H extu c H extuc\nout H ext\nB\nKK K HK K HKKK HK K HKKK H\ngf\nh\n, \n            (8) \n        2sin31 4cos5322 cos 2122 uc\nout H extB\nD KKK Hgk\n. (9) \n  \nIn our case , the investigated  (Ga,Mn)As epilayers are in -plane magnets (i.e., θ  π/2), the \nexternal magnetic field is applied in the sample plane (i.e., θH  π/2), and the precession \ndamping is relatively slow (i.e., 2  0) which yields  \n \n \n\n      \n2sin2 4cos2 cos4sin224cos32 cos2\nu c H extuc\nout H ext B\nK K HKKK H\nhgf\n  \n\n\n  , (10) \n \n         2sin31 4cos5322 cos 22uc\nout H extB\nD KKK Hgk\n. (11) \n \nEq. ( 10) express the sensitivity of the magnetization precession frequency to the magnetic \nanisotropy of the material that is a well -known effect which form the basis for the \ninterpretation of  FMR [3]. More interestingly, Eq. (11) shows that the precession damping kD, \nwhich is measured experimentally, depends not only on the Gilbert damping parametr   but \nalso on the sample anisotropy and on the mutual orientation of the external magnetic field and \nthe magnetization .   \n We note  that in previously reported magneto -optical pump -and-probe experiments [ 5, \n6, 19-23] the measured  experimental data were modeled by LLG equation in the form  \n  9 \n\n\n\n\n \n\n  \ndttMdtMMt HtMdttMd\nseff)()( )( )()( ,    (12) \n \nwhere Heff is the effective magnetic field . However, in (Ga,Mn) As the magnetic anisotropy is \nrather complex  and, therefore , modeling of MO signals by LLG in this form does not provide \nrealistic values of  because it is not possible to disentangle the effect  of magnetic anisotropy \nfrom  [see Eq. ( 11)]. We believe tha t this is one of the reason s why the dependence of  on \nMn concentration was so different in Ref. 5 and Ref 6. Similarly, the change of magnetic \nanisotropy of (Ga,Mn)As during the deposition of metal overlayer could be partially \nresponsible for the changes  of   that were reported in Ref. 23.  \n  \nEVALUATION  OF MAGNETIC ANISOTRO PY  \n  The dependence of the precession frequency on the magnetic anisotropy fields and on \nthe magnitude and orientation of external magnetic field [cf. Eq. (10)] enables to evaluate the  \nmagnetic anisotropy from the experimentally me asured precession frequencies very similarly \nas in the case of FMR [ 3]. In particular, for a sufficiently strong external magnetic field  = \nH and the following equations can be used to fit the precession fre quencies measured   \na) for Hext along  the [110] crystallographic direction (i.e., H = /4): \n  u c ext c out extBK K HK K Hhgf 2 2 2   \n     (13) \n \nb) for Hext along  the [010] crystallographic direction (i.e., H = /2): \n c ext u c out extBK HK K K Hhgf 2 2 2   \n     (14) \n \nc) for Hext along  the [-110] crystallographic direction (i.e., H = 3/4): \n  u c ext u c out extBK K HK K K Hhgf 2 2 2 2   \n    (15) \n \nAs an example, in Fig. 4 we show the measured dependences f (Hext) and their fits for two \norientations of Hext. To increase the precision of the magnetic anisotropy deter mination even \nfurther, for all the investigated samples we supplemented this method by two additional  10 experimental techniques that provide information about the samples magnetic anisotropy  – \nnamely, the probe -polarization dependence of the MO precession si gnal amplitude and \nSQUID  magnetometry . \n0 100 200 300 400 500 60005101520[010]f (GHz)\n0Hext (mT)[110]1\n234\n \nFig. 4. Dependence of the precession frequency f on external magnetic field Hext applied along the [010] and \n[110] crystallographic directions in Ga1-xMn xAs epilayer with x = 5.2% (points);  the lines are fits by Eqs. ( 14) \nand ( 13), respectively , with Kc = 31 mT, Ku = 27.5 mT, and Kout = -190 mT . \n \nIn (Ga,Mn)As there two MO effects that are responsible for the measured rotation of \nthe polarization plane  of the reflected linearly polarized light  at normal incidence [ 24]. The \nfirst of the MO effects is the well -known polar Kerr effect (PKE), where  occurs due to the \ndifferent index of refraction for  + and  - circularly polarized light propagating parall el to \nthe direction of magnetization  M. The p olarization rotation  due to PKE is proportional to the \nprojection of magnetization to the direction of light propagation , it is linear in magnetization \n(i.e., its sign is changed when the direction of magnetizat ion is reversed) , and it is independent \non the orientation of the input linear polarization  (see Fig. 5(b) for the angle definition ) \n[24]. The second MO effect is the magnetic linear dichroism (MLD), which originates from \ndifferent absorption (reflection ) coefficient for light linearly polarized parallel and \nperpendicular to M, that occurs if the light propagates perpendicular to the direction of \nmagnetization M. The p olarization rotation  due to MLD is proportional to the projection of \nmagnetization to th e direction perpendicular to the direction of light propagation, it is \nquadratic in magnetization (i.e., its sign is not changed when the direction of magnetization is \nreversed) and it varies as sin(2 β) [24]. In Fig.  5(a) we show  the MO signals measured by \nprobe pulses with different orientations  for identical pumping conditions. The measured \ndynamical MO signal δMO , which  is a function of the time delay between pump and probe \npulses t and the probe  polarization orientation  β, can be fitted well by the phenomenological \nequation [ 24],  11  \n   p Gt t\ne C eΦtf A t MO       2cos ,\n,    (16)  \n \nwhere A and C are the amplitudes of the oscillatory and pulse function, respectively, f is the \nferromagnetic moment precession freque ncy, Φ is the phase factor, τG is the Gilbert damping \ntime, and τp is the pulse function decay time. All the measured data in Fig. 5(a) can be fitted \nwell by Eq. ( 16) with a one set of parameters f, τG and τp. The dependence A() obtained by  \n0 500 1000 1500-40-2002040MO signal ( rad)\nTime delay (ps)(a)  = +30 deg\n = +120 deg = +75 deg\n = -15 deg\n     \n  \nFig. 5. (a) Dynamics of the MO signal measured by probe pulses with different probe polarization orientations  \nin (Ga,Mn)As epilayer with x = 5.2% for 0Hext = 0 mT (points); lines are fits by Eq. ( 16) with parameters f = 3.2 \nGHz, τG = 360 ps and τp = 1050 ps. (b ) Definition of the angle  that describes the orientation of the probe \npolarization plane E.  \n-45 0 45 90 13501020\nAMLDAPKEA (rad)\n (deg)0\n \nFig. 6.  Probe -polarization dependence of the oscillatory part A of the MO signal that was obtained by fi tting the \ndynamics shown in Fig. 5(a) by Eq.  (16); the values of A at time delay of 200 ps are shown (points). Lines are \nfits of A() by a sum of a polarization -independent signal due to PKE and a polarization -dependent signal due to \nMLD (Eq. (2) in Ref. 24). The vertical arrow depicts the deduced easy axis position in the sample without the \npump pulse, 0. \n \nthis fitting procedure is displayed in Fig. 6. The position of the maximum in the dependence \nA() at   120° corresponds to the equilibrium position o f the easy axis in the sample - i.e., \nthe in -plane position of magnetization without the pump pulse 0 [24]. The position of the \neasy axis in the sample plane is given by the relative magnitude of the cubic ( Kc) and uniaxial  12 (Ku) anisotropy fields. Therefo re, by measuring 0 without external magnetic field applied,  we \nare directly measuring  the ratio  Kc / Ku. \nThe in -plane anisotropy constants can be obtained also from magnetization loops \nmeasured by SQUID magnetometry. For any external magnetic field the or ientation of \nmagnetization is determined by the minimum of the energy [cf. Eq. ( 5)]. If the orientation of \nmagnetization as function of external magnetic field is known, the projection of the \nmagnetization into the measurement axis can be easily numericall y evaluated for every point \nof the magnetization loop. To obtain the anisotropy constants, we fitted the experimental data \nmeasured by SQUID until we obtained the best agreement between the data and the calculated \nmagnetization – see Fig. 7. It is worth no ting that this model does not describe the switching \nmechanism (governed by the domain wall physics which is not treated in our single domain \ndescription), so the parts of hysteresis loops containing the switching were not used in the \nanalysis. Moreover, i n the case of uniaxial systems (K u>K c) an analytical expression for the \nmagnetization measured along the hard axis can be utilized to analyze the data [ 1]. \n-400 -200 0 200 400-10-50510Magnetization (10-6 emu)\n0Hext (mT)1\n234\n \nFig. 7. Evaluation of the magnetic anisotropy from the SQUID magnetometr y. The SQUID measurement along \n[110] crystallographic direction in (Ga,Mn)As epilayer with x = 5.2% (points) is compared with the calculated \nmagnetization projection  for Kc = 31 mT and Ku = 27.5 mT  (line).  \n \nTo sum up, by a simultaneous fitting of the measu red dependence of the precession \nfrequency on an external magnetic field (Fig. 4), of the MO signal precession amplitude on a  \nprobe -polarization  (Fig. 6), and  of the data measured by  SQUID magnetometry  (Fig. 7) we \nevaluated very precisely the magnetic anis otropy f or all the investigated samples . The \nexample of the obtained in -plane angular dependence of the free energy in (Ga,Mn)As \nepilayer with Mn concentration x = 5.2% is shown in Fig. 8.  13 \n0 45 90 135 180-2-101\n120 130 140 150-1.98-1.96 Free energy F (arb. units)\n (deg)\n[-100][-110][010][110][100]\nF\n (deg) \nFig. 8. In-plane angular dependence of the free energy [Eq. ( 5)] in (Ga,Mn)As epilayer with x = 5.2% ; anisotropy \nfields Kc = 31 mT and  Ku = 27.5 mT . \n \nDETERMINATION OF GIL BERT DAMPING COEFFIC IENT  \n For numerical modeling of the measured MO data, we first computed from the LLG \nequatio n (Eqs. ( 3) and (4) with the measured magnetic anisotropy  fileds ) the time -dependent \ndeviations of the spherical angles [ (t) and (t)] from the corresponding equilibrium values \n(0, 0). Then we calculated how such changes of  and  modify the static magneto -optica l \nresponse of the sample MOstat, which is the signal that we detect experimentally  [24]: \n \n     0\n00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO\n.  (17) \n \nThe first two terms in Eq. ( 17) are connected with the out-of-plane and in -plane movement of \nmagnetization , and the last term describe s a change of the static magneto -optical response of \nthe sample due to the laser -pulses induced demagnetization [ 24]. PPKE and  PMLD are MO \ncoefficients that describe the MO response of the sample which we measured independently \nin a static MO experiment f or all the samples – see Fig. 9 for MO spectra measured in s ample \nwith x = 5.2%.   14 \n1.2 1.4 1.6 1.8 2.0 2.2 2.4-4-20246\n5laserMLDStatic MO signal (mrad)\nEnergy (eV)PKE\n1\n234 \nFig. 9. Spectral dependence of static PKE and MLD  in (Ga,Mn)As epilayer with x = 5.2% , the arrow indicate the \nspectral position of the laser pulses  used in the time -resolved experiment shown in Fig. 5 and Fig. 10; note that \nthe data for MLD are multiplied by 5 for clarity.  \n \n The example s of the fitting of the dynamical MO optical data are shown in Fig. 10. \nThe measured data can be fitted well by LLG  for time delays longer than  150 ps, which is a \ntime that it takes to establish the quasi -equilibrium conditions in the sample. We stress that \nthe only fitting parameters in our modeling are the Gilbert damping coefficient , the initial \ndeviation of the  spherical angles from the corresponding equilibrium values, and the \nparameters describing the in -plane movement of the easy axis and the demagnetization signal, \nwhich are apparent as the non -oscillatory signal in the measured dynamics [ 24]. The obtained \ndependence of  on Hext is shown in Fig. 11(a) for two different orientations of Hext. For Hext \napplied along the [010] direction,  decreases monotonously with Hext. On the contrary, for \nHext applied along [110] direction,  is a non -monotonous  function of  Hext reaching a similar \nvalues of  for 0 mT and 100 mT. However, this non -monotonous dependence is a \nconsequence of the field -induced frequency decrease (see Fig. 4) when the magnetic field is \napplied along the magnetically hard [110] direction (see Fig.  8). When  is plotted as a \nfunction of the precession frequency (rather than the external field) we do not observe any \nsignificant difference between the different crystallographic directions – see Fig. 11(b). A \nfield dependent damping parameter was repor ted in various magnetic ma terials and a variety \nof underl ying mechanisms responsible for it were suggested as an explanation [ 25-29]. We \nnote that t he damping parameter  extracted from the fits should be regarded as  a \nphenomenolo gical parameter that accou nts for combined effect s of a (frequency independent) \nintrinsic Gilbert damping, an inhomogeneous broadening, a two magnon scattering, and \nvarious propagation spin wave processes resulting from the nonuniform spatial profile of the \nexcited precession.  We a lso note that the rate of decrease of  with f is sample dependent and,  15 therefore, we show in the main paper the doping dependence of the frequency -independent \npart of . \n0 500 1000 1500 2000-40-30-20-100MO signal ( rad)\nTime delay (ps)(a) 10 mT1\n234\n \n0 500 1000 1500 2000-20-15-10-50MO signal ( rad)\nTime delay (ps)(b) 400 mT1\n234  \nFig. 10. Dynamics of the MO s ignal measured for  external magnetic field  (a) 0Hext = 10 mT and (b) 0Hext = \n400 mT applied along the [010] crystallographic direction in (Ga,Mn)As epilayer x = 5.2%  (points); lines are fits \nby LLG . \n0 200 400 6000246(a) (10-2)\n0Hext (mT)[110]\n[010]\n \n0 5 10 15 200246(b) (10-2)\nf (GHz)[010], [110]  \nFig. 11. (a) Dependence of the Gilbert damping coefficient α on external magnetic field Hext applied along the \n[010] and [110] crystallographic directions in (Ga,Mn)As epilayer with x = 5.2% . (b) Same data as in ( a) but as a \nfunction of the precession fre quency  f. \n \nDETERMINATION OF SPI N STIFFNESS  \nAs we show in Fig. 5 of the main paper, we observed more than one precession mode \nin (Ga,Mn)As epilayers with a sufficient thickness.  These precession modes are the spin \nwave resonances (SWRs) – i.e., spin waves (or magnons) that are selectively amplified by \nfulfilling the boundary conditions of the thin magnetic film [ 18, 30]. Up to now, SWRs in \n(Ga,Mn)As were investigated mainly in a frequency -domain where they are apparent as \nmultiple absorption peaks in the FM R spectra  [3, 10 - 12]. The existence of multiple  16 resonances in FMR reveal that there exist several external magnetic fields at which the \nLarmor precession frequency in the sample coincides with the microwave frequency. The \nresonant field for the n-th mode  (Hn) is obtained by solving the LLG  equation with a term \ncorresponding to exchange interactions in the material and by considering the appropriate \nboundary condition  [11]. In homogeneous thin films with a thickness L, only the \nperpendicular standing waves  with a wave vector k fulfilling the resonant condition \nn kL\nare amplified; the mode with n = 0 denotes the uniform magnetization precession \nwith zero k vector. In principle, there exist two symmetric  boundary conditions which  are \nschemat ically illustrated in Fig. 12. The position of n-th SWR mode in the FMR spectrum Hn \nis given by the Kit tel relation [ 11] and the following equation applies  \n \n22\n2\n0L gDn H H H\nBn n\n\n ,       (18) \n \nwhere n is an integer, D is the exchange spin stiffness cons tant, B is the Bohr magneton, g is \nthe g -factor, and L is the sample thickness  \n \n \n              \n  \n    \nFig. 12. Spin wave resonances in homogeneous thin magnetic films with a thickness L that have a node (a) or \nmaximum (b) at the surface; n is the mode numb er. \n \nIn FMR only the modes with odd n are observed  [11] and the corresponding resonant \nfields are smaller than that of the uniform magnetization precession (i.e., Hn > 0). In the \nmagneto -optical pump -and-probe experiment , the external magnetic field is ke pt constant \nduring the measurement of any dynamical MO trace. Consequently, the SWRs are apparent as \nadditional frequencies that are larger than that of the uniform magnetization precession. \nUltrafast optical pulses also excite all resonant modes without a ny k selectivity  [18, 30]. \nConsequently, for a homogeneous magnetic film with a given thickness, a higher number  of \nSWRs is detectable in the MO dynamical traces than in the FMR spectra. This is particularly \nimportant for (Ga,Mn)As that is magnetically hom ogeneous only when prepared in a form of  17 rather thin films and, therefore, where only a limited number of SWRs is present within a \ndetectable range of the precession frequencies. For an external magnetic field Hext applied in \nthe sample plane, the angular frequency of the n-th SWR mode  fn is given by  [3, 31] \n \n\n   n u c H extn uc\nout H ext B\nn\nH K K HH KKK H\nhgf\n   \n\n\n \n   \n2sin2 4cos2 cos4sin224cos32 cos2\n, \n           (19) \n \nwhich enables to convert the experimentally measured frequency spacing of individual modes \nto the field differences Hn from which the magnitude of the spin s tiffness D can be evaluated \nusing Eq. ( 18) (see Fig. 5 in the main paper) .  \nAs we illustrate  in the following  chapter, t he magnetic homogenei ty of the \ninvestigated epilayer is  absolutely essential for a correct determination of D from the \nmeasured SWR spac ing. Therefore , the experimental results obtained  in samples that had \nbeen prepared by etching the original 48 nm thick (Ga,Mn)As epilayer  down to the thickness \n39, 29 a nd 15 nm are of fundamental importance.  In Fig. 13 we show  the corresponding FFT \nspectr a of the measured oscillatory MO signal.  Clearly, the frequency f0 of the lowest SWR \ndoes not depend on the film thickness. This confirms that the lowest observed SWR really \ncorresponds to the uniform precession of magnetization and, moreover, it proves th at this film \nis magnetically homogeneous. Also the spacing H1 shows the expected [see Eq. (2)] linear \ndependence on n2 and 1/ L2 (see Fig. 5 in the main paper) that enables a reliable determination \nof the value of D. In Fig. 14 we compare the experimental data for 48 nm and 15 nm thick \nepilayers from which the FFT spectra depicted in Fig.  13 were computed. Clearly, the  etching \nof the sample from 48 nm to 15 nm not only suppressed the higher  SWRs, which is apparent \nfrom the purely sinusoidal shape of the dat a for the 15 nm film , but it also increased the \nprecession damping, which is probably a consequence of a slight variation of the etched film  \nthickness within the laser spot size of 25 m. These data illustrate that the magneto -optical \npump -and-probe experi ment is a very sensitive diagnostic tool not only of the magnetic but \nalso of the structural quality  of thin magnetic films.   18 \n0 5 10 15 20FFT (arb. units)\nf (GHz)48 nm\n39 nm\n29 nm\n15 nmf0\nf1\nf2 \nFig. 13. Fourier spectra of oscillatory MO signals measured for 0Hext = 20 mT applied along the [010] \ncrystallographic direction in samples prepared by etching from 48 nm thick (Ga,Mn)As epilayer with x = 7% \n(points), the curves are labeled by the film thicknesses , normalized  and vertically shifted for clarity; the lines are \nfits by a sum of Lorentzian pea ks. \n0 500 1000 1500 200048 nmMO signal (arb. units)\nTime delay (ps)15 nm1\n234\n \nFig. 14. Comparison of oscillatory parts of MO signals measured in the original 48 nm thick epilayer and in the \nepilayer that was etched down to 15 nm; the curves are normalized and  vertically shifted for clarity. \nExperiment al conditions are described in Fig. 13. \n \nDEMONSTRATION OF INA PPLICABILITY OF SPIN STIFFNESS \nMEASUREMENT  IN THICK (GA,MN)AS EPILAYERS  \nFinally, we illustrate the significance of the film magnetic homogeneity for a correct \nevaluation of the spin stiffness. F or this purpose we selected a 500 nm thick (Ga,Mn)As \nepilayer with 7% Mn (i.e, a sample with the same nominal Mn doping as the one used in \nexperiments depicted in Fig. 13 and Fig. 14). In Fig. 15 we show the temperature dependent \nmagnetization projections to several crystallographic directions  measured in the as -grown and \nannealed sample s. In the as -grown sample , the temperature dependence of magnetization \nprojections is strongly non -monotonous [see Fig. 15(a)]. Moreover,  the Curie temperature  Tc  19 is only  60 K that is very low for a material with 7% Mn. This is a consequence of a high \nconcentration of unintentional interstitial Mn impurities in the sample that compensate both \nthe local moment and the holes produced by substitutional Mn atoms  [32]. The amoun t of \ninterstitial Mn impurities in the sample can be reduced by a thermal annealing  [33]. However, \neven very long annealing times are not sufficient for obtaining a high quality sample from the \nthick epilayer due to the formation of the surface oxide that controls the outdiffusion of \ninterstitial Mn impurities  [33]. Therefore, the 40 h long annealing at 200 °C led to an increase \nof Tc but only to 90 K, which is still substantially lower than Tc  150 K observed in thin \nsamples with the same nominal concentr ation of Mn. Simultaneously, the temperature \ndependence of magnetization does not show the expected sharply vanishing magnetization at \nTc (cf. Fig. 2 in the main paper  for the data in optimized epilayers ).  \n0 25 50 75 10005101520\n [110]\n [-110]\n [010]Magnetization (emu/cm3)\nTemperature (K)(a) as-grown\n1\n234\n \n0 25 50 75 10005101520\n [110]\n [-110]\n [010]Magnetization (emu/cm3)\nTemperature (K)(b) annealed\n1\n234  \nFig. 15. Temperature dependence of the magnetization projections to different crystallographic directions \nmeasured by SQUID  in 500 nm thick (Ga,Mn)As epilayer with x = 7%. (a)  As-grown sample. (b) Sample  \nannealed for 40 hours at 200°C . \n0 50 100 150 200 2506789 (10-3cm)\nTemperature (K)(a)\n1\n234\n \n0 50 100 150 200 250024d/dt (10-5cm/K)\nTemperature (K)(b)\n1\n234  \nFig. 16. Temperature dependence of the resistivity  (a) and its temperature derivative d/dT (b) measured in \n500 nm thick (Ga,Mn)As epilayer with x = 7% annealed for 40 hours at 200°C.  \n \nIn Fig. 16 we show the t emperature dependence of the resistivity and its temperature \nderivative measured in the annealed sample. Clearly, there is no sharp Curie point singularity  20 in the temperature derivative of the resistivity which is the fingerprint of a high magnetic \nquality of (Ga,Mn)As epilayer (cf. Fig. 1(a) in the main paper)  \nIn Fig. 17 we show the time -resolved magneto -optical signals measured in this 500 nm \nthick epilayer. In the as -grown sample two precession modes can be identified . In the \nannealed sample the improved magnetic quality leads to a strong suppression of the \nmagnetization precession damping with respect to that observed in the as -grown sample. For \nexample, the data shown in Fig. 17(a) and (b) for the lowest modes correspond to damping \ntimes of 210 ps and 46 0 ps for the as -grown and annealed sample, respectively. In addition, \nthe annealing led to a considerable increase of the number of observed SWR modes in the \nmeasured TRMO signal. However, their identification is a rather complicated task.  \n-60-30030\n0204060\n0 1000 2000 3000-50050100\n0 500 1000-60060120  (a) as-grown\n  (b) annealedMO signal  ( rad)\nTime delay (ps)MO signal  ( rad)\nTime delay (ps)\n \nFig. 17. Time -resolved magneto -optical signals (points) measured in as -grown (a) and annealed (b) 500 nm thick \n(Ga,Mn)As epilayer with x = 7%; note the different  x-scales in (a) and (b). The lines in the upper parts of the \nfigures are a sum of  damped harmonic functions and the corresponding precession modes are plotted in the lower \nparts of the figures.  External magnetic field of 10  mT was applied along the [010] crystallographic direction.  \n \nIn Fig.  18 we show the FFT spectrum of the oscillato ry MO signal s measured in the \nannealed sample for external magnetic fields  of 10 mT and 20 mT . Even though the magnetic \nfield change was rather  small, the FFT spectra were changed dramatically. In particular, at 10 \nmT there are 3 peaks with comparable inte nsities (and 5 peaks in total) while at 20 mT there \nis only 1 strong peak (and 4 peaks in total).  In Fig. 19 we show the dependence of the \nfrequency of SWR modes on the external magnetic field – at least for the first sight , it is not \napparent how to assig n the observed modes to mode numbers defined by Eq. ( 18), which is \nthe basic requirement for an evaluation of the spin stiffness from the measured data .  \n  21 \n0 5 10FFT (arb. units)\nf (GHz)(a) 10 mT\n1\n234\n0 5 10FFT (arb. units)\nf (GHz)(b) 20 mT\n1\n234 \nFig. 18. Fourier spectrum of the oscillatory p art of the MO signal measured in the annealed sample for external \nmagnetic fields of 10  mT (a) and 20 mT (b) applied along the [010] crystallographic direction (points); the red \nline is a fit by a sum of Lorentzian peaks (green lines) and the arrows indica te positions of the peak frequencies.  \n \n0 10 20 30 40 5002468fn (GHz)\n0Hext (mT)1\n234\n \nFig. 19. Dependence of the precession frequency fn on Hext measured in the annealed sample for external \nmagnetic field applied along the [010] crystallographic direction.  \n \nIn Fig. 20 (a) we  show a plausible assignment  of the measured frequencies to four  \nSWRs described in the previous chapter  and one non -propagating surface mode [ 11]. We note \nthat t he identification  of the lowest mode  for fields below 15 mT as the surface mode is based \non the  analysis reported in Ref. 11 – in particular, due to the observations that this mode is \napparent  only at certain external magnetic fields and that it has a smaller amplitude that the \none assigned to the homogeneous precession [see Fig. 18(a)]. Following t he analysis reported \nin the previous chapter , we can now proceed to the evaluation of the spin stiffness. In Fig.  \n20 (b) the deduced values of Hn are plotted as a function of n2. The observed mode spacing \ndeviates significantly from that expected for SWR s in a magnetically homogeneous film  [see \nEq. ( 18)] which is another fingerprint of the mag netic inhomogeneity in this 500  nm thick \nepilayer [9-12]. Consequently, despite a large number of SWRs detected in this sample,  they \ncannot be used for a direct determination of the spin stiffness. \n  22 \n0 10 20 30 40 5002468\nSMn = 3\nn = 2\nn = 1\nn = 0(a)fn (GHz)\n0Hext (mT)1\n234\n0 2 4 6 8 10050100150\n(b)Hn (mT)\nn21\n234 \nFig. 20. (a) Dependence of the precession frequency fn on Hext measured in the annealed sample for external \nmagnetic field applied along the [010] crystallographic dir ection re-plotted from Fig. 19 with the depicted \nassignment of precession frequencies to the individual SWRs  and to the surface mode , SM (points ). Lines are fits \nby Eq. ( 19). 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K., Grimsditch, M., and Merlin, R. \nUltrafast op tical study of magnons in the ferromagnetic  semiconductor GaMnAs . \nSuperlatt. Microstruct.  41, 372 (2007).  \n[32] Jungwirth, T. et al.  Systematic study of Mn -doping trends in optical properties of \n(Ga,Mn)As . Phys. Rev. Lett.  105, 227201 (2010) and its Supplem entary material ,  \narXiv: 1007.4708.  \n[33] Olejník, K., Owen, M. H. S. , Novák, V., Mašek, J., Irvine, A. C. , Wunderlich, J., and \nJungwirth, T. Enhanced annealing, high Curie temperature, and low -voltage gating in \n(Ga,Mn)As:  A surface oxide control study . Phys. Rev. B  78, 054403 (2008),  \narXiv: 0802.2080.  47"    },    {        "title": "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. Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H.\nShima and K. Fukamichi, J. App. Phys. 91, 8037 (2002).\n8K.Yu. Guslienko, X.F. Han, D.J. Keavney, R. Divan and\nS.D. Bader, Phys. Rev. Lett. 96, 067205 (2006).\n9A.R. V´ olkel, F.G. Mertens, A.R. Bishop and G.M. Wysin,\nPhys. Rev. B 43, 5992 (1991).\n10G.M. Wysin, F.G. Mertens, A.R. V¨ olkel and A.R. Bishop,\ninNonlinear Coherent Structures in Physics and Biology ,\np. 177, K.H Spatschek and F.G. Mertens, editors, (Plenum\nPress, New York, 1994) (ISBN 0306448033).\n11G.M. Wysin, Phys. Rev. B 54, 15156 (1996).\n12B.A. Ivanov, H.J. Schnitzer, F.G. Mertens and G.M.\nWysin, Phys. Rev. B 58, 8464 (1998).\n13G.M. Wysin and A.R. V¨ olkel, Phys. Rev. B 54, 12921\n(1996).\n14K.L. Metlov and K.Yu. Guslienko, J. Mag. Magn. Mater.\n242–245 , 1015 (2002).\n15G.M. Wysin, J.Phys.: Condens.Matter 22, 376002 (2010).\n16A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n17D.L. Huber, Phys. Rev. B 26, 3758 (1982).\n18T.S. Machado, T.G. Rappoport and L.C. Sampaio, Appl.\nPhys. Lett. 100, 112404 (2012).\n19B.A. Ivanov et al., JETP Lett. 91, 178 (2010).20J.L. Garc´ ıa-Palacios and F.J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n21U. Nowak, in Annual Reviews of Computational Physics\nIX, p. 105, edited by D. Stauffer (World Scientific, Singa-\npore, 2000).\n22Zhongyi Huang, J. Comp. Math. 21, 33 (2003).\n23D. Suessa, J. Fidler and T. Schrefl Handbook of Magn.\nMater.1641 (2006).\n24G. Gioia and R.D. James Proc. R. Soc. London Ser. A453\n213 (1997).\n25C.J. Garc´ ıa-Cervera, “Magnetic Domains and Magnetic\nDomain Walls,” Ph.D. thesis, New York University (1999).\n26C.J. Garc´ ıa-Cervera, Z. Gimbutas and E. Weinan E J.\nComp. Phys. 18437 (2003).\n27G.M. Wysin, W.A. Moura-Melo, L.A.S. M´ ol and A.R.\nPeriera, J. Phys.: Condens. Matter 24296001 (2012).\n28J. Sasaki and F. Matsubara J. Phys. Soc. Japan 66, 2138\n(1997).\n29George Marsaglia and Arif Zaman, Computers in Physics\n8, No. 1, 117, (1994).\n30G.M. Wysin and F.G. Mertens, in Nonlinear Coherent\nStructures in Physics and Biology , Springer-Verlag Lec-\nture Notes in Physics, M. Remoissenet and M. Peyrard,\neditors, (Springer-Verlag, Berlin, New York, 1991) (ISBN\n0387548904).\n31K.Yu. Guslienko, V. Novosad, Y. Otani and K. Fukamichi,\nAppl. Phys. Lett. 78, 3848 (2001).\n32K.Y. Guslienko, A.N. Slavin, V. Tiberkevich, et al., Phys.\nRev. Lett. 101, 247203 (2008).\n33C.E. Zaspel, B.A. Ivanov, P.A. Crowell, et al., Phys. Rev.\nB72, 024427 (2005).\n34B.A. Ivanov, E.G. Galkina and A.Yu. Galkin, Low Temp.\nPhys.36, 747 (2010)."    },    {        "title": "1207.6686v1.Ultrafast_optical_control_of_magnetization_in_EuO_thin_films.pdf",        "content": "1  \nUltrafast optical control of magnetization in EuO thin films \nT. Makino1,*, F. Liu2,3, T. Yamasaki4, Y. Kozuka2, K. Ueno5,6, A. Tsukazaki2, T. Fukumura6,7, Y. Kong3, and \nM. Kawasaki1,2 \n1 Correlated Electron Research Group (CERG) and Cr oss-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institut e, \nWako 351-0198, Japan \n2 Quantum Phase Electronics Ce nter and Department of Applied Physics, Un iversity of Tokyo, Tokyo 113-8656, Japan, \n3 School of Physics, Nankai Un iversity, Tianjin 300071, China  \n4 Institute for Materials Research, T ohoku University, Sendai, 980-8577, Japan \n5Graduate School of Arts and  Sciences, University of  Tokyo, Tokyo 153-8902, Japan \n6PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan, \n7Department of Chemistry, Univers ity of Tokyo, Tokyo 113-0033, Japan, \n \nAll-optical pump-probe detection of magnetization precession has been performed for ferromagnetic EuO \nthin films at 10 K. We demonstrate that the circ ularly-polarized light can be used to control the \nmagnetization precession on an ultrafast time scale. This takes place within the 100 fs duration of a single \nlaser pulse, through combined contribution from two nonthermal photomagnetic effects, i.e., enhancement \nof the magnetization and an inverse Faraday effect. From the magnetic field dependences of the frequency \nand the Gilbert damping parameter, the intrinsic Gilbert damping coefficient is evaluated to be α ≈ 3×10-3. \nPACS numbers: 78.20.Ls, 42 .50.Md, 78.30.Hv, 75.78.J \n 2 Optical control of the spin in magnetic materials has \nbeen one of the major issues in the field of spintronics, \nmagnetic storage technology, and quantum computing1. \nOne type of the spin controls is based on the directional manipulation in the spin moments\n2. This yields in \nobservations of spin precession (reorientation) in \nantiferromagnets and ferromagne ts when magnetization is \ncanted with respect to an external field3–14. In many \nprevious reports, the spin precession has been driven with \nthe thermal demagnetization induced with the photo-irradiation. Far more intriguing is the ultrafast \nnonthermal control of magnetization by light\n8,10,14, which \ninvolves triggering and suppression of the precession. The precession-related anisotropy is expected to be \nmanipulated through laser-induced modulation of \nelectronic state because the anisotropy field originates \nfrom the magnetorcrystalline anisotropy based on the \nspin-orbit coupling. Recently, the spin precession with the \nnon-thermal origin has been observed in bilayer manganites due to a hole-concentration-dependent \nanisotropic field in competing magnetic phases\n15. Despite \nthe success in triggering the reorientation by ultrafast laser pulses, the authors have not demonstrated the possibility \nof the precessional stoppage. \n On the other hand, photomagnetic switch of the \nprecession has been reported in ferrimagnetic  garnets with \nuse of helicity in light\n8,10. The authors attributed the \nswitching behavior to long-lived photo-induced \nmodification of the magnetocrystalline anisotropy16 \ncombined with the inverse Faraday effects17,18. The \nunderlying mechanism for the former photo-induced effect \nis believed to be redistribution in doped ions16. This is too \nunique and material-dependent, which is not observed in wide variety of magnets. For establishing the universal \nscheme of such “helicity-controllable” precession, it \nshould be more useful to rely on more generalized mechanisms such as the carrier-induced ferromagnetism \nand the magnetic polarons\n19. A ferromagnet should be a \nbetter choice than a ferrimagnet or an antiferromagnet, e.g., for aiming a larger-amplitude modulation by making \nuse of its larger polarization-rotation angle per unit length . \nWe have recently reported  the optically-induced \nenhancement of magnetization in ferromagnetic EuO \nassociated with the optical transition from the 4 f to 5 d \nstates\n20. This enhancement was attributed to the \nstrengthened collective magnetic ordering, mediated with \nthe magnetic polarons. The helicity-controllable \nprecession is expected to be observed in EuO by combining the photo-induced magnetization \nenhancement20 with the inverse Faraday effect17,18 because \nthe magnetization is related to the magnetic anisotropy. \nThe occurrence of the inverse Faraday effects is expected \nbecause of the high crystalline symmetry in EuO17,18. The \nmagnetic properties of EuO are represented by the \nsaturation magnetization of 6.9 μB/Eu, the Curie \ntemperature of 69 K, and the strong in-plane \nanisotropy21,22. \n In this article, we report observation of the \nphotomagnetic switch of the spin precession with the \nnonthermal origin in a EuO thin film  for the first time to \nthe best of our knowledge. Due to the above-mentioned \nreasons, our findings deserve the detailed investigations \nsuch as the dependence on the circularly polarized lights, the frequency of precession, the Gilbert damping constants, \nand the magnitudes of the photo-induced anisotropic field. \n EuO films were deposited on YAlO\n3 substrate using a \npulsed laser deposition system with a base pressure lower \nthan 8×10-10 Torr22. The EuO films were then capped with \nAlO x films in-situ . EuO and AlO x layers have thicknesses \nof 310 and 30 nm, respectively. The film turned out to be \ntoo insulating to be quantified by a conventional transport measurement method. The all-optical experiments have \nbeen performed using a standard optical set-up with a \nTi:sapphire laser combined with a regenerative amplifier (accompanied with optical parametric amplifier). The \nwavelength, width, and repetition rate of the output pulse \nwere 650 nm, ≈100 fs, and 1 kHz, respectively.  The \npump and probe pulses were both incident on the film at \nangles of θ\nH ≈ 45 degree from the direction normal to the \nfilm plane as shown in inset of Fig. 1. The direction of the probe beam is slightly deviated from that of the pump so \nas to ensure the sufficient spatial separation of the \nreflected beams. The angle between the sample plane and the external field is approximately 45 degree. The \npolarization rotation of the reflected probe pulses due to \nthe Kerr effect was detected using a Wollaston prism and a balanced photo-receiver. The pump fluence was \napproximately 0.5 mJ/cm\n2. A magnetic field was applied \nusing a superconducting electromagnet cryostat. The maximum applied magnetic field was μ\n0H ≈ 3 T. All the \nmeasurements were performed at 10 K. \n Figure 1 shows a magneto-optical Kerr signal as a \nfunction of the pump-probe delay time for a EuO film at \nμ\n0H = 3.2 T under the irradiation of right-circularly 3 polarized ( σ+) light. Its time trace is composed of \ninstantaneous increase and d ecay of the Kerr rotation, and \nsuperimposed oscillation20. The oscillatory structure \ncorresponds to the precession of magnetization. A solid (black) curve in Fig. 1 shows the result of fit to the \nexperimental data using an exponentially decaying \nfunction and a damped oscillatory function. The precession is observed even wi th the linearly polarized \nlight, which is consistent with the fact that EuO is a \nferromagnet at this temperature. \n \nFIG. 1 (color online). Time-resolved Kerr signals recorded \nfor a EuO thin film at a magnetic field of 3.2 T, and a \ntemperature of 10 K for ri ght circularly-polarized ( σ\n+) \nlight. The inset schematically shows the experimental arrangement. Experimental data are shown by (red) \nsymbols, while the result of fit was shown by a full (blue) \nline. \n \nFor the detailed discussion of the precession properties, \nwe subtracted the non-oscillatory part from the Kerr signal as a background. The results are shown in Fig. 2 for nine \nmagnetic fields and for σ\n+ and left-circularly polarization \n(σ¯). The subtracted data were then fitted with the damped \nharmonic function in the form of Aexp(−t/τ) sin(2πft+φ), \nwhere A and φ are the amplitude and the phase of \noscillation, respectively. The amplitude of the precession was not found to depend on the plane of the linear \npolarization of the pump pulse. There is a linear \nrelationship between the amplitude of precession and the pump fluence for the excitation intensity range measured. \nIt is also noticed in Fig. 2 that the precession amplitudes \nare different each other for the two helicities ( σ\n+ and σ¯) \neven at the same magnetic fields. The magnetic field \ndependence of the amplitude is summarized in Fig. 3(d). The minimum precession amplitude appears at around μ0H \n= +0.4 T for the σ¯, while the minimum is observed at \nμ0H = −0.4 T for the σ+ as indicated by the shaded regions. \nTo explain such disappearance of the precession and the triggering of precession even  with a linearly-polarized \nlight, it is necessary to take two effects into account. One \nof the effects that we seek should be odd with respect to the helicity of light. An effective magnetic field through \nthe inverse Faraday effect is plausible to interpret this \nphenomenon because this satisfies the above requirements \n[H\nF\n// (black arrows) in Figs. 3(a) and 3(b)]. While the \nnormal Faraday effect causes difference in the refractive \nindices for the left and right circularly polarized lights propagating in a magnetized medium, it is also possible to \ninduce the inverse process where circularly polarized \nlights create a magnetization or an effective field\n17,18. The \nfield associated with the inverse Faraday effect changes its \nsign when the circular polarization is changed from \nleft-handed to right-handed. \n \nFIG. 2 (color online). A series of precession signals under various \nmagnetic fields for right- and left-circularly polarized ( σ+ and σ¯) \nlights. Solid circles show the e xperimental data for which the \nnon-oscillatory background is s ubtracted, while solid curves \nrepresent the calculated data as described in the text.  \n \nThe other effect involved is considered to be the \nphotoinduced enhancement of the anisotropic field \n(magnetization) associated with the 4 f →5d optical \ntransition [ ΔM (purple arrows) in Figs. 3(a) and 3(b)]20. \nOur previous work quantified the photoinduced \n4 enhancement of the magnetization to be ΔM/M ≈  0.1%20. \nThe amplitude of precession is determined from \ncombination of ΔM with the component of the \ninverse-Faraday field ( HF\n//) approximately projected onto \nthe easy-axis direction. For example, no precession is \ntriggered for μ0H of +0.4 T ( −0.4 T) and σ¯ (σ+), which is \ndue to the balance of these two effects [Fig. 3(a)]. On the other hand, constructive contribution of these effects leads \nto a change in the direction of the magnetization [two \ndashed lines and a red arrow in Fig. 3(b)], which enhances the precession amplitude. The strength of the \nphotoinduced field H\nF can be estimated to be \napproximately 0.2 T at the laser fluence of 0.5 mJ/cm2. \nThe derivation was based on Eq. (17) of Ref. 10. For more \nquantitative discussion for the suppression and \nenhancement of precession, the effect of the perpendicular component of inverse Faraday field is necessary to be \ntaken into account. Such analysis is not performed here \nbecause this goes beyond the scope of our work. \n \nFIG. 3 (color online). Graphical  illustrations of the magnetic \nprecession; its suppression (a) and enhancement (b). M is a \nmagnetization (green), H the external magnetic field (blue), Heff the \neffective magnetic field (red), ΔM a photo-induced magnetization \nenhancement (purple), and the HF\n// the inverse Faraday field (black). \nThe situations of suppression correspond to the conditions of 0.4 T for σ¯ and −0.4 T for σ+. The situations of enhancement are for \nopposite cases. Magnetic field depe ndences of the magnetization \nprecession related quantities for σ+ and σ¯; precession frequency f \n(c), amplitude (d), and effective Gilbert damping αeff (e) (f). \n \nFor the derivation of the precession-related parameters, \nwe plot the frequency ( f) and the amplitude of the \nmagnetization precession for two different helicities as a \nfunction of H with closed symbols in Figs. 3(c) and 3(d). \nTo deduce the Landé g-factor g, we calculated f(H) using \na set of Kittel equations for taking the effect of tilted \ngeometry into account as12,23: \n12 f HH                             ( 1 )  \n2\n1e f f cos( ) cosH HH M               ( 2 )  \n2e f f cos( ) cos 2H HH M               ( 3 )  \nHere, γ is the gyromagnetic ratio ( gμB/h), μB the Bohr \nmagneton,  h Planck’s constant, and θH an angle between \nthe magnetic field and direction normal to the plane. Meff \nis the effective demagnetizing field given as Meff = MS-2K\n⊥/MS, where MS is the saturation magnetization and K⊥ is \nthe perpendicular magnetic anisotropy constant. θ is an \nequilibrium angle for the magnetization, which obeys the \nfollowing equation: \neff sin 2 (2 / )sin( )H HM                ( 4 )  \nA solid (black) line in Fig. 3( c) corresponds to the result \nof the least-square fit for the frequency f. The values of \nparameters are g ≈ 2 and μ0Meff  ≈ 2.4 T. The g value is \nconsistent with the one derived from the static ferromagnetic resonance measurement\n24. \n \nHaving evaluated the precession-related parameters \nsuch as g and Meff, we next discuss H dependence of an \neffective Gilbert damping parameter αeff. This quantity is \ndefined as: \neff1\n2f                              ( 5 )  \nFigures 3(e) and 3(f) show the effective Gilbert \ndamping parameter αeff derived from the decay time \nconstant ( τ) for σ+ and σ¯, respectively. Despite relatively \nstrong ambiguity shown with ba rs in Figs. 3(e) and 3(f), \nthe damping parameters αeff is not independent of the \nmagnetic field. It is rather a ppropriate to interpret that for \n5 αeff for low fields are larger than those at higher fields. \nSuch dependence on magnetic field is consistent with \nthose in general observed for a wide range of the \nferrimagnets and ferromagnet s. Two-magnon scattering \nhas been adopted for the explanation of this trend25. When \nthe magnitude or direction of the magnetic anisotropy \nfluctuates microscopically, magnons can couple more efficiently to the precessional motion\n25. Such may cause \nan additional channel of relaxation. Due to the suppressed \ninfluence of the abovementioned two-magnon scattering, the higher-field data correspon d to an intrinsic Gilbert \ndamping constant α ≈ 3×10\n-3, as shown with a dashed \n(black) line in Figs. 3(e)  and 3(f). This value is \ncomparable with that reported in Fe26,27,28,29 and \nsignificantly larger than that of yttrium iron garnet, which \nis known for intrinsically low magnetic damping8,10,14. \n \nIn conclusion, we have reported the observation of \nmagnetization precession and the dependence on light helicity in ferromagnetic EuO films. We attribute it to the \nphoto-induced magnetization enhancement combined with \nthe inverse Faraday effect. The magnetic field dependence of the precession properties al lowed us the evaluation of \nthe Gilbert damping constant to be ≈3×10\n-3. \nAcknowledgements—the authors thank K. Katayama, \nM. Ichimiya, and Y. Takagi for helpful discussion. This \nresearch is granted by the Japan Society for the Promotion \nof Science (JSPS) throug h the “Funding Program for \nWorld-Leading Innovative R&D on Science and \nTechnology (FIRST Program),” initiated by the Council \nfor Science and Technology Policy (CSTP) and in part supported by KAKENHI (Grant Nos. 23104702 and \n24540337) from MEXT, Japan (T. M.). \n \nREFERENCES \n* tmakino@riken.jp  \n1 A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. \nPisarev, A. M. Balbashov, and Th. Rasing, Nature \n435, 655 (2005).  \n2 A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. \nPhys. 82, 2731 (2010).  \n3 C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, \nD. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 (1999).  \n4 Q. Zhang, A. V. Nurmikko, A. Anguelouch, G. Xiao, \nand A. Gupta, Phys. Rev. Lett. 89, 177402 (2002).  \n5 M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, \nL. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. \nRev. Lett.  88, 227201 (2002).  6 R. J. 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B 84, 134425 (2011)  \n "    },    {        "title": "1208.0318v1.Artificial_Neural_Network_Based_Prediction_of_Optimal_Pseudo_Damping_and_Meta_Damping_in_Oscillatory_Fractional_Order_Dynamical_Systems.pdf",        "content": " \n   \n 1 \nArtificial Neural Network Based Prediction of \nOptimal Pseudo-Damping and Meta-Damping in \nOscillatory Fractional Order Dynamical Systems \nSaptarshi Das 1, Indranil Pan 1,2  \n1. Department of Power Engineering, Jadavpur \nUniversity, Salt-Lake Campus, LB-8, Sector 3, Kolka ta-\n700098, India. Email: saptarshi@pe.jusl.ac.in  \n2. MERG, Energy, Environment, Modelling and Mineral s \n(E 2M2) Research Section, Department of Earth Science \nand Engineering, Imperial College London, Exhibitio n \nRoad, London SW7 2AZ, UK. Email:  \ni.pan11@imperial.ac.uk , indranil.jj@student.iitd.ac.in  Khrist Sur 1,3 , Shantanu Das 4 \n3. Center for Soft Computing Research, Indian Stati stical \nInstitute, 203 Barrackpore Trunk Road, Kolkata-7001 08, \nIndia. Email: khrist@ieee.org , khrist@isical.ac.in  \n4. Reactor Control Division, Bhabha Atomic Research  \nCentre, Mumbai-400085, India. \nEmail: shantanu@magnum.barc.gov.in \n \n \nAbstract —This paper investigates typical behaviors like \ndamped oscillations in fractional order (FO) dynami cal systems. \nSuch response occurs due to the presence of, what i s conceived as, \npseudo-damping and meta-damping in some special cla ss of FO \nsystems. Here, approximation of such damped oscilla tion in FO \nsystems with the conventional notion of integer ord er damping \nand time constant has been carried out using Geneti c Algorithm \n(GA). Next, a multilayer feed-forward Artificial Ne ural Network \n(ANN) has been trained using the GA based results t o predict the \noptimal pseudo and meta-damping from knowledge of t he \nmaximum order or number of terms in the FO dynamica l system. \nKeywords— Artificial Neural Network (ANN); fraction al order \nlinear systems; meta-damping; pseudo-damping; Genet ic Algorithm   \nI.   INTRODUCTION  \nFractional order dynamical systems which are govern ed by \nfractional order differential equations have got re newed interest \nin the science and engineering community in recent past for its \nhigher capability and flexibility in modeling of na tural \nprocesses [1], [2]. It is well known from basics of  control \ntheory that second order stable oscillatory dynamic al systems \nor higher order stable oscillatory systems, which c an also be \napproximated as second order transfer function mode ls, decay \nwith an exponential envelope upon step or impulse t ype \nexcitation [3]. In other words, a physical system, governed by \nsecond order differential equation of the form (1),  with \nexcitation ()u t and response ()y t shows oscillatory time \nresponse for ()0,1 ξ∈. \n() ()( ) ( )\n( ) ( )( ) ( )2\n2 2 \n2\n2\n2\n22\n2d y t dy t y t u t dt dt \nd y t dy t y t u t dt dt ξω ω ω \nτ ξτ + + = \n⇒ + + =             (1)  \nHere, parameters {}, , ξ ω τ represent the system’s damping \nratio, natural frequency and time constant respecti vely with 1τ ω =. For step and impulse type excitation, the dynamic al \nsystem governed by (1) exhibit damped time response s with an \nexponential envelope, represented by (2) and (3) re spectively. \n( )2\n2 1 \n211 sin 1 tan \n1tey t t ξω ξω ξ ξ ξ−\n−    −    = − − +     −          (2) \n( ) ()2\n2sin 1 \n1tey t t ξω ωω ξ \nξ−\n= − \n−            (3) \nIn contrast, a dynamical system, governed by a two term \nfractional order differential equation (4) can also  show \noscillatory damped time response for ()1,2 α∈, although there \nis no explicit damping term, containing ξin (4). This typical \nbehavior of fractional order systems lead to the co ncept of \n“Pseudo-damping” which can not be visualized with t he \nconventional theory of integer order calculus for d escribing the \ndynamics of physical systems. \n()()()taD y t by t u t α+ =              (4) \nLaplace transform of (4) with zero initial conditio n gives the \nsystem’s transfer function as (5) which again produ ces its \nimpulse response as the Green’s function (6) upon i nverse \nLaplace transformation [1] for the two-term FO syst em (4). \n( )()\n( ) ( )21 1 1 Y s G s U s as b a s b a α α   = = =     + +             (5) \n( )1\n2 , 1 bg t t E t a a α α \nα α −  = −                 (6) \nIn (6), ,Eα β represents the two-parameter Mittag-Leffler \nfunction which is a higher transcendental, encompas sing a \nlarge family of conventional transcendental functio ns like \ntrigonometric, inverse circular, exponential, logar ithmic, \nhyperbolic etc. [1]-[2]. The series representation of two  \n   \n 2 \nparameter Mittag-Leffler function is given by (7) w hich is a \ngeneralized template and reduces to an exponential function for \n1, 1 α β = = [1]-[2]. Also, from (6) it can be observed that the  \nenvelope is guided by a power law instead of an exp onential \none in (3). In this system, a Mittag-Leffler type o scillation \ntakes place in contrast to the sinusoidal oscillati on in (3). \n( )( ),\n0: , 0, 0 k\nkzE z kα β α β α β ∞\n== > > Γ + ∑           (7) \nThe present paper firstly attempts to approximate t he \noscillations, produced due to step excitation of (4 ) with an \nequivalent template given by (1) using GA i.e. find ing optimal \npseudo/meta (FO)-damping or time constants, associa ted with \nthe oscillatory time response of the FO system. Nex t the \noptimal FO damping and time constants are predicted  using a \nmultilayer feed-forward ANN. This approach reduces the \ncomputational load, associated with running GA ever y time for \nfinding out the equivalent optimal FO damping for a ny \narbitrary FO system within this range and such an a pplication is \njustified from the point that multilayer feed-forwa rd ANN is \ngenerally very good function approximator [4]-[5]. In [6], the \nconcept of optimal fractional order damping was fir st proposed \nwith a specific need for faster stabilization of os cillatory \nsystems using the concept of FO damping with respec t to some \nintegral performance indices like Integral of Squar ed Error \n(ISE), Integral Time weighted Squared Error (ITSE) etc. This \npaper gives a new concept of finding the optimal in teger order \nequivalence of FO damping using ISE/ITSE as perform ance \nindices and also proposes their ANN based predictio n. \nRest of the paper is organized as follows. Section II briefly \nintroduces the basics of pseudo-damping and meta-da mping in \nsome special class of oscillatory FO dynamical syst em. Section \nIII describes time domain simulation of pseudo/meta -damping \nand their GA based optimal time domain approximatio n. \nSection IV presents the ANN based training and pred iction \nperformance for these optimal FO-damping. The paper  ends \nwith the conclusion as section V, followed by the r eferences. \nII.  CONCEPT OF PSEUDO AND META -DAMPING IN \nFRACTIONAL ORDER LINEAR DYNAMICAL SYSTEMS  \nIn [1]-[2], it has been reported that time and freq uency \ndomain representations of few special functions, re lated to \nfractional calculus like R-function and G-function are given by: \n()\n( )( )1 1 \n1\n01n v v n \nns a t \ns a n v α\nαα+ − − ∞\n−\n=  =  − Γ + −   ∑ L                    (8) \n( )( )( ) ( ){ } ( )( )\n( ) ( )( )1\n1\n01 1 \n1jr j v v\nr\njr r j r a t s\nj r j v s a α\nα α+ − − ∞\n−\n=  − − − − − −   =  Γ + Γ + − −  ∑⋯L  (9) \nIf Laplace transform of excitation and response of a \nfractional order transfer function (FOTF) ()G s be ()U s and \n()Y s respectively, then simple treatments as in (10) giv es its \nstep and impulse response. Therefore, to find out t he step and \nimpulse response for few classes of fractional orde r systems, \n1v= − and 0v=need to be considered respectively in (8) and (9). Time domain simulation for FO systems using (8 )-(9) \nneeds evaluation of few convergent infinite series at each \ndiscrete time step ( t). For numerical implementation the \ninfinite series have been evaluated in MATLAB at ea ch t, with \nthe order of accuracy being 0.001. \n()()()\n( ) ( ) ( )\n( )\n( )1\n1for step response \nfor impulse response Y s G s U s \ny t g t u t \nG s s \nG s −\n−=\n⇒= ∗ \n    =    L\nL        (10) \nUnder this condition, expressions (8) and (9) repre sents the step \nresponse of stable FOTFs with the replacement of ( a) by unity \nand ( b) by ( b−) in structures like (4). i.e. \n( )1 2 1 1 ,rP P s b s b αα= = + +          (11) \nIt has been seen from (4) that the step response gi ves \nsustained oscillation for 0α=. But the response becomes \ndamped for 1 2 α< < . Hence, a fractional order system of the \nform (4) having no explicit damping term in it, als o exhibits \ndamped oscillation in time response for 1 2 α< < . Such an \noscillation has been approximated by using a second  order \nsystem of the form (1) while minimizing few integra l error \nindices. The FO system of the structure (11) with 1 2 α< < , \ncan be modified with normalized frequency to unity ( 1b=) as: \n1 2 2 1 1 \n1 2 1 Ps s s ατ τξ =+ + + ɶ≃                  (12) \nwhere, {},τ ξ can be termed as the optimal pseudo-time \nconstant and pseudo-damping respectively with respe ct to some \nintegral error index. \nThe second class of FO systems in (11) exhibits dif ferent \ntype of oscillations if different combinations of o rders are used \nin the expansion of the polynomials, although the h ighest order \nof the models are same and only the number of fract ional order \nterms varies in the model. It is well known that or der of a FO \nLTI system is determined by the maximum order prese nt in the \ndenominator polynomial. If it be assumed that 2rα=, the \nsystem governed by (9) becomes a different class of  fractional \nsecond order system (13) which can again be represe nted by \nequivalent second order approximation with {},τ ξ being the \noptimal meta-time constant and meta-damping respect ively. \nSimilar treatment of normalizing the frequency to u nity yields: \n( )2 2 2 2 1 1 \n2 1 1Ps s sα ατ τξ =+ + +ɶ≃                 (13) \nThe following examples put more light on the behavi or of \nsuch systems with meta-damping. Simple modification  of (13) \ngives first and second order transfer functions lik e (14). It is \ninteresting to note that though the leading order r emains one \nand two in these models, the number of fractional o rder \nelements increase upon binomial expansion for the t erms with \nhigher powers. These additional number of FO terms puts extra  \n   \n 3 \ndamping to the FO system which is defined as the me ta-\ndamping in FO dynamical system of the form (13). Th us a FO \nsystem represented by (14) is distinctly characteri zed by the \nnumber of FO elements present in it and not by the leading FO \norder unlike (12). This typical behavior is the mot ivation \nbehind defining two different class of FO damping i .e. pseudo-\ndamping for system (12) and meta-damping for system  (13). \n( ) ( )( )( )( )\n( ) ( )( )( )( )1 5 4 2 10 0.2 0.25 0.5 0.1 \n2 10 8 5 4 0.2 0.25 0.4 0.5 1 1 1 1 1 , , , , \n1 1 1 1 1 \n1 1 1 1 1 , , , , \n1 1 1 1 1 s s s s s \ns s s s s αα\nαα=\n+ + + + + \n=\n+ + + + + ⋯\n⋯(14) \nIt is therefore clear that the pseudo-damping is as sociated \nwith the reduction in the highest order of a FO sys tem whereas \nmeta-damping is associated with the increase in fra ctional order \nelements within a FO model though the highest order  of the \nplant remains the same. \nIII.  TIME DOMAIN SIMULATION OF FRACTIONAL ORDER \nSYSTEMS WITH PSEUDO -DAMPING AND META -DAMPING  \nMATLAB based codes have been developed using the \ninfinite series representations of such FOTF i.e. ( 8)-(9) under \nimpulse/step excitation. Time domain simulation usi ng (8)-(9) \noften gives poor result above 30 seconds. This is d ue to the fact \nthat gamma function in the denominator of (8)-(9) a pproaches \ntowards a very large value which can not be compute d using \nmost of the scientific programming languages, due t o buffer \noverflow. Thus it is recommended to reliably use ex pression \n(8)-(9) for time domain simulation of the special c lass of FO \nsystems only up to 30 seconds. Simulation of first order system \nwith meta-damping and 0.9 α<also becomes computationally \ninfeasible due to blowing up of the associated gamm a \nfunctions. Similarly, second order systems with met a-damping \nand 0.9 α<gives reliable time response up to 25 seconds, \nbelow which the results are unreliable as also repo rted in \nHartley and Lorenzo [6] in the context of optimal F O damping. \nA.  Step Response Characteristics \n \nFigure 1.  Step response of FO system with pseudo-damping. \nFO systems given in (12)-(13) have now been subject ed to \nstep input excitations and evaluated at each discre te time step using (8)-(9) and shown in Fig. 1-3. Fig. 1 shows t hat the \noscillations become more damped with decrease in th e order \n(α) of FO system (12). Similar behaviors can be found  for FO \nsystems with meta-damping with leading order being 2 (Fig. 2) \nand 1 (Fig. 3) respectively. It is interesting to n ote that even \nfirst order systems in the presence of other FO ele ments may \nexhibit oscillations as shown in Fig. 3. In Fig. 1 the decaying \nenvelope may be guided by a power law as reported i n (6), but \nthe nature of oscillations for meta-damping in Fig.  2-3 are more \ncomplex to be represented as closed form solutions unlike (6). \n \nFigure 2.  Step response characteristics of fractional second order system \nwith.meta-damping. \n \nFigure 3.  Step response characteristics of fractional first o rder system with \nmeta-damping. \nB.  Impulse Response Characteristics \nThe impulse responses have been shown next for the above \ndiscussed three classes of FO systems. As expected in Fig. 6, \nthe oscillations start from a value of unity for fi rst order \nsystems (with additional FO elements causing meta-d amping). \nFig. 5 shows the oscillations starting from zero co nfirming the \npreservation of the second order behavior of the FO  system. In \nFig. 4 showing FO systems with pseudo-damping mixed  \nbehaviors can be observed regarding the initial val ue of the \nimpulse response which indicates that the classical  notion of \njudging the order of the system by looking only at the impulse \nresponse characteristics is not valid for pseudo-da mping. \n  \n   \n 4 \n \nFigure 4.  Impulse response of FO system with pseudo-damping. \n \nFigure 5.  Impulse response characteristics of fractional seco nd order system \nwith.meta-damping. \n \nFigure 6.  Impulse response characteristics of fractional firs t order system \nwith meta-damping. \nC.  Genetic Algorithm Based Approach for Finding Optima l \nPseudo-Damping and Meta-Damping \nGenetic algorithm (GA) is a stochastic optimization  process \nwhich can be used to minimize a chosen objective fu nction. A \nsolution vector is initially randomly chosen from t he search \nspace and undergoes reproduction, crossover and mut ation, in \neach iteration to give rise to a better population of solution vectors in the next iteration. Reproduction implies  that \nsolution vectors with higher fitness values can pro duce more \ncopies of themselves in the next generation. Crosso ver refers \nto information exchange based on probabilistic deci sions \nbetween solution vectors. In mutation a small rando mly \nselected part of a solution vector is occasionally altered, with a \nvery small probability. This way the solution is re fined \niteratively until the objective function is minimiz ed below a \ncertain tolerance level or the maximum number of it erations \nare exceeded. In the present study the number of po pulation \nmembers in GA is chosen to be 20. The crossover and  \nmutation fraction are chosen to be 0.8 and 0.2 resp ectively for \nminimization of the following objective functions. \n( ) ( )2 2 \n0 0 ,ISE ITSE J e t dt J t e t dt ∞ ∞ \n= = ⋅ ∫ ∫         (15) \nEvaluation of the objective functions have been don e in each \ngeneration of GA within the finite time horizon of 25 seconds \nas discussed earlier. Minimization of error index ( 15) between \nthe respective FO systems and a second order approx imation \nas in (13) and (14) gives the optimum values of pse udo/meta-\ndamping and time constant. The ISE and ITSE based \noptimization results have been reported in Table I- III for the \nthree test cases, as in Fig. 1-3. \nTABLE I.  GA  BASED RESULTS FOR OPTIMAL PSEUDO -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS  \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.0054 0.3485 1.3152 0.0235 0.47 0.9848 \n1.2 0.0168 0.5246 0.8094 0.0647 0.6467 0.6887 \n1.3 0.0287 0.6587 0.596 0.0979 0.7659 0.5537 \n1.4 0.0379 0.7634 0.4672 0.1153 0.8457 0.4635 \n1.5 0.0429 0.8432 0.374 0.1176 0.8998 0.3878 \n1.6 0.0429 0.9029 0.2965 0.1085 0.9374 0.3146 \n1.7 0.0378 0.9463 0.2247 0.0927 0.9647 0.239 \n1.8 0.0277 0.976 0.1527 0.074 0.9837 0.1597 \n1.9 0.0127 0.993 0.0771 0.0448 0.9947 0.0786 \nTABLE II.  GA  BASED RESULTS FOR OPTIMAL META -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS WITH LEADING ORDER = 1 \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.0057 0.3463 1.1946 0.02 0.4006 1.0538 \n1.2 0.0147 0.342 1.067 0.0512 0.4933 0.7747 \n1.3 0.0273 0.4182 0.7884 0.0769 0.5761 0.6247 \n1.4 0.0415 0.4793 0.6322 0.0956 0.6399 0.5336 \n1.5 0.0571 0.5317 0.5308 0.1105 0.6906 0.4667 \n1.6 0.0748 0.5996 0.4393 0.1271 0.7352 0.4091 \n1.7 0.097 0.6597 0.3708 0.1568 0.7784 0.3529 \n1.8 0.13 0.7233 0.3086 0.2382 0.8253 0.2913 \n1.9 0.1996 0.7959 0.2422 0.5806 0.8828 0.2131  \n   \n 5 \nTABLE III.  GA  BASED RESULTS FOR OPTIMAL META -DAMPING FOR \nFRACTIONAL ORDER SYSTEMS WITH LEADING ORDER = 2 \nISE Based ITSE Based Fractional \nOrder ( α) Jmin  τ ξ J min  τ ξ \n1.1 0.005 1.0426 0.7299 0.0462 1.0837 0.712 \n1.2 0.0138 1.0499 0.574 0.1053 1.0939 0.5732 \n1.3 0.0215 1.0452 0.4668 0.1359 1.0809 0.4809 \n1.4 0.0266 1.0362 0.3845 0.1407 1.0609 0.4066 \n1.5 0.0291 1.0263 0.3153 0.1315 1.0409 0.3395 \n1.6 0.029 1.0171 0.2529 0.117 1.0241 0.2741 \n1.7 0.0266 1.0094 0.1931 0.1038 1.012 0.2083 \n1.8 0.0217 1.0042 0.1325 0.0935 1.0047 0.1405 \n1.9 0.0119 1.0012 0.0679 0.0657 1.0012 0.0701 \nIV.  ANN  BASED PREDICTION OF OPTIMAL PSEUDO -\nDAMPING AND META -DAMPING  \nA.  Multi-Layer Feedforward Neural Network Architecture  \nThe standard neural network architecture consists o f an \ninput layer, one or more hidden layers with multipl e \nperceptrons and an output layer. The number of perc eptrons in \nthe hidden layer and the number of hidden layers ar e generally \nproblem specific and depend on the choice of the us er. In the \npresent study, the number of hidden layers is varie d from 1 to 2 \nand for each case the number of neurons in each lay er is varied \nfrom 5 to 25 in incremental steps of 5. The ANN is fully \nconnected, i.e., the output from each input and hid den neuron is \ndistributed to all the neurons of the subsequent la yer. Also a \nfeed-forward architecture is used, i.e. the data fl ows and is \nprocessed sequentially through the input, hidden an d output \nlayers and are not fed-back to the previous layers,  unlike the \nrecurrent ANN structure.  Hyperbolic tangent sigmoid ( tansig ) \nand logarithmic sigmoid ( logsig ) type activation functions and \ntheir combinations are used to create different ANN  \narchitectures for comparing the relative effectiven ess of these \nstructures at capturing the nonlinear relationship between the \ninput ( α) and output ( ,τ ξ ) data. \nB.  Training Performance of ANN and Time Domain \nPerformance of the Predicted Outputs \nMultilayer feed-forward ANN has now been employed t o \npredict the optimal pseudo/meta damping/time consta nts from \nthe knowledge of the fractional order of the dynami cal system. \nTables IV-VI gives the GA based optimal pseudo/meta  \ndamping ( ξ) and time constant ( τ) of the FO systems in terms \nof equivalent second order systems considering the ITSE \ncriterion, since ITSE puts more penalties on the er ror at later \nstages unlike ISE producing better accuracy. The AN Ns are \ntrained with Levenberg-Marquardt back-propagation a lgorithm \nwhich is a gradient based method and often gets stu ck in local \nminima. To check the consistency of the ANNs for ca pturing \nthe input-output relationship, the Mean Squared Err or (MSE) \nof 25 independent runs has been chosen as the perfo rmance \nmeasure. This is justified from the fact that often  large size \nmultilayer ANNs accurately establishes arbitrary no nlinear \nrelation between any input-output data but they mig ht not show consistency in the mapping [4]-[5]. Hence there is always a \ntrade-off between size of the ANN and the predictio n accuracy. \nTABLE IV.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF PSEUDO -DAMPING FOR 25  \nINDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.009 5 \nlogsig 0.0061 \ntansig 0.07 10 \nlogsig 0.0326 \ntansig 0.1009 15 \nlogsig 0.026 \ntansig 0.1609 20 \nlogsig 0.0418 \ntansig 0.1216 1 \n25 \nlogsig 0.0559 \ntansig/tansig 0.0185 \ntansig/logsig 0.0148 \nlogsig/tansig 0.0238 5 \nlogsig/logsig 0.0153 \ntansig/tansig 0.0453 \ntansig/logsig 0.0169 \nlogsig/tansig 0.0218 10 \nlogsig/logsig 0.0133 \ntansig/tansig 0.0714 \ntansig/logsig 0.0245 \nlogsig/tansig 0.0779 15 \nlogsig/logsig 0.0346 \ntansig/tansig 0.1075 \ntansig/logsig 0.0269 \nlogsig/tansig 0.1492 20 \nlogsig/logsig 0.0375 \ntansig/tansig 0.141 \ntansig/logsig 0.0341 \nlogsig/tansig 0.2229 2 \n25 \nlogsig/logsig 0.0357 \n \nFigure 7.  ANN prediction of pseudo-damping for FO system (12) .  \n   \n 6 \nTABLE V.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF META -DAMPING IN FO  SYSTEMS \nWITH LEADING ORDER = 1 FOR 25  INDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.0109 5 \nlogsig 0.0038 \ntansig 0.0487 10 \nlogsig 0.0232 \ntansig 0.0942 15 \nlogsig 0.0257 \ntansig 0.1113 20 \nlogsig 0.026 \ntansig 0.1002 1 \n25 \nlogsig 0.0694 \ntansig/tansig 0.0126 \ntansig/logsig 0.0117 \nlogsig/tansig 0.0086 5 \nlogsig/logsig 0.0074 \ntansig/tansig 0.0385 \ntansig/logsig 0.021 \nlogsig/tansig 0.0392 10 \nlogsig/logsig 0.0121 \ntansig/tansig 0.0763 \ntansig/logsig 0.0257 \nlogsig/tansig 0.0414 15 \nlogsig/logsig 0.0314 \ntansig/tansig 0.0985 \ntansig/logsig 0.0201 \nlogsig/tansig 0.0934 20 \nlogsig/logsig 0.0596 \ntansig/tansig 0.0991 \ntansig/logsig 0.0303 \nlogsig/tansig 0.0973 2 \n25 \nlogsig/logsig 0.0281 \n \n \nFigure 8.  ANN based prediction of meta-damping for fractional  second order  \nsystem. TABLE VI.  TRAINING PERFORMANCE FOR VARIOUS ANN  \nCONFIGURATIONS FOR THE PREDICTION OF META -DAMPING IN FO  SYSTEMS \nWITH LEADING ORDER = 2 FOR 25  INDEPENDENT RUNS  \nNumber of \nlayers Number of neurons \nin each hidden layer Activation \nfunction Average \nMSE \ntansig 0.0032 5 \nlogsig 0.0014 \ntansig 0.0274 10 \nlogsig 0.0091 \ntansig 0.0686 15 \nlogsig 0.0093 \ntansig 0.0431 20 \nlogsig 0.019 \ntansig 0.0654 1 \n25 \nlogsig 0.0436 \ntansig/tansig 0.0058 \ntansig/logsig 0.0059 \nlogsig/tansig 0.0038 5 \nlogsig/logsig 0.0043 \ntansig/tansig 0.014 \ntansig/logsig 0.0099 \nlogsig/tansig 0.0126 10 \nlogsig/logsig 0.0057 \ntansig/tansig 0.0451 \ntansig/logsig 0.006 \nlogsig/tansig 0.0353 15 \nlogsig/logsig 0.0154 \ntansig/tansig 0.0451 \ntansig/logsig 0.0304 \nlogsig/tansig 0.0339 20 \nlogsig/logsig 0.0139 \ntansig/tansig 0.0372 \ntansig/logsig 0.0137 \nlogsig/tansig 0.0351 2 \n25 \nlogsig/logsig 0.0206 \n \n \nFigure 9.  ANN based prediction of meta-damping for fractional  first order \nsystem.  \n   \n 7 \nTABLE VII.  ANN  BASED PREDICTED VALUES OF PSEUDO /M ETA -\nDAMPINGS AND TIME CONSTANTS FOR THREE CLASS OF FO  SYSTEMS  \nType of the \nFO System Minimum \nMSE Fractional \nOrder ( α) τ ξ \n1.1 0.469965 0.984699 \n1.2 0.69043 0.673145 \n1.3 0.760275 0.595913 \n1.4 0.819033 0.531421 \n1.5 0.898188 0.387535 \n1.6 0.937233 0.314444 \n1.7 0.966328 0.242967 \n1.8 1.009048 0.130724 with pseudo-\ndamping 6.1938×10 -4 \n1.9 0.986992 0.061456 \n1.1 0.400597 1.053798 \n1.2 0.495297 0.751861 \n1.3 0.576102 0.624702 \n1.4 0.639898 0.5336 \n1.5 0.688546 0.46988 \n1.6 0.7352 0.4091 \n1.7 0.779042 0.350727 \n1.8 0.8253 0.2913 First order \nwith meta-\ndamping 9.6422×10 -5 \n1.9 0.916861 0.207589 \n1.1 1.091173 0.707394 \n1.2 1.088813 0.573404 \n1.3 1.076325 0.4878 \n1.4 1.062211 0.402436 \n1.5 1.040513 0.339679 \n1.6 1.019453 0.278533 \n1.7 1.012378 0.208497 \n1.8 1.006116 0.13578 Second order \nwith meta-\ndamping 1.4428×10 -5 \n1.9 1.002982 0.070353 \n \nIn IV-VI the ANN with 5 neurons in the single hidde n layer \nwith logsig  activation function has been found to produce \nconsistently good prediction of the optimal FO damp ing in \nterms of minimum average MSE for 25 runs. Subsequen t \nincrease in number of layers or different combinati ons of \nactivation functions may result in lower MSE in som e \nparticular cases. But considering the consistency o f ANNs, \nthese complicated structures have been found to pro duce \nalways a higher value of average MSE. Table VII rep orts the \npredicted values and MSE of three different classes  of FO \nsystems for different fractional orders while consi dering single \nhidden layer containing 5 neurons and logsig  activation \nfunction. The simulations and results justify the a rgument that \ncomplicated ANN topology may establish arbitrary ma pping \nbut the consistency of such mapping reduces with la rge size of \nthe network [4]-[5]. Time response curves for the s ystems with \nthe predicted pseudo-meta damping and time constant s \ncorresponding to that presented Table VII and the G A based \noptimum results in Tables I-III has been shown in F ig. 7-9 for \nstep input excitation. In Fig 7-9 it is clear that the ANN based \npredicted results (dashed lines) are very close to the GA based \noptimum values (continuous lines). The prediction i s very accurate for first and second order meta-damping th an the \npseudo-damping and also for low value of fractional  order ( α). \nIn contemporary literatures like [7]-[9], the conce pt of \npseudo-damping was first proposed with a different second \norder like structure of the FO systems with commens urable \norders. In the present paper, pseudo-damping refers  to the \ndamping introduced in the system with decrease in i ts leading \norder which is different from that reported in [6]- [9]. The \ncontribution of the present paper is firstly to giv e systematic \ndefinition of pseudo and meta-damping in FO systems  and \nsecondly their ANN based prediction from the GA bas ed ITSE \noptimum results. \nV.  CONCLUSION  \nIn this paper, the concepts of pseudo-damping and m eta-\ndamping are introduced for some special class of fr actional \norder dynamical systems. Genetic algorithm is used to obtain \nthe equivalent second order damping characteristics  of these \nFO systems. An ANN based approach is used next to m odel \nthis arbitrary (nonlinear) relationship and elimina te the \nrequirement of running the computationally intensiv e GA every \ntime. Extensive parametric study have been done to find out the \nmultilayer feed-forward ANN architecture that is ca pable to \noptimally capture the nonlinearity, by minimizing t he MSE and \nsimultaneously avoiding the pitfall of over-fitting . Consistency \nof ANN structures for mapping the fractional order to optimal \nFO damping and time constants are judged by conside ring \naverage MSE of 25 independent runs. The time domain  \ncomparison of the GA based optimum results with the  ANN \nbased predicted results is found to be very close. \nREFERENCES  \n[1]  Igor Podlubny, “Fractional order differential equat ions”, Academic \nPress, 1999. \n[2]  Shantanu Das, “Functional fractional calculus”, Spr inger-Verlag, 2011. \n[3]  Katsuhiko Ogata, “Modern control engineering”, Pren tice-Hall, 1997. \n[4]  Moshe Leshno, Vladimir Ya. Lin, Allan Pinkus, and Sh imon Schocken, \n“Multilayer feedforward networks with a nonpolynomi al activation \nfunction can approximate any function”, Neural Networks , vol. 6, no. 6, \npp. 861-867, 1993. \n[5]  Franco Scarselli and Ah Chung Tsoi, “Universal Appro ximation Using \nFeedforward Neural Networks: A Survey of Some Existi ng Methods, \nand Some New Results”, Neural Networks , vol. 11, no. 1, pp. 15-37, Jan. \n1998. \n[6]  Tom T. Hartley and Carl F. Lorenzo, “A frequency-domai n approach to \noptimal fractional-order damping”, Nonlinear Dynamics , vol. 38, no. 1-\n4, pp. 69-84, 2004. \n[7]  Rachid Malti, Xavier Moreau, Firas Khemane, and Ala in Oustaloup, \n“Stability and resonance conditions of elementary f ractional transfer \nfunctions”, Automatica , vol. 47, no. 11, pp. 2462-2467, Nov. 2011. \n[8]  Rachid Malti, Xavier Moreau, and Firas Khemane, “Re sonance and \nstability conditions for fractional transfer functi ons of the second kind”, \nIn: Dumitru Baleanu, Ziya Burhanettin Guvenc, and J.A.  Tenreiro \nMachado, New Trends in Nanotechnology and Fractional  Calculus \nApplications , Springer-Verlag, pp. 429-444, 2010.  \n[9]  Jocelyn Sabatier, Olivier Cois, and Alain Oustaloup , “Modal placement \ncontrol method for fractional systems: application to a testing bench”, \nProceedings of DETC ’03, ASME 2003 Design Engineering Te chnical \nConferences and Computers and Informatics in Enginee ring \nConference , Sept. 2003, Chicago, Illinois, USA. \n "    },    {        "title": "1208.1462v1.Observation_of_Coherent_Helimagnons_and_Gilbert_damping_in_an_Itinerant_Magnet.pdf",        "content": "arXiv:1208.1462v1  [cond-mat.str-el]  7 Aug 2012ObservationofCoherent HelimagnonsandGilbertdamping in anItinerant Magnet\nJ. D. Koralek1,∗,†, D. Meier2,∗,†, J. P. Hinton1,2, A. Bauer3, S. A. Parameswaran2,\nA. Vishwanath2, R. Ramesh1,2R. W. Schoenlein1, C. Pfleiderer3, J. Orenstein1,2\n1Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA\n2Department of Physics, University of California, Berkeley , California 94720, USA and\n3Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n(Dated: DatedAugust 30, 2018)\nWe study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy\nto Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse\nto lower frequency as the applied magnetic field is increased ; the fingerprint of collective modes unique to\nhelimagnets,knownashelimagnons. Theuseoftime-resolve dspectroscopyallowsustoaddressthefundamen-\ntal magnetic relaxation processes by directly measuring th e Gilbert damping, revealing the versatility of spin\ndynamics inchiralmagnets.\nTheconceptofchiralitypervadesallofscience,havingpro -\nfound implications in physics, chemistry and biology alike .\nIn solids, relativistic spin-orbit coupling can give rise t o the\nDzyaloshinskii-Moriya (DM) interaction,2,3imparting a ten-\ndency for the electron spins to form helical textures with a\nwell-definedhandednessincrystalslackinginversionsymm e-\ntry. Helical spinorderisespeciallyinterestingwhenthem ag-\nnetismarisesfromthesameelectronsresponsibleforcondu c-\ntion as is the case in doped FeSi which displays unconven-\ntional magnetoresitence,4,5helimagnetism,6and the recently\ndiscovered Skyrmion lattice.7,8The excitations of helimag-\nnets have been studied over the past 30 years, culminating\nrecently in a comprehensive theory of spin excitations call ed\nhelimagnons.9,10Signatures of helimagnons have been ob-\nserved in neutron scattering11and microwave absorption,12\nyet little is known about their magnetodynamics and relax-\nation phenomenaon the sub-picosecondtimescales on which\nmagnetic interactions occur. Understanding the dynamics,\nhowever,isofgreatimportanceregardingspintransfertor que\neffects in chiral magnets, and related proposed spintronic s\napplications.13–15\nIn this work we study the dynamics of collective spin ex-\ncitationsin the itineranthelimagnetFe 0.8Co0.2Si. Ouroptical\npump-probemeasurementsidentifyanomalousmodesatzero\nwavevector ( q=0) which we identify unmistakably as heli-\nmagnons. These helimagnons manifest as strongly damped\nmagnetization oscillations that follow a characteristic s caling\nrelation with respect to temperature and magnetic field. The\nsub-picosecond time resolution of our technique enables de -\nterminationof the intrinsic Gilbert dampingparameterwhi ch\nis foundto be oneorderof magnitudelargerthan in localized\nsystems, revealing the versatility of the spin-lattice int erac-\ntions available in the emergent class of DM-driven helimag-\nnets.\nDespite being a non-magnetic insulator, FeSi is trans-\nformed into an itinerant magnet upon doping with cobalt.4,16\nWe have chosen Fe 0.8Co0.2Si for our study because it can\neasily be prepared in high quality single crystals17with a\nreasonably high magnetic ordering temperature TN, and its\nexotic equilibrium properties are well characterized, ope n-\ning the door for non-equilibrium dynamical studies. Small-\nangle neutron scattering8was used to determine the phase\ndiagram and has revealed helimagnetic spin textures belowTN=30 K that emerge from the interplay between the fer-\nromagnetic exchange and DM interactions. In zero magnetic\nfield the spins form a proper helix with a spatial period of\n≈350˚A,18whereasfinitefieldscantthespinsalongthehelix\nwavevector, kh, (see Fig. 1(c))inducinga conicalstate witha\nnet magnetization. Sufficiently high fields, H≥Hc, suppress\nthe conical order in favor of field alignment of all spins. In\ntheexperimentsreportedhere,femtosecondpulsesoflinea rly\npolarized 1.5 eV photons from a Ti:Sapphire oscillator were\nused to excite a (100) oriented single crystal of Fe 0.8Co0.2Si\nat near normal incidence. The changes induced in the sam-\nple by the pump pulse were probed by monitoringthe reflec-\ntion and Kerr rotation of time-delayed probe pulses from the\nsame laser. In order to minimize laser heating of the sam-\nple the laser repetition rate was reduced to 20 MHz with an\nelectro-optic pulse picker, ensuring that thermal equilib rium\nwasreachedbetweensuccessivepumppulses. Signaltonoise\nwas improved by modulation of the pump beam at 100 KHz\nandsynchronouslock-indetectionofthereflectedprobe. Ke rr\nrotation was measured using a Wollaston prism and balanced\nphotodiode. All temperature and field scans presented in thi s\nwork were performed from low to high TandH||(100)after\nzero-fieldcooling.\nFig. 1 shows the transient reflectivity, ΔR/R, as a function\nof temperature and magnetic field. At high temperature we\nobserve a typical bolometric response from transient heati ng\nof the sample by the pump pulse (Fig 1 (a)).19This is char-\nacterized by a rapid increase in reflectivity, followed by tw o-\ncomponent decay on the fs and ps timescales, corresponding\nto the thermalizationtimes between differentdegreesof fr ee-\ndom (electron, spin, lattice, etc.).20As the sample is cooled\nbelowTN, the small thermal signal is beset by a much larger\nnegative reflectivity transient (Fig. 1 (b)) with a decay tim e\nof roughly τR≈175 ps at low temperature (Fig. 3 (b)). A\nnatural explanation for this is that the pump pulse weakens\nthe magnetic order below TN, which in turn causes a change\nin reflectivity via the resulting shift of spectral weight to low\nenergy.21Thetemperaturedependenceof the peak ΔR/Rval-\nues is plotted in Fig. 1 (c) for several applied fields, showin g\nonlyweakfielddependence.\nToaccessthemagnetizationdynamicsmoredirectlywean-\nalyze the polarizationstate of the probe pulses, which rota tes\nby an angle θKupon reflection from the sample surface, in2\nFIG.1: Timedependence ofthepump-inducedtransientreflec tivityΔR/Rinthe(a)paramagneticand(b)helimagneticstates. Thetem perature\ndependence of the maximum ΔR/Ris plottedin(c)for several applied magnetic fields.\nproportion to the component of the magnetization along the\nlight trajectory. The change in Kerr rotation induced by the\noptical pump, ΔθK, is shown in Fig. 2 as a function of tem-\nperature and field. The upper panels show temperature scans\nat fixedmagneticfield,while afieldscan atfixedtemperature\nisshowninpanel(d). Weobservethat ΔθKchangessignas H\nisreversed(notshown),andgoestozeroas Hgoesto zeroor\nas temperatureis raised above TN. Oscillationsof the magne-\ntization are clearly visible in the raw data below 25 K in the\nhelimagneticphase.\nIn order to analyze the magnetization dynamics, we use a\nsimple phenomenological function that separates the oscil la-\ntory and non-oscillatorycomponentsseen in the data. It con -\nsists ofadecayingsinusoidaloscillation,\nΔθK=e−t\nτK[A+Bsin(ωt)] (1)\nwitha timedependentfrequency,\nω(t)=2πf0/bracketleftBig\n1+0.8/parenleftBig\ne−t\nτK/parenrightBig/bracketrightBig\n(2)\nwhich decays to a final value ω0. We emphasize that there is\nonly a single decay time τKdescribing the magneto dynam-\nics, and it is directly related to the Gilbert damping parame -\nterα=(2πf0τK)−1. This function produces excellent fits to\nthe data as illustrated in Fig. 3 (a), allowing accurate extr ac-\ntion of the oscillation frequencies and decay times shown in\nFigs. 3 (b)-(d). The oscillation frequency is reduced as ei-\nther field or temperature is increased, while the decay time\nτKis roughly constant and equal to τRbelow 20 K. As the\ntemperature is raised towards the phase transition, the rel ax-\nation time τKdiverges, which can be understood in terms of\na diverging magnetic correlation length due to the presence\nof a critical point. The similarity between the decay times τR\nandτKwithin the ordered phase reflects strongly correlated\ncharge and spin degrees of freedom, and supports the notion\nthatΔR/Risdeterminedbythemagneticorder.\nThemagneticoscillationfrequencyreaches f0≈4.8GHzat\nlowtemperature,whichcorrespondstoaLarmorprecessiono f\nspinssubjectedtoafieldof170mT,whichisroughlythecrit-\nical field Hcrequiredto destroy the spin helix. This, togetherwith the fact that the oscillation frequencyis nonzero only in\nthehelicalstate,suggeststhattheoscillationsarecomin gfrom\nexcitations unique to the helical structure. It is well know n\nthat magnetization oscillations can be optically induced b y\nultrafast generation of coherent magnons,24–26however, or-\ndinary magnons cannot explain our data as their frequency\nwouldincreasewith H,oppositetowhatisseenin Fig. 3(c).\nBased on these observations, we propose the following in-\nterpretation of our results: In the helical magnetic phase, the\npump photons weaken the magnetic order through the ultra-\nfast demagnetization process.27As described above, this re-\nduction in magnetic order gives rise to a decrease in the re-\nflectivity at 1.5 eV which is nearly field independent. As a\nmagnetic field is applied the spins become canted along the\nhelix wavevector,giving rise to a macroscopic magnetizati on\nwhichweobserveinKerrrotationviaitscomponentalongthe\nprobelight trajectory. The demagnetizationfromthe pumpi s\nresponsible for the initial peak seen in the ΔθKtime traces,\nand is captured by the exponential component of our fitting\nfunction (green curve in Fig 3 (a)). The pump photons also\nlaunch a coherentspin wave, giving rise to the oscillations in\nΔθK(red curve in Fig. 3 (a)). The form of the oscillatory\ncomponent goes like sin (ωt)rather than [1−cos(ωt)], sug-\ngesting impulsive stimulated Raman scattering as the mecha -\nnism of excitation.25The anomalousfield dependenceshown\nin Fig. 3 (c) leads to the unambiguous conclusion that the\noptically excited spin waves are the fundamental modes of\nhelimagnetstermedhelimagnons.10Specifically,theoptically\naccessible helimagnon mode consists of the constituents of\nthe spin helix precessing in-phase about their local effect ive\nfield. Since this local effective field is reduced during the u l-\ntrafast demagnetizationprocess, the oscillation frequen cyde-\ncreases as a function of time delay as the field recovers, ne-\ncessitatingthetimedependentfrequencyinEq. 1. Theabili ty\nto resolve helimagnons with femtosecond time resolution at\nq=0isuniquetoouropticalprobe,andcomplimentsneutron\nscatteringwhichisrestrictedtomappinghelimagnonbands at\nhigherq. This regionof reciprocalspace is particularlyinter-\nestinginthecaseofhelimagnetsastheperiodicityintrodu ced\nbythehelicalspintexturegeneratesbandsthatarecentere dat\nq=±khand therefore have finite frequency modes at q=03\nFIG. 2: (a),(b),(c) Time dependence of the pump-induced cha nge in Kerr rotation, ΔθK, as a function of temperature for several applied\nmagneticfields. (d) ΔθKasafunctionofmagneticfieldforseveraltemperatures. Cur vesareoffsetforclarity. Alsoshownisaschematicphase\ndiagram, adapted from Reference 8,withredarrows illustra tingthe temperature and fieldscans usedin(a)-(d).\neven in the absence of a gap. This is in contrast to ordinary\nmagnonsinwhichthebandsaregenerallycenteredat q=0so\nthattheassociatedmodehaszerofrequency. Wenotethatour\nobservationsare in agreementwith previouswork on the col-\nlectivemodesofskyrmions28whichcoexistwithhelimagnons\nin theA-Phase(see Fig. 3).12Theappearanceofthese modes\nis not expected in our data as their corresponding oscillati on\nperiodsexceedtheobserveddampingtimein Fe 0.8Co0.2Si.\nInordertoquantitativelytestthehelimagnoninterpretat ion\nwetaketheexpressionforthe q=0helimagnonfrequencyin\nanexternalmagneticfield,\nf0=gµBHc/radicalbig\n1+cos2θ (3)\nwheregistheeffectiveelectron g-factor,µBistheBohrmag-\nneton,andπ\n2−θistheconicalanglei.e. theamountthespins\nare canted away from khby the applied field H. Ignoringde-\nmagnetization effects of the spin waves themselves, we can\nwrite sinθ=H\nHc, whereHcis the critical field at which the\nspinsall alignwiththe field andthe helimagnonceasesto ex-\nist asa well-definedmode. Thenweobtain,9\nf0=gµBHc/radicalBigg\n1−1\n2/parenleftbiggH\nHc/parenrightbigg2\n(4)\nwhich expresses the magnon frequency as a function of ap-\nplied field. This expression fits the data remarkably well as\nshown in Fig. 3 (c), capturingthe decrease in frequencywith\nincreasing Hwhich is unique to helimagnons. However, due\ntothefactthattheoscillationperiodexceedsthedampingt imeforfieldsabove75mT,itisnotpossibletoextractthevalueo f\nthecritical field Hcinthis system. The solidline in Fig. 3(d)\nisafittotheform f0∝/radicalBig\n1−T\nTNwhichgives TNasafunction\nofHin reasonableagreementwithpublisheddata.8\nThe Gilbert damping parameter can be directly obtained\nfrom the measured decay times through the relation α=\n(2πf0τK)−1, which gives a value of α≈0.4 for the heli-\nmagnetic phase of Fe 0.8Co0.2Si. This is an order of magni-\ntude larger than what was seen in insulating Cu 2OSeO3,12\nwhere helimagnetism arises from localized rather than itin -\nerant spins. The contrast in dynamics between these systems\nis critical in the context of potential spintronic applicat ions\nbasedonhelimagnetismwherethereisatradeoffbetweenfas t\nswitching which requires large damping, and stability whic h\nreliesonlowdamping.\nIn summary, this work demonstrates ultrafast coherent op-\ntical excitation of spin waves in an itinerant DM-driven spi n\nsystem and reveals the underlying spin dynamics. We iden-\ntifytheseexcitationsashelimagnonsthroughtheiranomal ous\nfield dependence and explain our observations with a com-\nprehensive model. Our experiments directly yield the intri n-\nsic Gilbertdampingparameter,revealingastrikingdiffer ence\nin spin relaxationphenomenabetweenitinerant andlocaliz ed\nhelimagnets. The results elucidate the dynamicsof collect ive\nmodes common to the actively studied B20 transition metal\ncompounds that codetermine their performance in potential\nspinbasedapplications.\nAcknowledgments: The work in Berkeley was supported\nby the Director, Office of Science, Office of Basic Energy4\nFIG.3: (a) Exemplary ΔθKoscillation data (blue circles) and fit (black line) using th e model described inthe text. The fitis decomposed into\nan exponential term (green curve) and an oscillatory term (r ed curve). The fitting function uses a single time constant τKfor all terms which\nis plotted in panel (b) as a function of temperature and field. For comparison we also plot the decay time of the reflectivity ,τR, averaged over\nall fields. The solid lines are guides to the eye. Panels (c) an d (d) show the reduced magnetization oscillation frequency for field scans and\ntemperature scans respectively, andsolidlines are fitstot he data as described inthe maintext.\nSciences,MaterialsSciencesandEngineeringDivision,of the\nU.S. Department of Energy under Contract No. DE-AC02-\n05CH11231. C.P. and A.B. acknowledge support through\nDFGTRR80(FromElectronicCorrelationstoFunctionality) ,\nDFG FOR960 (Quantum Phase Transitions), and ERC AdG\n(291079, TOPFIT). A.B. acknowledges financial support\nthrough the TUM graduate school. D.M. acknowledges sup-portfromtheAlexandervonHumboldtfoundationandS.A.P.\nacknowledgessupportfrom the SimonsFoundation. C.P. and\nA.B. also thank S. Mayr, W. M¨ unzer, and A. Neubauer for\nassistance.\n∗Theseauthorscontributedequallytothiswork.\n†Email address: jdkoralek@lbl.gov and meier@berkeley.edu\n2I. E.Dzyaloshinskii, Sov. Phys.JETP 5, 1259 (1957).\n3T. Moriya, Phys. Rev. 120, 91(1960).\n4N. Manyala etal., Nature404, 581 (2000).\n5N. 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Kapeliovich and T. L. Perelman, Sov. Phy s.\nJETP39, 375 (1975).\n21F.P.Mena etal.,Phys. Rev. B 73, 085205 (2006).\n22S.A.Brazovskii,S.G.Dmitriev,Sov.Phys.JETP 42,497(1976).\n23M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii ,\nP.B¨ oni, andC.Pfleiderer,arXiv:1205.4780v1(2012).\n24M. van Kampen etal.,Phys. Rev. Lett. 88, 227201 (2002).\n25A. M.Kalashnikova et al.,Phys.Rev. B 78, 104301 (2008).\n26D. Talbayev et al.,Phys. Rev. Lett. 101, 097603 (2008).\n27A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n28M. Mochizuk, Phys. Rev. Lett. 108, 017601 (2012)."    },    {        "title": "1209.3120v1.Skyrmion_Dynamics_in_Multiferroic_Insulator.pdf",        "content": "arXiv:1209.3120v1  [cond-mat.str-el]  14 Sep 2012Skyrmion Dynamics in Multiferroic Insulator\nYe-Hua Liu,1You-Quan Li,1and Jung Hoon Han2,3,∗\n1Zhejiang Institute of Modern Physics and Department of Physi cs,\nZhejiang University, Hangzhou 310027, People’s Republic of China\n2Department of Physics and BK21 Physics Research Division,\nSungkyunkwan University, Suwon 440-746, Korea\n3Asia Pacific Center for Theoretical Physics, Pohang, Gyeong buk 790-784, Korea\n(Dated: October 31, 2018)\nRecent discovery of Skyrmion crystal phase in insulating mu ltiferroic compound Cu 2OSeO 3calls\nfor new ways and ideas to manipulate the Skyrmions in the abse nce of spin transfer torque from\nthe conduction electrons. It is shown here that the position -dependent electric field, pointed along\nthe direction of the average induced dipole moment of the Sky rmion, can induce the Hall motion\nof Skyrmion with its velocity orthogonal to the field gradien t. Finite Gilbert damping produces\nlongitudinal motion. We find a rich variety of resonance mode s excited by a.c. electric field.\nPACS numbers: 75.85.+t, 75.70.Kw, 76.50.+g\nSkyrmions are increasingly becoming commonplace\nsightings among spiral magnets including the metallic\nB20 compounds[1–5] and most recently, in a multiferroic\ninsulator Cu 2OSeO3[6]. Both species of compounds dis-\nplaysimilarthickness-dependentphasediagrams[5,6] de-\nspitetheircompletelydifferentelectricalproperties,high-\nlighting the generality of the Skyrmion phase in spiral\nmagnets. Along with the ubiquity of Skyrmion matter\ncomes the challenge of finding means to control and ma-\nnipulatethem, inadevice-orientedmannerakintoefforts\nin spintronics community to control the domain wall and\nvortex motion by electrical current. Spin transfer torque\n(STT) is a powerful means to induce fast domain wall\nmotion in metallic magnets[7, 8]. Indeed, current-driven\nSkyrmion rotation[9] and collective drift[10], originating\nfrom STT, have been demonstrated in the case of spiral\nmagnets. Theory of current-induced Skyrmion dynam-\nics has been worked out in Refs. [11, 12]. In insulating\ncompounds such as Cu 2OSeO3, however, the STT-driven\nmechanism does not work due to the lack of conduction\nelectrons.\nAs with other magnetically driven multiferroic\ncompounds[13], spiral magnetic order in Cu 2OSeO3is\naccompanied by finite electric dipole moment. Recent\nwork by Seki et al.[14] further confirmed the mecha-\nnism of electric dipole moment induction in Cu 2OSeO3\nto be the so-called pd-hybridization[15–17]. In short, the\npd-hybridization mechanism claims the dipole moment\nPijfor every oxygen-TM(transition metal) bond propor-\ntional to ( Si·ˆeij)2ˆeijwhereiandjstand for TM and\noxygen sites, respectively, and ˆ eijis the unit vector con-\nnecting them. Carefully summing up the contributions\nof such terms over a unit cell consisting of many TM-\nO bonds, Seki et al.were able to deduce the dipole\nmoment distribution associated with a given Skyrmionic\nspin configuration[14]. It is interesting to note that the\nnumerical procedure performed by Seki et al.is pre-\ncisely the coarse-graining procedure which, in the text-book sense of statistical mechanics, is tantamount to the\nGinzburg-Landau theory of order parameters. Indeed we\ncan show that Seki et al.’s result for the dipole moment\ndistribution is faithfully reproduced by the assumption\nthat the local dipole moment Piis related to the local\nmagnetization Siby\nPi=λ(Sy\niSz\ni,Sz\niSx\ni,Sx\niSy\ni) (1)\nwith some coupling λ. A similar expression was pro-\nposed earlier in Refs. [18, 19] as the GL theory\nof Ba 2CoGe2O7[20], another known pd-hybridization-\noriginated multiferroic material with cubic crystal struc-\nture. Each site icorresponds to one cubic unit cell of\nCu2OSeO3with linear dimension a∼8.9˚A, and we have\nnormalized Sito have unit magnitude. The dimension of\nthe coupling constant is therefore [ λ] = C·m.\nHaving obtained the proper coupling between dipole\nmoment and the magnetizaiton vector in Cu 2OSeO3one\ncan readily proceed to study the spin dynamics by solv-\ning Landau-Lifshitz-Gilbert (LLG) equation. Very small\nvalues of Gilbert damping parameter are assumed in the\nsimulation as we are dealing with an insulating magnet.\nA new, critical element in the simulation is the term aris-\ning from the dipolar coupling\nHME=−/summationdisplay\niPi·Ei=−λ\n2/summationdisplay\niSi\n0Ez\niEy\ni\nEz\ni0Ex\ni\nEy\niEx\ni0\nSi,(2)\nwhere we have used the magneto-electric coupling ex-\npression in Eq. (1). In essence this is a field-dependent\n(voltage-dependent) magnetic anisotropy term. The to-\ntal Hamiltonian for spin is given by H=HHDM+\nHME, whereHHDMconsists of the Heisenberg and the\nDzyaloshinskii-Moriya (DM) exchange and a Zeeman\nfield term. Earlier theoretical studies showed HHDMto\nstabilize the Skyrmion phase[1, 21–24].2\nTwo field orientations can be chosen independently in\nexperiments performed on insulating magnets. First, the\ndirection of magnetic field Bdetermines the plane, or-\nthogonal to B, in which Skyrmions form. Second, the\nelectric field Ecan be applied to couple to the induced\ndipole moment of the Skyrmion and used as a “knob” to\nmove it around. Three field directions used in Ref. [14]\nand the induced dipole moment in eachcase areclassified\nas (I)B/bardbl[001],P= 0, (II) B/bardbl[110],P/bardbl[001], and\n(III)B/bardbl[111],P/bardbl[111]. One can rotate the spin axis\nappearing in Eq. (1) accordingly so that the z-direction\ncoincides with the magnetic field orientation in a given\nsetup and the x-direction with the crystallographic[ 110].\nIn each of the cases listed above we obtain the magneto-\nelectric coupling, after the rotation,\nH(I)\nME=−λ\n2/summationdisplay\niEi([Sy\ni]2−[Sx\ni]2),\nH(II)\nME=−λ\n2/summationdisplay\niEi([Sz\ni]2−[Sx\ni]2),\nH(III)\nME=−λ\n2√\n3/summationdisplay\niEi(3[Sz\ni]2−1).(3)\nIn cases (II) and (III) the E-field is chosen parallel to the\ninduced dipole moment P,Ei=EiˆP, to maximize the\neffect of dipolar coupling. In case (I) where there is no\nnet dipole moment for Skyrmions we chose E/bardbl[001] to\narrive at a simple magneto-electric coupling form shown\nabove.\nSuppose now that the E-field variation is sufficiently\nslow on the scale of the lattice constant ato allow the\nwriting of the continuum energy,\nHME=−λd\na/integraldisplay\nd2rE(r)ρD(r). (4)\nIt is assumed that all variables behave identically along\nthe thickness direction, oflength d. The “dipolarcharge”\ndensityρD(r)couplestotheelectricfield E(r)inthesame\nwayasthe conventionalelectric chargedoes tothe poten-\ntial field in electromagnetism. The analogy is also useful\nin thinking about the Skyrmion dynamics under the spa-\ntially varying E-field as we will show. The continuum\nform of dipolar charge density in Eq. 4 is\nρ(I)\nD(ri) =1\n2a2([Sy\ni]2−[Sx\ni]2),\nρ(II)\nD(ri) =1\n2a2([Sz\ni]2−[Sx\ni]2−1),\nρ(III)\nD(ri) =√\n3\n2a2([Sz\ni]2−1). (5)\nDivision by the unit cell area a2ensures that ρD(r) has\nthe dimension of areal density. Values for the ferromag-\nnetic case, Sz\ni= 1, is subtracted in writing down the def-\ninition (5) in order to isolate the motion of the Skyrmion\nFIG. 1: (color online) (a) Skyrmion configuration and (b)-(d )\nthe corresponding distribution of dipolar charge density f or\nthree magnetic field orientations as in Ref. 14. (b) B/bardbl[001]\n(c)B/bardbl[110] (d) B/bardbl[111]. For each case, electric field is\nchosen as E/bardbl[001],E/bardbl[001] and E/bardbl[111], respectively. See\ntext for the definition of dipolar charge density. As schemat i-\ncally depicted in (a), the Skyrmion executes a Hall motion in\nresponse to electric field gradient.\nrelative to the ferromagnetic background. Due to the\nsubtraction, the dipolar charge is no longer equivalent to\nthe dipole moment of the Skyrmion. The distribution of\ndipolar charge density for the Skyrmion spin configura-\ntion in the three cases are plotted in Fig. 1. In case (I)\nthe total dipolar charge is zero. In cases (II) and (III)\nthe net dipolar charges are both negative with the re-\nlation,Q(II)\nD/Q(III)\nD=√\n3/2, where QD, of order unity,\nis obtained by integrating ρD(r) over the space of one\nSkyrmion and divide the result by the number of spins\nNSkinside the Skyrmion. If the field variation is slow on\nthe scale of the Skyrmion, then the point-particle limit is\nreached by writing ρD(r) =QDNSk/summationtext\njδ(r−rj) whererj\nspans the Skyrmion positions, and identical charge QDis\nassumed for all the Skyrmions. We arrive at the “poten-\ntial energy” of the collection of Skyrmion particles,\nHME=−λQDNSkd\na/summationdisplay\njE(rj). (6)\nA force acting on the Skyrmion will be given as the gra-\ndientFi=−∇iHME. Inter-Skyrmion interaction is ig-\nnored.\nThe response of Skyrmions to a given force, on the\nother hand, is that of an electric charge in strong\nmagnetic field, embodied in the Berry phase action3\n(−2πS¯hQSkd/a3)/summationtext\nj/integraltext\ndt(rj×˙rj)·ˆz, whereQSkis the\nquantized Skyrmion charge[12, 25], and Sis the size of\nspin. Equation of motion follows from the combination\nof the Berry phase action and Eq. (6),\nvj=λ\n4πS¯ha2NSkQD\nQSkˆz×∇jE(rj), (7)\nwherevjis thej-th Skyrmion velocity. Typical Hall ve-\nlocity can be estimated by replacing |∇E|with ∆E/lSk,\nwhere ∆Eis the difference in the field strength between\nthe left and the right edge of the Skyrmion and lSkis its\ndiameter. Taking a2NSk∼l2\nSkwe find the velocity\nλlSk\n4πS¯h∆E∼10−6∆E[m2/V·s], (8)\nwhich gives the estimated drift velocity of 1 mm/s for the\nfield strength difference of 103V/m across the Skyrmion.\nExperimental input parameters of lSk= 10−7m, and\nλ= 10−32C·m were taken from Ref. [14] in arriving at\nthe estimation, as well as the dipolar and the Skyrmion\nchargesQD≈ −1 andQSk=−1. We may estimate the\nmaximum allowed drift velocity by equating the dipolar\nenergy difference λ∆Eacross the Skyrmion to the ex-\nchange energy J, also corresponding to the formation en-\nergy of one Skyrmion[24]. The maximum expected veloc-\nity thus obtained is enormous, ∼104m/s forJ∼1meV,\nimplying that with the right engineering one can achieve\nrather high Hall velocity of the Skyrmion. In an encour-\naging step forward, electric field control of the Skyrmion\nlattice orientation in the Cu 2OSeO3crystal was recently\ndemonstrated[26].\nResults of LLG simulation is discussed next. To start,\na sinusoidal field configuration Ei=E0sin(2πxi/Lx) is\nimposed on a rectangular Lx×Lysimulation lattice with\nLxmuch larger than the Skyrmion size. In the absence\nof Gilbert damping, a single Skyrmion placed in such\nan environment moved along the “equi-potential line” in\nthey-direction as expected from the guiding-center dy-\nnamics of Eq. (7). In cases (II) and (III) where the\ndipolar charges are nonzero the velocity of the Skyrmion\ndrift is found to be proportional to their respective dipo-\nlar charges QDas shown in Fig. 2. The drift velocity\ndecreased continuously as we reduced the field gradient,\nobeying the relation (7) down to the zero velocity limit.\nThe dipolar charge is zero in case (I), and indeed the\nSkyrmion remains stationary for sufficiently smooth E-\nfield gradient. Even for this case, Skyrmions can move\nfor field gradient modulations taking place on the length\nscale comparable to the Skyrmion radius, for the reason\nthat the forces acting on the positive dipolar charge den-\nsity blobs (red in Fig. 1(a)) are not completely canceled\nby those on the negative dipolar charge density blobs\n(blue in Fig. 1(a)) for sufficiently rapid variations of thefield strength gradient. A small but non-zero drift veloc-\nityensues, asshowninFig. 2. Longitudinalmotionalong\nthe field gradient begins to develop with finite Gilbert\ndamping, driving the Skyrmion center to the position of\nlowest “potential energy” E(r). For the Skyrmion lat-\ntice, imposing a uniform field gradient across the whole\nlattice may be too demanding experimentally, unless the\nmagnetic crystal is cut in the form of a narrow strip the\nwidth of which is comparable to a few Skyrmion radii.\nIn this case we indeed observe the constant drift of the\nSkyrmions along the length of the strip in response to\nthe field gradient across it. The drift speed is still pro-\nportional to the field gradient, but about an order of\nmagnitude less than that of an isolated Skyrmion under\nthe same field gradient. We observed the excitation of\nbreathing modes of Skyrmions when subject to a field\ngradient, and speculate that such breathing mode may\ninterfere with the drift motion as the Skyrmions become\nclosed-packed.\n0 500 1000 1500 2000−30−25−20−15−10−505\nty(ab. unit)\n  \nv(I)\nHall=−7×10−4\nv(II)\nHall=−1.32×10−2\nv(III)\nHall=−1.49×10−2\nv(II)\nHall\nv(III)\nHall≈Q(II)\nD\nQ(III)\nD=√\n3\n2case (I)  : B||[001]\ncase (II) : B||[110]\ncase (III): B||[111]\nFIG. 2: (color online) Skyrmion position versus time for cas es\n(I) through (III) for sinusoidal electric field modulation ( see\ntext) with the Skyrmion center placed at the maximum field\ngradient position. The average Hall velocities (in arbitra ry\nunits) in cases (II) and (III) indicated in the figure are ap-\nproximately proportional to the respective dipolar charge s, in\nagreement with Eq. (7). A small velocity remains in case\n(I) due to imperfect cancelation of forces across the dipola r\ncharge profile.\nSeveral movie files are included in the supplementary\ninformation. II.gif and III.gif give Skyrmion motion for\nEi=E0sin(2πxi/Lx) onLx×Ly= 66×66 lattice for\nmagneto-electric couplings (II) and (III) in Eq. (3). III-\nGilbert.gif gives the same E-field as III.gif, with finite\nGilbert damping α= 0.2. I.gif describes the case (I)\nwhere the average dipolar charge is zero, with a rapidly\nvarying electric field Ei=E0sin(2πxi/λx) andλxcom-\nparable to the Skyrmion radius. The case of a narrow\nstrip with the field gradient across is shown in strip.gif.\nMochizuki’s recent simulation[27] revealed that inter-4\nnal motion of Skyrmions can be excited with the uniform\na.c. magnetic field. Some of his predictions were con-\nfirmed by the recent microwave measurement[29]. Here\nwe show that uniform a.c. electric field can also ex-\ncite several internal modes due to the magneto-electric\ncoupling. Time-localized electric field pulse was applied\nin the LLG simulation and the temporal response χ(t)\nwas Fourier analyzed, where the response function χ(t)\nrefersto(1 /2)/summationtext\ni([Sy\ni(t)]2−[Sx\ni(t)]2), (1/2)/summationtext\ni([Sz\ni(t)]2−\n[Sx\ni(t)]2), and (√\n3/2)/summationtext\ni[Sz\ni(t)]2for cases (I) through\n(III), respectively. (In Mochizuki’s work, the response\nfunction was the component of total spin along the a.c.\nmagnetic field direction.)\nIn case (I), the uniform electric field perturbs the ini-\ntial cylindrical symmetry ofSkyrmion spin profileso that/summationtext\ni([Sx\ni(t)]2−[Sy\ni(t)]2) becomes non-zero and the over-\nall shape becomes elliptical. The axes of the ellipse then\nrotates counter-clockwise about the Skyrmion center of\nmass as illustrated in supplementary figure, E-mode.gif.\nThere are two additional modes of higher energies with\nbroken cylindrical symmetry in case (I), labeled X1 and\nX2 in Fig. 3 and included as X1-mode.gif and X2-\nmode.gif in the supplementary. The rotational direction\nof the X1-mode is the same as in E-mode, while it is the\nopposite for X2-mode.\nAs in Ref. [27], we find sharply defined breathing\nmodesin cases(II) and(III) atthe appropriateresonance\nfrequency ω, in fact the same frequency at which the a.c.\nmagnetic field excites the breathing mode. The verti-\ncal dashed line in Fig. 3 indicates the common breath-\ning mode frequency. Movie file B1-mode.gif shows the\nbreathing mode in case (III). Additional, higher energy\nB2-mode (B2-mode.gif) was found in cases (II) and (III),\nwhich is the radial mode with one node, whereas the B1\nmode is nodeless.\nIn addition to the two breathing modes, E-mode and\nthe two X-modes are excited in case (II) as well due\nto the partly in-plane nature of the spin perturbation,\n−(λE(t)/2)/summationtext\ni([Sz\ni(t)]2−[Sx\ni(t)]2). In contrast, case\n(III), where the perturbation −(√\n3λE(t)/2)/summationtext\ni[Sz\ni(t)]2\nis purely out-of-plane, one only finds the B-modes. As\na result, case (I) and (III) have no common resonance\nmodes or peaks, while case (II) has all the peaks (though\nX1 and X2 peaks are small). Compared to the magnetic\nfield-induced resonances, a richer variety of modes are\nexcited by a.c. electric field. In particular, the E-mode\nhas lower excitation energy than the B-mode and has a\nsharp resonance feature, which should make its detection\na relatively straightforward task. Full analytic solution\nof the excited modes[28] will be given later.\nIn summary, motivated by the recent discov-\nery of magneto-electric material Cu 2OSeO3exhibiting\nSkyrmion lattice phase, we have outlined the theory of\nSkyrmion dynamics in such materials. Electric field gra-\ndient is identified as the source of Skyrmion Hall motion.00.20.40.60.81Imχ(ω) (ab. unit)\n  \nB1\nR1\nR2(a)B||z,Bω||x,y\nB||z,Bω||z\n00.05 0.10.15 0.20.25 0.30.3500.20.40.60.81\nωImχ(ω) (ab. unit)\n  \nE\nX1X2B1\nB2(b)case (I)  : B||[001],Eω||[001]\ncase (II) : B||[110],Eω||[001]\ncase (III): B||[111],Eω||[111]\nFIG. 3: (color online) (a) Absorption spectra for a.c. unifo rm\nmagnetic field as in Mochizuki’s work (reproduced here for\ncomparison). (b) Absorption spectra for a.c. uniform elect ric\nfield in cases (I) through (III). In case (I) where there is no\nnet dipolar charge we find three low-energy modes E, X1, and\nX2. For case (III) where the dipolar charge is finite we find\nB1 and B2 radial modes excited. Case (II) exhibits all five\nmodes. Detailed description of each mode is given in the text .\nSeveralresonantexcitationsbya.c. electricfieldareiden-\ntified.\nJ. H. H. is supported by NRF grant (No. 2010-\n0008529, 2011-0015631). Y. Q. L. is supported by NSFC\n(Grant No. 11074216). J. H. H. acknowledges earlier\ncollaboration with N. Nagaosa, Youngbin Tchoe, and J.\nZang on a related model and informative discussion with\nY. Tokura.\n∗Electronic address: hanjh@skku.edu\n[1] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[2] W. M¨ unzer, A. Neubauer, T. Adams, S. M¨ uhlbauer, C.\nFranz, F. Jonietz, R. Georgii, P. B¨ oni, B. Pedersen, M.\nSchmidt, A. Rosch, and C. Pfleiderer, Phys. Rev. B 81,\n041203(R) (2010).\n[3] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[4] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nature\nMat.10, 106 (2011).\n[5] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K.\nOhoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and\nY. Tokura, Phys. Rev. Lett. 106, 156603 (2011).\n[6] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science5\n336, 198 (2012).\n[7] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[8] D. C. Ralph and M. Stiles, J. Magn. Mag. Mat. 320,\n1190 (2008).\n[9] F. Jonietz, S. M¨ uhlbauer, C. Pfleiderer, A. Neubauer, W.\nM¨ unzer, A. Bauer, T. Adams, R. Georgii, P. B¨ oni, R. A.\nDuine, K. Everschor, M. Garst, and A. Rosch, Science\n330, 1648 (2010).\n[10] T. Schulz, R. Ritz, A. Bauer, M. Halder, M.Wagner, C.\nFranz, C. Pfleiderer, K. Everschor, M. Garst, and A.\nRosch, Nat. Phys. 8, 301 (2012).\n[11] K. Everschor, M. Garst, R. A. Duine, and A. Rosch,\nPhys. Rev. B 84, 064401 (2011).\n[12] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).\n[13] Y. Tokura and S. Seki, Adv. Mat. 21, 1 (2009).\n[14] S. Seki, S. Ishiwata, and Y. Tokura, arXiv:1206.4404v1\n(2012).\n[15] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 74, 224444 (2006).\n[16] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 76, 144424 (2007).\n[17] Taka-hisa Arima, J. Phys. Soc. Jpn. 76, 073702 (2008).\n[18] Judit Romh´ anyi, Mikl´ os Lajk´ o, and Karlo Penc, Phys.Rev. B84, 224419 (2011).\n[19] J. H. Han, unpublished note.\n[20] H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and\nY. Tokura, Phys. Rev. Lett. 105, 137202 (2010).\n[21] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989); A. Bogdanov and A. Hubert, J. Magn.\nMagn. Mater. 138, 255 (1994).\n[22] U. K. Roßler, A. N. Bogdanov, and C. Pfleiderer, Nature\n442, 797 (2006).\n[23] Su Do Yi, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. 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Lett. 109, 037603 (2012)."    },    {        "title": "1209.3669v2.Nonlinear_emission_of_spin_wave_caustics_from_an_edge_mode_of_a_micro_structured_Co2Mn0_6Fe0_4Si_waveguide.pdf",        "content": "arXiv:1209.3669v2  [cond-mat.mes-hall]  12 Dec 2012Nonlinear emission of spin-wave caustics from an edge mode o f a\nmicro-structured Co 2Mn0.6Fe0.4Si waveguide\nT.Sebastian∗and T.Br¨ acher\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany and\nGraduate School Materials Science in Mainz,\nGottlieb-Daimler-Straße 47, 67663 Kaiserslautern, Germa ny\nP.Pirro, A.A.Serga, and B.Hillebrands\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nT.Kubota\nWPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nKatahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan\nH.Naganuma, M.Oogane, and Y.Ando\nDepartment of Applied Physics, Graduate School of Engineer ing,\nTohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Jap an\n(Dated: August 30, 2018)\nAbstract\nMagnetic Heusler materials with very low Gilbert damping ar e expected to show novel magnonic\ntransport phenomena. We report nonlinear generation of hig her harmonics leading to the emission\nofcausticspin-wave beamsinalow-damping, micro-structu redCo 2Mn0.6Fe0.4SiHeuslerwaveguide.\nThe source for the higher harmonic generation is a localized edge mode formed by the strongly\ninhomogeneous field distribution at the edges of the spin-wa ve waveguide. The radiation character-\nistics of the propagating caustic waves observed at twice an d three times the excitation frequency\nare described by an analytical calculation based on the anis otropic dispersion of spin waves in a\nmagnetic thin film.\n1In the last years nonlinear spin dynamics in magnetic microstructure s made of metallic\nferromagnetic thin films or layer stacks have gained large interest.[1 –5] The intrinsically\nnonlinear Landau-Lifshitz and Gilbert equation (LLG), that govern s the spin dynamics,\ngives rise to a variety of nonlinear effects.[6, 7]\nAmong the metallic ferromagnets, the class of Cobalt-based Heusle r materials is very\npromising for future magnon spintronics devices and the observation of new phenomena of\nmagnonic transport. The reasons for the interest in these mater ials are the small magnetic\nGilbert damping, the high spin-polarization, and the high Curie temper ature.[8, 9]\nAs shown recently, the full Heusler compound Co 2Mn0.6Fe0.4Si (CMFS) is a very suitable\nmaterial to be used as a micro-structured spin-wave waveguide du e to the increased decay\nlength which was observed for wave propagation in the linear regime.[ 10] The reason for\nthis observation is the low Gilbert damping of α= 3×10−3of CMFS compared to Ni 81Fe19\nwithα= 8×10−3, which is the material commonly used in related studies.[9] The decre ased\nmagnetic losses not only lead to an increase of the decay length but a lso to large precession\nangles of the magnetic moments and, thus, to the occurrence of n onlinear effects. Regarding\nfuture applications, the investigation and the thorough understa nding of phenomena related\nto the spin-wave propagation in the nonlinear regime in Heusler compo unds is crucial.\nIn this Letter, we report nonlinear higher harmonic generation fro m a localized edge\nmode [11, 12] causing the emission of caustic spin-wave beams [13, 14 ] in a micro-structured\nHeusler waveguide. Spin-wave caustics are characterized by the s mall transversal aperture\nof a beam, which practically does not increase during propagation, a nd the well-defined\ndirection of propagation.\nThe investigated sample is a 5 µm wide spin-wave waveguide structured from a 30nm\nthick film of the Heusler compound CMFS. Details about the fabricatio n and the material\nproperties can be found in Refs.15 and 16. The microfabrication of the waveguide was per-\nformedusing electron-beam lithography andion-milling. Forthe excit ation ofspin dynamics\nin the waveguide the shortened end of a coplanar waveguide made of copper was placed on\ntop of it. The Oersted field created by a microwave current in this an tenna structure can be\nused to excite spin dynamics in the Gigahertz range. The antenna ha s a thickness of 400nm\nand a width of ∆ x= 1µm.\nAll observations have been carried out using Brillouin light scattering microscopy\n(µBLS).[17] µBLS is a powerful tool to investigate spin dynamics in microstructur es with a\n2FIG. 1. (Color online) Sketch of the sample design. The short ened end of a coplanar waveguide\nis used as an antenna structure to excite spin dynamics in a 5 µm wide CMFS waveguide with a\nthickness of 30nm. The waveguide is positioned in the x-y-pl ane with its long axis pointing along\nthe x-direction. The external magnetic field is applied tran sversely to the waveguide in y-direction.\nThe figure includes a µBLS spectrum taken at a distance of 4.5 µm from the antenna in the center\nof the waveguide (see laser beam in the sketch) for an excitat ion frequency of fe= 3.5GHz, a\nmicrowave power of 20mW, and an external field µ0Hext= 48mT.\nspatial resolution of about 250nm and a frequency resolution of up to 50MHz.\nIn the following description, the waveguide is positioned in the x-y-pla ne with the long\naxis pointing in x-direction. The origin of the coordinate system is give n by the position\nof the antenna between x=−1µm andx= 0µm. An external magnetic field of µ0Hext=\n48mT was applied transversely to the waveguide in y-direction result ing in Damon-Eshbach\ngeometry [18] forspin waves propagatingalong thewaveguide. Ask etch of thesample layout\nis shown in Fig.1.\nIn addition, Fig.1 includes a spectrum taken by µBLS for an excitation frequency of\nfe= 3.5GHz and a microwave power of 20mW at a distance of 4.5 µm from the antenna in\nthe center of the waveguide. The spectrum shows not only a peak a tfe= 3.5GHz but also\nat 2fe= 7.0GHzand 3 fe= 10.5GHz. Furthermore, the intensity of the directly excited spin\nwaveat3.5GHzislowerthanforthehigherharmonicsatthepointofo bservation. Aswewill\nsee, the higher harmonics are excited resonantly by nonlinear magn on-magnon interactions\nand the intensity distribution is a consequence of the different prop agation characteristics\nof the observed spin-wave modes.\nA two-dimensional intensity distribution for the detection frequen cyfd=fe= 3.5GHz\nas well as the calculated dispersion relation [19] for the center of th e CMFS waveguide are\n3012345\ny position ( µm)123456789x position ( µm)fd=fe\n0123456\nwavevector ( µm−1)3456789101112frequency (GHz)\nfe2fe3fe\n(a) (b)\nFIG. 2. (Color online) (a) µBLS intensity distribution for the detection frequency fdbeing equal\nto the excitation frequency fe= 3.5GHz. The observed intensity of the edge mode is maximal near\nthe edges of the CMFS waveguide. Please note that the excitin g antenna is positioned between\nx=−1µm andx= 0µm. The blue dots in the graph indicates the position of the mea surement\npresented in Fig.1. (b) Calculated dispersion relation for the center of the CMFS waveguide\naccording to Ref.19 as well as excitation frequency and high er harmonics (dashed lines).\nshown in Fig.2. Figure2(a) reveals a strong localization of the intens ity at the edges of the\nwaveguide. The non-vanishing intensity close to the antenna and be tween the edges of the\nwaveguide ( y= 1−4µm) can be attributed to nonresonant, forced excitation by the Oe rsted\nfield created by the microwave current. Figure2(b) shows the spin -wave dispersion for the\nCMFS waveguide calculated according to Ref.19 assuming a homogen eous magnetization\noriented in y-direction by the external field. The material paramet ers used in all our calcula-\ntions were determined experimentally on the unstructured film via fe rromagnetic resonance\n(MS= 1003kA/m and Hani= 1kA/m) and via µBLS in the micro-structured waveguide\n(Aex= 13pJ/m) following a method described in [9]. The effective field of µ0Heff= 46mT\nused in the calculations was obtained by micromagnetic simulations. As can be seen, the\nlower cut-off frequency f= 6.9GHz is well above the excitation frequency of fe= 3.5GHz\n(see dashed line in Fig.2(b)). In the center of the waveguide, the a ssumption of a homoge-\nneous magnetizationis a very goodapproximation. Close to theedge s, demagnetizing effects\nare responsible for a strongly decreased effective field and an inhom ogeneous magnetization\nconfiguration. The inhomogeneity of the magnetization does not allo w for a quantitative\n45 10 15 20\nmicrowave power (mW)BLS intensity (arb. u.)fd=fe\nfd=2fe\nfd=3fe\nFIG. 3. (Color online) Power dependence of the BLS intensity for the detection frequencies fd=\n3,5GHz, 7GHz, and 10.5GHz for the fixed excitation frequency fe= 3.5GHz. Please note the\nlog-log presentation of the data. The lines in the graph corr espond to fits according to Eq.1. A\nleast square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nmodeling of spin dynamics close to edges.[20] However, from previous work it is known that\nthis field and magnetization configuration allow for the existence of lo calized spin waves -\ncommonly referred to as edge modes - energetically far below the sp in-wave dispersion for\npropagating modes in the center.[11, 12] Therefore, we conclude t hat the spin-wave mode\natfd=fe= 3.5GHz is excited resonantly by the microwave field.\nAs can be seen from the spectrum in Fig.1, the applied microwave pow er of 20mW is\nsufficiently high to observe the nonlinear generation of higher harmo nics of the excitation\nfrequency fein theµBLS spectra. These resonant frequency multiplications to f= 2fe=\n7.0GHz and f= 3fe= 10.5GHz result in the excitation of propagating spin-wave modes in\nthe waveguide energetically above the cut-off frequency of the dis persion shown in Fig.2(b).\nFigure3 shows the dependence of the directly excited mode and the higher harmonics\non the applied microwave power. This data has been acquired close to the position of the\nedge mode and near the antenna at x= 0.7µm andy= 0.8µm. The data is presented on\na log-log scale with fits according to\nIn(p) =Anpsn+b, (1)\nwhereInis theµBLS intensity, Ana coupling parameter, pthe applied microwave power,\nandbthe noise-level in our measurement. As expected, these process es do not show a\n5threshold power level, but reliable detection on the background of t he noise is not possible\nfor powers below 5mW for fd= 3fe= 10.5GHz. The different slopes of the curves for the\ndifferent spin-wave modes nare caused by the different power-laws specified by the exponent\nsn. A least square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nThese experimental findings close to the integer values 1, 2, and 3 a re in accordance with\nboth reported experimental data and theoretical predictions fo r the nonlinear generation of\nhigher harmonics.[3, 7]\nThe observation of the second harmonic can be understood qualita tively by considering\nthe strong demagnetizing fields caused by the out-of-plane compo nentmz(t) during the\nmagnetization precession. Due to the demagnetizing fields, the mag netization precession\nM(t) around the y-direction (defined by the effective field) follows an ellip tical trajectory\nrather than a circular one. In contrast to the case of a circular pr ecession, the resulting\nprojection of Mon the y-axis is time dependent and oscillating with the frequency 2 fe. The\nresulting dynamic dipolar field |hy(t)| ∝m2\nx−m2\nzcan be regarded as the source for the\nfrequency doubling. Similar considerations lead to the observation o f higher harmonics. A\nfull quantitative derivation of higher harmonic generation and othe r nonlinear effects based\non the expansion of the LLG in terms of the dynamic magnetization ca n be found in Ref.7.\nFigures4(a)and(b)showintensity mapsforthedetectionfreque nciesfd= 2fe= 7.0GHz\nandfd= 3fe= 10.5GHz, respectively. In both cases the intensity radiated from the\nposition of the edge mode is strongly directed, has a small transver sal aperture, and shows\nnondiffractive behavior. These radiation characteristics recorde d for the three spin-wave\nmodes at f= 3.5, 7.0 and 10.5GHz presented in Figs.2(a) and 4 are responsible for t he\nintensity distribution shown in the spectrum in Fig.1. The position of t his measurement\nis indicated in the corresponding intensity maps with a circle. Since the spin wave at\nfe= 3.5GHz is localized at the edges of the waveguide, its intensity is compar ably weak in\nthe center. In contrast, the higher harmonics have frequencies above the cut-off frequency of\nthe spin-wave dispersion and can propagate in the center of the wa veguide. The propagation\nanglesofthese modessupport theintensity distributionrecorded inourmeasurement. While\nan increased intensity can be found already for fd= 2fe, forfd= 3fethe two beams starting\nfrom both edges of the spin-wave waveguide even intersect at the measurement position\nresultinginthehighestintensityatthispoint. Becauseofthewell-de finedpropagationangles\nof the higher harmonics and the localization of the edge mode, the int ensity distribution is\n6012345\ny position ( µm)123456789x position ( µm)θ=78.0◦\nHextvG\nθfd=2fe\n012345\ny position ( µm)123456789x position ( µm)θ=67.0◦\nHextvG\nθfd=3fe\n(a) (b)\nFIG. 4. (Color online) µBLS intensity distribution for (a) fd= 2fe= 7.0GHz and (b) fd=\n3fe= 10.5GHz. Both intensity maps show strongly directed spin-wave beams along the angle\nθ=∠(Hext,vG). The lines in the maps are guides to the eye to identify the pr opagation angle θ.\nThe blue dots in the graphs indicate the position of the measu rement presented in Fig.1.\nstrongly depending on the measurement position.\nThis observation of spin-wave beams with small transversal apert ure is reminiscent of the\nresults in Refs.21 and 22, where nonlinear three-magnon scatter ing in yttrium iron garnet\nis reported. However, in that case, the propagation direction of t he nonlinearly generated\nspin wave is given by momentum conservation in the scattering proce ss. In contrary, in\nour case, due to the strong localization of the edge mode, the assu mption of a well-defined\ninitial wavevector and, thus, a strict momentum conservation is no t justified. In particular,\nit is not possible to find an initial wavevector that allows for the nonline ar generation of the\nsecond and third harmonic at the same time still respecting momentu m conservation.\nIn the following, we will describe the observed propagationcharact eristics using the prop-\nerties of the anisotropic spin-wave dispersion in a magnetic thin film.[13 , 14, 19] Because\nof this anisotropy, the direction of the flow of energy, which is given by the direction of\nthe group velocity vG= 2π∂f(k)/∂kof the investigated spin waves, can differ significantly\nfrom the direction of its wavevector k. To estimate the relevant range in k-space in our\nexperiment, we have to consider the lateral dimensions of the sour ce for the nonlinear pro-\ncesses. Since the excitation by the oscillating Oersted field is most effi cient directly below\nthe antenna, the edge mode has the highest intensity in this region g iven by the width of\nthe antenna of ∆ x= 1µm. The spread of the edge mode in y-direction can be estimated\n7from the intensity map in Fig.2(a) to be smaller than 1 µm. Because of this localization, the\nedge mode in our measurement at f= 3.5GHz can be regarded as a source for the nonlinear\nemission of the higher harmonics with lateral dimensions of approxima tely 1×1µm2. As a\nfirst approximation, the Fourier transformation of this geometry lets us estimate the maxi-\nmum wavevector that can be excited by the edge mode to be kmax≈6.3µm−1. As we will\nsee, the direction of the group velocity can be assumed to be const ant for most wavevectors\nthat can be excited. This finally leads to the formation of the caustic s in our experiment.\nFor given frequency and external field, the iso-frequency curve f(kx,ky) =constcan be\ncalculated analytically from the dispersion relation. Calculations for f= 2fe= 7.0GHz and\nf= 3fe= 10.5GHz are illustrated in Fig.5(a), where kyis shown as a function of kx. Using\nthis data, we calculate the direction θof the flow of energy of the spin waves relative to the\nexternally applied field by:\nθ=∠(Hext,vG) = arctan( vx/vy) = arctan( dky/dkx). (2)\nFigure5(b)showsthecalculatedpropagationangle θintheCMFSwaveguideasafunction\nofky. The most important feature in the trend of θis the small variation of ∆ θ≤2◦in the\nrange of ky= 2−7µm−1for both frequencies f= 2feandf= 3fe. While the wavevector\nkchanges, the direction of vG- and, thus, the flow of energy - keeps almost constant as\na function of k. In this range, which includes the maximum wavevector kmax≈6.3µm−1\nthat can be emitted from the edge mode (see considerations above ), the calculations yield\nθcalc(2f) = 79◦andθcalc(3f) = 66◦as mean values, respectively. The dash-dotted lines in\nFig.5(b) represent the propagation angles θof the spin-wave beams observed experimentally\nas shown in Fig.4 ( θexp(2f) = 78◦andθexp(3f) = 67◦). The comparison of experimental\nfindings and analytical calculations shows an agreement within the ex pected accuracy of\nour measurement setup and is, therefore, supporting our conclu sion. Higher harmonics with\nky≤2µm−1are emitted with strongly varying directions from the edge mode and can be\nregarded as a negligible background in our measurement.\nIn summary, we reported nonlinear higher harmonic generation fro m a localized source\nin a micro-structured CMFS waveguide leading to the emission of stro ngly directed spin-\nwave beams or caustics. This observation results from the complex interplay of different\nphenomena in magnonic transport in magnetic microstructures. Th e localization of an edge\nmode due to demagnetizing fields in the waveguide leads to the format ion of a source for\n80 2 4 6 8\nkx (µm−1)02468ky (µm−1)\nk\nvG\nHextθf=2fe\nf=3fe\n0 2 4 6 8\nky (µm−1)657075808590θ (degree)f=2fe\nf=3fe\n(a) (b)\nFIG. 5. (Color online) Analytical calculations according t o Ref.19. (a) Iso-frequency curves\nfconst=f(kx,ky) forf= 2fe= 7.0GHz and f= 3fe= 10.5GHz. Based on these calculations ex-\nemplary directions for k,vG,Hextand the propagation angle θ=∠(Hext,vG) = arctan( dky/dkx)\nare shown in the graph. (b) Radiation direction θcalculated from the iso-frequency curves shown\nin (a). Dash-dotted lines correspond to the angles θexpobserved in the experiment.\nthe following nonlinear processes. The nonlinear higher harmonic gen eration results in the\nresonant excitation and emission of propagating spin waves at 2 feand 3fein a wavevector\nrange corresponding to the localization of the edge mode. The expe rimentally observed\npower dependencies of the different spin-wave modes show the exp ected behavior for direct\nresonant excitation and nonlinear higher harmonic generation. As s hown by our calculation,\nthe anisotropic spin-wave dispersion yields a well-defined direction of the flow of energy\nof the emitted spin waves in the relevant range in k-space. The calcu lation is not only\nqualitatively in accordance with our experimental findings but does a lso show quantitative\nagreement.\nWe gratefully acknowledge financial support by the DFG Research U nit 1464 and the\nStrategic Japanese-German Joint Research from JST: ASPIMATT . Thomas Br¨ acher is sup-\nported by a fellowship of the Graduate School Materials Science in Ma inz (MAINZ) through\nDFG-fundingoftheExcellence Initiative(GSC266). Wethankourco lleaguesfromthe Nano\nStructuring Center of the TU Kaiserslautern for their assistance in sample preparation .\n∗tomseb@physik.uni-kl.de\n[1] H.Schultheiss, X.Janssens, M. van Kampen, F.Ciubotaru , S.J.Hermsdoerfer, B.Obry,\n9A.Laraoui, A.A.Serga, L.Lagae, A.N.Slavin, B.Leven, B.Hi llebrands, Phys. Rev. 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B79, 144428 (2009).\n11"    },    {        "title": "1210.5647v1.Radiative_damping_of_surface_plasmon_resonance_in_spheroidal_metallic_nanoparticle_embedded_in_a_dielectric_medium.pdf",        "content": "arXiv:1210.5647v1  [cond-mat.mes-hall]  20 Oct 2012Radiative damping of surface plasmon resonance in spheroid al metallic nanoparticle\nembedded in a dielectric medium\nNicolas I. Grigorchuk∗\nBogolyubov Institute for Theoretical Physics, National Ac ademy of Sciences of Ukraine,\n14-b Metrologichna Str., Kyiv-143, Ukraine, 03680\n(Dated: September 13, 2021)\nThe local field approach and kinetic equation method is appli ed to calculate the surface plasmon\nradiative damping in a spheroidal metal nanoparticle embed ded in any dielectric media. The radia-\ntive damping of the surface plasmon resonance as a function o f the particle radius, shape, dielectric\nconstant of the surrounding medium and the light frequency i s studied in detail. It is found that\nthe radiative damping grows quadratically with the particl e radius and oscillates with altering both\nthe particle size and the dielectric constant of a surroundi ng medium. Much attention is paid to\nthe electron surface-scattering contribution to the plasm on decay. All calculations of the radiative\ndamping are illustrated by examples on the Au and Na nanopart icles.\nPACS numbers: 78.67.Bf; 68.49.Jk; 73.23.-b; 78.67.-n; 52. 25.Os; 36.40.Vz; 25.20.Dc; 52.25.Os; 78.47.+p.\nI. INTRODUCTION\nThe surface plasmons (SPs) excited by an electro-\nmagnetic radiation falling on a metal nanoparticle (MN)\nare still of great fundamental interest1–3since their pro-\nnounced local resonances, whose position, shape and in-\ntensity can be tuned over wide spectral range by varying\nthe size and shape of the MNs or by changing the sur-\nrounding medium.\nOnce excited, plasmon oscillations can damp non-\nradiatively by absorption caused by electron-phonon in-\nteractions, and/or radiatively by the resonant scatter-\ning process.4,5The electron-phonon interaction with de-\ncreasing MN size becomes more and more ineffective\ndue to kinematical restrictions imposed on the energy-\nmomentum conservation laws.6Nevertheless, the line\nbroadening has been observed in experiments with MNs\nscattering and absorption.3,7–14This means that besides\nelectron-phonon interaction the other damping mecha-\nnisms should be studied as well in order to interpret the\nobserved SP line broadening.\nUsually, both the surface and the radiative damping\nmechanisms play an important role in the SP decay. In\nRef. 14 it was shown that both the electron surface scat-\ntering and radiative damping can make significant con-\ntribution to the linewidth Γ (the full width at a half of\nmaximum of the surface plasmon resonance (SPR)).\nIn the MNs of a smaller radii, the penetration depth\nof the plasmon field reduces and becomes more localized\nnearthe surface.5This is due to the fact that the electron\nscreening is increased as the particle radius is reduced.\nAs a result, the bulk-induced loss processes play only a\nminorroleandtheelectronicexcitationsgeneratedbythe\nsurface potential dominate. On the other hand, if elec-\ntrons are confined in nanoparticles, where the mean free\npath (MFP) of the electrons becomes comparable with\n∗email: ngrigor@bitp.kiev.uathe size of the particles, the electron-surface scattering\ncomes into play5and the surface acts as an additional\nscatterer.\nThe radiative damping of the SPR is an important\nparameter since it would help to analyze the specificity\nof the transformationof the collective electron oscillation\nenergy into the optical far field.\nThe interest to treat of the SPR and radiative damp-\ning is maintained as well by the investigation of the local\nfield enhancement, an effect which increases the inten-\nsity of the incident light near the MNs surface by sev-\neral orders of magnitude.4,5Since a lot of devices in-\ncorporating MNs gains from this effect, Γ is treated as\na main parameter in applications such as field concen-\ntration for nanopatterning with nanowires,15plasmonic\nnanolithography,16and near-field optical microscopy,17\nastigmatic optical tweezers,18surface enhanced Raman\nscattering19etc.\nThe damping of SPR in MNs previously has been well\nstudied theoretically.20–26The influence of a nearby sur-\nface on the plasmon resonance of a metallic nanoparticle\nof finite size has been studied in the work 20 with an\naccount for the effects of both radiative and evanescent\nsurface-reflected waves. In Ref. 21 the oscillations of\nthe plasmon linewidth as a function of the radius of the\nnanoparticles were obtained from numerical calculations\nbasedon the time dependent localdensity approximation\nwithout regard for radiative damping. The temperature\neffect on the radiative lifetime of the surface plasmon\nwas studied in the work 22. To evaluate size and shape\ndependent dielectric functions and extinction spectra for\nMNs with various shape, the different empirical formulas\nwere proposed for electron MFP.23\nThe radiative damping calculated in Refs. 24–26 is\nproportionalto the product of the polarizability (propor-\ntional to particle volume) times k3= (2π/λ)3, wherekis\nwave number and λis the wavelength. The calculations\nhave been conducted in the electrostatic approximation,\nassuming V/λ3≪1, where Vis the MN volume.\nIn the works mentioned above was dropped from ac-2\ncount the fact that the role of an external electric field\n(which causes the dipole oscillations of electrons and the\nradiative damping) can play the inner electric field in\nMN. Besides, in these works the surface and radiative\ndecay have not been considered simultaneously for non-\nspherical MN.\nThe purpose of the present paper is to calculate the\nradiative damping of the surface plasmon under the ef-\nfect of inner electric field, for the case when the MFP\nof the electrons is larger than the particle size and their\nscatteringon the particlesurfaceplaysanimportantrole.\nWithin the framework of a local field approach and\nthe kinetic equation method, we have found that the\nlinewidth of SPR increases quadratically with the parti-\ncle radius and exhibits oscillations as a function of the\nnanoparticle radius and the refractive index of a sur-\nrounding medium.\nTherestofthepaperisorganizedasfollows. InSection\nIIthetheoreticalbackgroundtotheproblemispresented.\nSection III is devoted to the study of radiative damping.\nSection IV contains the calculation of the conductivity\ntensor and the discussion of obtained results. Section V\ncontains the summary and conclusions.\nII. SURFACE PLASMON LINEWIDTH\nAn interaction of light with a MN embedded in a\nmedium is studied in the framework of classical optics,\nassuming that the particle and the medium are contin-\nuous, homogeneous, and characterized by their dielec-\ntric function. To overcome the problem connected with\nan inhomogeneous line broadening (due to the size and\nshape distribution within the particle ensemble), we re-\nstrict ourselves only to a singleMN, which directly yields\nthe homogeneous linewidth.9,27\nIn general, the linewidth can be decomposed into con-\ntributions from the bulk dielectric constant, surface scat-\ntering, and radiative damping:4Γ = Γb+Γs+Γrad.\nIn the classical case of free electrons in bulk metal,\nthe damping Γ b(≡ν) is due to the inelastic scattering of\nthe electrons on phonons, lattice defects, or impurities ( ν\nrefers to the electron collision frequency), which shorten\nthe MFP.28In this case, the relation Γ b=υF/l∞holds,\nwhereυFis the Fermi velocity and l∞is the MFP of\nconduction electrons in the bulk (when the MN radius\ntends to infinity).\nSimilarly, to estimate the surface effect in electron-\nsurface scattering, the following empirical relation5,23\nΓs=AυF\nleff(1)\nis often used, where Ais a phenomenological factor and\nleffis a reduced effective MFP. For the sphere, the clas-\nsical theory gives leff=Rfor an isotropic scattering or\nleff= 4R/3 for a diffusive scattering, while the quantum\ntheory in a box model yields leff= 1.16Rorleff= 1.33R,\nwhereRis the radius of a spherical MN.23The bulk scattering does not cause appreciable attenu-\nation for the high frequency case, when ωτ≫1, where τ\nis the relaxation time. On the other hand, the contribu-\ntion to damping from surface scattering is negligible only\nifl∞/2R≪√\n1+ω2τ2. Thus, the surface contribution\nis important if ωτ≫max(1,l∞/2R).\nThe radiative contribution can be estimated with the\nhelp of the following expression9\nΓrad= 2/planckover2pi1kV, (2)\nwhereVis the nanoparticle volume and kis an another\nphenomenological constant to be taken from the experi-\nmental data.\nTo account both the surface and the radiative ef-\nfects for a MN embedded in a homogeneous, transparent\nmedium, the dielectric permittivity14,23\nǫ′(ω) =ǫinter(ω)+/parenleftBig\n1−ω2\npl\nω2+(Γb+Γs+Γrad)2/parenrightBig\n,\nǫ′′(ω) =ω2\npl\nωΓb+Γs+Γrad\nω2+(Γb+Γs+Γrad)2.(3)\nis usually used, where ǫ′andǫ′′are, respectively, the real\nand imaginary parts of the dielectric function of the par-\nticle material, ǫinteraccounts for the interband electron\ntransitions, ω2\npl= 4πnee2/m, andnerefers to the elec-\ntron concentration. The expression in the parentheses is\napplied for the intraband electron transitions.29\nOne can see that the effect of the surface is reduced\nsimply to the addition of a term γs≡γs(leff) in the\ndenominator of Eq. (3) in the form of Eq. (1), and the\nradiative effect is accounted simply by the term γrin the\nsame denominator. Formulae (1) and (2) can be applied\nonly to the MNs of a spherical shape in the case when\nthe MFP of electrons lis smaller than the particle size\nd. If the shape of MN differs from the spherical one or\nthe inequality l >2Rtakes place, then the expressions\n(1) and (2) can no longer be used.\nWe will consider the MNs with a moderate sizes for\nwhose, on the one hand, the condition l >2Ris still\nfulfilled (the collisions of the conduction electrons with\nthe particle surface remains to be an important relax-\nation process), and on the other hand, with the sizes\nenough large to account for the dissipation of the elec-\ntron energy due to the emission of electromagnetic waves\nby plasmons, so called radiation damping.\nIII. RADIATIVE DAMPING\nThe problem of a damping of the electron energy due\nto the radiation of a portion of the collective electron\noscillation energy into the optical far field has been ex-\ntensively studied in the literature.5,9,11,22,25\nAs we can see from Eq. (1) the surface damping de-\npends on a particle size. The relative contributions from\nthe radiative damping through the resonant scattering\nand absorption also strongly depend on the particle size.3\nIn particular, it is known4,5that the plasmon absorption\nis the only process taken place in small particles, whereas\nboth the absorption and the scattering are present in\nlarge particles, with the latter becoming more dominant\nas the particle size increases. The phenomenon is based\non an interplay of the dissipative and radiative damping.\nAn another assumption that the wavelength of the ab-\nsorbing light λis far above the characteristic size of the\nnanoparticle (about 25 nm for gold particles5) allows us\nto treat the MN as being immersed in a spatial uniform,\nbut oscillating in time electric field. This implies that\nEM field around the MN can be considered as homoge-\nneous and across the particle as uniform, such that all\nthe conduction electrons move in-phase producing only\ndipole-type oscillations.\nThe external electric field E(0)exp(−iωt) induces an\ninner (potential) electric field Eininside the particle\nwhich is coordinate independent. The field Eincan be\nlinearly expressed in terms of E(0)by employing the de-\npolarization tensor. In terms of principal axes, the depo-\nlarization tensor, which coincide with with the principal\naxes of the ellipsoid, the relation between components of\nexternal and inner fields look as4\nE(0)\nj=Ej,in[1+Lj(ǫ/ǫm−1)], (4)\nEj,inare the components of the electric field inside the\nMN and Ljare the principal value of the j-th compo-\nnent of the depolarization tensor that is also known as\na geometric factors. The explicit expressions of Ljfor\na MN with a particular shape can be found elesewhere\n(see, e.g., Refs. [30], [4], and [31]). The complex dielec-\ntric permittivity of the particle material is denoted by\nǫ(=ǫ′+ǫ′′) andǫmrefers to the dielectric constant of\nthe surrounding medium.\nElectrons are accelerated in the presence of an electric\nfield inside the MN. It is well known from the classical\nelectrodynamics32that accelerated charges emit electro-\nmagnetic radiation in all directions. To calculate the line\nbroadening that is entirely caused by an increase of Γ rad\ndue to the radiative effect, we will use the time depen-\ndence of a classical dipole oscillator. The force of a decel-\nerative radiation of a dipole under an inner electric field\n(Eq. (4)) can be represented as\nFrad(t) =−2e\n3c3√ǫm[1+L(ǫ/ǫm−1)]...\nd(t),(5)\nwheredis the MN dipole moment. The minus sign\nmeans that this force is opposit to the dipole moment\ndirection. In the case of one electron and a medium\nwithεm=ε= 1, Eq. (5) transforms into the well-\nknown expressions from the classical electrodynamics.32\nThe linewidth due to the radiative damping of dipole vi-\nbrations can be expressed through the Fby means of\nΓrad=e\nmIm/bracketleftBigg\nFrad(t)\n˙d(t)/bracketrightBigg\nN. (6)whereN=Vneis the number of free electrons in the\nMN and Vrefers to the particle volume. It is necessary\nto underline here that the radiation damping rate is pro-\nportional to the total number of oscillating electrons in\nMN.\nSupposing d(t) =d0exp(−iωt), we obtain for j-th\ncomponent of a radiative linewidth the following expres-\nsion:\nΓj,rad=2\n3e2ω2\nmc3N√ǫmIm[1+Lj(ǫjj/ǫm−1)].(7)\nPutting here ǫ′′\njj≡ǫm, we get known expression, e.g.,\nfrom Ref. 22.\nFor the sake of simplicity, we will assume that the di-\nelectric matrix has no influence on the MN and can be\ncharacterized by\nǫ′\nm(ω) =const≡ǫ′\nm, ǫ′′\nm(ω) = 0,(8)\ni.e., the dielectric constant of the surrounding medium is\nassumed to be frequency independent. However, it may\nhappen in some actual cases that the dielectric medium\nis strongly absorptive at frequencies below the ωpl. If\nthat is the case, then the ǫ′′\nmis strongly dependent on\nfrequency and contributes to the attenuation of the oscil-\nlations. With accounting for the dielectric matrix prop-\nerties given by Eq. (8), Eq. (7) can be rewritten as\nΓj,rad(ω) =2\n3e2ω2\nmc3NLjǫ′′\njj(ω)\n√ǫm. (9)\nIf one will consider the case of the frequency close to\nthe frequency of the bulk plasma oscillations of electrons\nin metal ωpl, then the imaginary part of the dielectric\nfunction tensor for free electron gas can be expressed\nwithin the Drude-Sommerfeld model as4,5,28\nǫ′′\njj(ω) = 4πσ′\njj(ω)\nω, (10)\nwhereσ′\njjis the principal components of the real part of\nthe conductivity tensor. Taking into account the expres-\nsion (10), we get\nΓj,rad(ω) =8π\n3e2ω\nmc3NLjσ′\njj(ω)\n√ǫm. (11)\nThe real part of the conductivity tensor can be ex-\npressed through the imaginary part of the polarizability\ntensorαjjby means of\nσ′\njj(ω) =ω\nVǫm|1+Lj(ǫ/ǫm−1)|2Imαjj(ω).(12)\nIn the case of the SPR, one can use in Eq. (3) for\nnonspherical MNs, the frequency\nω=ωsp=ωpl/radicalbig\nε∞+(1/Lj−1)n2.(13)4\nHeren2=ǫmandε∞≡1+ǫinteris the high frequency\ndielectric constant due to interband and core transitions\nof the inner electrons in a MN’s material.\nThere are different possibilities to calculate σ′\njj(ω) for\nthe different frequency regime. Below, we will demon-\nstrate how to calculate the σ′\njjas applied to the nanopar-\nticle case.\nIV. CALCULATION OF σ′\njj\nTocalculatethe tensor σ′\njjwewillusethe kineticequa-\ntions method. Benefit of this method is that it permits\none to study the effect of the particle shape on the mea-\nsuredphysicalvalues. Second, it enablesusto investigate\nthe particles whose sizes are those that the particle sur-\nface start to play an important role. A diffuse boundary\nscattering is assumed to be a good approximation in this\ncase.\nThe simplest form of a nonspherical shape is a\nspheroid. We restrict ourselves to the nanoparticles with\naspheroidal shape only. Applyingthementionedmethod,\nwe have found33that the components of the conductiv-\nity tensor for light polarized along ( /bardbl) or across ( ⊥) the\nrotation axis of a spheroidal MN are\nσ′\n(/bardbl\n⊥)(ω) =9ω2\npl\n16πRe\n1\nν−iωπ/2/integraldisplay\n0/parenleftbiggsinθcos2θ\n1\n2sin3θ/parenrightbigg\nΨ(θ)dθ\n\nυ=υF,\n(14)\nwhereνis the electron collision frequency and θis the\nangle between rotation axes of the spheroid and direc-\ntion of an electron velocity. Here and below, the upper\n(lower) symbol in the parentheseson the left-hand side of\nEq. (14) corresponds to the upper (lower) expression in\nthe parentheses on the right-hand side of this equation.\nThe subscript υ=υFmeans that the electron velocity\nin the final expressions should be taken on the Fermi\nsurface.\nThe complex Ψ function entering in Eq. (14) has the\nform\nΨ(q) = Φ(q)−4\nq2/parenleftbigg\n1+1\nq/parenrightbigg\ne−q, (15)\nwhere\nΦ(q) =4\n3−2\nq+4\nq3, q=2R\nυ′(ν−iω).(16)\nOne can see from Eq. (16) that the qis governed by\nthe ”deformed” electron velocity, which in the case of a\nspheroidal MN takes the form\nυ′=υR/radicalBigg/parenleftbiggsinθ\nR⊥/parenrightbigg2\n+/parenleftbiggcosθ\nR/bardbl/parenrightbigg2\n≡υ′(θ),(17)\nwhereυrefers to the electron velocity in a spherical par-\nticle with radius R;R/bardblandR⊥are the spheroid semi-\naxes directed alongand acrossthe spheroid rotation axis,respectively.34The semiaxes are connected to the radius\nof sphere Rof an equivalent volume through the relation\nR3=R/bardblR2\n⊥.\nThe last summand in Eq. (15) represents the oscilla-\ntion part of the Ψ function and the first one refers to its\nsmooth part. The Ψ function in Eq. (15) varies with the\nangleθbecause the parameter qbecomes dependent on\nthe angle θfor a spheroidal particle, namely\nq(θ) =2\nυFν−iω/radicalBig\ncos2θ\nR2\n/bardbl+sin2θ\nR2\n⊥. (18)\nA. Conductivity tensor in high-frequency limit\nLet us introduce the frequency of electron oscillations\nbetween particle walls as\nνs=υF\n2R. (19)\nDepending on sizes of MN, its shape and temperature,\nthe variety of relations between frequencies νs,νandωpl\ncan be achieved. For example, for the Na nanoparticle\nwith the radius of R <2˚A,νs≃ωpl. On the other\nhand, with R >126˚A, the electron oscillation frequency\nbecomes νs< ν, whereν≃4.24·1013s−1as is estimated\nfor the Na at 3000K. This leads to different expressions\nforσ(ω), which can be used in Eq. (11) for calculation of\nthe plasmon linewidth.\nBelow, we will consider the case when the contribution\nofthebulkdampingtotheradiativeplasmonlinewidthof\nSPR is neglected. The components of the conductivity\ntensor for a spheroidal MN in the highfrequency (HF)\nlimit (ω≫νs) andνs≫ν, can be represented as35\nσ′\n(/bardbl\n⊥)(ω) =9\n32π/parenleftBigωpl\nω/parenrightBig2υF\nR⊥/parenleftbiggη(ep)\nρ(ep)/parenrightbigg\n,(20)\nwhereR⊥(=Rx1/3,x=R⊥/R/bardbl) is a spheroid semi-\naxis directed across to the spheroid rotation axis, and\nη(ep) andρ(ep) are some smooth functions36dependent\nonly on the spheroid eccentricity ep=√\n1−x2(a prolate\nspheroid), or ep=√\nx2−1 (an oblate one).\nOne can use the following asymptotic expressions for\nfunctions η(x)andρ(x)inthecasesofboththeextremely\nsmall or the large axial ratio, respectively:\nη(x)≃\n\nπ/8+3πx2/16,for a prolate spheroid\nx/2+1/(4x),for an oblate spheroid,\nρ(x)≃\n\n3π/16+πx2/32,for a prolate spheroid\nx\n4+−1+4ln2 x\n8x,for an oblate spheroid.\n(21)\nIf one consider the nanowires and nanorods, which can\nbe reasonably approximated as prolate spheroids, then5\none can put η≃π/8 andρ≃3π/16 with a sufficient\ndegree of accuracy. In the case of MNs with a spherical\nshapeη=ρ= 2/3.\n1. Radiative damping for spheroidal MN in HF limit\nIn the highfrequency limit, when Eq. (20) can be ap-\nplied, fortwocomponentsofthe linewidth ofa spheroidal\nMN embedded in a medium with ǫm, we obtain the fol-\nlowing equation\nΓ(/bardbl\n⊥),rad,sp=3\n4e2ω2\npl\nmωc3NυF\nR⊥√ǫmL(/bardbl\n⊥)/parenleftbiggη(ep)\nρ(ep)/parenrightbigg\n,(22)\nwhere explicit expressions for η(ep) andρ(ep) can be\nfound in Ref. 36. Accounting for\nN=4\n3πR/bardblR2\n⊥ne=m\n3e2ω2\nplR/bardblR2\n⊥,(23)\nEq. (22) at the resonance frequency (13) takes the form\nΓ(/bardbl\n⊥),rad,sp=ω3\npl\n4c3υFR/bardblR⊥L(/bardbl\n⊥)/radicalbigg\n2n2+ǫ∞\nn2/parenleftbiggη(ep)\nρ(ep)/parenrightbigg\n.\n(24)\nThe depolarization coefficients L(/bardbl\n⊥)for spheroidal MN\ncan be found elsewhere.4,5In the case of MNs with a\nvery elongated shape ( x≪1), they can be expressed as\nL/bardbl(x)≃x2/bracketleftbigg\nln/parenleftbigg2\nx−x\n4/parenrightbigg\n−1/bracketrightbigg\n, L⊥(x) = [1−L/bardbl(x)]/2.\n(25)\n2. Radiative damping for a spherical MN in HF limit\nIn the case of MN with a spherical shape R/bardbl=R⊥≡\nR, and one can put the depolarization factor equal to\nL/bardbl=L⊥= 1/3 in Eqs. (4), (5), (7)–(13), (11), and (22).\nEq. (24) for MN with a spherical shape can be rewrit-\nten as\nΓrad,sp=1\n18/parenleftBigωpl\nc/parenrightBig3\nυFR2/radicalbigg\n2n2+ε∞\nn2.(26)\nThe increase in linewidth from the radiative damping\n(as we can see from Eqs. (24) and (26)) is proportional to\nthe MN surface area. This can be understood from the\nfact that the surface scattering is a solely mechanism for\nthe change of the electron velocity and acceleration. We\ndo not account any other radiative mechanisms in our\ntheory as, for instance, the electron-electron collisions.\nThe estimations of Γ for the spherical Au ( ǫ∞= 9.84)\nand Na ( ǫ∞= 1.14)37particles with 2 R= 400˚Aembed-\nded in the vacuum ( n= 1), in accordance with Eq. (26),\ngive: 6.7 and 0.795 meV (or 98 and 1209fs), respectively.\nIf the MN is embedded in the dielectric media with\nǫm>1, then the environment effect ought to be taken\ninto account.B. Environment effect\nBecause the effect of an electric field on the embedded\nnanoparticles becomes weaker in a dielectric media pro-\nportionally to its refractive index, the environment effect\nplays an important role. The spectral peculiarities of an\nenvironment effect recently were investigated for the Ag\nand Au nanoparticles, for instance, in Refs. [26,38].\nIn order to study the significance of the radiative\nlinewidth behavior in more general situations as above\npresented, it is necessary to perform the numerical cal-\nculation in Eq. (11) with the use of a general expression\nfor the conductivity tensor given by Eq. (14). Then, the\nradiative linewidth of SPR with an account for Eqs. (13)\nand (23) can be expressed in the form\nΓrad,(/bardbl\n⊥)=8π\n9n/parenleftBigωpl\nc/parenrightBig3R/bardblR2\n⊥L(/bardbl\n⊥)σ′\n(/bardbl\n⊥)/radicalBig\nǫ∞+(1/L(/bardbl\n⊥)−1)n2.(27)\nIt allows to account an other important factor, namely\nthe dependence of the radiative damping on the particle\nshape. As already was outlined,4,5,39any change of the\nnanoparticle shape from a sphere, that introducing of\nan anisotropy, results in the splitting of the SPR into\ntwo modes: a transverse one (Γ ⊥, perpendicular to the\nspheroid axis of a revolution) and a longitudinal one (Γ /bardbl,\nparallel to this axis).\nIn Fig. 1 we illustrate the behavior of the radiative\ndamping components as a function of the medium re-\nfractive index for a fixed particle axes ratio.\nThe calculations were conducted for the Au nanopar-\nticle with the use of Eqs. (27) and (14), and the fol-\nlowing parameters37:ne≃5.9×1022cm−3,υF=\n1.39×108cm/s,ωpl= 1.37×1016s−1andǫ∞= 9.84.\n1 2 3 4 5 64681012\nRefractive index,n (m eV  ) Γ ⊥\n||2 4 6 8 10 12 140.50.60.70.8\nR e f r a c t i v e  i n d e x , nΓ ( m eV ) Na\nR = 200 A\nR = 180 A0\n0Au\nFIG. 1. (Color online) The radiative linewidth of the surfac e plas-\nmon resonance components (longitudinal /bardbland transverse ⊥) vs\nthe medium refractive index for a prolate Au nanoparticles w ith\nthe axes ratio R⊥/R/bardbl= 0.618. The dashed line corresponds to the\nspherical Au particle with the radius R= 200˚A. The inset shows\nthe same dependence for two spherical Na nanoparticles with the\nradii of 180 and 200 ˚A(solid lines). The dashed lines represent\nthe Γ(n) in the case, when the oscillation terms in Eq. (30) are6\nneglected.\nWe observe from Fig. 1 that both radiative damping\ncomponents are decreased with increasing the medium\nrefractive index, in other words, the effect is weaker for\nhigher dielectric constant of environment. The more of\nMN radius or the less the refractive index is, the higher\nis the calculated radiative damping.\nIn general, the resonance plasmon damping in the pro-\nlateAu nanoparticle was found to be weaker along the\nspheroidrevolutionaxisthantheoneacrossthisaxis. For\ntheoblateAu nanoparticle, on the contrary, the damping\nalong the revolution axis was stronger than that across\nthis axis. This result holds regardless of whether the\nphoto-excitation is close to the SPR or far from it.\nThe resonance peak position is shifted with changing\nthe refractive index of the surrounding medium. The\nspectral direction of the shift depends on a number of\nfactors, well studied in earlier publications.4,5,14,26,38\n1. Radiative plasmon linewidth for a spherical MN in a\nmore general case\nWe consider here the case υ′=υ, when the Ψ function\nceases to depend on the angle θ. Then the integration\noverθin Eq. (14) gives 1/3 and we derive for σ′\nσ′\nsph=3ω2\npl\n16πRe/parenleftbiggΨ(q)\nν−iω/parenrightbigg\n, (28)\nwithq= 2R(ν−iω)/υtaken at υ=υF.\nLet us choose the case ν≪νs, for illustration. After\nsimple algebraic calculations with the use of Eqs. (15)\nand (16), we obtain in this case:\nσ′\nsph≃3\n8πνsω2\npl\nω2/bracketleftbigg\n1−2νs\nωsinω\nνs+2ν2\ns\nω2/parenleftbigg\n1−cosω\nνs/parenrightbigg/bracketrightbigg\n.\n(29)\nSubstituting Eqs. (29) and (23) into Eq. (11), we get\nΓsph≃ωpl\n9nξ/parenleftbiggRωpl\nc/parenrightbigg3/bracketleftbigg\n1−2\nξsinξ+2\nξ2(1−cosξ)/bracketrightbigg\n,\n(30)\nwith\nξ≡ξ(n,R) =2Rωpl\nυF√\n2n2+ε∞. (31)\nOne can see from Eq. (30) that the radiative damping\nin a spherical MN oscillates with altering of both the\nradiusofMNand/ortherefractiveindexofanembedding\nmedium. Eq. (30) at ξ≫1 transforms into Eq. (26).\nThe inset at the Fig. 1 shows the behavior of the SPR\nlinewidth for spherical Na nanoparticles with different\nradii. For calculation we use Eq. (30) and the following\nparameters: ne≃2.65×1022cm−3,υF= 1.07×108cm/s,\nωpl= 9.18×1015s−1, andǫ∞= 1.14.The dashed (smooth) curves in the inset in Fig. 1 cor-\nrespond to the simple case when both the second and\nthe last terms are neglected in the square brackets of\nEq. (30). In this simple case, the radiative linewidth of\nthe SPR is decreasedwith enhancing ofarefractiveindex\nnof the environment. This takes place because the ra-\ndiative plasmon decay is weakened in the media with the\nhigher refractive index owing to decrease of an external\nEM field inside the MN.\nAs one can see from the inset in Fig. 1, the oscillations\nof Γ around its smooth curve make significant correc-\ntions to the smooth picture, especially at large n. The\noscillations are well pronounced for the Na nanoparticles\nwith the small radii and are disappeared for NP with\na larger radii. This can be connected with the number\nof an electron oscillations between particle walls, which\nis decreased as the particle radius is increased. These\noscillations is damped markedly with ndecreasing and\npractically disappeared at n <2. The damping enhances\nas the radius of MN becomes larger.\n2. Effect of the particle size\nFigure2depicts the behaviorofthe radiativelinewidth\nof SPR as the function of MN radius for some fixed\nmedium refractiveindices. The calculations werefulfilled\nfor the Au nanoparticle with the use of Eq. (30) and the\nsame numerical parameters as given above.\nAs we can see in Fig. 2 and from Eq. (30), the Γ is\nincreased quadratically with R(dashed lines). We sup-\npose, this is due to the fact that the radiative damping in\nthe MN with a sphericalshape is proportionalto the area\nof a sphere. The growth in Γ occurs slower in the media\nwith a higher refractive index. The redaction of radiat-\ningdampinginthemediawith higher nimplies areduced\ndephasing of the plasmon mode. Because the effect of an\nexternal electric field on the embedded nanoparticle be-\ncomes weaker in a dielectric media with higher refractive\nindex, the resonanceradiative damping tends to decrease\nwithin the nanoparticle.\nThe Γ oscillates around a smooth curves with an in-\ncrease of the particle radius. The period of these oscilla-\ntions is enhanced for the MNs embedded in the medium\nwith a higher refractive index. The magnitude of these\noscillations is the greater the higher refractive index is.\nThe oscillations of Γ with Rfollow the quadratic depen-\ndence as well. The inset at the Fig. 2 shows the same\ndependence for Na nanoparticle embedded in the media\nwithn= 1 and n= 9.\nOn the oscillations of the SP lifetime as a function of\nnanoparticle size it was pointed out for the first time\nin Ref. 21, where the semiclassical theory was used to\nevaluate the SPR in MNs.7\n0 100 200 300 4000123456\nr a d i u s   \nP a r t i c l e ,R/aBn = 9n = 1.33Γ ( m eV )0 100 200 300 4000.00.20.40.60.8\n  \nP a r t i c l e r a d i u s ,R/aBΓ ( m eV )\nn = 9n = 1 NaAu\nFIG. 2. (Color online) The radiative linewidth of SPR vs radi us of\nspherical Au nanoparticle (in units of the Bohr radius aB= 0.53)\nembedded in the water n≃1.33 or in the medium with refractive\nindexn= 9. The dashed lines represent the Γ( R) in the case, when\nthe oscillation terms in Eq. (30) are neglected. The inset sh ows the\nsame dependence for Na nanoparticle.\nThe behavior of Γ at small particle radii R/aB<40\nin our approach practically does not depend on both the\nparticle radius and the medium refractive index and can\nbe described more precisely in the frame of the quantum\ntheory.\nSo far we haveconsideredonly radiativeprocesses. Be-\nlow, we dwell shortly on the nonradiative processes when\nthe electron scattering in the MN dissipates oscillation\nenergy into heat.\nC. Linewidth for a nonradiative processes\nAnother way for calculation of Γ j,radsupposes the\nknowledge of the value of a nonradiative damping\nΓj,nonrad. That is\nΓj,rad=σsca\nσabsΓj,nonrad, (32)\nwhereσscaandσabsare the light scattering and absorp-\ntion cross sections, respectively.\nThe nonradiative damping rate of a SP can be ex-\npressed as39\nΓj,nonrad(ω) =4πLj\nǫ′m+Lj(1−ǫ′m)σ′\njj(ω),(33)\nwith the σ′\njjgiven in the Section IV. Eq. (33) defines\nthe linewidth or, correspondingly, the decay time of the\nplasmon resonance due to electron scattering both from\nthe bulk and from surfaces of the particle.\nInthe caseofmediumwith ǫ′\nm→1, Eq.(33)isreduced\nmerely to\nΓj,nonrad(ω) = 4πLjσ′\njj(ω). (34)So, the decay time of the plasmon resonance is the opti-\ncal conductivity of the MN at the light frequency multi-\nplied by a geometrical factor. To study the nonradiative\nlinewidth for MNs with a given Lj, it is enough to calcu-\nlate the real part of the conductivity tensor as a function\nof frequency.\nAs one can see from Eq. (32), the radiative damping\nrate start to dominate as the light scattering cross sec-\ntion by MN exceeds the light absorption one in it. The\ncalculationsforsphericalMNinvacuumgivesthatradius\nof MN for which both cross sections become comparable\neach with other. In the high-frequency limit, we found\nthat\nRHF≃/parenleftbigg3\n2/parenrightbigg3/4\n(cυF)1/4/radicalbiggc\nωωpl. (35)\nFor instance, R≃100˚Afor Au particle at the plasmon\nfrequency ω=ωpl/√\n3.\nIn the case of low frequencies, we have\nRLF≃c\nωpl/radicalBigg\n6c\nυF√ǫm. (36)\nThe latter formula gives R≃7860˚Afor Au particle em-\nbedded in a medium with ǫm= 1.\nIn order to take into account the radiative damping to-\ngetherwithcollisionsoffreecarrierswiththeMNsurface,\nthe effective damping rate Γ eff= Γnonrad+Γradmust be\nintroduced. For understanding of the decay mechanism\nof the electron plasma oscillations the knowledge of the\ndecay time is of central importance.\nV. SUMMARY AND CONCLUSIONS\nWe used a local field approach and a kinetic equation\nmethod to study the plasmon resonance linewidth for\nmetalnonsphericalnanoparticlesembeddedinanydielec-\ntric media. It enables to calculate the radiativelinewidth\nfor MNs with different geometry with an account for the\nlight scattering from the particle surfaces.\nThe general formula is proposed for a damping rate or\na decaytime due to electron scatteringfrom the bulk and\nparticle surfaces. By means of this formula one can eval-\nuate the linewidth directly through the tensor of optical\nconductivity of the MN.\nWith changing the MNs shape from spherical to the\nspheroidal one, the single radiative plasmon resonance\nsplits into two components: the longitudinal and a trans-\nverse one to the spheroid rotation axis. We found that\nboth the radiative damping components for Au nanopar-\nticle are decreased with increasing the medium refractive\nindex. The resonanceplasmondampingin the prolateAu\nnanoparticle was found to be weaker along the spheroid\nrevolution axis than that across this axis.\nFor Na nanoparticle, we detect the oscillations of Γ\nwith the refractive index increasing. The amplitude of8\nthese oscillations enhances for media with higher dielec-\ntricconstant. TheoscillationsarewellpronouncedforNa\nnanoparticleswiththesmallradiiandaredisappearedfor\nNa nanoparticles with a larger radii.\nWe clearly show for spherical MNs that the radiative\nlinewidth of SPR enhances quadratically with the parti-\ncle radius increasing. The growth in Γ occurs slower in\nthe media with a higherrefractiveindex. The Γ oscillates\nas well with changing in the particle radius. The mag-\nnitude of these oscillations is enhanced markedly for the\nMNs embedded in the medium with a higher refractiveindex.\nThe effects of both the particle radius and the environ-\nment on the radiative plasmon resonance linewidth are\nillustrated by the example of Au and Na nanoparticles.\nThe contribution of the nonradiative plasmon decay is\ndiscussed as well.\nOur theoretical results should be important for the\nanalysis of the experimental data on the optical and\ntransport properties of MNs embedded in various dielec-\ntric media.\n1T.K. Sau, A.L. Rogach, F. J´ ackel, T.A. Klar and J. Feld-\nmann, ”Properties and applications of colloidal nonspher-\nical noble metal nanoparticles,” Advanced Materials, 22,\n1805 (2010).\n2H. Mertens and A. Polman, ”Strong luminescence quan-\ntum efficiency enhancement near prolate metal nanoparti-\ncles: dipolar versus higher-order modes,” J. Appl. Phys.\n105, 044302 (2009).\n3C. Langhammer, M. Schwind, B. Kasemo, and I. 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Hall, ”Dynamic modifications to the\nplasmon resonance of a metallic nanoparticle coupled\nto a planar waveguide: beyond the point-dipole limit,”\nJ. Opt. Soc. Am. B 19, 1195 (2002).\n21R.A. Molina, D. Weinmann, and R.A. Jalabert, ”Oscilla-\ntory size dependence of the surface plasmon linewidth in\nmetallic nanoparticles,” Phys. Rev. B 65,155427 (2002);\n”Oscillatory behaviorand enhancementof thesurface plas-\nmon linewidth in embedded noble metal nanoparticles,”\nEur. Phys. J. D 24,127 (2003).\n22M. Liu, M. Pelton, and P. Guyot-Sionnest, ”Reduced\ndamping of surface plasmons at low temperatures,” Phys.\nRev. B79, 035418 (2009).\n23E.A. Coronado, G.C. Schatz, ”Surface plasmon broadening\nfor arbitrary shape nanoparticles: A geometrical probabil -\nity approach,” J. Chem. Phys. 119, 3926 (2003).\n24A. Wokaun, J.P. Gordon, and P.F. Liao, ”Radiation damp-\ning in surface enhanced Raman scattering,” Phys. Rev.\nLett.48, 957 (1982).\n25M. Meier and A. 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Jackson, Classical Electrodynamics (Willey, New\nYork, 2001).\n33N.I. Grigorchuk, P.M. Tomchuk, ”Optical and transport\nproperties of spheroidal metal nanoparticles with account\nfor the surface effect,” Phys. Rev. B 84, 085448 (2011).\n34For prolate spheroid ( a > b=c):R/bardbl≡aandR⊥≡b, butfor oblate one ( a=b > c):R/bardbl≡bandR⊥≡a.\n35N.I. Grigorchuk and P.M. Tomchuk, ”Theory for ab-\nsorption of ultrashort laser pulses by spheroidal metallic\nnanoparticles,” Phys. Rev. B 80, 155456 (2009).\n36P.M. Tomchuk and N.I. Grigorchuk, ”Shape and size ef-\nfects on the energy absorption by small metallic particles, ”\nPhys. Rev. B 73, 155423 (2006).\n37Ch.Kittel, Introduction to Solid State Physics (Wiley, New\nYork, 2005).\n38M.M. Miller and A.A. Lazarides, ”Sensitivity of metal\nnanoparticle plasmon resonance band position to the di-\nelectric environment as observed in scattering,” J. Opt. A:\nPure Appl. Opt., 8, S239 (2006).\n39N.I. Grigorchuk, ”Plasmon resonant light scattering on\nspheroidal metallic nanoparticle embedded in a dielectric\nmatrix,” Eur. Phys. Lett. 97, 45001 (2012)."    },    {        "title": "1211.0492v1.Dynamic_Spin_Injection_into_Chemical_Vapor_Deposited_Graphene.pdf",        "content": "Page 1 of 13 \n Dynamic Spin Injection into Chemical Vapor Deposited  \nGraphene  \nA. K. Patra1,a), S. Singh2,a), B. Barin2, Y. Lee3, J.-H. Ahn3, E. del Barco2,b), E. R. \nMucciolo2 and B. Özyilmaz1,4,5,6 ,b) \n1Department of Physics, National University of Singapore, 2 Science Dri ve 3, Singapore 117542  \n2Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n3 School of Advanced Materials Science & Engineering, SKKU Advanced Institute of \nNanotechnology  (SAINT) , Sungkyunkwan University, Suwon, Republic  Kore a 440746  \n4NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n5Graphene Research Center, National University of Singapore, Singapore 117542  \n6NUS Graduate School for Integrative Sciences and Engineering (NGS), National Univer sity of \nSingapore, Singapore 117456  \n \nWe demonstrate dynamic spin injection into chemical vapor deposition (CVD)  grown \ngraphen e by s pin pumping from permalloy  (Py) layer s. Ferromagnetic resonance \nmeasurements at room temperature reveal a strong enhancement of the  Gilbert damping \nat the Py/graphene interface , exceeding that observed in even Py/platinum  interfaces.  \nSimilar results are als o shown on Co/graphene layers . This enhancement  in the Gilbert \ndamping is understood as the consequence of spin pumping at t he interface d riven by \nmagnetization dynamics .  Our observation s suggest a  strong enhancement of spin -orbit \ncoupling in CVD graphene , in agreement with earlier spin valve measurements.    \na)A. K. Patra and S. Singh contributed equally to this work.  \nb) Autho rs to whom correspondence should be addressed. Electronic mail s: delbarco@physics.ucf.edu  and \nphyob@nus.edu.sg  \n \nPACS numbers: 72.25. -b, 76.50.+g  Page 2 of 13 \n In spintronic s, where the electron’s  spin degree of freedom , rather than its charge , is \nemployed to process in formation , the efficient generation of the large spin current s stands as  a \nkey requirement for future spintronic device s and applications. Several approaches to generate \npure spin currents  have been proposed and  are being widely investigated, namely, non-local spin \ninjection  [1], spin Hall effect  [2-4], and spin pumping  [5,6]. Among these , spin pumping offers  \nthe advantage  of producing  spin current s over large ( mesoscopic ) areas [7-13] at \nferromagnet ic/non-magnetic  (FM/NM)  interface s. In addition, dynamical  spin pumping is \ninsensitive to a potential impedance mismatch at the FM/NM interface  [14], a problem  \nubiquitous  in the non -local spin injection approach.  Dynamical s pin pumping consists of  \ngenerating pure spin current (i.e., with no net charge current) away from  a ferromagnet  into a \nnon-magnetic material,  induced by the  coherent precession of the magnetization  upon application \nof microwave stimuli of frequency matching the  ferromagnetic resonance (FMR)  of the system  \n[15]. Since pure spin currents carry awa y spin angular momentum, in an FMR experiment the \ntransfer of angular momentum from the FM into the NM layer results in an  enhance ment of the \nGilbert damping in the ferromagnet [5-15]. Most  studies of dynamical spin pumping on FM/NM  \ninterfaces have made us e of Pt and Pd NM layers, since the large spin -orbit coupling in these \nsystems enables the conversion of the injected spin current into an electric voltage  across the N M \nlayer, a phenomenon known as i nverse spin Hall effect  (ISHE) . Recently, spin pumping h as been \nexperimentally demonstrated in FM/semiconductor interfaces (e.g. , GaAs  [13] and p -type Si  \n[15]). However, there is no experimental report on spin pumping in FM/graphene interface s, \nthough graphene  [16] (a two -dimensional layer  of carbon atoms ), pos sesses unique electronic \nproperties (e.g. high mobility  and gate-tunable charge carrier , among others ), and stands as an \nexcellent material for spin transport due to its large spin coherence length [17]. Page 3 of 13 \n In this Letter we report experimental FM R studies of Py and Co  films and  polycrystalline \ngraphene grown by chemical vapor deposition on Cu foils [ 18,19] (henceforth, Co/Gr and Py/Gr , \nrespectively ) performed in  a broad -band microwave coplanar waveguide (CPW)  spectrometer . \nThe observation of a remarkable  broadening of the FMR absorption peaks in the Py/ Gr (88%)  \nand Co/Gr (133%) films demonstrate a strong  increase  of the Gilbert damping in the FM layer  \ndue to spin pumping at the FM/Gr interface and the consequent loss of angular momentum  \nthrough spin inject ion into the CVD graphene  layer . To account for such a remarkable absorption  \nof angular momentum , the spin orbit coupling in CVD graphene must be orders of magnitude \nlarge r than what is predicted for pristine, exfoliated graphene . \nTo prepare the FM/Gr sample s, single layer CVD grown graphene [18, 19] was first \ntransferred onto a Si substrate with 300 nm thick SiO 2 layer . The sample was then anneal ed in a \nH2/Ar environment at 300 °C for 3 hour to remove all organic residue s. For the Py layer we chose \nNi80Fe20, a material extensively  used for magnetic thin film  studies because  of its low \nmagneto crystalline anisotropy  and its insensitivity to strain.  The FM layer  (Py/14nm, Co/15nm)  \nwas deposited on top of the graphene layer lying over the SiO 2/Si substrate by electron-beam \nevaporat ion at a base pressure of 310-7 Torr. For the purpose of FMR comparison experiments, \na control FM film of the same thickness was deposited simultaneously on the same SiO 2 wafer in \nan area where graphene was no t present . The schematic of th e FM/Gr samples  is shown in \nFig. 1-a, together with the Raman spectr um of the  CVD  graphene  before the deposition of Py  \n(Fig. 1-c). The high intens ity of the  2D peak , when  compared to the G peak , and the  weakness of \nthe D peak , suggests that graphene is sin gle layer and of high quality ( i.e. low degree of \ninhomogeneity/defects ).   Page 4 of 13 \n \n \nFig. 1 : (Color online) (a) Schematic of the FM/Gr film sample. (b) Schematic of the FMR measurement \nsetup, with the sample placed up -side-down on top of the micro -CPW . (c) Raman spectrum of CVD \ngraphene.  \nFMR measurements were carried out at room temperature with a high -frequency \nbroadband (1 -50 GHz) micro -coplanar -waveguide ( -CPW) [20] using the flip -chip method [ 21-\n23], by which the sample is placed up -side-down covering the cen tral part of the CPW  (as shown \nin Fig. 1-b), where the transmission line is constricted to increase the density of the microwave \nfield and enhance sensitivity . The CPW was covered with a 100nm -thick insulating layer of \nPMMA resist, hardened by electron bea m exposure, to avoid any influence of the CPW, made \nout of gold, on the sample dynamics. A 1.5  Tesla rotatable  electromagnet was employed to vary \nthe applied field direction from the in -plane ( = 0o) to normal -to-the film plane (  = 90o) \ndirections . Fig. 2-a shows t he angular dependence of the FMR field measured at 10  GHz for both \nPy and Py/Gr films. The rotation plane is chosen to keep the dc magnetic field , H, perpendicular \nto the microwave field felt by the sample at all times, as shown in Fig. 1-b. The resonance fiel d \nincreases as the magnetic field is directed away from the film plane (i.e. increasing ), as \nexpected for a thin film ferromagnet with in-plane shape magneto -anisotropy.  The angular \ndependence of the  FMR field ( HR) can be fitted using the resonance frequency condition given \nby the Smit and Beljers formula [ 23,24], Page 5 of 13 \n \n21HH,                                                                       (1)  \nwhere \nf2  is the angular frequency, \n/Bg  the gyromagnetic ratio, and H1 and H2 are \ngiven by  \nH1=Hcos(-)-4Meffsin2\nH2=Hcos(-)4Meffcos22K2\nMssin2\n ,                                       (2) \nwhere  is the magnetization  angle , \n 2\n2 1 cos 4 2 4 4S S S eff MK MK M M   is the \neffective demagnetization field,  Ms is the saturation magnetization, and K1 and K2 are the first \nand second order anisotropy energies, respectively . The best fit s to the data in Fig. 2-a are given \nby the parameters shown  in the third  column of Table 1 , together with the corresponding \nparameters extracted from equivalent measurements on the Co and Co/Gr films (not shown) . \n \n \nFig. 2 : (color online) (a) Angular dependence of the FMR fi elds measured on both Py and Py/Gr samples \nat f = 10 GHz with the dc magnetic field, H, applied in a plane perpendicular to the microwave field \ngenerated by  the CPW at the sample position. (b) In-plane frequency dependence  of the FMR fields for \nboth Py and  Py/Gr samples. The intercepts with the x-axis give the effective demagnetizing fields of the \nsamples.  Page 6 of 13 \n It is useful to study the resonant behavior by applying the magnetic field at  = 0o \n(parallel configuration) and   = 90 (perpendicular configuration),  since the frequency behavior \nof the FMR fields are given respectively by,  \n       \n\n\n\n\n \n\n\n\n,//,2\n//\n4) 4 (\neff Reff R R\nM HM HH\n ,             (3) \nwhere \nBg  is the gyromagnetic ratio , \n1 //,4 4A S eff H M M   , \n2 1 ,4 4A A S eff H H M M \n, with \nS A MK H1 12  and \nS A MK H2 24  the first and second \norder anisotropy fields, respectively , which relate to su rface, interface and/or magnetoelastic \nanisotropy. Note that K 1 > 0 (>> K2) provides out -of-plane anisotropy, competing with the in -\nplane shape anisotropy . Consequently, a graphical representation of the in - and out -of-plane \nfrequency response of the FMR f ields , conforming to Eq . (3), results in a linear behavior from \nwhich the slope and intercept with the magnetic field axis give  and the effective \ndemagnetiz ation field s, respectively.  The results obtained for the Py and Py/Gr samples are \nshown in Fig. 2-b and 2-c, and the extracted parameters are listed  in the third  column of Table  1, \ntogether with those extracted from the Co and Co/Gr . Note that the anisotropy fields depend on \nthe selection of the saturation magnetization, w ith theoretical values MS,Py = 9.27 kG (attending \nto a 20/80 -Ni/Fe ratio  and assuming identical densities) , and MS,Co = 17.59 kG. For the Co and \nCo/Gr films, the effective saturation magnetization ( Meff = 17.7 kG) is similar to the one \nexpected from theo ry, hence there is negligible ou t-of-plane anisotropy (K 1 ~ 0), in agreement \nwith previous studies [ 25]. The situation is different in the case of the Py and Py/Gr, where the \nsmall Py anisotropy field HA1 = 1.98 kG grows significantly in the Py/Gr ( HA1 = 3.60 kG), \nsuggesting an increase of the Py surface anisotropy due to the presence of the graphene layer ( i.e. Page 7 of 13 \n interface effect). Nevertheless, the magnetization remains in the plane of the film  for all samples . \nTheory  Sample  HR vs.  , f Damping  Changes  \nPy: Ni80Fe20 \ngeff = 2.10  \nFeFe\nS NiNi\nSFe\nSNi\nS\neffg M g MM Mg\n2.0 8.02.0 8.0\n\n \n \nMs = 9.27 kG \nFe\nSNi\nS M M Ms 2.0 8.0\n \n \nwith \n21.2 094.6  NiNi\nS g kG M\n \n0.2 016.22  FeFe\nS g kG M\n \n Py g = 2.110 \n = 0.0 113 \nG = 0.311 GHz \nK1 increases  \n(interface)  \n \nDamping increases \nby ~88% Meff = 7.30 kG \nH1 = 1.98 kG \nK1 = 0.73106 erg/cc  \nPy/Gr  g = 2.107 \n = 0.0 213 \nG = 0.585 GHz Meff,// = 5.70 kG \nHA1 = 3.60 kG \nK1 = 1.32106 erg/cc  \nCo \ng = 2.145 \nMs = 17.59  kG Co g = 2. 149  = 0.0 210 \nG = 1.11 GHz (no K1)  \nDamping increases \nby ~133% Meff = 17.7 kG \nCo/Gr  g = 2. 149  = 0.0 489 \nG = 2.59 GHz Meff = 17.5 kG \n \nTABLE  I: Parameters extracted from the analysis of the data reported in this work.  \n \nWe now focus on the FMR linewidth and its frequency dependence when the magnetic \nfield is applied parallel to the film (  = 0o), from which information about the Gilbert damping \n(i.e., spin relaxation dynamics)  can be directly extracted. The inset to  Fig. 3 shows a field \nderivate of the CPW S 21 transmission parameter obtained when exciting the FMR at 10  GHz in \nboth Py and Py/Gr samples , with HR = 1.28 kG and 1.55 kG, respectively . The peak -to-peak \ndistance repre sents the linewidth , H, of the FMR, whose behavior as a function of frequency is \nshown for both samples in the main panel of Fig. 3. A remarkable in crease of the FMR linewidth \nby 88% is observed in the Py/Gr sample , and even higher (133%) in the Co/ Gr fil ms. The change \nin the linewidth  must be attributed to a substantial enhancement of the Gilbert damping in the Page 8 of 13 \n FM film due to the influence of the graphene directly underneath.  The frequency dependence of \nthe FMR linewidth can be written as a contribution f rom two parts : \nf H H\n\n34\n0\n,                                                               ( 4) \nwhere  is the parameter of the Gilbert damping \nSM G . The first term, \n0H , accounts for \nsample -dependent in homogeneous broadening of the linewidth and is independent of frequenc y, \nwhile the second term represents the dynamical  broadening of the FMR  and scales linearly with \nfrequency .  \n \n \nFig. 3: (color online) Frequency dependence of the FRM linewidth for Py , Py/Gr , Co and Co/Gr  films \nobtained with the magnetic field applied at  = 0 (in-plane configuration). The inset shows the field \nderivatives of CPW S 21 transmission parameter (at 10  GHz)  of the Py and Py/Gr samples , from which the \nlinewidth, H, is calculated as the peak -to-peak distance.  \n Page 9 of 13 \n As observed in Fig. 3, the measured l inewidth for  both FM and FM/Gr samples increase s \nlinearly with frequency, with negligible  inhomogen eous broadening,  indicat ing that damping in \nthe FM film can be properly  explained by the phenomenological Landau-Lifshitz -Gilbert \ndamp ing model . A similar br oadening of the FMR linewidth is observed in both samples when \nthe field is applied perpendicular to the plane, excluding frequency -dependence inhomogeneous \nbroadening (e.g. two -magnon scattering produced by changes in morphology of the FM surface \n[26]), as its possible source. By fitting the data in Fig. 3 to Eq. (4) (using \n0H  = 0), the damping \nparameter s  and G are determined and given in the fourth  column of Table  1 for all studied \nsamples. The Gilbert damping increases  substantially in the FM/Gr films  as a result of the \nincreased linewidth, when compared to the values  obtained in the FM sample s (which are \ncomparable  with values given in the literature  for similar Py  and Co  films  [9,22]). This is our key \nfinding.  Remarkably , the change in the damping para meter  in the Py/Gr sample \n(\nPy GrPy /  = 0.01 ) is even  more pronounced  than those observed in  Py/Pt systems, in \nwhich the thick (when compared to graphene) heavy transition metal  Pt layer provides the  large \nspin-orbit coupling necessary to absorb (i.e. , relax ) the spin accumulation  pumped away from the \nferromagnet.  The efficiency of spin injection is usually cataloged by means of the interfacial \nspin-mixing conductance, which is proportional to the additional damping parameter, , as \nfollows:   \n FMSdMg4\n,                                                               (5) \ngiving \ng = 5.261019 m-2 for our Py/Gr sample  with the thickness of the Py film dFM = 14 nm. \nThe Py/Gr  value is substantially larger than those found in other Py/NM systems with a metallic Page 10 of 13 \n NM layer, e.g. , \ng = 2.191019 m-2 in Py(Ni 81Fe19:10nm)/Pt(10nm)  [9] or \ng = 2.11019 m-2 in \nPy(Ni 80Fe20:15nm)/Pt(15nm)  [11]. Note that in the cited experiments, the spin -diffusion length \nof the non -magnetic layer (~10  nm for Pt) is smaller than the layer thickness. This is sign ificant \nsince it explains how the Pt layer is capable of dissipating the spin accumulation generated by the \ndynamical spin pumping , and account for the loss of angular momentum in the Py . In the case of \ngraphe ne, the enhancement of the damping parameter  is more complicated to understand. In a \nstandard FM/NM metallic system , the spin current injected in to the NM layer decays mainly \nperpendicularly to the interface [ 27], causing the enhancement of the damping parameter to \ndepend on the ratio between the layer  thickness and the spin -diffusion length in the NM . \nHowever, graphene  has effectively zero thickness  and, at least theoretically , a very weak intrinsic \nspin-orbit coupling. Therefore, the spin current must  decay in a FM/Gr film parallel and not \nperpendicul ar to the interface . Furthermore, some sizable spin -orbit coupling must exist in CVD \ngraphene  films . The lat ter may also explain the generally observed very short spin relaxation \ntimes in lateral CVD graphene spin valves  [28]. Recently, small levels of hyd rogen  [29] and \ncopper adatoms [30] have been predicted to lead to a strong e nhancement of the spin-orbit \ncoupling, bringing it into  meV range. Cu adat oms are  certainly likely to be present  in the CVD  \nsamples utilized in our experiments, pointing at a possi ble explanation for the large spin \npumping effect observed in our FM/Gr films .  \n  \n \n Page 11 of 13 \n  \nReferences:  \n[1] F. J. Jadema, A. T. Filip, and B. J. van Wees: Nature  (London) 410, 345  (2001)  \n[2] 2.T. Kimura, Y. Ot ani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett.  98, 156601 \n(2007).  \n[3] S. O. Valenzuela and M. Tinkham, Nature (London ) 442, 176 (2006 ). \n[4] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. 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"    },    {        "title": "1211.3611v2.Spin_transport_and_tunable_Gilbert_damping_in_a_single_molecule_magnet_junction.pdf",        "content": "Spin transport and tunable Gilbert damping in a single-molecule magnet junction\nMilena Filipović,1Cecilia Holmqvist,1Federica Haupt,2and Wolfgang Belzig1\n1Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n2Institut für Theorie der Statistischen Physik, RWTH Aachen, D-52056 Aachen, Germany\n(Dated: October 24, 2019)\nWe study time-dependent electronic and spin transport through an electronic level connected\nto two leads and coupled with a single-molecule magnet via exchange interaction. The molecular\nspin is treated as a classical variable and precesses around an external magnetic field. We derive\nexpressions for charge and spin currents by means of the Keldysh nonequilibrium Green’s functions\ntechniqueinlinearorderwithrespecttothetime-dependentmagneticfieldcreatedbythisprecession.\nThe coupling between the electronic spins and the magnetization dynamics of the molecule creates\ninelastic tunneling processes which contribute to the spin currents. The inelastic spin currents, in\nturn, generate a spin-transfer torque acting on the molecular spin. This back-action includes a\ncontribution to the Gilbert damping and a modification of the precession frequency. The Gilbert\ndamping coefficient can be controlled by the bias and gate voltages or via the external magnetic\nfield and has a nonmonotonic dependence on the tunneling rates.\nPACS numbers: 73.23.-b, 75.76.+j, 85.65.+h, 85.75.-d\nI. INTRODUCTION\nSingle-molecule magnets (SMMs) are quantum mag-\nnets, i.e., mesoscopic quantum objects with a perma-\nnent magnetization. They are typically formed by\nparamagnetic ions stabilized by surrounding organic\nligands.1SMMs show both classical properties such\nas magnetization hysteresis2and quantum properties\nsuch as spin tunneling,3coherence,4and quantum phase\ninterference.2,5They have recently been in the center of\ninterest2,6,7in view of their possible applications as in-\nformation storage8and processing devices.9\nCurrently, a goal in the field of nanophysics is to\ncontrol and manipulate individual quantum systems, in\nparticular, individual spins.10,11Some theoretical works\nhave investigated electronic transport through a molecu-\nlarmagnetcontactedtoleads.12–19Inthiscase, thetrans-\nport properties are modified due to the exchange inter-\naction between the itinerant electrons and the SMM,20\nmaking it possible to read out the spin state of the\nmolecule using transport currents. Conversely, the spin\ndynamics and hence the state of an SMM can also be\ncontrolled by transport currents. Efficient control of the\nmolecule’s spin state can be achieved by coupling to fer-\nromagnetic contacts as well.21\nExperiments have addressed the electronic trans-\nport properties through magnetic molecules such as\nMn12and Fe 8,22,23which have been intensively stud-\nied as they are promising candidates for memory\ndevices.24Various phenomena such as large conduc-\ntance gaps,25switching behavior,26negative differen-\ntial conductance, dependence of the transport on mag-\nnetic fields and Coulomb blockades have been exper-\nimentally observed.22,23,27,28Experimental techniques,\nincluding, for instance, scanning tunneling microscopy\n(STM),22,23,29–31break junctions,32and three-terminal\ndevices,22,23,27have been employed to measure elec-\ntronic transport through an SMM. Scanning tunnelingspectroscopy and STM experiments show that quantum\nproperties of SMMs are preserved when deposited on\nsubstrates.29The Kondo effect in SMMs with magnetic\nanisotropyhasbeeninvestigatedboththeoretically14and\nexperimentally.33,34It has been suggested35and experi-\nmentally verified36that a spin-polarized tip can be used\nto control the magnetic state of a single Mn atom.\nIn some limits, the large spin Sof an SMM can be\ntreated as a classical magnetic moment. In that case,\nthe spin dynamics is described by the Landau-Lifshitz-\nGilbert (LLG) equation that incorporates effects of ex-\nternal magnetic fields as well as torques originating from\ndamping phenomena.37,38In tunnel junctions with mag-\nnetic particles, LLG equations have been derived using\nperturbativecouplings39,40andthenonequilibriumBorn-\nOppenheimer approximation.16Current-induced magne-\ntization switching is driven by a generated spin-transfer\ntorque (STT)41–44as a back-action effect of the elec-\ntronic spin transport on the magnetic particle.16,45–47\nA spin-polarized STM (Ref. 36) has been used to\nexperimentally study STTs in relation to a molecular\nmagnetization.48This experimental achievement opens\nnew possibilities for data storage technology and appli-\ncations using current-induced STTs.\nThe goal of this paper is to study the interplay be-\ntween electronic spin currents and the spin dynamics of\nan SMM. We focus on the spin-transport properties of\na tunnel junction through which transport occurs via a\nsingle electronic energy level in the presence of an SMM.\nThe electronic level may belong to a neighboring quan-\ntum dot (QD) or it may be an orbital related to the\nSMM itself. The electronic level and the molecular spin\nare coupled via exchange interaction, allowing for inter-\nactionbetweenthespinsoftheitinerantelectronstunnel-\ning through the electronic level and the spin dynamics of\nthe SMM. We use a semiclassical approach in which the\nmagnetization of the SMM is treated as a classical spin\nwhose dynamics is controlled by an external magneticarXiv:1211.3611v2  [cond-mat.mes-hall]  22 Jul 20132\nfield, while for the electronic spin and charge transport\nwe use instead a quantum description. The magnetic\nfield is assumed to be constant, leading to a precessional\nmotion of the spin around the magnetic field axis. The\nelectronic level is subjected both to the effects of the\nmolecular spin and the external magnetic field, generat-\ningaZeemansplitofthelevel. Thespinprecessionmakes\nadditional channels available for transport, which leads\nto the possibility of precession-assisted inelastic tunnel-\ning. During a tunnel event, spin-angular momentum may\nbe transferred between the inelastic spin currents and the\nmolecular spin, leading to an STT that may be used to\nmanipulate the spin of the SMM. This torque includes\nthe so-called Gilbert damping , which is a phenomenolog-\nically introduced damping term of the LLG equation,38\nand a term corresponding to a modification of the pre-\ncession frequency. We show that the STT and hence the\nSMM’s spin dynamics can be controlled by the external\nmagnetic field, the bias voltage across the junction, and\nthe gate voltage acting on the electronic level.\nThe paper is organized as follows: We introduce our\nmodel and formalism based on the Keldysh nonequilib-\nrium Green’s functions technique49–51in Sec. II, where\nwe derive expressions for the charge and spin currents in\nlinear order with respect to a time-dependent magnetic\nfield and analyze the spin-transport properties at zero\ntemperature. In Sec. III we replace the general magnetic\nfield of Sec. II by an SMM whose spin precesses in an\nexternal constant magnetic field, calculate the STT com-\nponents related to the Gilbert damping, and the modifi-\ncationoftheprecessionfrequency, andanalyzetheeffects\nof the external magnetic field as well as the bias and gate\nvoltages on the spin dynamics. Conclusions are given in\nSec. IV.\nII. CURRENT RESPONSE TO A TIME\nDEPENDENT MAGNETIC FIELD\nA. Model and Formalism\nFor the sake of clarity, we start by considering a junc-\ntion consisting of a noninteracting single-level QD cou-\npled with two normal, metallic leads in the presence of an\nexternal, time-dependent magnetic field (see Fig. 1). The\nleads are assumed to be noninteracting and unaffected\nby the external field. The total Hamiltonian describing\nthe junction is given by ^H(t) = ^HL;R+^HT+^HD(t).\nThe Hamiltonian of the free electrons in the leads reads\n^HL;R=P\nk;\u001b;\u00182fL;Rg\u000fk\u001b\u0018^cy\nk\u001b\u0018^ck\u001b\u0018, where\u0018denotes the\nleft (L) or right ( R) lead, whereas the tunnel cou-\npling between the QD and the leads can be written as\n^HT=P\nk;\u001b;\u00182L;R[Vk\u0018^cy\nk\u001b\u0018^d\u001b+V\u0003\nk\u0018^dy\n\u001b^ck\u001b\u0018]. The spin-\nindependent tunnel matrix element is given by Vk\u0018. The\noperators ^cy\nk\u001b\u0018(^ck\u001b\u0018)and ^dy\n\u001b(^d\u001b)are the creation (an-\nnihilation) operators of the electrons in the leads and\nthe QD, respectively. The subscript \u001b=\";#denotes\neVB(t)\nFIG. 1: (Color online) A quantum dot with a single electronic\nlevel\u000f0coupled to two metallic leads with chemical potentials\n\u0016Land\u0016Rinthepresenceofanexternaltime-dependentmag-\nnetic field~B(t). The spin-transport properties of the junction\naredeterminedbythebiasvoltage eV=\u0016L\u0000\u0016R, theposition\nof the level \u000f0, the tunnel rates \u0000Land\u0000R, and the magnetic\nfield.\nthe spin-up or spin-down state of the electrons. The\nelectronic level \u000f0of the QD is influenced by an ex-\nternal magnetic field ~B(t)consisting of a constant part\n~Bcand a time-dependent part ~B0(t). The Hamiltonian\nof the QD describing the interaction between the elec-\ntronic spin ^~ sand the magnetic field is then given by\n^HD(t) = ^Hc\nD+^H0(t), where the constant and time-\ndependent parts are ^Hc\nD=P\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~Bcand\n^H0(t) =g\u0016B^~ s~B0(t). The proportionality factor gis the\ngyromagnetic ratio of the electron and \u0016Bis the Bohr\nmagneton.\nThe average charge and spin currents from the left lead\nto the electronic level are given by\nIL\u0017(t) =q\u0017\u001cd\ndt^NL\u0017\u001d\n=q\u0017i\n~\n\u0002^H;^NL\u0017\u0003\u000b\n;(1)\nwhere ^NL\u0017=P\nk;\u001b;\u001b0^cy\nk\u001bL(^\u001b\u0017)\u001b\u001b0^ck\u001b0Lis the charge and\nspin occupation number operator of the left contact. The\nindex\u0017= 0corresponds to the charge current, while\n\u0017=x;y;zindicates the different components of the spin-\npolarized current. The current coefficients q\u0017are then\nq0=\u0000eandq\u00176=0=~=2. In addition, it is useful to\ndefine the vector ^\u001b\u0017= (^1;^~ \u001b), where ^1is the identity\noperator and ^~ \u001bconsists of the Pauli operators with ma-\ntrix elements (^~ \u001b)\u001b\u001b0. Using the Keldysh nonequilibrium\nGreen’s functions technique, the currents can then be ob-\ntained as50,51\nIL\u0017(t) =\u00002q\u0017\n~Re\u0002\ndt0Tr\b\n^\u001b\u0017[^Gr(t;t0)^\u0006<\nL(t0;t)(2)\n+^G<(t;t0)^\u0006a\nL(t0;t)]\t\n;\nwhere ^Gr;a;<are the retarded, advanced, and lesser\nGreen’s functions of the electrons in the QD with the ma-\ntrix elements Gr;a\n\u001b\u001b0(t;t0) =\u0007i\u0012(\u0006t\u0007t0)hf^d\u001b(t);^dy\n\u001b0(t0)gi\nandG<\n\u001b\u001b0(t;t0) =ih^dy\n\u001b0(t0)^d\u001b(t)i, while ^\u0006r;a;<\nL(t;t0)\nare self-energies from the coupling between the QD\nand the left lead. Their nonzero matrix elements3\nare diagonal in the electronic spin space with re-\nspect to the basis of eigenstates of ^sz, given by\n\u0006r;a;<\nL(t;t0) =P\nkVkLgr;a;<\nkL(t;t0)V\u0003\nkL. The Green’s func-\ntionsgr;a;<\nkL(t;t0)are the retarded, advanced and lesser\nGreen’s functions of the free electrons in the left lead.\nThe retarded Green’s functions ^Gr\n0of the electrons in\nthe QD, in the presence of the constant magnetic field\n~Bc, are found using the equation of motion technique,52\nwhile the lesser Green’s functions ^G<\n0are obtained from\nthe Keldysh equation ^G<\n0=^Gr\n0^\u0006<^Ga\n0, where multipli-\ncation implies internal time integrations.51The time-\ndependent part of the magnetic field can be expressed\nas~B0(t) =P\n!(~B!e\u0000i!t+~B\u0003\n!ei!t), where~B!is a com-\nplex amplitude. This magnetic field acts as a time-\ndependent perturbation that can be expressed as ^H0(t) =P\n!(^H!e\u0000i!t+^Hy\n!ei!t), where ^H!is an operator in the\nelectronic spin space and its matrix representaton in the\nbasis of eigenstates of ^szis given by\n^H!=g\u0016B\n2\u0012\nB!zB!x\u0000iB!y\nB!x+iB!y\u0000B!z\u0013\n:(3)\nApplying Dyson’s expansion, analytic continuation rules\nand the Keldysh equation,51one obtains a first-order ap-\nproximation of the Green’s functions describing the elec-\ntrons in the QD that can be written as\n^Gr\u0019^Gr\n0+^Gr\n0^H0^Gr\n0; (4)\n^G<\u0019^Gr\n0^\u0006<^Ga\n0+^Gr\n0^H0^Gr\n0^\u0006<^Ga\n0+^Gr\n0^\u0006<^Ga\n0^H0^Ga\n0:\nThe expression for the currents in this linear approxima-\ntion is given by\nIL\u0017(t) =\u00002q\u0017\n~Re Tr\b\n^\u001b\u0017[^Gr\n0^\u0006<\nL+^G<\n0^\u0006a\nL (5)\n+^Gr\n0^H0^Gr\n0^\u0006<\nL+^Gr\n0^H0^G<\n0^\u0006a\nL+^G<\n0^H0^Ga\n0^\u0006a\nL]\t\n:\nEq. (5) is then Fourier transformed in the wide-\nband limit, in which the level width function, \u0000(\u000f) =\n\u00002 Imf\u0006r(\u000f)g, is constant, Ref\u0006r(\u000f)g= 0, and one can\nhence write the retarded self-energy originating from the\ndot-lead coupling as \u0006r;a(\u000f) =\u0007i\u0000=2. From this trans-\nformation, one obtains\nIL\u0017(t) =Idc\nL\u0017+X\n![IL\u0017(!)e\u0000i!t+I\u0003\nL\u0017(!)ei!t]:(6)\nUsing units in which ~= 1, the dc part of the\ncurrents51Idc\nL\u0017and the time-independent complex com-\nponentsIL\u0017(!)are given by\nIdc\nL\u0017=q\u0017\u0002d\u000f\n\u0019\u0000L\u0000R\n\u0000[fL(\u000f)\u0000fR(\u000f)] Tr Imf^\u001b\u0017^Gr\n0(\u000f)g(7)\nand\nIL\u0017(!) =\u0000iq\u0017\u0002d\u000f\n2\u0019\u0000L\u0000R\n\u0000n\n[fL(\u000f)\u0000fR(\u000f)] (8)\n\u0002Trf^\u001b\u0017[^Gr\n0(\u000f+!)^H!^Gr\n0(\u000f) + 2iImf^Gr\n0(\u000f)g^H!^Ga\n0(\u000f\u0000!)]g\n+X\n\u0018=L;R\u0000\u0018\n\u0000R[f\u0018(\u000f\u0000!)\u0000fL(\u000f)] Tr[^\u001b\u0017^Gr\n0(\u000f)^H!^Ga\n0(\u000f\u0000!)]o\n:In the above expressions, f\u0018(\u000f) = [e(\u000f\u0000\u0016\u0018)=kBT+ 1]\u00001is\nthe Fermi distribution of the electrons in lead \u0018, where\nkBis the Boltzmann constant. The retarded Green’s\nfunction ^Gr\n0(\u000f)is given by ^Gr\n0(\u000f) = [\u000f\u0000\u000f0\u0000\u0006r(\u000f)\u0000\n(1=2)g\u0016B^~ \u001b~Bc]\u00001.16\nThe linear response of the spin current with respect\nto the applied time-dependent magnetic field can be ex-\npressed in terms of complex spin-current susceptibilities\ndefined as\n\u001fL\n\u0017j(!) =@IL\u0017(!)\n@B!j; j =x;y;z: (9)\nThe complex components IL\u0017(!)are conversely given by\nIL\u0017(!) =P\nj\u001fL\n\u0017j(!)B!j. By taking into account that\n@^H!=@B!j= (1=2)g\u0016B^\u001bjand using Eq. (8), the current\nsusceptibilities can be written as\n\u001fL\n\u0017j(!) =\u0000iq\u0017g\u0016B\u0002d\u000f\n4\u0019\u0000L\u0000R\n\u0000n\n[fL(\u000f)\u0000fR(\u000f)](10)\n\u0002Trf^\u001b\u0017[^Gr\n0(\u000f+!)^\u001bj^Gr\n0(\u000f) + 2iImf^Gr\n0(\u000f)g^\u001bj^Ga\n0(\u000f\u0000!)]g\n+X\n\u0018\u0000\u0018\n\u0000R[f\u0018(\u000f\u0000!)\u0000fL(\u000f)]Tr[^\u001b\u0017^Gr\n0(\u000f)^\u001bj^Ga\n0(\u000f\u0000!)]o\n:\nThe components obey \u001fL\n\u0017j(\u0000!) =\u001fL\u0003\n\u0017j(!). In other\nwords, they satisfy the Kramers-Kronig relations53that\ncan be written in a compact form as\n\u001fL\n\u0017j(!) =1\ni\u0019P\u00021\n\u00001\u001fL\n\u0017j(\u0018)\n\u0018\u0000!d\u0018; (11)\nwithPdenoting the principal value.\nFor anyi;j;k =x;y;zsuch thatj6=kandj;k6=\ni, whereiindicates the direction of the constant part\nof the magnetic field ~Bc=Bc~ ei, the complex current\nsusceptibilities satisfy the relations\n\u001fL\njj(!) =\u001fL\nkk(!) (12)\nand\u001fL\njk(!) =\u0000\u001fL\nkj(!); (13)\nin addition to Eq. (11). The other nonzero compo-\nnents are\u001fL\n0i(!)and\u001fL\nii(!). In the absence of a constant\nmagnetic field, the only nonvanishing components obey\n\u001fL\nxx(!) =\u001fL\nyy(!) =\u001fL\nzz(!).\nFinally, the average value of the electronic spin in the\nQD reads~ s(t) =h^~ s(t)i= (1=2)P\n\u001b\u001b0~ \u001b\u001b\u001b0h^dy\n\u001b(t)^d\u001b0(t)i=\n\u0000(i=2)P\n\u001b\u001b0~ \u001b\u001b\u001b0^G<\n\u001b0\u001b(t;t)and the complex spin suscep-\ntibilities are defined as\n\u001fs\nij(!) =@si(!)\n@B!j: (14)\nThey represent the linear responses of the electronic spin\ncomponentstotheappliedtime-dependentmagneticfield\nand satisfy the relations similar to Eqs. (11), (12), and\n(13) given above.4\nB. Analysis of the spin and current responses\nWe start by analyzing the transport properties of the\njunction at zero temperature in response to the exter-\nnal time-dependent magnetic field ~B(t). The constant\ncomponent of the magnetic field ~Bcgenerates a Zeeman\nsplit of the QD level \u000f0, resulting in the levels \u000f\";#, where\n\u000f\";#=\u000f0\u0006g\u0016BBc=2in this section. The time-dependent\nperiodic component of the magnetic field ~B0(t)then cre-\nates additional states, i.e., sidebands, at energies \u000f\"\u0006!\nand\u000f#\u0006!(see Fig. 2). These Zeeman levels and side-\nbandscontributetotheelastictransportpropertiesofthe\njunction when their energies lie inside the bias-voltage\nwindow ofeV=\u0016L\u0000\u0016R.\nHowever, energylevelsoutsidethebias-voltagewindow\nmay also contribute to the electronic transport due to in-\nelastic tunnel processes generated by the time-dependent\nmagnetic field. In these inelastic processes, an electron\ntransmitted from the left lead to the QD can change its\nenergy by!and either tunnel back to the left lead or\nout into the right lead. If this perturbation is small, as is\nassumed in this paper where we consider first-order cor-\nrections, the transport properties are still dominated by\nthe elastic, energy-conserving tunnel processes that are\nassociated with the Zeeman levels.\nThe energy levels of the QD determine transport prop-\nerties such as the spin-current susceptibilities and the\nspin susceptibilities, which are shown in Fig. 3. The\nimaginary and real parts of the susceptibilities are plot-\nted as functions of the frequency !in Figs. 3(a) and 3(c).\nIn this case, the position of the unperturbed level \u000f0is\nsymmetric with respect to the Fermi surfaces of the leads\nand a peak or step in the spin-current and spin suscepti-\nbilities appears at a value of !, for which an energy level\nis aligned with one of the lead Fermi surfaces. In Figs.\n3(b) and 3(d), the susceptibilities are instead plotted as\nfunctions of the bias voltage, eV. Here, each peak or\nstep in the susceptibilities corresponds to a change in the\nnumber of available transport channels. The bias volt-\nage is applied in such a way that the energy of the Fermi\nsurface of the right lead is fixed at \u0016R= 0while the en-\nergy of the left lead’s Fermi surface is varied according\nto\u0016L=eV.\nIII. SPIN-TRANSFER TORQUE AND\nMOLECULAR SPIN DYNAMICS\nA. Model with a precessing molecular spin\nNow we apply the formalism of the previous section\nto the case of resonant tunneling through a QD in the\npresence of a constant external magnetic field and an\nSMM [see Fig. 4(a)]. An SMM with a spin Slives in\na(2S+ 1)-dimensional Hilbert space. We assume that\nthe spinSof the SMM is large and neglecting the quan-\ntum fluctuations, one can treat it as a classical vector\n\nµLµR=00↑+ω↑↑−ω↓+ω↓−ω↓FIG. 2: (Color online) Sketch of the electronic energy levels of\nthe QD in the presence of a time-dependent magnetic field. In\na static magnetic field, the electronic level \u000f0(solid black line)\nsplits into the Zeeman levels \u000f\";#(solid red and blue lines). If\nthe magnetic field in addition to the static component also in-\ncludes a time-dependent part with a characteristic frequency\n!, additionallevelsappearatenergies \u000f\"\u0006!(dottedredlines)\nand\u000f#\u0006!(dotted blue lines). Hence, there are six channels\navailable for transport.\nwhose end point moves on a sphere of radius S. In the\npresence of a constant magnetic field ~Bc=Bc~ ez, the\nmolecular spin precesses around the field axis according\nto~S(t) =S?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, where\nS?is the projection of ~Sonto thexyplane,!L=g\u0016BBc\nis the Larmor precession frequency and Szis the projec-\ntion of the spin on the zaxis [see Fig. 4(b)]. The spins\nof the electrons in the electronic level are coupled to the\nspin of the SMM via the exchange interaction J. The\ncontribution of the external magnetic field and the pre-\ncessional motion of the SMM’s spin create an effective\ntime-dependent magnetic field acting on the electronic\nlevel.\nThe Hamiltonian of the system is now given by ^H(t) =\n^HL;R+^HT+^HD(t) +^HS, where the Hamiltonians ^HL;R\nand ^HTare the same as in Sec. II. The Hamiltonian\n^HS=g\u0016B~S~Bcrepresents the interaction of the molecu-\nlar spin~Swith the magnetic field ~Bcand consequently\ndoes not affect the electronic transport through the junc-\ntion but instead contributes to the spin dynamics of the\nSMM. The Hamiltonian of the QD in this case is given by\n^HD(t) =^Hc\nD+^H0(t). Here, ^Hc\nD=P\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~Bc\ne\u000b\nis the Hamiltonian of the electrons in the QD in the pres-\nence of the constant part of the effective magnetic field,\ngiven by~Bc\ne\u000b=\u0002\nBc+J\ng\u0016BSz\u0003\n~ ez. The second term of the\nQD Hamiltonian, ^H0(t) =g\u0016B^~ s~B0\ne\u000b(t), represents the in-\nteraction between the electronic spins of the QD, ^~ s, and\nthe time-dependent part of the effective magnetic field,\ngiven by~B0\ne\u000b(t) =JS?\ng\u0016B\u0002\ncos(!Lt)~ ex+ sin(!Lt)~ ey\u0003\n. The\ntime-dependent effective magnetic field can be rewritten\nas~B0\ne\u000b(t) =~B!Le\u0000i!Lt+~B\u0003\n!Lei!Lt, where~B!Lconsists\nof the complex amplitudes B!Lx=JS?=2g\u0016B,B!Ly=5\n0.00.51.01.52.02.5-6-4-2024\nw@e0DcijL@10-3mBD\nmL=e+wHaL\nmL=e¯+w\nImczzLReczzLImcxyLRecxyLImcxxLRecxxL\n0.00.51.01.52.0-1012\neV@e0DcijL@10-2mBD\ne¯-wHbL\ne¯\ne¯+w\ne-w\ne\ne+w\nImczzLReczzLImcxyLRecxyLImcxxLRecxxL\n0.00.51.01.52.02.5-2-101\nw@e0Dcijs@10-2mBe0-1DHcL\nImczzsReczzsImcxysRecxysImcxxsRecxxs\n0.00.51.01.52.0-4-20\neV@e0Dcijs@10-1mBe0-1DHdL\nImczzsReczzsImcxysRecxysImcxxsRecxxs\nFIG. 3: (Color online) (a) Frequency and (b) bias-voltage dependence of the spin-current susceptibilities. (c) Frequency and\n(d) bias-voltage dependence of the spin susceptibilities. In (a) and (c), the chemical potential of the left lead is \u0016L= 2\u000f0, while\nin (b) and (d) the frequency is set to != 0:16\u000f0. All plots are obtained at zero temperature with ~Bc=Bc~ ez, and the other\nparameters set to \u0016R= 0; \u000f\"= 1:48\u000f0; \u000f#= 0:52\u000f0;\u0000 = 0:02\u000f0, and \u0000L= \u0000R= 0:01\u000f0.\niJS?=2g\u0016B, andB!Lz= 0. The time-dependent pertur-\nbation can then be expressed as ^H0(t) = ^H!Le\u0000i!Lt+\n^Hy\n!Lei!Lt, where ^H!Lis an operator that can be written,\nusing Eq. (3) and the above expressions for B!Li, as\n^H!L=JS?\n2\u0012\n0 1\n0 0\u0013\n: (15)\nThe time-dependent part of the effective magnetic field\ncreates inelastic tunnel processes that contribute to the\ncurrents. The in-plane components of the spin current\nfulfill\nILx(!L) =\u0000iILy(!L) (16)\n=JS?\n2g\u0016B[\u001fL\nxx(!L) +i\u001fL\nxy(!L)];\nwhere~Bcis replaced by ~Bc\ne\u000b. Thezcomponent vanishes\nto lowest order in H0(t).54Therefore, the inelastic spin\ncurrent has a polarization that precesses in the xyplane.\nThe inelastic spin-current components, in turn, exert an\nSTT (Refs. 41-44) on the molecular spin given by\n~T(t) =\u0000[~IL(t) +~IR(t)]; (17)thus contributing to the dynamics of the molecular spin\nthrough\n_~S(t) =g\u0016B~Bc\u0002~S(t) +~T(t): (18)\nUsing expressions (6), (8), and (15), the torque of Eq.\n(17) can be calculated in terms of the Green’s functions\n^Gr\n0(\u000f)and ^Ga\n0(\u000f)as\nTi(t) =\u0000JS?\n2\u0002d\u000f\n2\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015\n\u0000[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\n(19)\n\u0002Imf(^\u001bi)#\"Gr\n0;\"\"(\u000f)Ga\n0;##(\u000f\u0000!L)e\u0000i!Ltg;\nwith\u0015=L;R. Here (^\u001bi)#\",Gr\n0;\"\"(\u000f), andGa\n0;##(\u000f)are\nmatrix elements of ^\u001bi,^Gr\n0(\u000f)and ^Ga\n0(\u000f)with respect to\nthe basis of eigenstates of ^sz. This STT can be rewritten\nin terms of the SMM’s spin vector as\n~T(t) =\u000b\nS_~S(t)\u0002~S(t) +\f_~S(t) +\r~S(t):(20)\nThe first term in this back-action gives a contribution to6\nzBceVS⊥(t)(a) (b) \nFIG. 4: (Color online) (a) Resonant tunneling in the pres-\nence of an SMM and an external, constant magnetic field.\nThe electronic level of Fig. 1 is now coupled with the spin\nof an SMM via exchange interaction with the coupling con-\nstantJ. The dynamics of the SMM’s spin ~Sis controlled by\nthe external magnetic field ~Bcthat also affects the electronic\nlevel. (b)PrecessionalmotionoftheSMM’sspininaconstant\nmagnetic field ~Bcapplied along the zaxis.\nthe Gilbert damping, characterized by the Gilbert damp-\ning coefficient \u000b. The second term acts as an effective\nconstant magnetic field and changes the precession fre-\nquency of the spin ~Swith the corresponding coefficient\n\f. The third term cancels the zcomponent of the Gilbert\ndamping term, thus restricting the STT to the xyplane.\nThe coefficient of the third term \ris related to \u000bby\n\r=\u000b =!LS2\n?=SSz. Expressing the coefficients \u000band\n\fin terms of the current susceptibilities \u001f\u0018\nxx(!L)and\n\u001f\u0018\nxy(!L)results in\n\u000b=\u0000JSz\ng\u0016B!LSX\n\u0018[Ref\u001f\u0018\nxx(!L)g\u0000Imf\u001f\u0018\nxy(!L)g];\n(21)\n\f=J\ng\u0016B!LX\n\u0018[Imf\u001f\u0018\nxx(!L)g+ Ref\u001f\u0018\nxy(!L)g]:(22)\nBy inserting the explicit expressions for Gr\n0;\"\"(\u000f)and\nGa\n0;##(\u000f\u0000!L), one obtains\n\u000b=J2S2\nz\n!LS\u0002d\u000f\n8\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\u0002(23)\n1\n[(\u0000\n2)2+ (\u000f\u0000\u000f\")2][(\u0000\n2)2+ (\u000f\u0000\u000f#\u0000!L)2];\n\f=\u0000J\n!L\u0000\u0002d\u000f\n4\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\u0002(24)\n(\u0000\n2)2+ (\u000f\u0000\u000f\")(\u000f\u0000\u000f#\u0000!L)\n[(\u0000\n2)2+ (\u000f\u0000\u000f\")2][(\u0000\n2)2+ (\u000f\u0000\u000f#\u0000!L)2];\nwhere\u000f\";#=\u000f0\u0006g\u0016BBc\ne\u000b=2 =\u000f0\u0006(!L+JSz)=2are the\nenergies of the Zeeman levels in this section. In the small\nprecession frequency regime, !L\u001ckBT,\r!0and in\nthe limit of Sz=S!1the expression for the coefficient\n\u000bis in agreement with Ref. 16.\n\nµLµR=0↑↓↑−ωL↓+ωLFIG. 5: (Color online) Sketch of the electronic energy levels\nof the QD in the presence of a molecular spin precessing with\nthe frequency !Laround an external, constant magnetic field.\nThe corresponding Zeeman levels are \u000f\";#. The precessional\nmotion of the molecular spin results in emission (absorption)\nof energy corresponding to a spin flip from spin up (down) to\nspin down (up). Hence, there are only four channels available\nfor transport.\nB. Analysis of the spin-transfer torque\nIn the case of resonant tunneling in the presence of\na molecular spin precessing in a constant external mag-\nnetic field, one also needs to take the exchange of spin-\nangular momentum between the molecular spin and the\nelectronic spins into account in addition to the effects\nof the external magnetic field. Due to the precessional\nmotion of the molecular spin, an electron in the QD emit-\nting (absorbing) an energy !Lalso undergoes a spin flip\nfrom spin up (down) to spin down (up), as indicated\nby the arrows in Fig. 5. As a result, the levels at en-\nergies\u000f\";#\u0006!Lare forbidden and hence do not con-\ntribute to the transport processes. Consequently, there\nare only four transport channels, which are located at\nenergies\u000f\";#\u0007!L. Also in this case, there are elastic\nand inelastic tunnel processes. Some of the possible in-\nelastic tunnel processes are shown in Fig. 6. These re-\nstrictions on the inelastic tunnel processes are also vis-\nible in Fig. 3(b), which identically corresponds to the\ncase of the presence of a precessing molecular spin with\n!L= 0:16\u000f0andJSz= 0:8\u000f0. Namely, from Eq. (16),\nwhich is equivalent to RefILx(!L)g= ImfILy(!L)g=\nJS?\n2g\u0016B[Ref\u001fL\nxx(!L)g\u0000Imf\u001fL\nxy(!L)g]andImfILx(!L)g=\n\u0000RefILy(!L)g=JS?\n2g\u0016B[Imf\u001fL\nxx(!L)g+ Ref\u001fxy(!L)g],\nand from the symmetries of the susceptibilities displayed\nin Fig. 3(b), it follows that there are no spin currents at\neV=\u000f\";#\u0006!L.\nAs was mentioned, the spin currents generate an STT\nacting on the molecular spin. A necessary condition\nfor the existence of an STT, and hence finite values of\nthe coefficients \u000band\fin Eqs. (23) and (24), is that\n~IL(t)6=\u0000~IR(t)[see Eq. (17)]. This condition is met\nby the spin currents generated, e.g., by the inelastic tun-\nnel processes shown in Figs. 6(b) and 6(c). These tunnel\nprocesses occur when an electron can tunnel into the QD,\nundergo a spin flip, and then tunnel off the QD into ei-7\nµLµL\nµLµLµL(a) \n(f) (e) (d) (c) (b) µL\nFIG. 6: (Color online) Sketch of the inelastic spin-tunneling\nprocesses in the QD in the presence of the precessing molec-\nular spin in the field ~Bc=Bc~ ezfor different positions of\nthe energy levels with respect to the chemical potentials of\nthe leads,\u0016Land\u0016R. Only transitions between levels with\nthe same color (blue or red) are allowed. Different colored\ncurved arrows (magenta, brown, or green) represent different\nprocesses.\nther lead. From these tunnel processes it is implied that\nthe Gilbert damping coefficient \u000band the coefficient \f\ncan be controlled by the applied bias or gate voltage as\nwell as by the external magnetic field. If a pair of QD\nenergy levels, coupled via spin-flip processes, lie within\nthe bias-voltage window, the spin currents instead fulfill\n~IL(t) =~IR(t), leading to a vanishing STT [see Fig. 6(d)].\nIn Figs. 6(e) and 6(f) the position of the energy levels of\nthe QD are symmetric with respect to the Fermi levels\nof the leads, \u0016Land\u0016R. When the QD level with energy\n\u000f\"is aligned with \u0016L, this simultaneously corresponds to\nthe energy level \u000f#being aligned with \u0016R[see Fig. 6(f)].\nAs a result, a spin-up electron can now tunnel from the\nleft lead into the level \u000f\", while a spin-down electron in\nthe level\u000f#can tunnel into the right lead. These addi-\ntional processes enhance the STT compared to that of\nthe case 6(e).\nThe two spin-torque coefficients \u000band\fexhibit a non-\nmonotonic dependence on the tunneling rates \u0000, as can\nbe seen in Figs. 7, 8, and 9. For \u0000!0, it is obvious\nthat\u000b;\f!0. In the weak coupling limit \u0000\u001c!L, the\ncoefficients \u000band\fare finite if the Fermi surface energy\nof the lead \u0018,\u0016\u0018fulfills either of the conditions\n\u000f#\u0014\u0016\u0018\u0014\u000f#+!L (25)\nor\u000f\"\u0000!L\u0014\u0016\u0018\u0014\u000f\" (26)\ninsuchawaythateachconditionissatisfiedbytheFermi\nenergy of maximum one lead. These conditions are re-\nlaxed for larger tunnel couplings as a consequence of the\nbroadening of the QD energy levels, which is also re-\nsponsible for the initial enhancement of \u000band\fwith in-\ncreasing \u0000. Notice, however, that \u000band\fare eventually\nsuppressed for \u0000\u001d!L, when the QD energy levels aresignificantly broadened and overlap so that spin-flip pro-\ncesses are equally probable in each direction and there\nis no net effect on the molecular spin. Physically, this\nsuppression of the STT can be understood by noticing\nthat for \u0000\u001d!La current-carrying electron perceives the\nmolecular spin as almost static due to its slow precession\ncompared to the electronic tunneling rates and hence the\nexchange of angular momenta is reduced. With increas-\ning tunneling rates, the coefficient \fbecomes negative\nbefore it drops to zero, causing the torque \f_~Sto oppose\nthe rotational motion of the spin ~S.\nIn Fig. 7, the Gilbert damping coefficient \u000band the\ncoefficient\fare plotted as functions of the applied bias\nvoltage at zero temperature. We analyze the case of the\nsmallestvalueof \u0000(redlines), assumingthat !L>0. For\nsmalleV, all QD energy levels lie outside the bias-voltage\nwindow and there is no spin transport [see Fig. 6(a)].\nHence\u000b;\f!0. AteV=\u000f#the tunnel processes in\nFig. 6(b) come into play, leading to a finite STT and the\ncoefficient\u000bincreases while the coefficient \fhas a local\nminimum. In the voltage region specified by Eq. (25) for\n\u0016L, the coefficient \u000bapproaches a constant value while\nthe coefficient \fincreases. By increasing the bias voltage\ntoeV=\u000f#+!Lthe tunnel processes in Fig. 6(c) occur,\nleading to a decrease of \u000band a local maximum of \f. For\n\u000f#+!L<eV <\u000f\"\u0000!L, the coefficients \u000b;\f!0[see Fig\n6(d)]. In the voltage region specified by Eq. (26) for \u0016L,\n\u000bapproaches the same constant value mentioned above\nwhile\fdecreases between a local maximum at eV=\n\u000f\"\u0000!Land a local minimum at eV=\u000f\", which approach\nthe same values as previously mentioned extrema. With\nfurther increase of eV, all QD energy levels lie within the\nbias-voltage window and the STT consequently vanishes.\nFigure 8 shows the spin-torque coefficients \u000band\fas\nfunctions of the position of the electronic level \u000f0. An\nSTT acting on the molecular spin occurs if the electronic\nlevel\u000f0is positioned in such a way that the inequalities\n(25) and (26) may be satisfied by some values of eV,\u000f0\nand!L. Again, we analyze the case of the smallest value\nof\u0000(red curve). For the particular choice of parameters\nin Fig. 8, there are four regions in which the inequalities\n(25) and (26) are satisfied. Within these regions, \u000bap-\nproaches a constant value while \fhas a local maximum\nas well as a local minimum. These local extrema occur\nwhen one of the Fermi surfaces is aligned with one of the\nenergy levels of the QD. For other values of \u000f0, both\u000b\nand\fvanish.\nThe coefficients \u000band\fare plotted as functions of the\nprecession frequency !Lin Fig. 9. Here, \u000f0=eV=2and\ntherefore the positions of the energy levels of the QD are\nsymmetric with respect to the Fermi levels of the leads,\n\u0016Land\u0016R. Once more, we focus first on the case of the\nsmallest value of \u0000(indicated by the red curve). The\nenergies of all four levels of the QD depend on !L, i.e.,\n~Bc. For!L>0, when the magnitude of the external\nmagnetic field is large enough, the tunnel processes in\nFig. 6(f) take place due to the above-mentioned sym-\nmetries. These tunnel processes lead to a finite STT,8\n0.00.51.01.52.01234567\neV@e0Da@10-4D\ne¯+wLHaL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\n0.00.51.01.52.02.5-8-6-4-2024\neV@e0Db@10-4D\ne¯+wLHbL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 7: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the applied bias voltage eV=\n\u0016L\u0000\u0016R, with\u0016R= 0, for different tunneling rates \u0000at zero temperature. Other parameters are \u0000L= \u0000R= \u0000=2,\u000f\"= 1:48\u000f0,\n\u000f#= 0:52\u000f0,S= 100,J= 0:01\u000f0,JSz= 0:8\u000f0, and!L= 0:16\u000f0. In the case of the smallest value of \u0000(red lines), \u000bapproaches\na constant value when \u0016Llies within the energy range specified by Eqs. (25) and (26). The coefficient \fhas one local minimum\nand one local maximum for the same energy range.\n-0.50.00.51.01.52.01234567\ne0@eVDa@10-4D\nmR=eHaL\nmR=e-wL\nmR=e¯mR=e¯+wL\nmL=e-wLmL=e\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\n-0.50.00.51.01.52.0-6-4-2024\ne0@eVDb@10-4D\nmR=eHbL\nmR=e-wLmR=e¯+wL\nmR=e¯\nmL=e\nmL=e-wL\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\nFIG. 8: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the position of the electronic\nlevel\u000f0for different tunneling rates \u0000at zero temperature. The applied bias voltage is eV=\u0016L\u0000\u0016R, with\u0016R= 0. Other\nparameters are \u0000L= \u0000R= \u0000=2,\u000f\"\u0000\u000f0= 0:24eV,S= 100,J= 0:005eV,JSz= 0:4eV, and!L= 0:08eV. In the case of the\nsmallest value of \u0000(red lines), there are four regions in which the Gilbert damping and the change of the precession frequency\noccur. In each of these regions \u000f0satisfies the inequalities (25) and (26), and \u000bapproaches a constant value, while \fhas one\nlocal maximum and one local minimum.\na maximum for the Gilbert damping coefficient \u000b, and\na negative minimum value for the \fcoefficient. As !L\nincreases, the inequalities of Eqs. (25) and (26) are sat-\nisfied and the tunnel processes shown in Fig. 6(e) may\noccur. Hence, there is a contribution to the STT, but\nas is shown in Eq. (23), the Gilbert damping decreases\nwith increasing precession frequency. At larger values of\n!L, resulting in \u000f#+!L=\u0016L, the Gilbert damping co-\nefficient has a step increase towards a local maximum,\nwhile the coefficient \fhas a local maximum, as a conse-\nquence of the enhancement of the STT due to additional\nspin-flip processes occurring in this case. For even larger\nvalue of!L, the conditions (25) and (26) are no longerfulfilled and both coefficients vanish. It is energetically\nunfavorable to flip the spin of an electron against the\nantiparallel direction of the effective constant magnetic\nfieldBc\ne\u000b. Hence, as !Lincreases, more energy is needed\nto flip the electronic spin to the direction of the field.\nThis causes \u000bto decrease with increasing !L. Addition-\nally, the larger the ratio !L=\u0000, the less probable it is that\nspin-angular momentum will be exchanged between the\nmolecular spin and the itinerant electrons. For !L= 0,\nthe molecular spin is static, i.e.,_~S= 0. In this case\n~T(t) =~0. The coefficient \u000bthen drops to zero while the\ncoefficient\freaches a negative local maximum which is\nclose to 0. Both \u000band\freach an extremum value for9\n-4-20240123\nwL@e0Da@10-4DHaL\nmL=e-wL\nmL=emL=e¯\nmL=e¯+wL\nG=5e0G=3e0G=0.2e0G=0.02e0\n-4-2024-6-4-20\nwL@e0Db@10-4DHbL\nmL=e¯+wLmL=emL=e¯\nmL=e-wL\nG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 9: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the precession frequency !L=\ng\u0016BBcof the spin ~Sof the SMM, with ~Bc=Bc~ ez, for different tunneling rates \u0000at zero temperature. The applied bias voltage\niseV=\u0016L\u0000\u0016R= 2\u000f0, with\u0016R= 0. The other parameters are the same as in Fig. 7. In the case of the smallest \u0000(red lines),\nthe coefficient \u000bhas a step increase towards a local maximum while the coefficient \fhas a local maximum or minimum at a\nvalue of!Lcorresponding to a resonance of \u0016Lwith one of the levels in the QD.\nlarge values of \u0000at this point. For !L<0and\u0000\u001cj!Lj\n(red lines), at the value of !Lfor which\u0016L=\u000f\"\u0000!L, the\ncoefficient\u000bhasastepincreasetowardsalocalmaximum\nwhilethecoefficient \fhasanegativelocalminimum. The\ncoefficient\u000bthen decreases with a further decrease of !L\nas long as\u000f#\u0014\u0016L\u0014\u000f\"\u0000!L. At the value of !Lfor\nwhich\u0016L=\u000f#,\u000bhas another step increase towards a\nlocal maximum while \fhas a maximum value. Accord-\ning to Eq. (23), the Gilbert damping also does not occur\nif~Sis perpendicular to ~Bc. In this case \f.0and the\nonly nonzero torque component \f_~S(t)acts in the oposite\ndirection than the molecular spin’s rotational motion.\nIV. CONCLUSIONS\nIn this paper we have first theoretically studied time-\ndependent charge and spin transport through a small\njunction consisting of a single-level quantum dot cou-\npled to two noninteracting metallic leads in the pres-\nence of a time-dependent magnetic field. We used the\nKeldysh nonequilibrium Green’s functions method to de-\nrive the charge and spin currents in linear order with\nrespect to the time-dependent component of the mag-\nnetic field with a characteristic frequency !. We then\nfocused on the case of a single electronic level coupled\nvia exchange interaction to an effective magnetic field\ncreated by the precessional motion of an SMM’s spin in\na constant magnetic field. The inelastic tunneling pro-\ncesses that contribute to the spin currents produce an\nSTT that acts on the molecular spin. The STT con-\nsists of a Gilbert damping component, characterized bythe coefficient \u000b, as well as a component, characterized\nby the coefficient \f, that acts as an additional effective\nconstant magnetic field and changes the precession fre-\nquency!Lof the molecular spin. Both \u000band\fdepend\non!Land show a nonmonotonic dependence on the tun-\nneling rates \u0000. In the weak coupling limit \u0000\u001c!L,\u000b\ncan be switched on and off as a function of bias and gate\nvoltages. The coefficient \fcorrespondingly has a local\nextremum. For \u0000!0, both\u000band\fvanish. Taking\ninto account that spin transport can be controlled by the\nbias and gate voltages, as well as by external magnetic\nfields, our results might be useful in spintronic applica-\ntions using SMMs. Besides a spin-polarized STM, it may\nbe possible to detect and manipulate the spin state of an\nSMM in a ferromagnetic resonance experiment56–59and\nthus extract information about the effects of the current-\ninduced STT on the SMM. Our study could be com-\nplemented with a quantum description of an SMM in a\nsingle-molecule magnet junction and its coherent prop-\nerties, as these render the SMM suitable for quantum\ninformation storage.\nAcknowledgments\nWe gratefully acknowledge discussions with Mihajlo\nVanevićandChristianWickles. 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Monsoriu∗\nCentro de Tecnolog´ ıas F´ ısicas, Universitat Polit` ecnic a de Val` encia,\nCam´ ı de Vera s/n, 46022, Val` encia, Spain.\n(Dated: December 19, 2012)\nAbstract\nThe mobile acceleration sensor has been used to in Physics ex periments on free and damped\noscillations. Results for the period, frequency, spring co nstant and damping constant match very\nwell to measurements obtained by other methods. The Acceler ometer Monitor application for\nAndroid has been used to get the outputs of the sensor. Perspe ctives for the Physics laboratory\nhave also been discussed.\n∗Electronic address: jmonsori@fis.upv.es\n1I. INTRODUCTION\nElectronic portable or every day-use devices offer increasing pers pectives for the Physics\nlaboratory at all teaching levels. This is the case of the digital camer as [1], web cams [2],\noptical mouse of computers [3, 4], XBee transducers [5], wiimote [6] a nd other game console\ncontrollers [7]. For instance, digital techniques have been widely use d to visualize Physics\nconcepts [8, 9]. By using a digital camera, a real process can be follo wed [10, 11]. Then,\nby analyzing the recorded video a lot of information can be obtained s uch as distances,\ntime intervals and trajectories of objects. In this way, the study and explanation of complex\nconcepts to the students can be enormously facilitated. More rec ently, many works related\nto the use of wireless devices (such as the wiimote) in Physics teachin g have been reported\n[6, 12, 13]. The use of the wiimote for Physics experiments is based on its three axis\naccelerometer which communicates with the game console using a blue tooth device. Pioneer\nworks about the use of accelerometers in the Physics laboratory c an be referred to Weltin\n[14] and Hunt [15, 16]. The wiimote includes an infrared image sensor ab le to follow up to\nfive objects simultaneously, a bluetooth device and three accelero meters.\nThe wiimote gives Physics teachers a low cost way to track motion in a v ariety of Physics\nexperiments [17], however, it is not a common device at the Physics lab oratories. In this\npaper,westudythepossibilityofusinganotherusualacceleromet erwhichalmostallstudents\nhave with them, although they may not realize it. We study free and d amped oscillations\nby using the mobile phone acceleration sensor and the air track. Mec hanical oscillations\nis a very important topic in most of undergraduate Physics courses . Many proposals of\nPhysics laboratory have been published in relation to this topic [18, 19 ]. For their study,\nthe air track is very useful since the friction force can be apprecia bly decreased by creating\na layer of air between the glider and the track [20]. Even there are ma ny works reporting\nexperiments for the study of oscillations, this is one the first ones t hat uses a mobile phone\nas an accelerometer in Physics teaching.\nThe outline of the paper is the following. In section II, the experimen tal set up is\ncommented. It includes the introduction to the general features of the free Acceleration\nMonitor mobile application. In section III, the free harmonic oscillatio ns are studied. This\nsection is divided into two parts: the first for the calculation of the p eriod of the oscillations\nandthesecondforthecalculationofthespringconstant. Insect ionIV,thedampingconstant\n2of the damped oscillations is calculated. Finally, in section V, some conc lusions are drawn.\nII. EXPERIMENTAL SET UP\nA photograph of the experimental set up is shown in figure 1. The mo bile is placed on\nthe glider of the air track and connected to a fixed end by a spring. O nce the air flow goes\nthrough the air track the friction is considerably decreased. Unde r this situation, the glider\nwith the mobile on can oscillate almost freely upon perturbation of the system. The mobile\nthat has been used for the measurements is a Smart phone, LG-E5 10 bearing an Android\nversion 2 .3.4. The mobile mass is 124 .0gand the air track glider mass is 180 .6g. The mass\nof the glider can be changed by adding weights at both sides. In this w ay, the frequency of\nthe free oscillations is changed.\nFor the interaction with the mobile sensor, the free Android applicat ion ”Accelerometer\nMonitor ver 1 .5.0” has been used. This application takes 348 kBof SD card memory and\nit can be downloaded from Google play website at ref. [21]. This applica tion reports the\nvibrationsofthemobilephoneinrealtimebyregistering theaccelera tioncomponentson x,y\nandz- axis at each time step. The component of the acceleration of grav ity can be removed\nfrom the data. The precision in the measurement of the acceleratio n isδa= 0.01197m/s2\nand for the time is δt= 0.02s. Since the oscillations take place along the y-axis, the values\nof the acceleration for the xandz-axis remain very close to zero. This application also\nallows saving the output data to a file, from which the data can used f or further analysis.\nOnce the application has been downloaded to the mobile device, a small test can be\ndone to check its well functioning. If the mobile is left quiet on a horizo ntal surface, the\napplication output curves for the acceleration should indicate value s very close to zero for\nall axis.\nFor the case of damped oscillations, some dissipation of the amplitude of the oscillations\ncan be obtained by decreasing the air flow at the air track.\nIII. FREE HARMONIC OSCILLATIONS\nTheexperimentalresultsoftheacceleration a(t)obtainedwiththeAccelerometerMonitor\nmobile application have been fitted to the following equation:\n3TABLE I: Parameters and their errors from the fitting of the ac celeration data for free oscillations.\n(m±0.0001)(kg) (A±δA)m/s2(ω0±δω0)rad/s(φ0±δφ0)rad R2\nm10.3045 1 ,082±0,008 24,747±0,010−0,797±0,015 0,9938\nm20.4043 1 ,204±0,006 21,546±0,005 2,479±0,009 0,9968\nm30.5004 1 ,598±0,006 19,408±0,004−0,377±0,007 0,9984\nm40.6084 1 ,171±0,007 17,669±0,006−0,397±0,011 0,9953\nm50.6285 0 ,856±0,007 17,371±0,006 2,553±0,016 0,9875\nm60.6961 1 ,542±0,008 16,526±0,007 2,830±0,011 0,9966\na(t) =Asin(ω0t+φ0) (1)\nwhereAistheaccelerationamplitude, ω0istheangularfrequency and φ0thephaseconstant.\nThe parameters have been obtained by using a least square fitting. In figure 2, the screen of\nthe Accelerometer Monitor application while displaying free harmonic o scillations has been\nshown.\nIn order to start the oscillations in the system, the glider with the mo bile was shifted to\nthe right and then left free. The six different masses were obtained by adding weights to\nboth sides of the glider. The output of the mobile application with the a cceleration data is\ncollected in an ASCII file. First, a heading with some information such a s ”saving start date\nand time”, ”sensor resolution” and ”sensor vendor”, can be foun d. Following the heading,\nthe sensor measurements are shown. The first three columns indic ate the acceleration in the\nthree perpendicular axis, x(perpendicular to the device, positive to the right), y(along the\ndevice, positive upward) and z(perpendicular to xandyaxis, with positive direction as\ncoming out perpendicularly from the device display). In figure 3, a fr agment of the output\nfile is shown.\nThe scatter of the data points and the fit curve are shown in figure 4. In table I, the\nparameters and their errors from the fitting to equation 1 are reg istered. The quality of the\nfit can be seen in the values of the regression coefficient R2whose values are close to 1.\n4TABLE II: Periods calculated from the fit (column 2) and the pe riod obtained with the photometer\n(column 3) are shown. In the fourth column, the percentage di screpancy has been included.\n(Tfit±δTfit)s(TPhoto±0.001)s D(%)\nm10,2539±0,0001 0 ,259 1 ,99\nm20,2916±0,0001 0 ,291 0 ,21\nm30,3237±0,0001 0 ,323 0 ,23\nm40,3556±0,0001 0 ,356 0 ,11\nm50,3617±0,0001 0 ,365 0 ,91\nm60,3802±0,0002 0 ,380 0 ,05\nA. Calculation of the period T\nIn table II, the calculation of the period by using the fitted frequen cy and the period\nobtained directed from the photometer are shown. Results indicat e a very good agreement\nbetween the values since discrepancies are less than 1 % in most of ca ses.\nB. Calculation of the spring constant k\nOnce the values of the mass ( m) and the frequency of the free oscillations ( ω0) have\nbeen obtained, the spring constant kfitcan be calculated. For that purpose, a least square\nlinear regression to the equation ω2\n0=kfitm−1has been performed using the values of table\nIII. The string constant has also been calculated by hanging a mass (m) from the spring\nand measuring the shift in the position ( x). For the values, m= (500,2±0.1)gand\nx= (2,6±0,1)cm, we obtained k=mg/x= (189±7)N/m. The results for the spring\nconstant kand their errors have been shown in table IV.\nIV. DAMPED HARMONIC OSCILLATIONS\nThe damped harmonic oscillations of the acceleration a(t) can be represented by the\nfollowing equation:\n5TABLE III: Data for the calculation of the spring constant.\n(m−1±δ(m−1))kg−1(ω2\n0±δω2\n0)rad2/s2\nm13,2841±0,0011 612 ,4±0,5\nm22,4734±0,0006 464 ,2±0,2\nm31,9984±0,0004 376 ,7±0,1\nm41,6437±0,0003 312 ,2±0,2\nm51,5911±0,0003 301 ,8±0,2\nm61,4366±0,0002 273 ,1±0,2\nTABLE IV: Results for the spring constant, from the fit (colum n 1) and using a weight (column\n2). In the third column, the percentage discrepancy between both values has been shown.\nFrom the fit Using a weight\n(kfit±δkfit)N/m(k±δk)N/m D(%)\n187.9±0.6 189 ±7 0,58\na(t) =De−γtsin(ωt+φ) (2)\nwhereDis the acceleration amplitude for t= 0s,γis the damping constant and φ, the\nphase.\nIn figure 5, the screen of the Accelerometer Monitor application wh ile displaying damped\nharmonic oscillations has been shown. In figure 6, the data points an d the fit curve are\nillustrated. In the table V, the parameters and their errors from t he fitting to equation 2\nusing the mass m6are registered. The value of R2which indicates the quality of the fitting\nhas also been included.\nThe relaxation time, τ, is the inverse of the damping constant, τ= 1/γ. It can also be\nderived from the following equation:\n6TABLE V: Parameters and their errors from the fitting of the ac celeration data for damped oscil-\nlations.\n(D±δD)m/s2(γ±δγ)s−1(ω±δω)rad/s(φ±δφ)rad R2\nm61,64±0,02 0,58±0,02 16,52±0,01−0,56±0,01 0,9816\nTABLE VI: Results for the relaxation times. The percentage d iscrepancy between both values has\nalso been included.\n(τfit±δτfit)s−1(τDa±δτDa)s−1D(%)\n1,71±0.02 1 ,70±0,04 0,59\nτ=TDaTfit\n2π1\n(T2\nDa−T2\nfit)1/2(3)\nwhere,Tfit, is the period of the free oscillations (see table II) and TDa= 2π/ω, the period\nof the damped oscillations. In table VI, the relaxation times, obtaine d from the fitting\n(τfit= 1/γ) and from equation 3 ( τDa), are shown. Results indicate a very good agreement.\nV. CONCLUSIONS\nFree and damped oscillations have been studied in a very simple way by u sing the mobile\nphone acceleration sensor. For this purpose, the free Accelerom eter Monitor application for\nAndroid operating system have been used. Results for the period, frequency and the spring\nconstant using the mobile sensor data show a very good agreement with the classical way of\nmeasuring this quantities in the Physics laboratory. For instance, t he period obtained from\nthefittingiscompared tothedirect measurement ofthephotomet er andthespring constant,\ncalculated from the fit, is compared to the spring constant calculat ed from a gravity method\nusing a weight. On the other hand, the damping constant as obtaine d from the fitting of the\nmobile phone data allows corroborating theoretical expressions on damped oscillations. This\nfact offers very good perspectives for the use of the acceleratio n mobile phone sensor in the\n7General Physics laboratory. For example, the mobile phone as acce lerometer sensor can be\nused to study the most common Physics laboratory pendulums such as the simple, physical\nand torsion pendulums. It can also be used for the study of two dime nsional harmonic\noscillations, for example, with the use of the air table [22].\nAcknowledgments\nAuthors would like to thank the Institute of Education Sciences, Un iversitat Polit` ecnica\nde Val` encia (Spain), for the support of the Teaching Innovation Group, MoMa.\n[1] J. A. Monsoriu, M. H. Gim´ enez, J. Riera and Ana Vidaurre, ”Measuring coupled oscillations\nusing an automated video analysis technique based on image r ecognition”, Eur. J. Phys. 26\n(2005) 1149-1155.\n[2] S. Shamim, W. Zia, and M. S. Anwar, ”Investigating viscou s damping using a webcam”, Am.\nJ. Phys. 78, 433 (2010).\n[3] O. O. Romulo and K. N. Franklin, ”The computer mouse as a da ta acquisition interface:\napplication to harmonic oscillators”, Am. J. Phys, 65 1115 ( 1997).\n[4] T. W. Ng and K. T. Ang, ”The optical mouse for harmonic osci llator experimentation”, Am.\nJ. Phys. 73793 (2005).\n[5] E. Ayars, E. Lai, ”Using XBee transducers for wireless da ta collection”, Am. J. Phys. 787\n(2010).\n[6] S. L. Tomarken, D. R. Simons, R. W. Helms, W. E. Johns, K. E. Schriver et al., ”Motion\ntracking in undergraduate physics laboratories with the Wi i remote”, Am. J. Phys. 80, 351\n(2012).\n[7] M. VannoniandS.Straulino, ”Low-cost accelerometers f orphysicsexperiments”, Eur.J.Phys.\n28(2007) 781–787.cm9tZXRlciJd\n[8] A. Vidaurre, J. Riera, M.H. Gim´ enez and J. A. Monsoriu, ” Contribution of digital simulation\nin visualizing physics processes, Comp. App. Eng. Educ. 1045 (2002).\n[9] J. Riera, M.H. Gim´ enez, A. Vidaurre and J.A. Monsoriu, ” Digital simulation of wave motion”,\nComp. App. Eng. Educ. 10161 (2002).\n8[10] H.C. Chung, J. Liang, S. Kushiyama and M. Shinozuka, ”Di gital image processing for non-\nlinear systems identification”, Int. J. Non-Linear Mech. 39691 (2004).\n[11] T. Greczylo and E. Debowska, ”Using a digital video came ra to examine coupled oscillations”,\nEur. J. Phys. 23441 (2002).\n[12] A. Kawam and M. Kouh, ”Wiimote Experiments: 3-D Incline d Plane Problem for Reinforcing\nthe Vector Concept”, Phys. Teach. 49, 508 (2011).\n[13] R. Ochoa, F. G. Rooney, and W. J. Somers, ”Using the Wiimo te in Introductory Physics\nExperiments”, Phys. Teach. 49, (2011).\n[14] H. Weltin, ”Accelerometer”, Am. J. Phys. 34, 825 (1966).\n[15] J. L. Hunt, ”Forced and dampedharmonic oscillator expe riment usingan accelerometer”, Am.\nJ. Phys. 53, 278 (1985).\n[16] J. L. Hunt, ”Accurate experiment for measuring the rati o of specific heats of gases using an\naccelerometer”, Am. J. Phys. 53, 696 (1985).\n[17] A. Skeffington and K. Scully, ”Simultaneous tracking of m ultiple points using a wiimote”,\nPhys. Teach. 50, 482 (2012).\n[18] P. Onorato, D. Mascoli, and A. De Ambrosis, ”Damped osci llations and equilibrium in a\nmass-spring system subject to sliding friction forces: Int egrating experimental and theoretical\nanalyses”, Am. J. Phys. 78, 1120 (2010).\n[19] G Flores-Hidalgo and F A Barone, ”The one-dimensional d amped forced harmonic oscillator\nrevisited”, Eur. J. Phys. 32(2011) 377–379.\n[20] J. Berger, ”On potential energy, its force field and thei r measurement along an air track”,\nEur. J. Phys. 947 (1988).\n[21] https://play.google.com/store/apps\n[22] N. C. Bobillo-Ares and Jose Fernandez-Nufiez, ”Two-dim ensional harmonic oscillator on an\nair table”, Eur. J. Phys. 16223 (1995).\nVI. FIGURE CAPTIONS\nFig.1. Photograph of the experimental set up used in the experimen ts. In the photograph\n(1) is the smart phone, (2) the glider, (3) the air track, (4) the st ring, (5) the photometer,\nand (6) the fixed end.\n9Fig. 2. Photographoftheexperimental setupusedintheexperime nts. Inthephotograph\n(1) is the smart phone, (2) the glider, (3) the air track, (4) the st ring, (5) the photometer,\nand (6) the fixed end.\nFig. 3. Fragment of the output data file of the Accelerometer Monit or mobile phone\napplication.\nFig. 4. As an example of the data points (scatter) and fit curves (s olid line) of the\nacceleration versustime, the case of the mass 3 has been shown.\nFig. 5. Photograph of the smart phone displaying a view of the Accele ration Monitor\napplication for the case of damped harmonic oscillations.\nFig. 6. Data points of the acceleration (scatter) versustime for damped oscillations.\nThe solid line represent the fit curve.\n10/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48 /s49/s46/s50/s53 /s49/s46/s53/s48 /s49/s46/s55/s53 /s50/s46/s48/s48 /s50/s46/s50/s53 /s50/s46/s53/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s32/s32/s65/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s40/s109/s47/s115/s50\n/s41\n/s116/s32/s40/s115/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s32/s65/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s40/s109/s47/s115/s50\n/s41\n/s116/s32/s40/s115/s41"    },    {        "title": "1301.2114v1.First_principles_calculation_of_the_Gilbert_damping_parameter_via_the_linear_response_formalism_with_application_to_magnetic_transition_metals_and_alloys.pdf",        "content": "arXiv:1301.2114v1  [cond-mat.other]  10 Jan 2013APS/123-QED\nFirst-principles calculation of the Gilbert damping param eter via the linear response\nformalism with application to magnetic transition-metals and alloys\nS. Mankovsky1, D. K¨ odderitzsch1, G. Woltersdorf2, and H. Ebert1\n1University of Munich, Department of Chemistry,\nButenandtstrasse 5-13, D-81377 Munich, Germany and\n2Department of Physics, Universit¨ at Regensburg, 93040 Reg ensburg, Germany\n(ΩDated: September 9, 2018)\nA method for the calculations of the Gilbert damping paramet erαis presented, which based on\nthe linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-\nRostoker band structure method in combination with the cohe rent potential approximation alloy\ntheory. To account for thermal displacements of atoms as a sc attering mechanism, an alloy-analogy\nmodel is introduced. This allows the determination of αfor various types of materials, such as\nelemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for\ndisordered magnetic alloys at T= 0 K and above. The effects of spin-orbit coupling, chemical a nd\ntemperature induced structural disorder are analyzed. Cal culations have been performed for the\n3dtransition-metals bcc Fe, hcp Co, and fcc Ni, their binary al loys bcc Fe 1−xCox, fcc Ni 1−xFex,\nfcc Ni 1−xCoxand bcc Fe 1−xVx, and for 5 dimpurities in transition-metal alloys. All results are in\nsatisfying agreement with experiment.\nPACS numbers: 72.25.Rb 71.20.Be 71.70.Ej 75.78.-n\nI. INTRODUCTION\nDuringthe lastdecadesdynamicalmagneticproperties\nhave attracted a lot of interest due to their importance in\nthe development of new devices for spintronics, in par-\nticular, concerning their miniaturization and fast time\nscale applications. A distinctive property of such devices\nis the magnetization relaxation rate characterizing the\ntime scale on which a system being deviated from the\nequilibrium returns to it, or how fast the device can be\nswitched from one state to another. In the case of dy-\nnamics of a uniform magnetization /vecMthis property\nis associated with the Gilbert damping parameter ˜G(M)\nused first in the phenomenologicalLandau-Lifshitz (LL)1\nand Landau-Lifshitz-Gilbert (LLG) theory2describing\nthe magnetization dynamics processes by means of the\nequation:\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n,(1)\nwhereMsis the saturation magnetization, γthe gy-\nromagnetic ratio and Heff=−∂MF[M(r)] being the\neffective magnetic field. Sometimes it is more conve-\nnient to use a dimensionless Gilbert damping parame-\nterαgiven byα=˜G/(γMs) (see, e.g.3–5). Safonov\nhas generalized the Landau-Lifshitz equation by intro-\nducing a tensorial form for the Gilbert damping parame-\nter with the diagonal terms characterising magnetization\ndissipation6. Beingintroducedasaphenomenologicalpa-\nrameter, the Gilbert damping is normally deduced from\nexperiment. In particular, it can be evaluated from the\nresonant line width in ferromagnetic-resonance (FMR)\nexperiments. The difficulty of these measurements con-\nsists in the problem that there exist several different\nsources for the broadening of the line width, which havebeen discussed extensively in the literature7–13. The line\nwidth that is observed in ferromagnetic resonance spec-\ntra is usually caused by intrinsic and extrinsic relax-\nation effects. The extrinsic contributions are a conse-\nquence of spatially fluctuating magnetic properties due\nto sample imperfections. Short range fluctuations lead\nto two magnon scattering while long range fluctuations\nlead to an inhomogeneous line broadening due a super-\nposition of local resonances14. In order to separate the\nintrinsic Gilbert damping from the extrinsic effects it is\nnecessary to measure the frequency and angular depen-\ndence of the ferromagnetic resonance line width, e. g.\ntwo magnon scattering can be avoided when the mag-\nnetization is aligned along the film normal11(perpen-\ndicular configuration). Usually one finds a linear fre-\nquency dependence with a zero frequency offset and one\ncan write ∆ H(ω) =αω\nγ+∆H(0). When such measure-\nments are performed over a wide frequency range the\nslope of ∆ Has a function of frequency can be used\nto extract the intrinsic Gilbert damping constant. In\nmetallic ferromagnets Gilbert damping is mostly caused\nby electron magnon scattering. In addition Gilbert-like\ndamping can be caused by eddy currents. The magni-\ntude of the eddy current damping is proportional to d2,\nwheredis the sample thickness10. In sufficiently thin\nmagnetic films ( d≤10 nm) the eddy current damping\ncan be neglected10. However, for very thin films relax-\nation mechanisms that occur at the interfaces can also\nincrease and even dominate the damping. Such effects\nare spin pumping15,16and the modified electronic struc-\nture at the interfaces. In the present work spin pumping\nand the modified interface electronic structure are not\nconsidered and we assume that bulk-like Gilbert damp-\ning dominates.\nMuch understanding of dynamical magnetic properties\ncould in principle be obtained from the simulation of2\nthese processes utilizing time-dependent first-principles\nelectronic structure calculations, that in turn would pave\nthe way to developing and optimizing new materials for\nspintronic devices. In spite of the progress in the de-\nvelopment of time-dependent density functional theory\n(TD-DFT) during the last decades17that allows to study\nvariousdynamicalprocessesin atomsand molecules from\nfirstprinciples, applicationstosolidsarerare. Thisisdue\nto a lack of universally applicable approximations to the\nexchange-correlation kernel of TD-DFT for solids. Thus,\natthemoment, atractableapproachconsistsintheuseof\nthe classical LLG equations, and employing parameters\ncalculated within a microscopic approach. Note however\nthat this approach can fail dealing with ultrafast mag-\nnetization dynamics, which is discussed, for instance, in\nRefs. [18 and 19], but is not considered in the present\nwork.\nMost of the investigations on the magnetization dissi-\npation have been carried out within model studies. Here\none has to mention, in particular, the so-called s-dor\np-dexchange model20–23based on a separate considera-\ntion of the localized ’magnetic’ d-electrons and delocal-\nizeds- andp-electrons mediating the exchange interac-\ntions between localized magnetic moments and responsi-\nble for the magnetization dissipation in the system. As\nwas pointed out by Skadsem et al.24, the dissipation pro-\ncess in this case can be treated as an energy pumping\nout of thed-electron subsystem into the s-electron bath\nfollowed by its dissipation via spin-flip scattering pro-\ncesses. This model gave a rather transparent qualita-\ntive picture for the magnetization relaxation in diluted\nalloys, e.g. magnetic semiconductors such as GaMnAs.\nHowever, it fails to give quantitative agreement with ex-\nperiment in the case of itinerant metallic systems (e.g.\n3d-metalalloys),wherethe d-statesareratherdelocalised\nand strongly hybridized with the sp-electrons. As a con-\nsequence the treatment of allvalence electrons on the\nsame footing is needed, which leads to the requirement\nof first-principles calculations of the Gilbert damping go-\ning beyond a model-based evaluation.\nVarious such calculations of the Gilbert damping pa-\nrameter are already present in the literature. They usu-\nally assume a certain dissipation mechanism, like Kam-\nbersky’sbreathingFermisurface(BFS)25,26, ormoregen-\neral torque-correlation models (TCM)3,27. These models\ninclude explicitly the spin-orbit coupling (SOC), high-\nlighting its key role in the magnetization dissipation pro-\ncesses. However, the latter methods used for electronic\nstructure calculations cannot take explicitly into account\ndisorder in the system that in turn is responsible for the\naforementioned spin-flip scattering process. Therefore,\nthis has to be simulated by using external parameters\ncharacterizing the finite lifetime of the electronic states.\nThisweakpointwasrecentlyaddressedbyBrataas et al.4\nwho described the Gilbert damping by means of scatter-\ning theory. This development supplied the formal basis\nfor the first parameter-free investigations on disordered\nalloys for which the dominant scattering mechanism ispotential scattering caused by chemical disorder5.\nTheoretical investigations of the magnetization dissi-\npation by means of first-principles calculations of the\nGilbert damping parameter already brought much un-\nderstanding of the physical mechanisms responsible for\nthis effect. First of all, key roles are played by two ef-\nfects: the SOC of the atomic species contained in the\nsystem and scattering on various imperfections, either\nimpurities or structural defects, phonons, etc. Account-\ning for the crucial role of scattering processes respon-\nsible for the energy dissipation, different types of scat-\ntering phenomena have to be considered. One can dis-\ntinguish between the ordered-compound or pure-element\nsystems for which electron-phonon scattering is a very\nimportant mechanism for relaxation, and disordered al-\nloys with dominating scattering processes resulting from\nrandomly distributed atoms of different types. In the\nfirst case, the Gilbert damping behavior is rather differ-\nent at low and high temperatures. At high temperature\natomic displacements create random potentials leading\ntoSOC-inducedspin-flipscattering. At lowtemperature,\nwhere the magnetization dissipation is well described via\nthe BFS (Breathing Fermi-surface) mechanism25,26, the\nspin-conserving electron-phonon scattering is required to\nbring the electronic subsystem to the equilibrium at ev-\nery step of the magnetization rotation, i.e. to reoccupy\nthe modified electronic states.\nIn this contribution we describe a formalism for the\ncalculation of the Gilbert damping equivalent to that\nof Brataas et al.4, however, based on the linear re-\nsponce theory28as implemented in fully relativistic mul-\ntiple scattering based Korringa-Kohn-Rostoker (KKR)\nformalism. It will be demonstrated that this allows to\ntreat elegantly and efficiently the temperature depen-\ndence ofαin pure crystals as well as disordered alloys.\nII. THEORETICAL APPROACH\nTo have direct access to real materials and to obtain\na deeper understanding of the origin of the properties\nobserved experimentally, the phenomenological Gilbert\ndamping parameter has to be treated on a microscopic\nlevel. This implies to deal with the electrons responsible\nforthe energydissipation in the magnetic dynamicalpro-\ncesses. Thus, one equates the corresponding expressions\nfor the dissipation rate obtained in the phenomenologi-\ncal and microscopic approaches ˙Emag=˙Edis. Although\na temporal variation of the magnetization is a required\ncondition for the energy dissipation to occur, the Gilbert\ndamping parameter is defined in the limit ω→0 (see\ne.g., Ref. [24]) and therefore can be calculated within the\nadiabatic approximation.\nIn the phenomenological LLG theory the time depen-\ndent magnetization M(t) is described by Eq. (1). Ac-\ncordingly, the time derivative of the magnetic energy is3\ngiven by:\n˙Emag=Heff·dM\ndτ=1\nγ2(˙ˆm)T[˜G(M)˙ˆm](2)\nwhereˆm=M/Msdenotes the normalized magnetiza-\ntion. To represent the Gilbert damping parameter in\nterms of a microscopic theory, following Brataas et al.4,\nthe energy dissipation is associated with the electronic\nsubsystem. The dissipation rate upon the motion of the\nmagnetization ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\n, is determined by the under-\nlying Hamiltonian ˆH(τ). Assuming a small deviation of\nthe magnetic moment from the equilibrium the normal-\nized magnetization ˆm(τ) can be expanded around the\nequilibrium magnetization ˆm0\nˆm(τ) =ˆm0+u(τ), (3)\nresulting in the expression for the linearized time depen-\ndent Hamiltonian for the system brought out of equilib-\nrium:\nˆH=ˆH0(ˆm0)+/summationdisplay\nµuµ∂\n∂uµˆH(ˆm0).(4)\nDue tothe smalldeviation from the equilibrium, ˙Ediscan\nbe obtained within the linearresponseformalism, leading\nto the expression4:\n˙Edis=−π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej),(5)\nwhereEFis the Fermi energy and the sums run over\nall eigenstates of the system. As Eq. (5) characterizes\nthe rate of the energy dissipation upon transition of the\nsystem from the tilted state to the equilibrium, it can\nbe identified with the corresponding phenomenological\nquantity in Eq. (2), ˙Emag=˙Edis. This leads to an ex-\nplicit expression for the Gilbert damping tensor ˜Gor\nequivalently for the damping parameter α=˜G/(γMs)\n(Ref. [4]):\nαµν=−/planckover2pi1γ\nπMs/summationdisplay\nij/summationdisplay\nµν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej),(6)\nwhere the summation is running over all states at the\nFermi surface EF.\nIn full analogy to the problem of electric\nconductivity29, the sum over eigenstates |ψi/angbracketrightmay be\nexpressed in terms of the retarded single-particle Green’s\nfunction Im G+(EF) =−π/summationtext\ni|ψi/angbracketright/angbracketleftψi|δ(EF−Ei). This\nleads for the parameter αto a Kubo-Greenwood-like\nequation:\nαµν=−/planckover2pi1γ\nπMsTrace\n/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc(7)with/angbracketleft.../angbracketrightcindicating a configurational average in case of\na disordered system.\nThe most reliable way to account for spin-orbit cou-\npling as the source of Gilbert damping is to evaluate\nEq. (7) using a fully relativistic Hamiltonian within the\nframework of local spin density formalism (LSDA)30:\nˆH=cα·p+βmc2+V(r)+βσ·ˆmB(r).(8)\nHereαiandβare the standard Dirac matrices, σde-\nnotes the vector of relativistic Pauli matrices, and pis\nthe relativistic momentum operator31. The functions\nV(r) andB=σ·ˆmB(r) are the spin-averaged and\nspin-dependent parts, respectively, of the LSDA poten-\ntial. The spin density ms(r) as well as the effective ex-\nchange field B(r) are assummed to be collinear within\nthe unit cell and aligned along the z-direction in the\nequilibrium (i. e. ms,0(r) =ms(r)ˆm0=ms(r)ezand\nB0(r) =B(r)ˆm0=B(r)ez). Tilting of the magnetiza-\ntion direction by the angle θaccording to Eq. (3), i.e.\nms(r) =ms(r)ˆm=ms(r)(sinθcosφ,sinθsinφ,cosθ)\nandB(r) =B(r)ˆmleads to a perturbation term in the\nHamiltonian\n∆V(r) =βσ·(ˆm−ˆm0)B(r) =βσ·uB(r),(9)\nwith (see Eq. (4))\n∂\n∂uµˆH(ˆm0) =βσµB(r). (10)\nThe Green’s function G+in Eq. (7) can be obtained in\na very efficient way by using the spin-polarized relativis-\ntic version of multiple scattering theory30that allows us\nto treat magnetic solids:\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(r,E)τnm\nΛΛ′(E)Zm×\nΛ′(r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn)/bracketrightbig\n.(11)\nHerer,r′refer to site nandm, respectively, where\nZn\nΛ(r,E) =ZΛ(rn,E) =ZΛ(r−Rn,E) is a function\ncentered at site Rn. The four-component wave functions\nZn\nΛ(r,E) (Jn\nΛ(r,E)) are regular (irregular) solutions to\nthe single-site Dirac equation labeled by the combined\nquantum numbers Λ (Λ = ( κ,µ)), withκandµbeing\nthe spin-orbit and magnetic quantum numbers31. The\nsuperscript ×indicates the left hand side solution of the\nDirac equation. τnm\nΛΛ′(E) is the so-called scattering path\noperator that transfers an electronic wave coming in at\nsiteminto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to\nthe following expression for the damping parameter:\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(12)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the4\ntotal magnetic moment µtot=µspin+µorb, ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and with the energy argument EFomit-\nted. The matrix elements in Eq. (12) are identical to\nthose occurring in the context of exchange coupling32:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(r)/bracketleftbigg∂\n∂uµˆH(ˆm0)/bracketrightbigg\nZn\nΛ(r)\n=/integraldisplay\nd3rZn×\nΛ′(r) [βσµBxc(r)]Zn\nΛ(r).(13)\nThe expression in Eq. (12) for the Gilbert-damping\nparameterαis essentially equivalent to the one obtained\nwithin the torque correlation method (see e.g. Refs. [33–\n35]). However, in contrast to the conventional TCM the\nelectronicstructureishererepresentedusingtheretarded\nelectronic Green function giving the present approach\nmuch more flexibility. In particular, it does not rely on\na phenomenological relaxation time parameter.\nThe expression Eq. (12) can be applied straightfor-\nwardly to disordered alloys. This can be done by de-\nscribing in a first step the underlying electronic struc-\nture (forT= 0 K) on the basis of the coherent po-\ntential approximation (CPA) alloy theory. In the next\nstep the configurational average in Eq. (12) is taken fol-\nlowing the scheme worked out by Butler29when dealing\nwith the electrical conductivity at T= 0 K or residual\nresistivity, respectively, of disordered alloys. This im-\nplies in particular that so-called vertex corrections of the\ntype/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat\naccount for scattering-in processes in the language of the\nBoltzmann transport formalism are properly accounted\nfor.\nOne has to note that the factorg\nµtotin Eq. (12) is sep-\narated from the configurational average /angbracketleft.../angbracketrightc, although\nboth values, gandµtot, have to represent the averageper\nunit cell doing the calculations for compounds and al-\nloys. This approximation is rather reasonable in the case\nof compounds or alloys where the properties of the ele-\nments of the system are similar (e.g. 3 d-element alloys),\nbut can be questionable in the case of systems containing\nelements exhibiting significant differences (3 d-5d-, 3d-4f-\ncompounds, etc), or in the case of non-uniform systems\nas discussed by Nibarger et al36.\nThermal vibrations as a source of electron scattering\ncan in principle be accounted for by a generalization of\nEqs. (7) – (13) to finite temperatures and by including\nthe electron-phonon self-energy Σ el−phwhen calculating\nthe Green’s function G+. Here we restrict our considera-\ntion to elastic scattering processes by using a quasi-static\nrepresentation of the thermal displacements of the atoms\nfrom their equilibrium positions. The atom displaced\nfrom the equilibrium position in the lattice results in a\ncorresponding variation ∆ tn=tn−tn\n0of the single-site\nscattering matrix in the global frame of reference37,38. A\nsingle-site scattering matrix tn(the underline denotes a\nmatrix in an angular momentum representation Λ) for\nthe atomndisplaced by the value sn\nνfrom the equilib-\nrium position in the lattice can be obtained using thetransformation matrices37,39\nUn\nLL′(sν,E) = 4π/summationdisplay\nL′′il′′+l−l′\n×CLL′L′′jl′′(sn\nν√\nE)YL′′(ˆsn\nν).(14)\nHeremeis the electron mass, jla spherical Bessel func-\ntion,CLL′L′′stands for the Gaunt coefficients, and a\nnon-relativistic angular momentum representation with\nL= (l,ml) has been used. Performing a Clebsch-Gordon\ntransformation for the transformation matrix Un\nLL′to\nthe relativistic Λ representation, the tmatrixtnfor the\nshifted atom can be obtained from the non-shifted one\ntn\n0from the expression\ntn\nν= (Un\nν)−1tn\n0Un\nν. (15)\nTreating for a discrete set of displacements sn\nνeach\ndisplacement as an alloy component, we introduce an\nalloy-analogy model to average over the set sn\nνthat\nis chosen to reproduce the thermal root mean square\naverage displacement/radicalbig\n/angbracketleftu2/angbracketrightTfor a given temperature\nT. This in turn may be set according to /angbracketleftu2/angbracketrightT=\n1\n43h2\nπ2mkΘD[Φ(ΘD/T)\nΘD/T+1\n4] with Φ(Θ D/T) the Debye func-\ntion,hthe Planck constant, kthe Boltzmann constant\nandΘ DtheDebyetemperature40. Ignoringthezerotem-\nperature term 1 /4 and assuming a frozen potential for\nthe atoms, the situation can be dealt with in full analogy\nto the treatment of disordered alloys on the basis of the\nCPA (see above).\nFor small displacements the transformation Eq. (14)\ncan be expanded with respect to sn\nν(see Ref. [39]) re-\nsulting in a linear dependence on sn\nνfor non-vanishing\ncontributions with ∆ l=|l−l′|=±1. This leads, in\nparticular, in the presence of atomic displacements for\ntransition-metals (TM), for which an angular momen-\ntum cut-off of lmax= 2 in the KKR multiple scattering\nexpansion is in general sufficient for an undistorted lat-\ntice, to an angular momentum expansion up to at least\nlmax= 3. However, this is correct only under the assump-\ntion of very small displacements allowing linearisation of\nthe transformation Uwith respect to the displacement\namplitudes. Thus, since the temperature increase leads\nto a monotonous increaseof s, the cut-off lmaxshould also\nbe increased.\nIII. MODEL CALCULATIONS\nIn the following we present results of calculations for\nwhich single parameters have artificially been manipu-\nlated in the first-principles calculations in order to sys-\ntematically reveal their role for the Gilbert-damping.\nThis approach is used to disentangle competing influ-\nences on the Gilbert-damping parameter.5\nA. Vertex corrections\nThe impact of vertex corrections is shown in Fig. 1\nfor two different cases: Fig. 1(a) represents the Gilbert\ndamping parameter for an Fe 1−xVxdisordered alloy as a\nfunction of concentration, while Fig. 1(b) gives the cor-\nresponding value for pure Fe in the presence of temper-\nature induced disorder and plotted as a function of tem-\nperature. Both figures show results calculated with and\nwithout vertexcorrectionsallowingforcomparison. First\nof all, a significant effect of the vertex corrections is no-\nticeable in both cases, although the variation depends on\nincreasingconcentrationofVinthebinaryFe 1−xVxalloy\nand the temperature in the case of pure Fe, respectively.\nSome differences in their behavior can be explained by\nthe differences of the systems under consideration. Deal-\ning with temperature effects via the alloy analogy model,\nthe system is considered as an effective alloy consisting\nof a fixed number of components characterizing different\ntypes of displacements. Thus, in this case the tempera-\nture effect is associated with the increase of disorder in\nthe system caused only by the increase of the displace-\nment amplitude, or, in other words – with the strength\nof scattering potential experienced by the electrons rep-\nresented by tn(T)−tn\n0. In the case of a random alloy\ntheA1−xBxvariation of the scattering potential, as well\nas the difference tn\nA−tn\nB, upon changing the concentra-\ntions is less pronounced for small amounts of impurities\nBandtheconcentrationdependenceisdeterminedbythe\namount of scatterers of different types. However, when\nthe concentration of impurities increases, the potentials\nof the components are also modified (this is reflected,\ne.g., in the shift of electronic states with respect to the\nFermi level, that will be discussed below) and this can\nlead to a change of the concentration dependence of the\nvertex corrections. An important issue which one has to\nstress that neglect of the vertex corrections may lead to\nthe unphysical result, α <0, as is shown in Fig. 1(a).\nIn terms of the Boltzmann transport formalism, this is\nbecause of the neglect of the scattering-in term41lead-\ning obviously to an incomplete description of the energy\ntransfer processes.\nB. Influence of spin-orbit coupling\nAswasalreadydiscussedabove,thespin-orbitcoupling\nfor the electrons of the atoms composing the system is\nthe main driving force for the magnetization relaxation,\nresultingintheenergytransferfromthemagneticsubsys-\ntem to the crystal lattice. Thus, the Gilbert damping pa-\nrameter should approach zero upon decreasing the SOC\nin the system. Fig. 2 shows the results for Py+15%Os,\nwhere√αis plotted as a function of the scaling param-\neter of the spin-orbit coupling42applied to all atoms in\nthe alloy. As one can see,√αhas a nearly linear depen-\ndence on SOC implying that αvaries in second order in\nthe strength of the spin-orbit coupling43.00.10.2 0.3 0.4 0.5\nconcentration xV02040α × 103without vertex corrections\nwith vertex correctionsFe1-xVx\n(a)\n0100200 300 400 500\ntemperature (K)051015202530α × 103without vertex corrections\nwith vertex correctionsbcc Fe\n(b)\nFIG. 1. The Gilbert damping parameter for (a) bcc Fe 1−xVx\n(T= 0 K) as a function of V concentration and (b) for bcc-Fe\nas a function of temperature. Full (open) symbols give resul ts\nwith (without) the vertex corrections.\n0 0.5 1 1.5 2\nSOC scaling parameter00.10.20.30.4  α1/2 Py+15%Os\nFIG. 2. The Gilbert damping parameter for Py+15%Os as\na function of the scaling parameter of spin-orbit coupling a p-\nplied to all atoms contained in the alloy. Red dashed line in\nplot – linear fit. The values 0 and 1 for the SOC scaling pa-\nrameter correspond to the scalar-relativistic Schr¨ oding er-like\nand fully relativistic Dirac equations, respectively.6\nIV. RESULTS AND DISCUSSIONS\nA. 3dtransition-metals\nWe have mentioned above the crucial role of scatter-\ning processes for the energy dissipation in magnetiza-\ntion dynamic processes. In pure metals, in the absence\nof any impurity, the electron-phonon scattering mecha-\nnism is of great importance, although it plays a different\nrole in the low- and high-temperature regimes. This was\ndemonstrated by Ebert et al.28using the alloy analogy\napproach,aswellasbyLiu44et al.usingthe ’frozenther-\nmal lattice disorder’ approach. In fact both approaches\nare based on the quasi-static treatment of thermal dis-\nplacements. However, while the average is taken by the\nCPA within the alloy analogy model the latter requires\na sequence of super-cell calculations for this purpose.\nAs a first example bcc Fe is considered here. The cal-\nculations have been performed accounting for the tem-\nperature induced atomic displacements from their equi-\nlibrium positions, according to the alloy analogy scheme\ndescribed in section II. This leads, even for pure systems,\nto a scattering process and in this way to a finite value\nforα(see Fig. 3(a)). One can see that the experimen-\ntal results available in the literature are rather different,\ndepending on the conditions of the experiment. In par-\nticular, the experimental results Expt. 2 (Ref. [45]) and\nExpt. 3 (Ref. [46]) correspondto bulk while the measure-\nments Expt. 1 (Ref. [47]) have been done for an ultrathin\nfilm with 2 .3 nm thickness. The Gilbert damping con-\nstant obtained within the present calculations for bcc Fe\n(circles,a= 5.44 a.u.) is compared in Fig. 3(a) with\nthe experiment exhibiting rather good agreement at the\ntemperature above 100 K despite a certain underestima-\ntion. One can also see a rather pronounced increase of\nthe Gilbert damping observed in the experiment above\n400 K (Fig. 3(a), Expt. 2 and Expt. 3), while the theo-\nretical value shows only little temperature dependent be-\nhavior. Nevertheless, the increase of the Gilbert damp-\ning with temperature becomes more pronounced when\nthe temperature induced lattice expansion is taken into\naccount, that can be seen from the results obtained for\na= 5.45 a.u. (squares). Thick lines are used to stress\nthe temperature regions for which corresponding lattice\nparameters are more appropriate. At low temperatures,\nbelow 100 K, the calculated Gilbert damping parameter\ngoes up when the temperature decreases, that was ob-\nserved only in the recent experiment47. This behavior is\ncommonly denoted as a transition from low-temperature\nconductivity-like to high-temperature resistivity-like be-\nhavior reflecting the dominance of intra- and inter-band\ntransitions, respectively3. The latter are related to the\nincrease of the smearing of electron energy bands caused\nby the increase of scattering events with temperature.\nNote that even a small amount of impurities reduces\nstrongly the conductivity-like behavior28,45, leading to\nthe more pronounced effect of impurity-scattering pro-\ncesses due to the increase of scattering events caused bychemical disorder. Large discrepancies between the lat-\nter experimental data47and theoretical results of the α\ncalculationsforbcc Fe arerelatedto the verysmall thick-\nness of the film investigated experimentally, that leads to\nan increase of spin-transfer channels for magnetization\ndissipation as was discussed above.\nResults for the temperature dependent Gilbert-\ndamping parameter αfor hcp Co are presented in Fig.\n3(b) which shows, despite certain underestimation, a rea-\nsonable agreement with the experimental results45. The\ngeneral trends at low and high temperatures are similar\nto those seen in Fe.\nThe results for pure Ni are given in Fig. 3(c) that show\nin full accordance with experiment a rapid decrease of α\nwith increasing temperature until a regime with a weak\nvariation of αwithTis reached.\nNote that in the discussions above we have treated α\nas a scalar instead of a tensorial quantity ignoring a pos-\nsible anisotropy of the damping processes. This approx-\nimation is reasonable for the systems considered above\nwith the magnetization directions along a three- or four-\nfold symmetry axis (see, e.g., the discussions in Ref. [48\nand 49]). For a more detailed discussion of this issue\nthe anisotropy of the Gilbert damping tensor α(M) has\nbeen investigated for bcc Fe. To demonstrate the depen-\ndence ofαon the magnetization direction M, the cal-\nculations have been performed for M= ˆz|M|with the\nˆzaxis taken along the /angbracketleft001/angbracketright,/angbracketleft111/angbracketrightand/angbracketleft011/angbracketrightcrystallo-\ngraphic directions. Fig. 4 presents the temperature de-\npendence of the diagonal elements αxxandαyy. As to be\nexpected for symmetry reasons, αxxdiffers from αyyonly\nin the case of ˆ z/bardbl/angbracketleft011/angbracketright. One can see that the anisotropic\nbehavior of the Gilbert damping is pronounced at low\ntemperatures. With an increase of the temperature the\nanisotropy nearly disappears, because of the smearing of\nthe energy bands caused by thermal vibrations49. A sim-\nilar behavior is caused by impurities with a random dis-\ntribution, aswasobservedforexamplefortheFe 0.95Si0.05\nalloy system. The calculations of the diagonal elements\nαxxandαyyfor two different magnetization directions\nalong/angbracketleft001/angbracketrightand/angbracketleft011/angbracketrightaxes giveαxx=αyy= 0.00123 in\nthe first case and αxx= 0.00123 and αyy= 0.00127 in\nthe second, i.e. the damping is nearly isotropic.\nThe damping parameter αincreases very rapidly with\ndecreasing temperature in the low temperature regime\n(T≤100 K) for all pure ferromagnetic 3 dmetals, Fe,\nCo, and Ni (see Fig. 3), leading to a significant discrep-\nancy between theoretical and experimental results in this\nregime. The observed discrepancy between theory and\nexperiment can be related to the exact limit ω= 0 taken\nintheexpressionfortheGilbertdampingparameter. Ko-\nrenmann and Prange13have analyzedthe magnon damp-\ning in the limit of small wave vector of magnons q→0,\nassuming indirect transitions in the electron subsystem\nand taking into account the finite lifetime τof the Bloch\nstates due to electron-phonon scattering. They discuss\nthe limiting cases of low and high temperatures showing\nthe analogy of the present problem with the problem of7\n0 200 400 600\nTemperature (K)0246810α × 103Theory: a = 5.42 a.u.\nTheory: Fe+0.1% Vac.\nTheory: a = 5.45 a.u.\nExpt. 1\nExpt. 2\nExpt. 3bcc Fe\n(a)\n0100200 300 400 500600\nTemperature (K)05101520α × 103Expt\nTheory\nTheory: Co + 0.03% Vac\nTheory: Co + 0.1% Vachcp Co\n(b)\n0100200 300 400 500\nTemperature (K)00.050.10.150.20.25 αExpt\nTheory\nTheory: Ni + 0.1%Vacfcc Ni\n(c)\nFIG. 3. Temperature variation of the Gilbert damping pa-\nrameter of pure systems. Comparison of theoretical results\nwith experiment: (a) bcc-Fe: circles and squares show the re -\nsults for ideal bcc Fe for two lattice parameters, a= 5.42 a.u.\nanda= 5.45 a.u.; stars show theoretical results for bcc Fe\n(a= 5.42 a.u.) with 0.1% of vacancies (Expt. 1 - Ref. [47],\nExpt. 2 - Ref. [45], Expt. 3 - Ref. [46]); (b) hcp-Co: circles\nshow theoretical results for ideal hcp Co, stars - for Co with\n0.03% of vacancies, and ’pluses’ - for Co with 0.1% of vacan-\ncies (Expt. Ref. [45]); and (c) fcc-Ni (Expt. Ref. [45]).extreme cases for the conductivity leading to the normal\nand anomalous skin effect. On the basis of their result,\nthe authors point out that the expression for the Gilbert\ndamping obtained by Kambersky25, withα∼τis cor-\nrect in the limit of small lifetime (i.e. qvFτ≪1, in their\nmodel consideration, where qis a magnon wave vector\nandvFis a Fermi velocity of the electron). In the low-\ntemperature limit the lifetime τincreases with decreas-\ningTand one has to use the expression corresponding\nto the ’anomalous’ skin effect for the conductivity, i.e.\nα∼tan−1(qvFτ)/qvF, leading to a saturation of αupon\nthe increase of τ.\nAnother possible reason for the low-temperature be-\nhavior of the Gilbert damping observed experimentally\ncan be structural defects present in the material. To\nsimulate this effect, calculations have been performed for\nfcc Ni and bcc Fe with 0.1% of vacancies and for hcp\nCo with 0.1% and 0.03% of vacancies. Fig. 3(a)-(c)\nshows the corresponding temperature dependence of the\nGilbert dampingparameterapproachingafinite value for\nT→0. The remaining difference in the T-dependent be-\nhavior can be attributed to the non-linear dependence of\nthe scattering cross section at low temperatures as is dis-\ncussed in the literature for transport properties of metals\nand is not accounted for within the present approxima-\ntion.\nB. 3dTransition-metal alloys\nAs is mentioned above, the use of the linear response\nformalism within multiple scattering theory for the elec-\ntronicstructurecalculationsallowsustoperformthenec-\nessary configurational averaging in a very efficient way\navoiding supercell calculations and to study with mod-\nerate effort the influence of varying alloy composition\nonα. The corresponding approach has been applied to\n0100200 300 400 500\nTemperature (K)051015202530α × 103Theory: z||<001>: αxx=αyy\nTheory: z||<011>: αxx\nTheory: z||<011>: αyy\nTheory: z||<111>: αxx=αyybcc Fe\nFIG. 4. Temperature variation of the αxxandαyycom-\nponents of the Gilbert damping tensor of bcc Fe with the\nmagnetization direction taken along different crystallogr aphic\ndirections: M= ˆz|M|/bardbl/angbracketleft001/angbracketright(circles), M/bardbl/angbracketleft011/angbracketright(squares),\nM/bardbl/angbracketleft111/angbracketright(diamonds).8\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nconcentration xCo0123456 α × 103bcc, CPA\nCsCl: CPA (partial ord)\nbcc, NL CPA (ord)\nbcc, NL CPA (disord)\nn(EF)Fe1-xCox\nn(EF)\nn(EF) (sts./Ry)\n102030405060\n0\n(a)\n-8-6-4 -2 0 2\nenergy (eV)00.511.52n↑tot(E) (sts./eV)00.511.52n↓tot(E) (sts./eV)x = 0.01\nx = 0.5EF\n(b)\nFIG. 5. (a) Theoretical results for the Gilbert damping pa-\nrameter of bcc Fe 1−xCoxas a function of Co concentration:\nCPA results for the bcc structure (full circles) describing\nthe random alloy system, results for the partially ordered\nsystem (opened square) for x= 0.5 (i.e. for Fe 1−xCox\nalloy with CsCl structure and alloy components randomly\ndistributed in two sublattices in the following proportion s:\n(Fe0.9Co0.1)(Fe0.1Co0.9), the NL-CPA results for random al-\nloy with bcc structure (opened circles) and the NL-CPA re-\nsults for the the system with short-range order within the\nfirst-neighbor shell (opened diamonds). The dashed line rep -\nresents the DOS at the Fermi energy, EF, as a function of Co\nconcentration. (b) spin resolved DOS for bcc Fe 1−xCoxfor\nx= 0.01 (dashed line) and x= 0.5 (solid line).\nthe ferromagnetic 3 d-transition-metal alloy systems bcc\nFe1−xCox, fcc Ni 1−xFex, fcc Ni 1−xCoxand bcc Fe 1−xVx.\nFig. 5(a) shows as an example results for the Gilbert\ndamping parameter α(x) calculated for bcc Fe 1−xCox\nforT= 0 K at different conditions. Full circles rep-\nresent the results of the single-cite CPA calculations\ncharacterizing the random Fe-Co alloy. These results\nare compared to those obtained employing the non-local\nCPA52,53(NL-CPA) assuming no short-range order in\nthe system (opened circles). Dealing in both cases (CPA\nand NL-CPA), with completely disordered system, the\nNL-CPA maps the alloy problem on that of an impurity\ncluster embedded in a translational invariant effective\nmedium determined selfconsistently, thereby accounting0 0.2 0.4 0.6\nconcentration xCo0246 α × 103Exp.\nTheoryFe1-xCox\n(a)\n0 0.2 0.4 0.6\nconcentration xCo010203040 α × 103Exp.\nTheory (Starikov et al.)\nTheoryNi1-xCox\n(b)\n0 0.2 0.4 0.6\nconcentration xFe01020 α × 103Theory (Starikov et al.)\nTheory\nExpt 1\nExpt 2Ni1-xFex\n(c)\n0 0.2 0.4 0.6\nconcentration xV012345 α × 103Expt.\nTheory, T = 0 K\nTheory, T = 300 KFe1-xVx\n(d)\nFIG. 6. The Gilbert damping parameter for Fe 1−xCox(a)\nNi1−xCox(b) and Ni 1−xFex(c) as a function of Co and Fe\nconcentration, respectively: present results within CPA ( full\ncircles), experimental data by Oogane50(full diamonds). (d)\nResults for bcc Fe 1−xVxas a function of V concentration:\nT= 0 K (full circles) and T= 300 K (open circles). Squares:\nexperimental data51. Open circles: theoretical results by\nStarikov et al.5.9\nfor nonlocal correlations up to the range of the cluster\nsize. The present calculations have been performed for\nthe smallestNL-CPAclusterscontainingtwositesforbcc\nbased system, accountingfor the short-rangeorder in the\nfirst-neighborshell. As onecan see, this results in a small\ndecrease of the αvalue in the region of concentrations\naroundx= 0.5 (opened diamonds), that is in agreement\nwith the results obtained for partially ordered system\n(opened square) for x= 0.5. The latter have been calcu-\nlated for the Fe 1−xCoxalloy having CsCl structure and\nalloycomponentsrandomlydistributed intwosublattices\nin the following proportions: (Fe 0.9Co0.1)(Fe0.1Co0.9).\nBecause the moments and spin-orbit coupling strength\ndo not differ very much for Fe and Co, the variation of\nα(x) should be determined in the concentrated regime\nessentially by the electronic structure at the Fermi en-\nergyEF. As Fig. 5(a) shows, there is indeed a close\ncorrelation with the density of states n(EF) that may be\nseen as a measure for the number of available relaxation\nchannels. The change of α(x) due to the increase of the\nCo concentration is primarily determined by an appar-\nent shift of the Fermi energy also varying with concen-\ntration (Fig. 5(b)). The alloy systems considered have\nthe common feature that the concentration dependence\nofαis governed by the concentration dependent density\nof statesn(EF).\nA comparison of theoretical αvalues with the experi-\nmentforbccFe 1−xCoxisshowninFig. 6(a),demonstrat-\ning satisfying agreement. In the case of Ni 1−xFexand\nNi1−xCoxalloysshown in Fig. 6, (b) and (c), the Gilbert\ndampingdecreasesmonotonouslywith the increaseofthe\nFe and Co concentration, in line with experimental data.\nAt all concentrations the experimental results are under-\nestimated by theory approximately by a factor of 2. The\ncalculated dampingparameter α(x) is found in verygood\nagreementwith theresultsbasedonthe scatteringtheory\napproach5demonstrating numerically the equivalence of\nthe two approaches. An indispensable requirement to\nachieve this agreement is to include the vertex correc-\ntions mentioned above. As suggested by Eq. (12) the\nvariation of α(x) with concentration xmay also reflect\nto some extent the variation of the averagemagnetic mo-\nment of the alloy, µtot.\nThepeculiarityoftheFe 1−xVxalloywhencomparedto\nthose discussed above is that V is a non-magnetic metal\nand has only an induced spin magnetic moment. De-\nspite that, the concentration dependence of the Gilbert\ndamping parameter at T= 0 K for small amounts of\nV (see Fig. 6(d)) displays the same trend as the pre-\nviously discussed alloys shown in Fig. 6(a)-(c). Taking\ninto account a finite temperature of T= 300 K changes\nαvalue significantly at small V concentrations leading\nto an improved agreement with experiment for pure Fe,\nwhile it still compares poorly with the experimental data\natxV= 0.27. One should stress once more that the con-\ncentration dependent behavior of the Gilbert damping\nparameter of the alloys discussed above is different for\nan increased amount of impurities (more than 10%), as aresultofadifferentvariationoftheDOS n(EF)causedby\na concentration dependent modification of the electronic\nstates and shift of the Fermi level.\nC. 5dimpurities in 3 dtransition metals\nAs discussed in our recent work28investigating the\ntemperature dependent Gilbert damping parameter for\npure Ni and for Ni with Cu impurities, αis primarily\ndetermined by the thermal displacement in the regime of\nsmall impurity concentrations. This behavior can also be\nseen in Fig. 7, where the results forFe with 5 d-impurities\nare shown. Solid lines represent results for T= 0 K for\nan impurity concentration of 1% (full squares) and 5%\n(full circles). As one can see, at smaller concentrations\nthe maximum of the Gilbert damping parameter occurs\nfor Pt. With increasing impurity content the αparame-\nter decreases in such a way that at the concentration of\n5% a maximum is observed for Os.\nThe reason for this behavior lies in the rather weak\nscattering efficiency of Pt atoms due to a small DOS\nn(EF) of the Pt states when compared for example for\nOsimpurities (see Fig. 9). This results in a slowdecrease\nofαat small Pt concentration when the BFS mechanism\nis mostly responsible for the energy dissipation. A con-\nsequence of this feature can be seen in the temperature\ndependence of α(T= 300 K, opened squares): a most\npronounced temperature induced decrease of the αvalue\nis observed for Pt and Au. When the concentration of\n5d-impurities is increased up to 5%, the maximum in α\noccurs for the element with the most efficient scatter-\ning potential resulting in spin-flip scattering processes\nresponsible for dissipation. The temperature effect at\nthis concentration is very small.\nConsidering in more detail the temperature dependent\nbehavior of the Gilbert damping parameter for Fe with\nOs and Pt impurities, shown in Fig. 8, one can also ob-\nserve the consequence of the features mentioned above.\nAt 1% of Pt impurities αdecreases much steeper upon\nincreasing the temperature, as compared to the case of\nOs impurities. Therefore, in the first case the role of the\nscattering processes due to atomic displacements is much\nmore pronounced than in the second case with rather\nstrong scattering on the Os impurities. When the con-\ncentration increases to 5% the dependence of αon the\ntemperature in both cases becomes less pronounced.\nThe previousresults can be comparedto the results for\nthe 5d-impurities in the permalloy Fe 80Ni20(Py), which\nhas been investigated also experimentally54. This system\nshows some difference in the concentration dependence\nwhen compared with pure Fe, because Py is a disordered\nalloy with a finite value of the αparameter. Therefore,\na substitution of 5 dimpurities leads to a nearly linear\nincrease of the Gilbert damping with impurity content,\njust as seen in experiment54.\nThe total damping for 10% of 5 d-impurities shown in\nFig.10(a)variesroughlyparabolicallyoverthe 5 dTMse-10\nTa W Re Os Ir Pt Au0246810 α × 103x = 0.05; T = 0 K\nx = 0.05; T = 300 K \nx = 0.01; T = 0 K\nx = 0.01; T = 300 K\nFIG. 7. Gilbert damping parameter for bcc Fe with 1%\n(squares) and 5% (circles) of 5 dimpurities calculated for\nT= 0K (full symbols) and for T= 300K (opened sysmbols).\nries. This variation of αwith the type of impurity corre-\nlateswellwith the densityofstates n5d(EF) (Fig. 10(b)).\nAgain the trend of the experimental data is well repro-\nduced by the calculated values that are however some-\nwhat too low.\nV. SUMMARY\nIn summary, aformulationforthe Gilbert dampingpa-\nrameterαin terms of linear response theory was derived\nthatledtoaKubo-Greenwood-likeequation. Thescheme\nwas implemented using the fully relativistic KKR band\nstructuremethod incombinationwiththe CPAalloythe-\nory. This allows to account for various types of scat-\ntering mechanisms in a parameter-free way, that might\nbe either due to chemical disorder or to temperature-\ninduced structural disorder (i.e. electron-phonon scat-\ntering effect). The latter has been described by using\nthe so-called alloy-analogy model with the thermal dis-\nplacement of atoms dealt with in a quasi-static manner.\nCorresponding applications to pure metals (Fe, Co, Ni)\naswellastodisorderedtransition-metalalloysledto very\ngood agreement with results based on the scattering the-\nory approach of Brataas et al.4and well reproduces the\nexperimental results. The crucial role of vertex correc-\ntions for the Gilbert damping is demonstrated both in\nthe case of chemical as well as structural disorder and\nthe accuracy of finite-temperature results is analyzed via\ntest calculations.\nFurthermore, the flexibility and numerical efficiency\nof the present scheme was demonstrated by a study\non metallic systems on a series of binary 3 d-alloys\n(Fe1−xCox, Ni1−xFex, Ni1−xCoxand Fe 1−xVx), 3d−5d\nTM systems, the permalloy-5 dTM systems. The agree-\nment between the present theoretical and experimental\nresults is quite satisfying, although one has to stress\na systematic underestimation of the Gilbert damping\nby the numerical results. This disagreement could be\ncaused either by the idealized system considered theoret-\nically (e.g., the boundary effects are not accounted for0100200 300 400 500\ntemperature (K)12345α × 103Fe0.99Me0.01Pt\nOs\n(a)\n0100200 300 400 500\ntemperature (K)22.533.54α × 103Fe0.95Me0.05\nPtOs\n(b)\nFIG. 8. Gilbert damping parameter for bcc Fe 1−xMxwith\nM= Pt (circles) and M= Os (squares) impurities as a func-\ntion of temperature for 1% (a) and 5% (b) of the impurities.\n-8-6-4 -2 0 2\nenergy (eV)00.81.62.43.2n↑(E) (sts./eV)00.81.62.43.2n↓(E) (sts./eV)Pt\nOsEF\nEF\nFIG. 9. DOS for Pt in Fe 1−xPtx(full line) and Os in\nFe1−xOsx(dashed line) for x= 0.01.11\nTa W Re OsIr Pt Au02468α × 102Expt.\nTheory\n(a)\nTa W Re OsIr Pt Au051015n5d(EF) (Sts/Ry)\n(b)\nFIG. 10. 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Physics 101,\n033911 (2007)"    },    {        "title": "1302.4744v1.Chirality_Sensitive_Domain_Wall_Motion_in_Spin_Orbit_Coupled_Ferromagnets.pdf",        "content": "Chirality Sensitive Domain Wall Motion in Spin-Orbit Coupled Ferromagnets\nJacob Linder1\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n(Dated: March 31, 2022)\nUsing the Lagrangian formalism, we solve analytically the equations of motion for current-induced\ndomain-wall dynamics in a ferromagnet with Rashba spin-orbit coupling. An exact solution for the\ndomain wall velocity is provided, including the e\u000bect of non-equilibrium conduction electron spin-\ndensity, Gilbert damping, and the Rashba interaction parameter. We demonstrate explicitly that\nthe in\ruence of spin-orbit interaction can be qualitatively di\u000berent from the role of non-adiabatic\nspin-torque in the sense that the former is sensitive to the chirality of the domain wall whereas\nthe latter is not: the domain wall velocity shows a reentrant behavior upon changing the chirality\nof the domain wall. This could be used to experimentally distinguish between the spin-orbit and\nnon-adiabatic contribution to the wall speed. A quantitative estimate for the attainable domain\nwall velocity is given, based on an experimentally relevant set of parameters for the system.\nPACS numbers:\nI. INTRODUCTION\nThe study of domain-wall motion in ferromagnetic ma-\nterials has attracted much interest in recent years. Be-\nsides its allure from a fundamental physics viewpoint,\nelectric control of magnetic textures is attractive in terms\nof potential new applications such as magnetic memory.\nA key concept in the context of domain wall motion is\nthe so-called spin-transfer torque [1{3]: in essence, it con-\nsists of a transfer of a transverse spin-current component\nto the ferromagnetic order parameter which may occur\nin a non-collinear magnetization con\fguration. Active\ncontrol over domain-wall motion is a chief objective in\nterms of realizing the \"magnetic race-track\" technology\nput forward in [4]. Other ways to manipulate domain wall\nmotion include energy redistribution in the presence of\nan external magnetic \feld, by applying microwave radia-\ntion [5{9], and by using magnons [10{13]. Moreover, the\nconcept of spin-transfer torque has recently been studied\nin antiferromagnets [14{19] including the possibility to\nmove domain-walls.\nInterestingly, it turns out that spin-orbit interactions\ncan substantially modify both the spin-transfer torque\nand the resulting domain wall motion [20, 21, 25]. The\ncombined in\ruence of a magnetic exchange \feld together\nwith spin-orbit interaction, typically taken in the Rashba\nform, can be shown to give rise to a non-equilibrium\nspin density perpendicular to the injected current \row.\nIn turn, this gives rise to an e\u000bective magnetic \feld\nwhich causes magnetization dynamics and, for su\u000e-\nciently strong current density, magnetization switching.\nThe e\u000bective \"spin-orbit torque\" arising in this manner is\nqualitatively di\u000berent from the conventional spin-transfer\ntorque due to the di\u000berent mechanism at hand: it does\nnot require the presence of non-collinear magnetic ele-\nments and will cause magnetization dynamics in a single\nferromagnetic layer [28]. In addition, it is important to\nnote that whereas di\u000berent types of domain walls behave\nin the same way in the absence of spin-orbit interactions,\nthe exact magnetization texture plays a key role whenspin-orbit coupling is present due to the coupling between\nthe electron motion and the spin torque.\nIn light of the above, several experimental and theoreti-\ncal works has recently explored the in\ruence of spin-orbit\ninteractions on magnetization dynamics in various mag-\nnetic systems [22{30]. Since the addition of spin-orbit\nterms in the equations of motion for the magnetization\ntexture complicates their solution, the large majority of\nthese works have relied on numerical methods to solve\nthe Landau-Lifshitz-Gilbert (LLG) equation. In this pa-\nper, we utilize the Lagrangian formalism in order to write\ndown and solve analytically the equations of motion for\na domain wall within a collective-coordinate description.\nThe analytical nature of this approach allows us to iden-\ntify a transparent expression for the domain wall velocity\nand how it depends on parameters such as the spin-orbit\ninteraction, exchange \feld, and Gilbert damping of the\nsystem. Alternatively, one could have derived this result\nvia the LLG equation, but the present formalism makes\nit easier to accommodate non-equilibrium spin-density\nterms and higher order corrections due to the spin-orbit\ninteraction. We show from the analytical expression that\nthe presence of spin-orbit interactions renders the domain\nwall velocity to be chirality sensitive, in e\u000bect depend-\ning on whether the wall changes magnetization direction\nfrom positive to negative or vice versa along the direction\nof the current. We provide an estimate for the magnitude\nof the domain wall velocity using a set of experimentally\nrelevant parameters and show that the velocity behaves\nqualitatively di\u000berently depending on the chirality of the\ndomain wall and displays reentrant behavior. Our \fnding\nsuggests a way to experimentally distinguish between the\nnon-adiabatic and spin-orbit contribution to the domain-\nwall speed.\nII. THEORY\nWe consider \frst a ferromagnetic domain wall with an\neasy (hard) axis of magnetic anisotropy [36] along thearXiv:1302.4744v1  [cond-mat.mes-hall]  19 Feb 20132\nTilt angleφˆz\nˆy(easy)\nˆx(hard)\nTilt angleφˆz(easy)\nˆy(hard)\nCurrent direction(a)\n(b)Current direction\nFIG. 1: (Color online) Sketch of the magnetization textures\nconsidered in this paper. The domain wall nanowire extends\nalong thex-axis. (a) Hard axis along current direction ( x) and\neasy axis along z-direction. In equilibrium, the magnetization\nof the domain wall rotates in the yz-plane exactly as in Ref.\n[21] and similarly to Ref. [28]. Out of equilibrium, i.e. under\na current bias, a \fnite component may be acquired along the\nx-axis as indicated by the tilt angle \u001e. This geometry is the\nmain focus of this article, relevant for nanowires with perpen-\ndicular magnetic anisotropy. (b) Hard axis along y-direction\nand easy axis along z-direction. In equilibrium, the magne-\ntization rotates in the xz-plane. Out of equilibrium, a \fnite\ncomponent may be acquired along the y-axis as indicated by\nthe tilt angle \u001e.\ny-axis (x-axis). An electric current is injected along the\nx-axis [see Fig. 1a)], and the inversion symmetry is as-\nsumed broken in the ^ z-direction. This gives rise to a\nRashba spin-orbit coupling term, which in\ruences the\nmagnetization dynamics. More precisely, it can be shown\n[20] that the spin-orbit coupling generates an e\u000bective\nmagnetic \feld\nHSOC/\u000bR^z\u0002je (1)\nwhere\u000bRis the Rashba interaction strength and jeis\nthe current density vector. In order to treat the domain\nwall as rigid in a collective-coordinate framework, thus\nmaking other modes of deformation apart from the tilt\nangle\u001eirrelevant, the easy axis anisotropy energy Kis\nassumed larger than its hard axis equivalent K?[31], i.e.\njKj\u001djK?j. As is normally done, we do not take into\naccount the e\u000bect of the end-boundaries of the nanowire,\nassuming thus that the domain-wall center is located suf-\n\fciently far away from these during its propagation.\nThe starting point is the Lagrangian for such a Blochdomain-wall which was derived in Ref. [21]. For this type\nof domain wall, the magnetization rotates in the plane\nperpendicular to the extension of the wire (and current\ndirection), similarly to Ref. [28] (the only di\u000berence from\nRef. [28] is which of the perpendicular axes that is the\neasy one, in both cases the hard axis is along the current\ndirection). It reads:\nL= (\u001e_x\u0000sin2\u001e)\u00002\u0015xJ\u0001\n\u0016\u0000syx_\u001e\u0000\u001eJ\u0010\u0001\n\u0016+F(\u0015)\u0011\n(2)\nwith the de\fnition\nF(\u0015) =~2\u00152\nmL2\u00101\n\u0016+2\n\u0001\u0011\n: (3)\nAbove,\u001eis the tilt angle of the domain wall (see Fig.\n1) whilexis the normalized position of the center of\nthe domain wall ( x=X=L whereLis the wall thick-\nness). The quantities \u0015andJare the Rashba interac-\ntion strength and the current density normalized against\nmL= ~2andevc, respectively, with vcbeing the drift ve-\nlocity of electrons at the intrinsic threshold current for\n\u0001=\u0016= 1 without Rashba interactions. Moreover, \u0001 is\nthe exchange splitting, \u0016is the Fermi energy while m\nis the electron mass. Finally, syrepresents the constant\nnon-equilibrium spin density induced by the applied elec-\ntric \feld generating the current.\nIII. RESULTS AND DISCUSSION\nWe will model dissipation in this system by a Rayleigh\ndissipation function of the form [31]\nW=\u000b\n2( _x2+_\u001e2): (4)\nThe Lagrange equations are obtained via\nd\ndt@L\n@_q\u0000@L\n@q=\u0000@W\n@_q; q2fx;\u001eg: (5)\nwheretis dimensionless time-coordinate normalized\nagainstl=vc. De\fning for convenience\nc=\u0001\n\u0016+F(\u0015) (6)\nwe obtain the following Lagrange-equations, which were\nstudied numerically in [21]:\n_\u001e+\u000b_x=sy_\u001e\u00002\u0015J\u0001\n\u0016;_x\u0000\u000b_\u001e=\u0000sy_x+ sin 2\u001e+cJ:\n(7)\nThe role of the dissipative spin-transfer torque (also\nknown as non-adiabatic or \f-torque) will be addressed\nlater on - we will see that it plays a similar role as the\nspin-orbit coupling, but with one important di\u000berence.3\n0 200 400−150−100−50050100150200250300350\nCurrent J[m/s]Domain wall velocity /angbracketleftVDW/angbracketright[m/s]\n0 200 400−150−100−50050100150200250300350\nCurrent J[m/s]Domain wall velocity /angbracketleftVDW/angbracketright[m/s]\n  |Λ|= 0\n1.0e-14 eV ·m\n5.0e-14 eV ·m\n1.0e-13 eV ·m\n2.0e-13 eV ·m\n1.0e-12 eV ·m\n1.0e-11 eV ·m(b) (a)\nFIG. 2: (Color online) Plot of the terminal domain wall ve-\nlocity using \u000b= 0:005,L= 75 nm,m= 0:04me,\u0016= 0:05\neV, \u0001 = 0:02 eV, and vc= 150 m/s. (a) Positive chirality\n\u0015>0. (b) Negative chirality \u0015<0.\nBy combining the above equations, we are able to elimi-\nnate the _x-dependence and obtain a \frst-order di\u000beren-\ntial equation for the tilt angle:\n_\u001e=\u0000\u000b\n1 +\u000b2\n1+sy\u0000sy\u0010sin 2\u001e\n1 +sy+cJ\n1 +sy+ 2J\u0015\u0001\n\u000b\u0016\u0011\n(8)This equation may be cast into integral form as follows:\nZ\nd\u001e\u000b\u00001(1 +sy)(sy\u00001\u0000\u000b2=(1 +sy))\na+ sin 2\u001e=t:(9)\nWe de\fne the quantities\na=cJ+2\u0015\u0001J(1 +sy)\n\u000b\u0016;\nb=\u0000\u000b\u0000\u000b\u00001[1\u0000(sy)2]: (10)\nThe formal solution of this integral is obtained after some\nalgebraic manipulation:\ntan\u001e=\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0);\n(11)\nwhereC0is an integration constant to be determined\nfrom the initial conditions. In particular, \u001e(t= 0) = 0\nand\u001e(t= 0) =\u0019=2 yieldC0= atan(1=p\na2\u00001) and\nC0=\u0019=2, respectively.\nHaving obtained an explicit expression for the tilt\nangle, we are now in a position to identify the time-\ndependence of the domain-wall center x, thus also ob-\ntaining the domain-wall velocity _ x. To do so, we \frst\nobtain _\u001efrom Eq. (11) as:\n_\u001e=(a2\u00001)\nabcos\u00002(b\u00001tp\na2\u00001 +C0)\n1 + [\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)]2: (12)\nSubstituting this into the \frst Lagrange-equation, one obtains an explicit expression for the domain-wall velocity vDW\nvDW=\u00002J\u0015\n\u000b\u0001\n\u0016\u0000(1\u0000sy)(a2\u00001)\nab\u000bcos\u00002(b\u00001tp\na2\u00001 +C0)\n1 + [\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)]2: (13)\nEqs. (11) and (13) determine an exact analytical expression for the time-dependence of the domain wall tilt-angle\nand velocity, respectively, which we will analyze in more detail below.\nBy integration, Eq. (13) may be used to identify the\ntime-evolution of the domain-wall center:\nx=\u00002Jt\u0015\n\u000b\u0001\n\u0016\u0000(1\u0000sy)\n\u000bZ\ndt_\u001e: (14)\nThe integral over _\u001eis evaluated by making use of the\nformula\nZdycos\u00002(y+\r)\n[1 + (\u0011+\u0010tan(x+\r))2]=\u0010\u00001atan[\u0010tan(x+\r) +\u0011];\n(15)\nresulting in the following equation describing the instan-taneous position of the domain-wall center\nx\u0000x0=\u00002Jt\u0015\n\u000b\u0001\n\u0016\u00001\u0000s\r\n\u000b\u0002\ntan\u00001[a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)\u0000a\u00001];\n(16)\nwithx0being an integration constant related to the ini-\ntial position of the domain-wall.\nIt is worth noting here that there exists a particularly\nsimple solution to the equation set Eq. (7) in the spe-\ncial case of a constant tilt angle, i.e. a domain-wall that\npreserves its shape and magnetization direction and thus4\nonly propagates. This amounts to setting _\u001e= 0, which\ndictates that the tilt angle must be constant:\nsin(2\u001e0) =\u0000cJ\u00002(1 +s\r)J\u0015\n\u000b\u0001\n\u0016; (17)\nand gives for the domain-wall velocity:\nvDW\u0011_x=\u00002J\u0015\n\u000b\u0001\n\u0016: (18)\nThis is identical to the \frst term in Eq. (13). Physically,\nthis implies a constant drift velocity of the domain-wall\nunder the application of a current. Interestingly, it is\nseen thatvDW= 0 in the absence of \u0015, meaning that the\nspin-orbit interaction is fully responsible for the domain-\nwall motion. In a more general scenario where the tilt\nangle is not restricted to being constant, i.e. allowing for\ndomain-wall deformation, the general expression for the\nvelocity is given by Eq. (13). As we shall discuss later,\nthis corresponds to the Walker breakdown threshold.\nThe necessity of spin-orbit coupling to drive the\ndomain-wall motion is a feature pertaining speci\fcally\nto the Bloch-domain wall with anisotropy axis chosen\nas described in the beginning of this section [see Fig.\n1a)]. In fact, for di\u000berent choices of anisotropy directions,\nspin-orbit coupling mainly contributes as a quantitative\ncorrection to the domain-wall velocity without being a\nprerequisite for its existence. To see this, one may con-\nsider instead a domain-wall where the magnetic easy and\nhard axes lie along the zandy-axes, respectively [see\nFig. 1b)]. In this case, the appropriate Lagrangian to\nconsider is [21]\nL= (\u001e_x\u0000sin2\u001e) + cos\u001e_\u001e\u0000\u0019sy\n2( _xcos\u001e\u0000sin\u001e_\u001e)\n\u0000szx_\u001e\u0000\u001eJ(\u0001\n\u0016\u0000~2\n2\u0001mL2) +\u0019\u0015sin\u001eJ\u0001=\u0016: (19)Using the same approach as above, one arrives at the\nfollowing set of integro-di\u000berential equations for the time-\nevolution of the domain-wall center and tilt angle:\n_\u001e+\u0019sy\n2sin\u001e_\u001e+sz_\u001e=\u0000\u000b_x (20)\nin addition to\nZ\nd\u001e\u000b\u00001[\u000b2+ (1 +\u0019sy\n2sin\u001e+sz)2]=[\u0019\u0015cos\u001eJ\u0001=\u0016\n\u0000sin 2\u001e\u0000J(\u0001\n\u0016\u0000~2\u00152\n2\u0001mL2)] =t\u0000t0: (21)\nUnlike the situation considered formerly, this equation\nset may be solved exactly analytically only under simpli-\nfying circumstances. For instance, by assuming that the\n\fnal term in the denominator of the integral equation\nabove dominates and moreover using s\r\u001c1,\r=fy;zg,\none identi\fes the domain-wall velocity in the strong-\ncurrent regime J\u001d1 as\nvDW=\u0000J\n1 +\u000b2\u0010\u0001\n\u0016\u0000~2\u00152\n2\u0001mL2\u0011\n: (22)\nAs seen, the presence of spin-orbit coupling in this case\nbrings about a minor correction to the \fnal velocity, es-\npecially for a strong ferromagnet where \u0001 =\u0016dominates\nthe expression in parantheses. The domain-wall is driven\ndirectly by the current Jwith a velocity that increases\nwith decreasing dissipation \u000b. The above result for the\ndomain wall velocity shows that the spin-orbit coupling\nhas a qualitatively di\u000berent e\u000bect on di\u000berent domain\nwall textures.\nIn the following, we focus on the more interesting case where the spin-orbit coupling in\ruences qualitatively the\ndomain-wall motion [Eqs. (11) and (13)] and investigate the precise dynamics using the derived analytical expressions.\nFor concreteness, we consider initial conditions such that the tilt angle of the domain-wall at t= 0 is zero, meaning\nthat we may write:\ntan\u001e=1\nahp\na2\u00001 tan\u0010tp\na2\u00001\nb+ atan\u00101p\na2\u00001\u0011\u0011\n\u00001i\n;\nvDW=\u00002J\u0015\n\u000b\u0001\n\u0016\u0000\u000b\u00001(1\u0000sy)(a2\u00001)\nab\u0002cos\u00002(b\u00001tp\na2\u00001 + atan(1=p\na2\u00001))\n1 +1\na2[p\na2\u00001 tan(b\u00001tp\na2\u00001 + atan(1=p\na2\u00001))\u00001]2: (23)\nIn the limit \u000b!1 , we \fnd that \u001e(t)!0 andvDW!0\nas expected. The analytical expression for vDWabove\nreveals that the velocity has non-monotonic behavior as\na function of time. In particular, there is a resonance\nconditiont=tresat which the velocity increases in mag-nitude:\ntres=(n+ 1=2)\u0019bp\na2\u00001\u0000b\na2\u00001; (24)\nassuming that a\u00151. In e\u000bect, vDWexhibits oscillations\nwhich persist even for its terminal behavior t!1 . It is5\ntherefore of interest to establish the average domain wall\nvelocity by averaging over one period T=\u0019b=p\na2\u00001:\nhvDWi=1\nTZT\n0dtvDW: (25)\nNote that for a<1, only the drift-term in vDWsurvives\nast!1 [since cos2(\u0006it)!1 in this limit]. Performingthe integral with this in mind, one \fnds for arbitrary a\nthat\nhvDWi=\u00002J\u0015\u0001\n\u000b\u0016\u00001\u0000sy\n\u000bbsignfagRefp\na2\u00001g;(26)\nwhich written out in terms of the original physical pa-\nrameters reads\nhvDWi=\u00002J\u0015\u0001\n\u000b\u0016+(1\u0000sy)signfJg\n\u000b2+ (1\u0000sy)\u0002signn2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0001\n\u0016+F(\u0015)o\n\u0002Re(sh2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0001\n\u0016+F(\u0015)i2\nJ2\u00001)\n(27)\nEq. (27) is the main result of this work and constitutes\na generally valid analytical expression for the terminal\ndomain wall velocity taking into account both Rashba\nspin-orbit coupling, the non-equilibrium spin density, and\nGilbert damping. For consistency, we have veri\fed that\na fully numerical solution of the equations of motion give\nidentical results as the above analytical expression for the\ndomain wall velocity.\nWe will proceed to analyze this velocity quantitatively\nfor a realistic set of parameters and investigate how it de-\npends in particular on the applied current and the mag-\nnitude of the spin-orbit coupling. Before doing so, one\nshould note that it follows from Eq. (27) that there exists\nboth a threshold current Jcfor which the second term in\nEq. (27) is non-zero:\njJcj=\f\f\f2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0010\u0001\n\u0016+~2\u00152\nmL2\u0016+2~2\u00152\nmL2\u0001\u0011\f\f\f\u00001\n:\n(28)\nIn the limiting case of zero spin-orbit interaction, one ob-\ntainsjJcj= (\u0001=\u0016)\u00001in agreement with previous studies.\nThe presence of spin-orbit interaction is seen from the\nanalytical expression of jJcjto reduce the threshold cur-\nrent monotonically with increasing \u0015, consistently with\nthe numerical study in Ref. [21]. This monotonic be-\nhavior appears also when tuning the chemical potential\n\u0016: increasing \u0016lowers the polarization and increases the\nthreshold current.\nEq. (28) is in fact the Walker threshold value which\nseparates the regimes of domain wall motion with a \fxed\npro\fle, i.e. _\u001e= 0 and the regime with a domain wall\nrotating its spatial pro\fle as time increases, i.e. _\u001e6= 0.\nTo see this, one may revert to the original equations of\nmotion in Eq. (7). There exists a tilt angle \u001ewhich\nsatis\fes _\u001e= 0 if the following equation is satis\fed:\nsin 2\u001e=\u0000\u00102\u0015\u0001(1 +sy)\n\u000b\u0016J+cJ\u0011\n: (29)Since the left-hand side varies between \u00061, one may \fnd\na solution if the following equation holds:\njJj\f\f\f2\u0015\u0001(1 +sy)\n\u000b\u0016+c\f\f\f<1: (30)\nwhich is completely equivalent to Eq. (28) after rewrit-\ning. For larger currents J, there exists no time-\nindependent solution \u001eand domain wall distortion _\u001eis\nnow inevitable past the Walker breakdown.\nIt has previously been suggested that the spin-orbit\ninteraction and non-adiabatic spin-torque in\ruence the\nmagnetization dynamics in the same manner, since the\nlatter may be included by substituting \u0015!\f+\u0015[21].\nHowever, it was noted in Ref. [25] that the chirality of the\ndomain wall determines the e\u000bective sign of the spin-orbit\ncoupling\u0015in the equations of motion. Formally, this\ncorresponds to taking the domain wall pro\fle represented\nwith polar angles \u001eand\u0012asM=M0(\u0000sin\u0012sin\u001e^x+\ncos\u0012^y+sin\u0012cos\u001e^z) and performing the transformations\n\u001e!(\u0000\u001e) and cos\u0012!(\u0000cos\u0012). The parameter \u001e=\n\u001e(t) is the time-dependent tilt angle, whereas \u0012is de\fned\nby\nsin\u0012= sech[(~x\u0000X(t))=LDW]; (31)\nwhere ~xis the position along the magnetic wire, X(t)\nis the time-dependent center position of the domain-\nwall, whereas LDWis the domain wall width. This sug-\ngests that the role of spin-orbit coupling is chirality-\nsensitive and in this regard di\u000bers qualitatively from non-\nadiabaticity. Below, we will investigate this e\u000bect and\nshow that the chirality indeed gives rise to highly di\u000ber-\nent behavior for the domain wall velocity. The chirality\nis changed in our analytical expression Eq. (27) by let-\nting\u0015!(\u0000\u0015). We note that our conclusions remain\nunchanged even when including the \f-term, as long as it\nis small (which is typically the case).\nIn the remainder of this paper, we \fx the following pa-\nrameters:\u000b= 0:005,L= 75 nm,m= 0:04me,\u0016= 0:056\neV, \u0001 = 0:02 eV, and vc= 150 m/s to model an exper-\nimentally realistic semiconductor system [32], where me\nis the bare electron mass. We restrict our attention to a\nscenario where the electron-spin density satis\fes sy\u001c1.\nIn Eq. (27), note that Jand\u0015are normalized quanti-\nties. To make a quantitative estimate for the domain wall\nvelocity, we de\fne the non-normalized current J=Jvc\nand spin-orbit interaction \u0003 = ~\u0015=(mL) which have units\nm/s and eV\u0001m, respectively. Similarly, we restore the di-\nmension of the terminal domain wall velocity by de\fning\nhVDWi=vchvDWiwith units m/s.\nWe show in Fig. 2 a plot of the terminal domain wall\nvelocity as a function of the applied current for several\nvalues of the spin-orbit interactions. To illustrate the ef-\nfect of the chirality, we plot in (a) the case \u0003 >0 while\nin (b) \u0003<0. The latter results are consistent with the\nnumerical study of Ref. [21], and shows that the spin-\norbit interaction can greatly enhance the domain wall\nvelocity at low currents up to the Walker breakdown. In\nthe former case, however, the domain wall velocity be-\nhaves di\u000berently when changing the current J. The spin\ntorque induced by the e\u000bective Rashba-\feld is now di-\nrected opposite to the conventional current-driven spin\ntorque and there is a competition between the two. For\nnon-zero \u0003, the domain wall velocity starts by moving in\nthe direction opposite to the current, whereas it eventu-\nally changes sign with increasing J(above the threshold\ncurrent). Thus, experimentally observing a sign-reversal\nof the domain-wall velocity with applied current would\nbe an indication of precisely this chirality sensitive spin-\norbit coupling e\u000bect. We note that this sign-reversal of\nthe wall velocity is di\u000berent from the one predicted in\n[25], which originated from a Slonczewski-like spin-orbit\ntorque proportional to \fand when considering a di\u000ber-ent type of domain wall pro\fle. In our case, the \feld-\nlike spin-orbit torque is su\u000ecient to cause the velocity-\nreversal.\nIV. SUMMARY\nIn conclusion, we have used the Lagrangian formalism\nto derive an exact analytical expression for the domain\nwall velocity in a spin-orbit coupled ferromagnet. 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Brandt1\n1Walter Schottky Institut, Technische Universit¨ at M¨ unch en, Am Coulombwall 4, 85748 Garching, Germany\n2Institut f¨ ur Halbleitertechnik, Technische Universit¨ a t Braunschweig,\nHans-Sommer-Straße 66, 38023 Braunschweig, Germany\n3Institut f¨ ur Quantenmaterie, Universit¨ at Ulm, 89069 Ulm , Germany\n4Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften,\nWalther-Meißner-Straße 8, 85748 Garching, Germany\n(Dated: October 31, 2018)\nA modeling approach for standing spin-wave resonances base d on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert equation is presented. In c ontrast to a previous study [Bihler et al.,\nPhys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical ma gnetization precession and\nmagnetic properties arbitrarily varying across the layer t hickness, including the magnetic anisotropy\nparameters, the exchange stiffness, the Gilbert damping, an d the saturation magnetization. To\ndemonstrate the usefulness of our modeling approach, we exp erimentally study a set of (Ga,Mn)As\nsamples grown by low-temperature molecular-beam epitaxy b y means of angle-dependent stand-\ning spin-wave resonance spectroscopy and electrochemical capacitance-voltage measurements. By\napplying our modeling approach, the angle dependence of the spin-wave resonance data can be re-\nproduced in a simulation with one set of simulation paramete rs for all external field orientations.\nWe find that the approximately linear gradient in the out-of- plane magnetic anisotropy is related\nto a linear gradient in the hole concentrations of the sample s.\nPACS numbers: 75.50.Pp, 76.50.+g, 75.70.-i, 75.30.Ds\nKeywords: (Ga,Mn)As; spin wave resonance; magnetic anisot ropy\nI. INTRODUCTION\nDue to their particular magnetic properties, in-\ncluding magnetic anisotropy,1–3anisotropic magneto-\nresistance4,5and magneto-thermopower,6in the past\nyears ferromagnetic semiconductors have continued to\nbe of great scientific interest in exploring new physics\nand conceptual spintronic devices.7–11The most promi-\nnentferromagneticsemiconductoris(Ga,Mn)As, wherea\nsmall percentage of Mn atoms on Ga sites introduces lo-\ncalizedmagneticmomentsaswellasitinerantholeswhich\nmediate the ferromagneticinteractionofthe Mn spins ( p-\ndexchange interaction).12Both theoretical and experi-\nmental studies have shown that the magnetic anisotropy,\ni.e., the dependence of the free energy of the ferromagnet\non the magnetization orientation, depends on the elas-\ntic strain and the hole concentration in the (Ga,Mn)As\nlayer,12,13openingupseveralpathwaystomanipulatethe\nmagnetic anisotropy of (Ga,Mn)As.14–16\nA common spectroscopic method to probe the mag-\nnetic anisotropy of ferromagnets and in particular\n(Ga,Mn)As, is angle-dependent ferromagnetic resonance\n(FMR),17–23where FMR spectra are taken as a func-\ntion of the orientation of the external magnetic field. If\nthe magnetic properties of the ferromagnet are homo-\ngeneous, a zero wave vector ( k= 0) mode of collec-\ntively, uniformally precessing magnetic moments couples\nto the microwavemagnetic field, e.g., in a microwavecav-\nity, allowing for a detection of the magnetization preces-\nsion. The resonance field of this mode, referred to as\nuniform resonance magnetic field, depends on the em-\nployedmicrowavefrequency and the magnetic anisotropyparameters. Thus, by recording FMR spectra at differ-\nent orientations of the external field with respect to the\ncrystal axes, the anisotropy parameters can be deduced\nfrom the experiment. However, if the magnetic prop-\nerties of a ferromagnetic layer are non-homogeneous or\nthe spins at the surface and interface of the layer are\npinned, non-propagating modes with k/negationslash= 0, referred to\nas standing spin-wave resonances (SWR), can be excited\nby the cavity field and thus be detected in an FMR ex-\nperiment. On one hand this can hamper the derivation\nof anisotropy parameters, on the other hand a detailed\nanalysis of these modes can elucidate the anisotropy pro-\nfile of the layer and the nature of spin pinning condi-\ntions. Furthermore, the excitation of spin waves is of\ntopical interest in combination with spin-pumping,24–27\ni.e., the generation of pure spin currents by a precessing\nmagnetization.28–30In this context, the exact knowledge\nof the magnetization precession amplitude as a function\nof the position coordinate within the ferromagnet is of\nparticular importance.24\nSeveral publications report on SWR modes in\n(Ga,Mn)As with a mode spacing deviating from what is\nexpected according to the Kittel model for magnetically\nhomogeneous films with pinned spins at the surface.31–36\nThese results have been attributed to an out-of-plane\nanisotropyfieldlinearly31,36orquadraticallyvarying33–35\nas a function of the depth into the layer, as well as to\nspecific spin pinning conditions at the surface and at the\ninterface to the substrate.35While most of these studies\nhave focused on the spacings of the resonancefields when\nmodeling SWR measurements, in Ref. 36 a more sophis-\nticated approach, based on a normal mode analysis,37,38\nwas employed to model resonance fields as well as rela-2\ntive mode intensities for the external field oriented along\nhigh-symmetry directions, assuming a circularly precess-\ning magnetization.\nIn this work, we present a more general modeling ap-\nproach for SWR, based on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert (LLG) equation. This ap-\nproach holds for any orientation of the external mag-\nnetic field and accounts for elliptical magnetization pre-\ncession [Sec. II]. It allows for a simulation of arbitrar-\nily varying profiles of the magnetic properties across the\nthickness of the film, including vatiations of the mag-\nnetic anisotropy parameters, the exchange stiffness, and\nthe Gilbert damping parameter. As the result ofthe sim-\nulation, we obtain the Polder susceptibility tensor as a\nfunction of the depth within the ferromagnet. Based on\nthis result, the absorbedpowerupon spin waveresonance\nandthe magnetizationprecessionamplitude asafunction\nof the depth can be calculated for any orientation of the\nexternal magnetic field.\nWe apply our modeling approach to a set of four\n(Ga,Mn)As samples epitaxially grown with different\nV/III flux ratios [Sec. III], motivated by the obser-\nvation that V/III flux ratios of /lessorsimilar3 lead to a gra-\ndient in the hole concentration p[Ref. 39], which in\nturn is expected to cause non-homogeneous magnetic\nanisotropyparameters.31,36Electrochemicalcapacitance-\nvoltage (ECV) measurements revealed a nearly linear\ngradient in pacross the thickness of the layers investi-\ngated. To show that our modeling approach is capa-\nble of simulating SWR spectra for arbitrary magnetic\nfield orientations, angle-dependent SWR data were taken\nand compared with the model using one set of magnetic\nparameters for each sample, revealing gradients in the\nuniform resonance magnetic fields. We discuss the in-\nfluence of the gradient in pon the observed uniform\nresonance field gradients as well as possible influences\nof strain and saturation magnetization gradients on the\nobserved out-of-plane anisotropy profile. It should be\nemphasized, however, that the objective of this work is\nto show the usefulness of our modeling approach, while\na detailed investigation of the origin of the gradient in\nthe out-of-plane magnetic anisotropy profile and there-\nfore a detailed understanding of the particular materials\nphysics of (Ga,Mn)As is beyond the scope of this study.\nFinally, we summarize our results and discuss further po-\ntential applications of this work [Sec. IV].\nII. THEORETICAL CONSIDERATIONS\nIn this section, we provide the theoretical framework\nnecessary to describe the full angle dependence of the\nspin-wave resonance spectra presented in Sec. III. Refer-\nring to the coordinate system depicted in Fig. 1, we start\nfrom the canonical expression for the free enthalpy den-\nsity (normalized to the saturation magnetization M) forφ0Θ0\n123m2\nm1m3≈1\nm\nx||[100]y||[010]z||[001]\nSubstrateFerromag net\nFIG. 1: (color online) Relation between the two coordinate\nsystems employed. The ( x,y,z) frame of reference is spanned\nby the cubic crystal axes, while the (1 ,2,3) coordinate sys-\ntem is determined by the equilibrium orientation of the mag-\nnetization (3-direction) and two transverse directions, t he 2-\ndirection being parallel to the film plane; the latter system is\nzandµ0Hdependent, as described in the text.\na tetragonally distorted (Ga,Mn)As film13,20,40,41\nG= const −µ0H·m+B001m2\nz+B4⊥m4\nz\n+B4/bardbl(m4\nx+m4\ny)+1\n2B1¯10(mx−my)2.(1)\nHere,µ0His a static external magnetic field, B001\nis a uniaxial out-of-plane anisotropy parameter, re-\nflecting shape and second-order crystalline anisotropy,13\nB4⊥,B4/bardbl, andB1¯10are fourth-order crystalline and\nsecond-order uniaxial in-plane anisotropy parameters,\nrespectively;1mx,my,mzdenote the components of the\nnormalized magnetization vector m(z) =M(z)/M(z)\nalong the cubic axes [100], [010], and [001], respectively.\nWe assume the magnetic properties of the layer to be ho-\nmogeneouslaterally(within the xyplane) and inhomoge-\nneous vertically (along z); the anisotropy parameters in\nEq. (1) and the magnetization are consequently a func-\ntion of the spatial variable z. To obtain the anisotropy\nparameters from Eq. (1) in units of energy density, it\nwould therefore be required to know the zdependence\nand the absolute value of M.\nThe minimum of Eq. (1) determines the equilibrium\norientation of the magnetization, given by the angles\nθ0=θ0(z) andφ0=φ0(z), cf. Fig. 1. To describe the\nmagnetization dynamics, we introduce a new frame of\nreference (1 ,2,3) shown in Fig. 1, in which the equilib-\nrium orientation of the magnetization m0coincides with\nthe axis 3. For small perturbations, the magnetization\nprecesses around its equilibrium with finite transverse\ncomponents of the magnetization mi(i= 1,2) as illus-\ntrated in the inset in Fig. 1. The transformation between\nthe two coordinate systems is given in the Appendix A\nby Eqs. (A1) and (A2). We write for the (normalized)\nmagnetization3\nm=\n0\n0\n1\n\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nm0+\nm1\nm2\n0\n+O(m2\n1,m2\n2).(2)\nThe evolution ofthe magnetizationunder the influence\nof an effective magnetic field µ0Heffis described by the\nLLG equation42,43\n∂tm=−γm×µ0Heff+αm×∂tm,(3)\nwhereγis the gyromagnetic ratio and αa phenomeno-\nlogical damping parameter. The effective magnetic field\nis given by36\nµ0Heff=−∇mG+Ds\nM∇2M+µ0h(t),(4)\nwhere∇m= (∂m1,∂m2,∂m3) isthe vectordifferentialop-\neratorwith respect to the componentsof m,Ds= 2A/M\nis the exchange stiffness with the exchange constant A,\n∇2is the spatial differential operator ∇2=∂2\nx+∂2\ny+∂2\nz,\nandh(t) =h0e−iωtis the externally applied microwave\nmagnetic field with the angular frequency ω;h(t) is ori-\nented perpendicularly to µ0H. Since the magnetic prop-\nerties are independent of xandy, Eq. (3) simplifies to\n∂tm=−γm×[−∇mG+Dsm′′+µ0h(t)]+αm×∂tm,\n(5)\nwithm′′=∂2\nzm, neglecting terms of the order of m2\ni(for\ni= 1,2). By definition of the (1 ,2,3) coordinate system,\nthe only non-vanishing component of ∇mGin the equi-\nlibrium is along the 3-direction. For small deviations of\nmfrom the equilibrium we find44\n∇mG=\nG11m1+G21m2\nG12m1+G22m2\nG3\n, (6)\nwhere we have introduced the abbreviations Gi=\n∂miG|m=m0andGij=∂mi∂mjG|m=m0; the explicit ex-\npressions for these derivatives are given in the Appendix\nA.\nInthefollowing,wecalculatethetransversemagnetiza-\ntion components assuming a harmonic time dependence\nmi=mi,0e−iωt. The linearized LLG equation, consider-\ning only the transverse components, reads as\n/parenleftbigg\nH11H12\nH21H22/parenrightbigg/parenleftbigg\nm1\nm2/parenrightbigg\n−Ds/parenleftbigg\nm′′\n1\nm′′\n2/parenrightbigg\n=µ0/parenleftbigg\nh1\nh2/parenrightbigg\n,(7)\nwherewe haveintroducedthe abbreviations H11=G11−\nG3−iαω/γ,H12=H∗\n21=G12+iω/γ, andH22=G22−\nG3−iαω/γ. We have dropped all terms which are non-\nlinear inmiand products of miwith the driving field.\nResonant uniform precession of the magnetization\n(m′′\ni= 0) occurs at the so called uniform resonance fieldµ0Huni(z), which is found by solving the homogeneous\n(h= 0) equation\nH11(z)H22(z)−H12(z)H21(z) = 0\n⇔(G11−G3)(G22−G3)−G2\n12=/parenleftbiggω\nγ/parenrightbigg2\n(8)\nforµ0H, neglecting the Gilbert damping ( α= 0). Equa-\ntion(8)canbeusedtoderiveanisotropyparametersfrom\nangle-dependent FMR spectra. As extensively discussed\nby Baselgia et al.44, using Eq. (8) is equivalent to using\nthe method of Smit and Beljers, which employs second\nderivatives of the free enthalpy with respect to the spher-\nical coordinates.41,45,46\nTo illustrate the role of the uniform resonance field in\nthe context of spin-wave resonances, we consider the spe-\ncial case where magnetization is aligned along the [001]\ncrystal axis ( θ0= 0), before we deal with the general\ncase of arbitrary field orientations. Neglecting the uniax-\nial in-plane anisotropy( B1¯10= 0) since this anisotropyis\ntypically weaker than all other anisotropies,13,41we find\nG3=−µ0H+2B001+4B4⊥andG11=G22=G12= 0,\nresulting in the uniform resonance field\nµ0H001\nuni(z) =ω/γ+2B001(z)+4B4⊥(z).(9)\nTo find the eigenmodes of the system, we consider the\nunperturbed and undamped case, i.e., α= 0 and h= 0\nin Eq. (7). With m2=im1= ˜mwe find the spin-wave\nequation\nDs˜m′′+µ0H001\nuni(z)˜m=µ0H˜m (10)\nin agreementwith Ref. 36. The relationofthe anisotropy\nparameters defined in Ref. 36 to the ones used here\nis given by B001=K100\neff/M+B1¯10,B1¯10=−K011\nu/M,\n2B4⊥=−K⊥\nc1/M, and 2B4/bardbl=−K/bardbl\nc1/M. Equation (10)\nis mathematically equivalent to the one-dimensional\ntime-independent Schr¨ odinger equation, where the\nuniform resonance field corresponds to the potential,\n˜mto the wave function, µ0Hto the energy, and Dsis\nproportional to the inverse mass. To calculate the actual\nprecession amplitude of the magnetization, the coupling\nof the eigenmodes of Eq. (10) to the driving field is\nrelevant, which is proportional to the net magnetic\nmoment of the mode.36,38In analogy to a particle in a\nbox, the geometry of the uniform resonance field as well\nas the boundary conditions determine the resonance\nfields and the spatial form of the precession amplitude.\nFor the remainder of this work, we assume the spins\nto exhibit natural freedom at the boundaries of the\nfilm, i.e.,∂z˜m= ˜m′= 0 at the interfaces,36,47since\nthese boundary conditions have been shown to describe\nthe out-of-plane SWR data of similar samples well.36\nTo graphically illustrate the influence of the uniform\nresonance field on the SWR modes, we consider in Fig. 2\na ferromagnetic layer with a thickness of 50 nm with\nconstant magnetic properties across the layer (a) and\nwith a linearly varying uniform resonance field (c); in4\n(a) (b)\n(c) (d) m (arb. u.)~ m (arb. u.)~\nFIG. 2: Simulation to demonstrate the influence of the uni-\nform resonance field µ0H001\nunion theSWRmodes for m0||[001],\nassuming circular precession. In (a), µ0H001\nuniis set to be con-\nstant across the layer, while in (c) it varies linearly (blue ,\ndashed lines), in analogy to a square potential and a trian-\ngular potential, respectively. The dotted black lines are t he\nresonance fields, calculated assuming boundary conditions of\nnatural freedom, see text. The solid red lines show the eigen -\nmodes of the system, i.e., the precession amplitude ˜ mof the\nmagnetization; for each mode the dotted line corresponds to\n˜m= 0. As can be seen in (a), for a constant uniform reso-\nnancefieldthefirstmodeoccursattheuniformresonance field\nand exhibits a constant precession amplitude across the lay er,\ni.e., an FMR mode. The second and third mode (higher-order\nmodes are not shown) exhibit a non-uniform magnetization\nprofile. In order to couple to the driving field the modes need\nto have a finite net magnetic moment. As can be seen in\n(a), the positive and negative areas of the second and third\nmode are equal, thus these modes are not visible in the SWR\nspectrum (b). This is in contrast to the case of the linearly\nvarying uniform resonance field (c) where the mode profile is\ngiven by Airy functions, which have a nonzero net magnetic\nmoment also for the second and third mode, resulting in a\nfinite SWR intensity of these modes (d). The spectra in (b)\nand (d) were calculated by integrating over the eigenmodes\n˜mand convoluting the square of the result with Lorentzians.\nboth cases we assume Ds= 13 Tnm−2, a similar value\nas obtained in previous studies.36For these conditions,\nwe numerically solve Eq. (10) by the finite difference\nmethod described in the Appendix B1, in orderto obtain\nthe resonance fields (eigenvalues) and the zdependence\nof the transverse magnetic moments (eigenfunctions).\nTo which amount a mode couples to the driving field is\ndetermined by the net magnetic moment of the mode,\nwhich is found by integrating ˜ m(z) over the thickness of\nthe film. For the magnetically homogeneous layer, the\nonly mode that couples to the driving field is the uniformprecession mode at µ0H001\nuni, since modes of higher order\nhave a zero net magnetic moment [Fig. 2 (a)], resulting\nin one resonance at the uniform resonance field, cf. Fig. 2\n(b). For the non-uniform layer, with µ0H001\nuni(z) linearly\nvarying across the film, the mode profile is given by Airy\nfunctions31,36,38and various non-uniform modes couple\nto the driving field, resulting in several spin-wave reso-\nnances with their amplitude proportional to the square\nof the net magnetic moment36,38of the corresponding\nmode, cf. Fig. 2 (c) and (d).\nWe now turn to the general case of arbitrary field ori-\nentations. Due to the magnetic anisotropy profile, the\nmagnetizationorientationisaprioriunknownandafunc-\ntion ofzandµ0H. Furthermore, the assumption of a\ncircularly precessing magnetization is not generally jus-\ntified. To solve Eq. (7) for arbitrary field orientations,\nwe employ a finite difference method as outlined in the\nAppendix B2. BysolvingEq.(7), weobtainthe zdepen-\ndent generalized Polder susceptibility tensor ¯ χ(µ0H,z),\nwhich relates the transverse magnetization components\nMi(z) =M(z)mi(z) with the components of the driving\nfield by\n/parenleftbigg\nM1\nM2/parenrightbigg\n= ¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg\n. (11)\nInamicrowaveabsorptionmeasurement, thecomponents\nMiwhich are out-of-phase with the driving field are de-\ntected. The absorbed power density is related to the\nimaginary part of ¯ χ(µ0H,z) and can be calculated by48\nP=ωµ0\n2z0Im/braceleftbigg/integraldisplay0\n−z0/bracketleftbigg/parenleftbigh∗\n1,h∗\n2/parenrightbig\n¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg/bracketrightbigg\ndz/bracerightbigg\n,\n(12)\nwherez0is the thickness ofthe ferromagneticlayer. Note\nthat the position coordinate zis negative in the film,\ncf. Fig. 1.\nTo obtain an impression of how gradients in differ-\nent anisotropy parameters influence the SWR spectra,\nwe plot in Fig. 3 simulated SWR spectra together with\nthe magnetization precession cone as a function of depth\nin the ferromagnetic layer. We assume a constant sat-\nuration magnetization (its value is not relevant for the\noutcome of the simulation), a constant exchange stiff-\nnessDs= 35 Tnm2unless otherwise specified, α= 0.09,\nandB001= 90 mT, B4||=−50 mT,B4⊥= 15 mT.\nIn Fig. 3 (a), we assume B001to vary across the layer\nthickness according to B001(z) =B001−b001×zwith\nb001=−0.8 mT/nm. Figure 3 (a i) shows the simu-\nlated SWR spectra calculated by taking the first deriva-\ntive of Eq. (12) with respect to µ0Hfor different angles\nψdefined in the inset in Fig. 3 (c iv). We observe sev-\neral SWR modes for µ0H||[001] which become less as\nµ0His tilted away from [001]. At ψ= 40◦only one\nmode is visible while for ψ= 0◦we again observe mul-\ntiple SWR modes. This observation can be understood\nby considering the uniform resonance fields as a func-\ntion of the depth for these orientations. In Fig. 3 (a5\n90\n03060\n90\n03060\n90\n03060\nSWR Intensi ty (arb . units)ψ (deg.) ψ (deg.) ψ (deg.)\n200 600 400\nµ0H (mT)ψµ0H\n[110][001](a i) (a ii) (a iii) (a iv)\n(b i) (b ii) (b iii) (b iv)\n(c i) (cii) (ciii) (c iv)Im(m1m2-m1m2) (arb. u.) * *\nFIG. 3: Atlas illustrating the influence of gradients in the a nisotropy parameters on SWR spectra. In (a) all anisotropy\nparameters are kept constant with the values given in the tex t, exceptB001which is varied linearly. Correspondingly, in (b)\nand (c)B4⊥andB4||were varied linearly, respectively. Panels (i) show the firs t derivative of simulations using Eq. (12) with\nrespect toµ0Hand panels (ii)-(iv) show the precession cone Im( m∗\n1m2−m1m∗\n2) in a color plot together with the uniform\nresonance field µ0Huni(z) (dashed blue lines) at three different external field orient ations; the black dotted lines indicate the\nresonance field positions of the modes. Panel (a i) additiona lly shows the influence of a linear gradient in the exchange st iffness\nparameter on the spin-wave spectra, see text for further det ails and discussion.\nii)-(a iv), we show the uniform resonance field (dashed\nblue line) for ψ= 0◦,ψ= 30◦, andψ= 90◦, respec-\ntively, together with the magnetization precession cone\nIm(m∗\n1m2−m1m∗\n2) in a contour plot as a function of\ndepth andµ0H. Atψ= 90◦, the uniform resonance field\nvaries strongly across the film, which can be understood\nby considering Eq. (9). This results in several spin wave\nmodes with their resonance fields indicated by dotted\nlines.\nFor other field orientations, the formula for the uni-\nform resonance field can also be derived but results in a\nlonger, more complex equation than Eq. (9). Important\nin this context is that positive values of B001lead to an\nincrease(decrease) ofthe resonancefield for the magneti-\nzation oriented perpendicular (parallel) to the film plane,\naccountingfor the reversedsign ofthe slopesof µ0Huniin\nFig. 3 (a ii) and (a iv). Consequently, in between those\ntwo extreme cases µ0Hunimust be constant across the\nlayer for some field orientation, in our case for ψ= 30◦,\nresulting in a single SWR mode, cf. Fig. 3 (a i) and (a\niii). In addition to the SWR simulations with constantDs, weplotinFig.3(ai)simulatedSWRspectrawith Ds\nvarying linearly across the film with Ds= 35−65 Tnm2\n(blue, dotted lines) and Ds= 35−5 Tnm2(green, dotted\nlines). A decreasing Dsleads to a decreasing spacing in\nthe modes and vice versa for an increasing Dsas can be\nseen, e.g., for µ0H||[001].\nIn Fig. 3 (b), we consider the case where all magnetic\nparameters are constant with the values given above, ex-\nceptB4⊥(z) =B4⊥−b4⊥×zwithb4⊥=−0.4 mT/nm.\nAs evident from Eq. (9), this results in the same slope of\nµ0Huniforψ= 90◦as in the case above where we varied\nB001only, cf. Fig. 3 (a iv) and (b iv). In contrast to the\ncasedepictedin(a), however,herefor ψ= 0◦theuniform\nresonancefield is constant. This can be understood when\nevaluating the parametersthat enter in the calculation of\nthe uniform resonance field [Eq. (8)]. If mis in the film\nplane, none of the parameters in Eqs. (A4)-(A6) depends\nonB4⊥, resulting in a constant uniform resonance field\nforψ= 0◦. Asmis tilted away from the film plane,\nB4⊥enters in some of the terms Eqs. (A4)-(A6). As a\nconsequence, µ0Hunivaries, first such that it increases6\n[cf. Fig. 3 (b iii)] and finally, such that it decreases as a\nfunction of depth [cf. Fig. (b iv)].\nFinally, we discuss the case where all parameters are\nconstant except B4||(z) =B4||−b4||×zwithb4||=\n−0.4 mT/nm [Fig. 3 (c)]. Here, µ0Huniis constant for\nψ= 90◦aspredictedbyEq.(9). As mistiltedawayfrom\n[001] a varying B4||leads to a varying uniform resonance\nfield as shown in Fig. 3 (c ii) and (c iii). Here, a sign\nreversal of the slope as it was the case in Fig. 3 (a) and\n(b) does not take place and multiple resonances occur,\nstarting from ψ= 60◦[Fig. 3 (c i)].\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\n(Ga,Mn)As samples with a nominal Mn concentration\nof≈4% were grown on (001)-oriented GaAs substrates\nby low-temperature molecular-beam epitaxy at a sub-\nstrate temperature of 220◦C using V/III flux ratios of\n1.1, 1.3, 1.5, and 3.5, referred to as sample A, B, C, and\nD, respectively. The layer thickness was 210-280 nm as\ndetermined from the ECV measurements, cf. Fig. 4. For\nsamples with V/III flux ratios /lessorsimilar3 a gradient in the hole\nconcentration has been reported,39hence this set of sam-\nples was chosen to study the influence of a gradient in p\non the out-of-plane magnetic anisotropy. Further details\non the sample growth can be found in Refs. 39 and 41.\nThe hole concentration profile of the as-grown\n(Ga,Mn)As layers were determined by ECV profiling us-\ning a BioRad PN4400 profiler with a 250 ml aqueous\nsolution of 2.0 g NaOH+9.3 g EDTA as the electrolyte.\nFor further details on the ECV analysis see Ref. 39. The\nresults of the ECV measurements for the layers investi-\ngated areshown in Fig. 4(a). Except for the sample with\nV/III=3.5, they reveala nearlylinearly varying hole con-\ncentration across the layer thickness with different slopes\nand with the absolute value of the hole concentration\nat the surface of the layer varying by about 20%. The\nprofiles are reproducible within an uncertainty of about\n15%.\nTo investigate the magnetic anisotropy profiles of the\nsamples, weperformedcavity-basedFMRmeasurements,\nusing a Bruker ESP300 spectrometer operating at a mi-\ncrowave frequency of 9.265 GHz ( X-band) with a mi-\ncrowave power of 2 mW at T= 5 K; we used magnetic\nfield modulation at a frequency of 100 kHz and an ampli-\ntude of3.2mT. Since wearemainlyinterested in the out-\nof-plane magnetic anisotropy, we recorded spectra for ex-\nternalmagneticfieldorientationswithinthe crystalplane\nspanned by the [110] and [001] crystal axes in 5◦steps,\ncf. the inset in Fig. 5. For each orientation, the field was\nramped to 1 T in order to saturate the magnetization\nand then swept from 650 mT to 250 mT; the spectra for\nthe samples investigated are shown in Fig. 5.\nWe start by discussing qualitative differences in the\nspectra. The samples A and B exhibit several pro-\nnounced resonances for the external field oriented along(a)\n(b)\nFIG.4: (a)Theholeconcentration inthedifferent(Ga,Mn)As\nsamples is shown as a function of the depth within the layers\nas determined by ECV profiling. (b) The uniform resonance\nfieldsµ0H001\nuni(z) for the four samples obtained from the sim-\nulations for the out-of-plane orientation of the external fi eld\n(ψ= 90◦) as a function of the depth.\n[001], which we attribute to standing spin-wave reso-\nnances [Fig. 5 (a) and (b)]. For these samples, the [001]\ndirection is the magnetically hardest axis since at this\norientation the resonance field of the fundamental spin-\nwavemode is largerthan at all other orientations. As the\nexternal field is rotated into the film plane, the resonance\nposition of this mode gradually shifts to lower field val-\nues as expected for a pronounced out-of-plane hard axis.\nIn contrast, the samples C and D exhibit the largest res-\nonance fields for a field orientation of 50-60◦[Fig. 5 (c)\nand (d)] pointing to an interplay of second- and fourth-\norder out-of-plane anisotropy with different signs of the\ncorresponding anisotropy parameters. These samples ex-\nhibit spin-wave resonances as well, however they are less\npronounced than for samples A and B.\nTo quantitatively model the spin-wave spectra we nu-\nmerically solve for each magnetic field orientation the\nspin-wave equation (7) by the finite difference method as\noutlined in the Appendix B2. Although this method al-\nlows for the modeling of the SWR for arbitrary profiles\nof the anisotropy parameters, the exchange stiffness, the\nGilbert damping parameter, and the saturation magne-\ntization, we assume the parameters to vary linearly as\na function of z. This approach is motivated by the lin-\near gradient in the hole concentration, which in first ap-\nproximation is assumed to cause a linear gradient in the7\nψ[001]\n[110]µ0Hext\nSimulationExperiment(a) (b)\n(d) (c)\nV/III=1.5 V/III=3.5V/III=1.3 V/III=1.1\nFIG. 5: The spin-wave resonance data (dotted, blue lines) ar e shown together with simulations (red, solid lines) using t he\nnumerical procedure described in the text and in the Appendi x B2. The data were obtained as a function of the external\nmagnetic field orientation and magnitude for samples with a V /III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The\nrotation angle ψis defined in the inset and the parameters used for the simulat ions are summarized in Tab. I.8\nanisotropy parameters, resulting in the spin-wave reso-\nnances observed in the samples.31,36In Tab. I, we have\nsummarized the parameters used in the simulation for\nthe different samples. The parameters in capital let-\nters denote the value at the surface of the sample while\nthe ones in lower-case letters denote the slope of this\nparameter; e.g., the zdependence of the second-order,\nuniaxial out-of-plane anisotropy parameter is given by\nB001(z) =B001−b001×z. The layer thickness used for\nthe simulationcanbe inferredfromFig. 4(a) andwasde-\ntermined from the ECV data under the assumption that\nat the position where the hole concentration rapidly de-\ncreases the magnetic properties of the layer abruptly un-\ndergo a transition from ferromagnetic to paramagnetic.\nFor the simulations, we divided each film into n= 100\nlayers with constant magnetic properties within each\nlayer. For the gyromagnetic ratio we used γ=gµB//planckover2pi1\nwithg= 2.21\nAs result of the simulation we obtain the Polder sus-\nceptibility tensor ¯ χ(µ0H,z) and the transverse magneti-\nzationcomponentsasafunctionof zandµ0H. Addition-\nally, weobtainthe zdependenceoftheuniformresonance\nfield by solving Eq. (8) for each field orientation. In an\nSWR absorption experiment with magnetic field mod-\nulation, the obtained signal is proportional to the first\nderivative of the absorbed microwave power with respect\nto the magnetic field. Thus, we calculate the absorbed\npowerusingthesimulatedsusceptibilityandEq.(12) and\nnumerically differentiate the result in order to compare\nthe simulated SWR spectra with the experiment. Addi-\ntionally, we use a global scaling factor, accounting, e.g.,\nfor the modulation amplitude, which is the same for all\nfield orientations, and we multiply all the simulated data\nwith this factor. In Fig. 5, we plot the experimental data\ntogether with the simulations using the parameters given\nin Tab. I, demonstrating that a reasonableagreementbe-\ntweentheoryandexperimentcanbefoundwithonesetof\nsimulation parameters for all magnetic field orientations\nfor each sample.\nWe will now exemplarily discuss the angle dependence\nof the SWR spectrum of sample A shown in Fig. 5(a)\nbased on the uniform resonance field and the resulting\nmagnetizationmodeprofileobtainedfromthesimulation.\nTo this end, we plot in Fig. 6 (a)-(c) the magnetization\nprecession amplitude Im( m∗\n1m2−m1m∗\n2) for selected ex-\nternal field orientations as a function of depth and ex-\nternal magnetic field in a contour plot, together with the\ncorresponding uniform resonance field. In Fig. 6 (d)-(f),\nwe show for each external field orientation a magnifica-\ntion of the corresponding SWR spectrum together with\nthesimulation. Notethatincontrasttothenormal-mode\napproach (Appendix B1) used to calculate the modes in\nFig. 2, where the coupling of each mode to the cavity\nfield has to be found by integration, the approach elabo-\nrated in the Appendix B2 directly yields the transverse\nmagnetization components, already accounting for the\ncoupling efficiency and the linewidth. Further, the ap-\nproach presented in the Appendix B2 is also valid whenthe differencein the resonancefields oftwomodesis com-\nparable with or smaller than their linewidth, in contrast\nto the normal-mode approach38.\nIf the external field is parallel to the surface normal\n(ψ= 90◦) the uniform resonance field varies by about\n350 mT across the film thickness [cf. the dashed line\nin Fig. 6 (a)], resulting in several well-resolved stand-\ning spin-wave modes. The spin-wave resonance fields are\nplotted as dotted lines in Fig. 6 (a); since the spacing of\nthe resonance fields is larger than the SWR linewidth,\nthe modes are clearly resolved, cf. Fig. 6 (a) and (d).\nIn the simulation two regions with different b001values\nwereused in orderto reproducethe spacingofthe higher-\norder spin-wave modes found in the experiment. Using\nthe same slope as in the first 100 nm for the entire layer\nwould lead to a smaller spacing between the third and\nhigher order modes. Instead of defining two regions with\ndifferent slopes b001, a gradient in the exchange stiffness\nwith positiveslopecouldalsobe usedto modelthe exper-\nimentally found mode spacing as discussed in the context\nof Fig. 3. Since the exchange interaction in (Ga,Mn)As is\nmediated by holes12andpdecreases across the layer, we\nrefrainfrom modeling ourresultswith a positivegradient\ninDs. Further, the results in Ref. 36 rather point to a\nnegative gradient in Dsin a similar sample. However, a\ndecreasing Mn concentration as a function of the depth\ncould lead to an increase of Ds.34\nFinally, we note, since B1¯10= 0 in the simulations,\nthe magnetization precesses circularly for ψ= 90◦and\nthus Im(m∗\n1m2−m1m∗\n2) = 2sin2τ,49with the precession\ncone angle τ. For all other orientations, mprecesses\nelliptically which is accounted for in our simulations. In\nthe simulations of the precession amplitudes, we have\nassumed an externally applied microwave magnetic field\nwithµ0h= 0.1 mT.\nAt an external field orientation of ψ= 50◦the uni-\nform resonance field is nearly constant across the layer,\nand consequently only one SWR mode is observed with\nan almost uniform magnetization precession across the\nlayer, cf. Fig. 6 (b). The precession amplitude is a mea-\nsureforthe SWR intensity. While the fundamental mode\natψ= 90◦exhibits a larger precession cone at the in-\nterface, it rapidly decays as a function of the depth, in\ncontrast to the nearly uniform precession amplitude for\nψ= 50◦. Since the entire layer contributes to the power\nabsorption, consequently, the SWR mode at ψ= 50◦is\nmore intense than the fundamental mode for ψ= 90◦,\nwhich is indeed observed in the experiment [cf. Fig. 6 (d)\nand (e)].\nFor the magnetic field within the film plane [ ψ= 0◦,\ncf.Fig.6(c)], the uniformresonancefieldagainvarieslin-\nearly across the film, however in a less pronounced way\nthan for the out-of-plane field orientation and with an\nopposite sign of the slope. The sign reversal of the slope\ncan be understood in terms of the uniaxial out-of-plane\nanisotropy parameter B001: positive values of these pa-\nrameters lead to an increase (decrease) of the resonance\nfield for the magnetization oriented perpendicular (par-9\nTABLE I: Simulation parameters and their zdependence of the samples under study as obtained by fitting t he simulations to\nthe SWR measurements. For the anisotropy parameters the cap ital letters denote the value at the surface of the film and the\nlower case letters the slope as described in the text. For sam ple A, the first value of b001was used for the first 100 nm and the\nsecond one for the remaining layer. In addition to the anisot ropy parameters, the saturation magnetization is also assu med to\nvary linearly across the layer, while its absolute value is u nknown and not important for the SWR simulations.\nSample V/III B001 b001 B4/bardblb4/bardblB4⊥b4⊥Dsα∂M(z)\n∂zM(0)\n(mT) (mT\nnm) (mT) (mT\nnm) (mT) (mT\nnm) (Tnm2) (1\nµm)\nA 1.1 90 -0.1, -0.3 -50 0.05 25 -0.3 35 0.09 -3\nB 1.3 130 -0.5 -50 0 0 0 20 0.06 -4\nC 1.5 75 -0.4 -55 -0.04 -15 0 40 0.11 -4\nD 3.5 91 -0.3 -55 -0.04 -15 0 20 0.09 -3\nallel) to the film plane, accounting for the slopes of the\nuniform resonance fields in Fig. 6. Since the gradient in\nthe uniform resonance field is less pronounced for ψ= 0◦\nthan forψ= 90◦, the spin-wave modes are not resolved\nforψ= 0◦, since their spacing is smaller than the SWR\nlinewidth, leading to one rather broad line [cf. Fig. 6 (c)\nand (f)]. A steeper gradient in B4||in combination with\na different Gilbert damping (or with an additional inho-\nmogeneous damping parameter) and amplitude scaling\nfactor, could improve the agreement of simulation and\nexperiment in the in-plane configuration, as discussed\nlater. A detailed study of the in-plane anisotropy pro-\nfile is however beyond the scope of this work. Given that\nthe presented simulations were obtained with one set of\nparameters, the agreement of theory and experiment is\nreasonably good also for the in-plane configuration, since\nsalient features of the SWR lineshape are reproduced in\nthe simulation.\nHaving discussed the angle-dependence of the SWR\nspectra, we turn to the zdependence of the out-of-plane\nanisotropy of sample A. Our simulations reveal that it\nis governed by the zdependence of both B001(z) and\nB4⊥(z). Assuming only a gradient in B001results in a\nreasonable agreement of theory and experiment for the\nexternal field oriented along [001] and [110], but fails to\nreproduce the spectra observed for the intermediate field\norientations, e.g., ψ= 50◦. This is illustrated by the\ndashed black line in Fig. 6 (e), which represent simu-\nlations with a constant B4⊥(z) forψ= 50◦. As can\nbe seen, this simulation produces several spin-wave res-\nonances, whereas in the experiment only one resonance\nis present, which is better reproduced by the simulation\nwith bothB001(z) andB4⊥(z) varying across the layer.\nWe will now discuss the anisotropy parameters of all\nsamples. In contrast to sample A, the out-of-plane\nanisotropy profile of all other samples appears to be gov-\nerned by a gradient in B001(z). As already discussed\nqualitatively, the hard axis of the samples is determined\nby an interplay of B001andB4⊥. For sample A and B\nB4⊥is positive and zero, leading to an out-of-plane hard\naxis. Incontrast,sampleCandDexhibitanegative B4⊥,\nleading to a hard axis between out-of-plane and in-plane.\nTheB4||parameter is negative and of similar magnitude\nfor all samples.Since the out-of-plane anisotropy profile of sample A\nis governed by B001(z) andB4⊥(z), a comparison of the\nout-of-plane anisotropy profile between all samples based\non anisotropy parameters is difficult. We therefore com-\npare the uniform resonance fields, where both anisotropy\nparameters enter. As evident from Fig. 6, the strongest\ninfluence of the magnetic inhomogeneity of the layers on\nthe uniform resonance fields is observed for the exter-\nnal field along [001]. To compare the hole concentration\nprofile in Fig. 4 (a) with the anisotropy profile, we there-\nfore plot in Fig. 4 (b) the zdependence of the uniform\nresonance field µ0H001\nunifor this field orientation. The\nfigure demonstrates that the gradient in µ0H001\nuniis cor-\nrelated with the gradient in p. For the sample with the\nstrongest gradient in pthe gradient in µ0H001\nuniis also\nmost distinct while the samples with a weaker gradient\ninpexhibit a less pronounced gradient in µ0H001\nuni. How-\never, for sample D, exhibiting a nearly constant p, we\nstillobservestandingspinwaveresonancesfor µ0H||[001]\n[Fig. 5 (d)], reflected in a slight gradient of µ0H001\nuni. This\nobservation suggests that aditionally other mechanisms\nlead to a variation of the anisotropy profile. One possi-\nbilitywouldbeagradientintheelasticstrainofthelayer,\ndue to a non-homogeneous incorporation of Mn atoms in\nthe lattice. However, x-ray diffraction measurements of\nthis sample, in combination with a numerical simulation\nbased on dynamic scattering theory, reveal a variation of\nthe vertical strain ∆ εzzas small as 3 ×10−5across the\nlayer. According to the measurements in Ref. 13, such a\nvariation in strain would lead to a variation of the B001\nparameter by a few mT only, insufficient to account for\nthe variation of µ0Huniby almost 100 mT across the\nlayer. A more likely explanation seems to be a varia-\ntion of the saturation magnetization, which should also\ninfluence the anisotropy parameters. In the simulation,\na non-homogeneous saturation was assumed, potentially\nexplaining also the observed gradient in the anisotropy\nparameters and therefore in the uniform resonance field.\nIn contrast to the out-of-plane anisotropy parameters,\nB4||was found to depend only weakly on z, for all sam-\nples except sample B where it was constant. Addition-\nally,B1¯10, typically of the order of a few mT,13might\nhave an influence and interplay with B4||in determining\nthe in-plane anisotropy. We here however focus on the10\n0°\n(b) (c)\n(a)Im(m1m2-m1m2) (10-5) * *\n0 1.2Im(m1m2-m1m2) (10-5) * *\n0 0.3Im(m1m2-m1m2) (10-5) * *\n0 0.53\n(d) (e) (f)001arb.(a)\nFIG. 6: Simulated magnetization mode profile and uniform res onance field of sample A. The contour plots show the magneti-\nzation precession amplitude Im( m∗\n1m2−m1m∗\n2) as a function of the position within the film and the external magnetic field\nfor the external field aligned (a) along [001], (b) at an angle of 50◦with respect to [110] (cf. the inset in Fig. 5) and (c) along\n[110]. The blue, dashed lines in (a)-(c) show the uniform res onance field, obtained by numerically solving Eq. (8) for eac h given\nfield orientation. The dotted black lines in (a) indicate the resonance magnetic fields. In (d)-(f), a magnification of the data\n(blue dotted lines) and simulation (red solid lines) from Fi g. 5 (a) is shown using the same scale for all orientations. In (e), a\nsimulation with a different set of parameters is shown for com parison (black, dashed line), see text.11\nout-of-plane anisotropy and therefore neglect B1¯10in our\nsimulations. An in-plane rotation of the external field\nwould be required for a more accurate measurement of\nB4||andB1¯10, but is outside the scope of this work.\nAccording to the valence-band model in Ref. 12, an\noscillatory behavior of the magnetic anisotropy parame-\nters is expected as a function of p. Therefore, depend-\ning on the absolute value of p, different values for, e.g.,\n∂B001/∂pare expected. In particular, there are regions\nwhere a anisotropy parameter might be nearly indepen-\ndent ofpand other regions with a very steep pdepen-\ndence. Since the absolute value of pis unknown, a quan-\ntitative discussion of the pdependence of the obtained\nanisotropy parameters based on the model in Ref. 12\nis not possible. In addition to p, thep-dexchange\nintegral,12which mayalsovary asa function ofthe depth\nin a non-homogneous film, also influences the anisotropy\nparameters,12further complicating a quantitative analy-\nsis.\nFor all samples, we used a constant exchange stiffness\nDsin our modeling. As alluded to above, there is some\nambiguity in this assumption, since the exchange stiff-\nness and the gradient in the anisotropy both influence\nthe mode spacing. For simplicity, however, we intended\nto keep as many simulation parameters as possible con-\nstant. The absolute values obtained for the exchange\nstiffness agree within a factor of 2 with the ones obtained\nin previous experiments36,50but are a factor of 2-4 larger\nthan theoretically predicted.51For the reasons discussed\nabove,thereisalargeuncertaintyalsointhederivationof\nthe absolute value of Dsfrom standing spin-wave modes\nin layers with a gradient in the magnetic anisotropy con-\nstants.\nIn order to use one parameter set for all field-\norientations, the Gilbert damping parameter was as-\nsumed to be isotropic in the simulations. The modeling\nof the SWR data could be further improved by assum-\ning a non-isotropic damping, its value being larger for\nµ0H||[110] than for µ0H||[001] [cf. Fig. 5]. This how-\never, only improves the result when assuming a field\norientation-dependent scaling factor for the amplitude,\nwhich could be motivated, e.g., by the assumption that\nthe microwave magnetic field present at the sample po-\nsition depends on the sample orientation within the\ncavity. The absolute values of αdetermined here are\ncomparable with the ones obtained by ultra-fast opti-\ncal experiments,52but are larger than the typical α=\n0.01...0.03 values found by frequency-dependent FMR\nstudies.53,54As already alluded to, inhomogeneous line-\nbroadening mechanisms may play a dominant role,54in\nparticular for as-grown samples.55We therefore assume\nthat the values for αobtained in this study overesti-\nmate the actual intrinsic Gilbert damping. A frequency-\ndependent SWR study would be required to determine\nthe intrinsic α. Such a study could possibly also reveal a\np-dependent αas theoretically predicted.55In our study,\nassuming a zdependent αdid not improve the agree-\nment between simulation and experiment, corroboratingthe conjecture that inhomogeneous broadening mecha-\nnisms dominate the linewidth and therefore obscure a\npossiblezdependence of α.\nIV. SUMMARY\nWehavepresentedafinitedifference-typemodelingap-\nproach for standing spin-wave resonances based on a nu-\nmerical solution of the LLG equation. With this generic\nformalism, SWR spectra can be simulated accounting for\nelliptical magnetization precession, for arbitrary orienta-\ntionsofthe externalmagneticfield, andforarbitrarypro-\nfiles of all magnetic properties, including anisotropy pa-\nrameters, exchange stiffness, Gilbert damping, and sat-\nuration magnetization. The approach is applicable not\nonly to (Ga,Mn)As but to all ferromagnets.\nFour(Ga,Mn)Assamples, epitaxiallygrownwithV/III\nflux ratios of 1.1, 1.3, 1.5, and 3.5 were investigated by\nECV and spin-wave resonance spectroscopy, revealing a\ncorrelation of a linear gradient in the hole concentration\nwith the occurrence of standing spin wave resonances, in\nparticularfortheexternalfieldorientedout-of-plane. Us-\ning the presented modeling approach, the SWR spectra\ncould be reproduced in a simulation with one parameter\nset for all external field orientations. The simulation re-\nsults demonstrate that the profileof the out-of-planeuni-\nformresonancefieldiscorrelatedwiththeholeconcentra-\ntion profile. However, our measurements and simulations\nshow,that anon-uniformholeconcentrationprofileisnot\nthe only cause that leads to the observed non-uniform\nmagnetic anisotropy; possibly, a variation in the satura-\ntion magnetization also influences the anisotropy param-\neters. To gain a quantitative understanding of this issue,\nmore samples with known hole concentrations would be\nrequired, where both the absolute values and the profiles\nofpare varied. Such a study was, however, outside the\nscope of this work.\nBesides the modeling of SWR intensities and\nlinewidths, the presented formalism yields the magne-\ntization precession amplitude as a function of the po-\nsition within the ferromagnet. It can therefore be used\nto investigate spin-pumping intensities in (Ga,Mn)As/Pt\nbilayers.27The spin-pumping signal, detected as a volt-\nage across the Pt layer, should be proportional to\nthe magnetization precession cone in the vicinity of\nthe (Ga,Mn)As/Pt interface. By measuring the spin-\npumping signal as well as the SWR intensities of\n(Ga,Mn)As/Pt and by using our modeling approach, it\nshould be possible to investigate to which extent a mag-\nnetization mode which is localized at a certain posi-\ntion within the (Ga,Mn)As layer contributes to the spin-\npumping signal.12\nAcknowledgments\nThis work was supported by the Deutsche Forschungs-\nGemeinschaft via Grant No. SFB 631 C3 (Walter Schot-\ntky Institut) and Grant No. Li 988/4 (Universit¨ at Ulm).\nAppendix A: Coordinate Transformation and Free\nEnthalpy derivatives\nThe transformation between the crystallographiccoor-\ndinate system ( x,y,z) and the equilibrium system (1,2,3)\nis given by\n\nmx\nmy\nmz\n=T\nm1\nm2\nm3\n, (A1)\nwith\nT=\ncosθ0cosφ0−sinφ0sinθ0cosφ0\ncosθ0sinφ0cosφ0sinθ0sinφ0\n−sinθ00 cos θ0\n.(A2)\nThe derivatives of the free enthalpy density Eq. (6) with\nrespect to the magnetization components are\nG3=∂m3G|m=m0=−µ0H3+2B001cos2θ0\n+B1¯10(sinθ0cosφ0−sinθ0sinφ0)2\n+ 4B4⊥cos4θ0\n+ 4B4/bardblsin4θ0(cos4φ0+sin4φ0) (A3)\nG21=G12=∂m1∂m2G|m=m0\n= cosθ0(1−2cos2φ0)[B1¯10\n+ 12B4/bardblsin2θ0cosφ0sinφ0] (A4)\nG11=∂m1∂m1G|m=m0= 2B001sin2θ0\n+ 12cos2θ0sin2θ0[B4⊥\n+B4/bardbl(cos4φ0+sin4φ0)]\n+B1¯10cos2θ0(cosφ0−sinφ0)2(A5)\nG22=∂m2∂m2G|m=m0= 2B1¯10(sinφ0+cosφ0)2\n+ 24B4/bardblsin2θ0cos2φ0sin2φ0. (A6)Appendix B: Finite Difference Method\nIn this Appendix, we describe how the spin-waveequa-\ntion can be numerically solved by the finite difference\nmethod. We start with the simple case of a circulary pre-\ncessing magnetization, neglecting Gilbert damping and\nthe driving field (Sec. B1). Then we turn to the gen-\neral case, where the magnetization precesses elliptically\nand the Gilbert damping as well as the driving field are\nincluded (Sec. B2).\n1. The One-Dimensional, Homogeneous,\nUndamped Case\nHere, we describe how the resonance fields and the\nspin-wave modes can be found, assuming a circularly\nprecessing magnetization m2=im1= ˜m, a constant ex-\nchangestiffness, and a zindependent equilibrium magne-\ntization. This case has been considered in Ref. 36 using a\nsemi-analytical approach to solve the spin-wave equation\nEq. (10). The approach considered here, is slightly more\ngeneral, as it is straight forward to determine resonance\nfields and eigenmodes of the system for an arbitrary z\ndependence of the uniform resonance field. To solve\nEq. (10), we divide the ferromagnetic film into a finite\nnumbernof layers with equal thickness land constant\nmagnetic properties within each of these layers. The z\ndependence of ˜ mandµ0H001\nuniis thus given by an index\nj= 1...n. Within each of these layers the uniform reso-\nnance field and ˜ m(z) are thus constant and given by the\nvaluesµ0H001,j\nuni=:Kjand ˜mj, respectively. The second\nderivative of ˜ mis approximated by\n˜m′′(z=j·l)≈˜mj−1−2˜mj+ ˜mj+1\nl2.(B1)\nConsequently, Eq. (10) is converted to the homogeneous\nequation system\n\n.........\n... Kj−1+2d−d 0...\n...−d Kj+2d−d ...\n...0 −d Kj+1+2d ...\n.........\n\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n=µ0H\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n, (B2)\nwith the abbreviation d=−Ds/l2. The boundary condi-\ntion of natural freedom36(von Neumann boundary con-dition) reads as ˜ m0= ˜m1and ˜mn−1= ˜mnand can be\nincorporated in Eq. (B2). Since the matrix on the left13\nhand side of Eq. (B2) is sparse, it can be efficiently diag-\nonalized numerically, yielding the resonancefields (eigen-\nvalues) and the corresponding modes (eigenvectors). Af-\nter diagonalizingthe matrix, the relevantresonancefields\nare found by sorting the eigenvalues and considering only\nthe modes with positive resonance fields, corresponding\nto the bound states in the particle-in-a-box analogon.\nThe SWR amplitude of each mode is proportional to its\nnet magnetic moment; thus, the amplitudes can be found\nby integrating the (normalized) eigenmodes. The mode\nprofile, the resonance fields, and the SWR intensities are\nillustrated in Fig. 2 for a constant and a linearly varying\nuniform resonance field. The finite linewidth of the SWR\nmodes can be accounted for by assuming a Lorentzian\nlineshape for each mode with a certain linewidth and\nwith the resonance fields and intensities calculated as\ndescribed above36. Note that this approach to derive\nresonance fields and intensities is only valid if the mode\nseparation is large compared with the linewidth of the\nmodes; this restriction does not apply to the model pre-\nsented in the Appendix B2.\n2. The General Case\nTosolveEq.(7)forarbitrary µ0Handarbitrarilyvary-\ning magnetic properties, we again divide the ferromag-netic film into a finite number nof layers with equal\nthicknessland constant magnetic properties within each\nof these layers. In contrast to the case in the Appendix\nB1, where only the uniform resonance field was varied\nacross the layer, here potentially all magnetic proper-\nties entering Eq. (7) can be assumed to be zdependent.\nAdditionally, the components of the driving field µ0hi\n(i= 1,2), can also vary as a function of z, since the\n(1,2,3) frame of reference is zdependent and thus the\nprojections of the driving field have to be calculated for\neach layer. The zdependence of the components mi\n(i= 1,2), of the parameters H11,H12,H21,H22(de-\nfined in Sec. II) and the exchange stiffness is thus given\nby the index j= 0...n; the second derivative of each of\nthe components miis approximated as in Eq. (B1).\nThe linearized LLG equation Eq. (7), is thus converted\ninto the inhomogeneous equation system\n\n..................\n...Hj−1\n11−2dj−1Hj−1\n12dj−10 0 0 ...\n... Hj−1\n21Hj−1\n22−2dj−10dj−10 0 ...\n... dj0Hj\n11−2djHj\n12dj0...\n... 0 djHj\n21Hj\n22−2dj0 dj...\n... 0 0 dj+10Hj+1\n11−2dj+1Hj+1\n12...\n... 0 0 0 dj+1Hj+1\n21Hj+1\n22−2dj+1...\n..................\n\n...\nmj−1\n1\nmj−1\n2\nmj\n1\nmj\n2\nmj+1\n1\nmj+1\n2...\n=µ0\n...\nhj−1\n1\nhj−1\n2\nhj\n1\nhj\n2\nhj+1\n1\nhj+1\n2...\n,(B3)\nwith the abbreviation dj=−Dj\ns/l2. At the boundaries\nof the magnetic film we again assume the spins to exhibit\nnatural freedom m0\ni=m1\niandmn\ni=mn+1\ni.\nTo simulate a spin-wave spectrum for a given orienta-\ntion of the external field and a given profile of the mag-\nnetic properties, we numerically sweep the magnetic fieldand calculate the equilibrium magnetization orientation\nfor all indices j= 0...nat a given external field. The\ninverse of the matrix in Eq. (B3), multiplied by µ0M(z),\nis the generalized Polder susceptibility tensor ¯ χ(µ0H,z),\nwhich relatesthe transversemagnetization with the driv-\ning field, cf. Eq. (11).\n∗Electronic address: dreher@wsi.tum.de\n1J. Zemen, J. Kucera, K. Olejnik, and T. Jungwirth, Phys.\nRev. B80, 155203 (2009).\n2X. Liu, W. L. Lim, Z. Ge, S. Shen, M. Dobrowolska, J. K.\nFurdyna, T. Wojtowicz, K. M. Yu, and W. Walukiewicz,Appl. Phys. Lett. 86, 112512 (2005).\n3L. Dreher, D. Donhauser, J. Daeubler, M. Glunk, C. Rapp,\nW. Schoch, R. Sauer, and W. Limmer, Phys. Rev. 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B 69,\n085209 (2004)."    },    {        "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": "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": "1304.5883v2.Constant_residual_electrostatic_electron_plasma_mode_in_Vlasov_Ampere_system.pdf",        "content": "arXiv:1304.5883v2  [physics.plasm-ph]  19 Nov 2013Constant residual electrostatic electron plasma mode in Vl asov-Ampere system\nHua-sheng XIE1,∗\n1Institute for Fusion Theory and Simulation, Department of P hysics,\nZhejiang University, Hangzhou, 310027, People’s Republic o f China\n(Dated: January 11, 2018)\nIn a collisionless Vlasov-Poisson (V-P) electron plasma sy stem, two types of modes for electric\nfield perturbation exist: the exponentially Landau damped e lectron plasma waves and the initial-\nvalue sensitive ballistic modes. Here, the V-P system is mod ified slightly to a Vlasov-Ampere\n(V-A) system. A new constant residual mode is revealed. Math ematically, this mode comes from\nthe Laplace transform of an initial electric field perturbat ion, and physically represents that an\ninitial perturbation (e.g., external electric field pertur bation) would not be damped away. Thus,\nthis residual mode is more difficult to be damped than the balli stic mode. [Physics of Plasmas 20,\n112108 (2013); doi: 10.1063/1.4831761]\nI. INTRODUCTION\nFor the evolution of linear electron plasma wave in\nan unmagnetized plasma, the one-dimensional linearized\nVlasov-Poisson (V-P) system is\n∂tδf=−ikvδf+δE∂vf0, (1a)\nikδE=−/integraldisplay\nδfdv, (1b)\nwhere time tand space xhave been normalized by the\ninverse plasma frequency ω−1\np= 1//radicalbig\nn0e2/ǫ0mand the\nDebye length λD=/radicalbig\nǫ0κBT/n0e2, respectively, and κB\nis Boltzmann’s constant. That is, ωp= 1 and λD= 1.\nThe perturbations have the harmonic dependence eikx,\nandf0=fM= exp(−v2/2)/√\n2π, wherevhas been nor-\nmalized by the electron thermal speed, is the Maxwellian\nequilibrium distribution.\nIn most cases, the plasma frequency is sufficiently high\ncompared with the electron collision frequency ν. The\ncollision can be neglected for time scale t≪ν−1. The\ntypical characteristic nonlinear time scale tNLis the\nbounce time tB=ω−1\nB= (kδE)−1of trapped parti-\ncles. Due to δE≪1,ω−1≃ω−1\np= 1≪ω−1\nB≃tNL,\nwe can also ignore the nonlinear effect, because we are\nmainly interested in the linear time scale in this paper,\ni.e.,t≪tNL. We have also assumed the immobile ion in\nEq.(1) because me/mi≤1/1836≪1.\nAs an initial value problem, for arbitrary initial δf(v),\nthe problem can be described by a superposition of\na complete set of eigen solutions, as shown by van\nKampen[1] and Case[2]. It is also well known that\nthis system can support exponentially Landau damped\nmode[3], which is the time asymptotic behavior. This\nasymptotic behavior can also be described by dispersion\nrelation[3, 6]. For convenience, we designate the disper-\nsion relation solution portion as the Landau part and\nthe initial-value sensitive [referring to the moments of\n∗Electronic mail: huashengxie@gmail.comthe perturbations, as shown below, or Eqs.(7) and (A3)]\nportion in Case-van Kampen (CvK) mode as the ballis-\ntic part, as the latter part is usually closely related to\nthe−ikvδfterm in Eq.(1). For f0=fM, CvK eigen\nsolutions are undamped in the sense that no solution of\nthe linear Vlasov-Poisson equation with ℑ(ω)<0 exist\nfor the harmonic perturbation ∼exp(−iωt), where ωis\nthe mode frequency. However, because of phase mixing,\nthe (moments with respect to velocity of the) perturba-\ntions are still damped when all such modes are consid-\nered, e.g., considering δE(t) instead of δf(v,t), which\ncan yield Landau’s solution for some initial δf(v) (espe-\ncially, entire function[3, 4], which is holomorphic overthe\nwholecomplex plane). That is, the ballistic modes (ifnot\nspecified, here and hereafter, we are referring to the mo-\nments of these modes) usually decay faster than Landau\ndamping. A comprehensive description of these modes\nand their relations in this V-P system can be found in\nRef.[6].\nFor example, for the initial perturbation δf(v,0) =\nA0exp[−(v−ua)2/u2\nb], where |ua| ≫(|ub|,|ωr/k|) with\nωrreal frequency of the least damping Landau solution,\nthe ballistic mode decays like exp( −ikuat−k2u2\nbt2) [5].\nOn the contrary, for example, a slow (relative to that of\nLandau) algebraic decay mode\n∝1\ntsin(kubt)exp(−ikuat), (2)\ncan be excited by the initial perturbation [5]\nδf(v,0) =/braceleftbigg\nA0,for|v−ua| ≤ub,\n0,otherwise ,(3)\nwhich is not holomorphic, and thus not an entire func-\ntion.\nIf we neglect the term proportional to the electric field\nin Eq.(1), the solution Eq.(2) will be straightforwardand\nprecise. However, the difference is that, for Eq.(2), the\nelectric field is kept, which holds for very large t, i.e.,\nt→ ∞.\nFor the linear collisionless initial value problem of the\nV-P system, the above pictures are complete.2\nIn this paper, we find that if we change the equations\nslightly to Vlasov-Ampere (V-A) equations, the domi-\nnant mode will be a new constant residual mode, and\nnot the Landau and ballistic modes.\nII. VLASOV-AMPERE SYSTEM\nFrom the charge continuity equation ∂tρ+∂xJ= 0,\nwhereρ=/integraltext\nδfdvandJ=/integraltext\nvδfdv, the V-A and V-P\nsystems are equivalent (the proof is straightforward) if\nthe Poisson’s equation is used to supplement the initial\ncondition in the V-A system.\nThe reasons below fueled our interest to obtain a more\ncomplete description of the linearized V-A system, where\nthe Poisson’s equation Eq.(1b) in the V-P system has\nbeen replaced by Ampere’s law Eq.(4b) and the whole\nsystem of Eq.(1) is changed to\n∂tδf=−ikvδf+δE∂vf0, (4a)\n∂tδE=/integraldisplay\nvδfdv, (4b)\nwhere the normalization and all other definitions are un-\nchanged.\nFirst, in the work of Horne and Freeman [10], they\nneeded to use the V-P equations for the first few steps\nof the numerical integration before continuing with the\nV-A equations. Without the Poisson start, recovering\nthe Landau damping solutions is difficult. How do we\nexplain this finding?\nSecond, in a toroidal system, a well-known resid-\nual mode called the Rosenbluth-Hinton residual zonal\nflow[11, 12], was found. The authors showed that\npoloidal flows driven by ion-temperature gradient (ITG)\nturbulence will not damp to zero by linear collisionless\nprocesses,andthefinalpoloidalvelocity up∝ne}ationslash= 0[11]. Usu-\nally, physics in complicated systems can be understood\nusing simple system. For example, the phase mixing in\nvelocity space can also be found in a real space non-\nuniform system (see e.g., [13, 14]), and rich physics are\nassociated. Thus, phasemixingisausefulconceptforun-\nderstanding continuum damping, from Alfv´ en wave[13]\nto geodesic acoustic mode[14]. Can a constant residual\nmode be found in a simple system?\nA. Initial value solutions\nBefore to discuss the new constant residual mode, one\ncan refer to the review of the derivations and the verifica-\ntion of the Landau and ballistic solutions in V-P system\nin the Appendix A, which will be used for comparison.\nSimilar to the V-P system, the dispersion relation of\nplasma waves for the V-A system can be derived as\nDVA(ω,k)≡ω+/integraldisplay\nCv∂f0/∂v\nω−kvdv= 0.(5)Eqs.(5) and (A1) are equivalent, so they should both\nyield the normal modes given by the dispersion relation,\nwhich (the asymptotic solution) is independent of the\ninitialcondition. Thatis, weshouldalsobeabletoobtain\nthe Landau damped solutions from the V-A equations\nwithout using the Poisson start. However, as mentioned,\nexisting simulations indicate that this process is difficult.\nApplying Laplace transform ( Lp) in time and Fourier\ntransform( Fr) in spacetothe V-Asystem, wecan obtain\nδE(t,k) =L−1\npδˆE(ω,k) (6)\n=/integraldisplay\nCωe−iωtdω\n2π/bracketleftBig/integraldisplay\ndvvδfk(0)\n(ω−kv)D(ω,k)−δEk(0)\nD(ω,k)/bracketrightBig\n,\nWe are interested in asymptotic behavior, so only the\nω=kvpole and the maximum- ℑωmnormal mode are\nkept. From Eq.(6) we have,\nlim\nt→∞δE=1\n2π/integraldisplay\nvLdv/bracketleftBigve−ikvtδfk(0)\nDVA(kv,k)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nballistic part+ve−iωmt\n(ωm−kv)∂ωmD/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nLandau part/bracketrightBig\n−/integraldisplay\nCωe−iωtdω\n2πδEk(0)\nDVA(ω,k)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\ninitialδEpart. (7)\nNoting the relation kDVA=ωDVP, the main differ-\nence between Eqs.(7) and (A3) is in the initial δEor\nδE(0), which is a constant. To determine the typical\nasymptotic behavior, we let DVA≈ωin theδEpart of\nthe integral and obtain the residual mode\nlim\nt→∞[δEVA(t)−δEVP(t)+δEPoisson(0)]∝δE(0).(8)\nEq.(8) indicates that for almost all kinds of initial per-\nturbations δf(v,0), the electricfield perturbation δEwill\nnot be damped away,except that when the initial δEsat-\nisfies the Poisson’s equation Eq.(1b). Therefore, in the\nV-A system, the dominant mode would be this constant\nresidual mode, and Landau damping will not be evident.\nSimilar to the ballistic mode, this mode is also from the\ninitial perturbation. We shall designate it as the E mode,\nto distinguish it from ballistic mode.\nThis result indicates that the linear residual mode can\nalso be found in simple systems. The issue on why we\ncan hardly find Landau damping in V-A system is also\nanswered.\nHowever, in contrast to the zero poloidal and toriodal\nmode numbers for residual zonal flow [ m=n= 0 with\nperturbationquantitiesfactor ei(mθ−nφ)], thewavevector\nkfor E mode is not necessarily zero.\nNow, we verify the above calculation using simulation.\nEq.(4) is solved numerically as an initial value problem.\nThe simulation scheme and the initial δfperturbation\nare similar to that in the Appendix A. Fig. 1 shows the\nresidual E mode, where δE(0) = 0.02.\nThe results for a V-A simulation of Landau damping\nusing a Poisson start is given in Fig. 2. The parameters3\n0510152025−30−25−20−15−10−50\ntln|δ E|ωtheory=1.7999−0.53455i\n  \nsimulation ln|Re[ δ E]|\nsimulation ln|Im[ δ E]|\ntheory\nFIG. 1: V-A simulation of the residual E mode, which re-\nmains as a non-zero constant at large t. The theoretical value\nωtheory= 1.7999−0.53455iand corresponding red dot line are\nthe Landau damping dispersion relation prediction, which i s\nsolved from Eq.(5).\n0510152025−30−25−20−15−10−50\ntln|δ E|ωtheory=1.7999−0.53455i\n  \nsimulation ln|Re[ δ E]|\nsimulation ln|Im[ δ E]|\ntheory\nFIG. 2: V-A simulation of Landau damping using Poisson\nstart. At this condition, the V-A system is completely equiv -\nalent to the V-P system and produces Landau damping as\nexpected.\nare the same as those in Fig. 1, except that a V-P run\nis added in the initial ittime step(s). In this linear sim-\nulation, it= 1 is sufficient. For nonlinear simulations,\none may need more time steps [10]. In Fig. 2, we can see\nthat both the real and imaginary parts of δEmatch the\nLandau result.\nIn the V-A simulation, the residual E mode is domi-\nnant and the initial perturbation Eq.(3) also cannot pre-\nvent it (not shown here).\nCompared with the V-P system, the free parame-\nterδE(0) in the V-A system change the picture com-\npletely. Mathematically, the residual mode comes from\nthe Laplacetransformofinitial electric field perturbation\nand physically represents that an initial mode (e.g., ex-\nternal electric field perturbation) would not be dampedaway. Here, we solved the initial value problem. Exper-\nimentally, we usually address the boundary value prob-\nlem, which means the electric field will not be zero but\nremains a constant at large distance. The solution for\nthis boundary value problem can be found in Landau[3].\nB. Eigenmodes in the V-A system\nOther phenomena are related to Landau damping. For\nexample, the Landau damped normal mode is not an\neigenmodein theV-Psystem, althoughagrowingnormal\nmodecanbeaneigenmode(see, e.g., Ref. 15). Theeigen-\nmode problem with collision term and the connections\nbetween collisionless case have been studied by several\nauthors[15–19]. Bratanov et al.[15, 16] reported new nu-\nmerical investigations regarding the connection between\nthe CvK eigenmode and the Landaunormal mode. Spec-\ntral density accumulation occurs around the real fre-\nquency of the Landau-damped mode.\nWe rewrite the governing equations into the matrix\nformM·/vectorF=ω/vectorF, with∂t=−iω. The eigenvec-\ntor/vectorFfor the V-P system is /vectorF={Fj}={δf(vj)},\nwherevj=vmin+(j−1)∆vandj= 1,2,3,...,Nv+ 1.\nBratanov[15] verified numerically that although all the\n(CvK) eigenvalue solutions of /vectorFare undamped, their in-\ntegral, which is related to δE, can yield Landau damping\nbecause of phase mixing.\nAquestionthenarises: whatif δE(themoment)isalso\ncontained in the eigenvector /vectorF? Will it give the Landau\nsolution directly, so that the Landau damped mode can\nalso be an eigenmode? The V-A system can combine δE\ninto/vectorFdirectly. The eigenvector of the V-A system then\nbecomes\n/vectorF={Fj}=/braceleftbigg\nδf(vj), j= 1,2,3,···,Nv+1,\nδE, j =Nv+2.(9)\nUsing Eqs.(1) and (4), we can easily find the elements of\nthe eigenvalue matrices Mfor the V-P and V-A systems,\nrespectively.\nThe solutions are shown in Fig. 3. No eigenmodes and\nthe normal modes are identical in panel (a). That is, our\nresult also indicates that the Landau damped mode in\nthe V-A system is not an eigenmode.\nThe only difference between the V-A and V-P sys-\ntems is in the spectral density around ωr= 0: an ex-\ntra accumulating point exists at ωr= 0 of the V-A sys-\ntem [see panel (b)]. Noting Eq.(7), we can attribute\nthis new accumulating point to the E mode, particu-\nlarly to the constant residual E mode. That is, two\nω= 0 solutions exist: one from the continuum CvK\nmode and another from the new residual E mode, which\ncauses the singularity at ωr= 0 in the spectral density\nfigure [panel (b)]. Panels (c) and (d) show the corre-\nsponding eigen functions δf(vj). As expected, the eigen\nfunction of the continuum CvK mode is singular[1, 2],\nwhereas the eigen function of the E mode is smoother4\n−5 05−3−2−10123\nωrωi(a) V−A eigenmode solutions\n  \n−10 0 101.151.21.25\nωrspectral density(b) V−A spectral density, ∆v=0.0625\n  \neigenmodes density\nLandau damping ωr\n−10 0 1000.050.10.15\nvjδ f(vj)(c) Eigenfunction of ω=0 E−mode\n  \nRe[δ f]\nIm[δ f]\n−10 0 10−0.200.20.40.60.8\nvjδ f(vj)(d) Eigenfunction of ω=0 CvK mode\n  eigenmodes\nLandau solutions\nRe[δ f]\nIm[δ f]\nFIG. 3: Eigenmodes in the V-A system. (a) compared with\nthe Landau damping normal modes (dispersion relation solu-\ntions), (b) the spectral density, (c) and (d) the correspond ing\neigen functions for ω= 0 E mode and continuum CvK mode.\n[Due to the numerical error ( ω≃10−16∝ne}ationslash= 0), we do\nnot know whether the δ-singularity at v= 0 in panel\n(c) is a correct structure or a numerical problem at\npresent. The values ∆+(δfvj=0)≡δf(∆v)−δf(0) or\n∆−(δfvj=0)≡δf(0)−δf(−∆v) of the peaks in panels\n(c) and(d) arealsosensitivetothediscretegridsize∆ v.].\nIII. SUMMARY\nIn this paper, we have presented a relatively complete\npicture for the modes in the linearized electrostatic 1D\nV-Asystem. BesidestheusualLandaumodeandthebal-\nlistic mode in the V-P system, a new constant residual\nmode is found, and this mode is usually dominant. An-\nalytical asymptotic solutions, eigenmode solutions, and\nlinear simulations are consistent. In contrast to ballis-\ntic mode, this residual mode is more robust, i.e., more\ndifficult to be damped.\nThe finding indicates that the residual mode would be\ncommon, and not merely exists in complicated system\nsuch as the Rosenbluth-Hinton residual zonal flow found\nin toroidal system.\nOne may also be interested in other applications of\nV-A system. A successful example of the application of\nextended V-A equation with collision, source, and sink is\nthe Berk-Breizman model[20, 21] for discussing the non-\nlinear single Alfv´ en mode driven by an energetic injected\nbeam, especially for Alfv´ en eigenmodes in tokamak.IV. ACKNOWLEDGEMENTS\nThe author would like to thank M. Y. Yu for his as-\nsistance in improving the manuscript and for the help-\nful discussions. Discussions with R. B. Zhang and com-\nments from Y. Xiao are also appreciated. We especially\nthank the two anonymous referees for their useful com-\nments and suggestions, which are helpful in improving\nthis work. This work is supported by Fundamental Re-\nsearch Fund for Chinese Central Universities.\nAppendix A: Vlasov-Poisson system\nFor Landau mode, the dispersion relation of the V-P\nsystem Eq.(1) is\nDVP(ω,k)≡k+/integraldisplay\nC∂f0/∂v\nω−kvdv= 0,(A1)\nwhereCis the proper integral contour[3, 5–8].\nSimilar to Chen[5] or Jackson[6], by applying Laplace\ntransform ( Lp) in time and Fourier transform ( Fr) in\nspace to the V-P system, we can obtain\nδE(t,k) =/integraldisplay\nCωe−iωtdω\n2π/bracketleftBig/integraldisplay\ndvδfk(0)\n(ω−kv)D(ω,k)/bracketrightBig\n.(A2)\nWe are interested in asymptotic behavior, so that only\ntheω=kvpole and the maximum- ℑωmnormal mode\nare kept. From Eq.(A2) we have\nlim\nt→∞δE=1\n2π/integraldisplay\nvLdv/bracketleftBige−ikvtδfk(0)\nDVP(kv,k)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nballistic part+e−iωmt\n(ωm−kv)∂ωmD/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nLandau part/bracketrightBig\n.\n(A3)\nFocusing on the ballistic part, one can easily solve for\nthe Gaussian perturbation or Eq.(3), and obatain the\n∝exp(−t2) or∝1/tsolutions.\nThe analytical solutions from Eqs.(A2) or (A3) are\nusually approximations, so we would like to verify them\nnumerically.\nWe solve Eq.(1) as an initial value problem from t= 0\ntotend=Nt∆tusing a 4th-order Runge-Kutta scheme.\nThe discrete velocity space is from vmintovmax. There\nareNvuniform grids of size ∆ v= (vmax−vmin)/Nv.\nAs mentioned, formost initial perturbations δf(t= 0),\nthe asymptotic behavior of δE(t) is determined by the\nnormal mode that is Landau damped. Using a Gaussian\ninitial perturbation δfwithA0= 0.05,ua= 1.0, and\nub= 1.0, we have successfully reproduced the Landau\ndamped solution (not shown here, or see Fig. 2). The\nelectric field δE(t= 0) and δE(t) are obtained from the\nPoisson equation using the initially given and the calcu-\nlatedδf, respectively. We should use a small ∆ vto avoid\nnon-physical recurrence effect (the Poincar´ e recurrence)\natTR= 2π/(k∆v), which is due to the discreteness of5\n−10 0 10−0.200.20.40.6\nvf(t)=f0+Re[ δ f](a) k=0.5, dv=0.0097656\n  \n−10 0 10−0.100.1\nvRe[δ f](b) t=200, dt=0.02\n  \n0 100 200−0.4−0.200.20.4(c) TR=1286.7964\ntδ E\n  \n0 100 20005001000\nt1/|δ E|(d) |δ E|−1 vs tf(t=0)\nf(t)δ f(t=0)\nδ f(t)\nRe[δ E]\nIm[δ E]\nFIG. 4: Simulation of the 1 /tdecay ballistic mode. (a) and\n(b) show the f(v,t) andδf(v,t) att= 0 and t= 200. (c)\nand (d) show the δE(t) and|δE(t)|−1versust. The∝1/t\nbehavior of δE(t) is clear in (d), where the red dashed line is\ny= 3.3xand is provided as reference.the velocity space [9]. The verificationalso indicates that\nour simulation scheme is feasible.\nThe 1/tdamped ballistic mode from the initial per-\nturbation Eq.(3) can be seen in Fig. 4 for A0= 0.05,\nua= 0.0, andub= 2.0. As reference, the red dashed\nline in panel (d) is for y= 3.3x.|δE|decays as c/t, as\nanalytically predicted in Eq.(2). Note that the constant\ncdepends on A0.\nNotably, our linear analysis and simulation are carried\nout in the k-space, i.e., all perturbation quantities have\naneikxfactor. However, one can obtain the correspond-\ning (x,v) phase space figure by direct mapping using the\nrelationf(x,v) =f0(v)+ℜ[δf(k,v)eikx].\nThe above analytical calculations and simulations\nshow a complete picture of the typical Landau and bal-\nlistic modes in the V-P system.\n[1] N. van Kampen, Physica 21, 949 (1955).\n[2] K. M. Case, Annals of Phys. 7, 349 (1959).\n[3] L. D. Landau, J. Phys. (USSR) 10, 25 (1946).\n[4] T. Stix, Waves in Plasmas (AIP, New York, 1992).\n[5] L. Chen, Waves and Instabilities of Plasmas (World Sci-\nentific, 1987).\n[6] J. D. Jackson, J. Nucl. Energy 1, 171 (1960).\n[7] N. Krall and A. Trivelpiece, Principles of Plasma Physics\n(McGraw-Hill, 1973).\n[8] D. R. Nicholson, Introduction to Plasma Theory (Wiley,\n1983).\n[9] C. Z. Cheng and G. Knorr, J. Comput. Phys. 22, 330\n(1976).\n[10] R. B. Horne and M. P. Freeman, J. Comput. Phys. 171,\n182 (2001).\n[11] M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett.,\n80, 724 (1998).\n[12] F. L. Hinton and M. N. Rosenbluth, Plasma Phys. Con-\ntrol. Fusion, 41, A653 (1999).\n[13] A. Hasegawa and L. Chen, Phys. Rev. Lett., 32, 454(1974).\n[14] Z. Qiu, F. Zonca and L. Chen, Plasma Science and Tech-\nnology,13, 257(2011).\n[15] V. Bratanov, “Landau and van Kampen Spectra in Dis-\ncrete Kinetic Plasma Systems”, Master thesis, LMU Mu-\nnich, 2012.\n[16] V. Bratanov, F. Jenko, D. Hatch and S. Brunner, Phys.\nPlasmas, 20, 022108 (2013).\n[17] C. S. Ng, A. Bhattacharjee and F. Skiff, Phys. Rev. Lett.\n83, 1974 (1999).\n[18] C. S. Ng, A. Bhattacharjee and F. Skiff, Phys. Rev. Lett.\n92, 065002 (2004).\n[19] P. P. Hilscher, K. Imadera, J. Q. Li and K. Kishimoto,\nPhys. Plasmas, 20, 082127 (2013).\n[20] H. L. Berk and B. N. Breizman, Phys. Fluids B, 2, 2226\n(1990);2, 2235 (1990); 2, 2246 (1990).\n[21] H. L. Berk, B. N. Breizman and M. Pekker, Phys. Rev.\nLett.,76, 1256 (1996)."    },    {        "title": "1304.6232v1.L2_L2_foreach_sparse_recovery_with_low_risk.pdf",        "content": "`2=`2-foreach sparse recovery with low risk\u0003\n(PRELIMINARY VERSION )\nANNA C. G ILBERTyHUNG Q. N GOzELYPORATxATRIRUDRAz\nMARTIN J. S TRAUSSy\nyUniversity of Michigan, fannacg,martinjs g@umich.edu\nzUniversity at Buffalo (SUNY), fhungngo,atrig@buffalo.edu\nxBar-Ilan University, porately@cs.biu.ac.il\nAbstract\nIn this paper, we consider the “foreach” sparse recovery problem with failure probability p. The goal\nof which is to design a distribution over m\u0002Nmatrices \band a decoding algorithm Asuch that for\nevery x2RN, we have the following error guarantee with probability at least 1\u0000p\nkx\u0000A(\bx)k26Ckx\u0000xkk2;\nwhereCis a constant (ideally arbitrarily close to 1) andxkis the bestk-sparse approximation of x.\nMuch of the sparse recovery or compressive sensing literature has focused on the case of either p= 0\norp= \n(1) . We initiate the study of this problem for the entire range of failure probability. Our two\nmain results are as follows:\n1. We prove a lower bound on m, the number measurements, of \n(klog(n=k) + log(1=p))for\n2\u0000\u0002(N)6p<1. Cohen, Dahmen, and DeV ore [5] prove that this bound is tight.\n2. We prove nearly matching upper bounds for sub-linear time decoding. Previous such results ad-\ndressed only p= \n(1) .\nOur results and techniques lead to the following corollaries: (i) the first ever sub-linear time decoding\n`1=`1“forall” sparse recovery system that requires a log\rNextra factor (for some \r < 1) over the\noptimalO(klog(N=k))number of measurements, and (ii) extensions of Gilbert et al. [7] results for\ninformation-theoretically bounded adversaries.\nCohen, Dahmen, and DeV ore [6] prove a \n(N)lower bound for the “forall” case (i.e. p= 0). Our\nlower bound technique is inspired by their geometric approach, and is thus very different from prior\nlower bound proofs using communication complexity. Our technique yields a stronger result which\nholds for the entire range of failure probability p, as well as provides a simpler, more intuitive proof of\nthe original result by Cohen et al. For the upper bounds, we provide several algorithms that span the\ntrade-offs between number of measurements and failure probability. These algorithms include several\ninnovations that may be of use in similar applications. They include a new combination of the recur-\nsive constructive sparse recovery technique of Porat and Strauss [22] and the efficiently decodable list\nrecoverable codes. These list recoverable codes focus on a different range of parameters than the ones\nconsidered in traditional coding theory. Our best parameters are obtained by considering a code defined\nby the algorithmic version of the Loomis-Whitney inequality due to Ngo, Porat, R ´e and Rudra [20].\n\u0003An extended abstract of this work will appear in the proceedings of ICALP 2013. ACG’s work is supported in part by NSF CCF\n1161233. HQN’s work is partly supported by NSF grant CCF-1161196. AR is supported by NSF CAREER grant CCF-0844796\nand NSF grant CCF-1161196. MJS is supported in part by NSF CCF 0743372 and NSF CCF 1161233.arXiv:1304.6232v1  [cs.DS]  23 Apr 20131 Introduction\nIn a large number of modern scientific and computational applications, we have considerably more data\nthan we can hope to process efficiently and more data than is essential for distilling useful information.\nSparse signal recovery [8] is one method for both reducing the amount of data we collect or process initially\nand then, from the reduced collection of observations, recovering (an approximation to) the key pieces of\ninformation in the data. Sparse recovery assumes the following mathematical model: a data point is a vector\nx2RN, using a matrix \bof sizem\u0002N, wherem\u001cN, we collect “measurements” of xnon-adaptively\nand linearly as \bx; then, using a “recovery algorithm” A, we return a good approximation to x. The error\nguarantee must satisfy\nkx\u0000A(\bx)k26Ckx\u0000xkk2; (1)\nwhereCis a constant (ideally arbitrarily close to 1) andxkis the bestk-sparse approximation of x. This\nis customarily called an `2=`2-error guarantee in the literature. This paper considers the sparse recovery\nproblem with failure probability p, the goal of which is to design a distribution over m\u0002Nmatrices \band a\ndecoding algorithm Asuch that for every x2RN, the error guarantee holds with probability at least 1\u0000p.\nThe reader is referred to [8] and the references therein for a survey of sparse matrix techniques for sparse\nrecovery, and to [2] for a collection of articles (and the references therein) that emphasize the applications\nof sparse recovery in signal and image processing.\nThere are many parameters of interest in the design problem: (i) number of measurements m; (ii) decoding\ntime, i.e. runtime of algorithm A; (iii) approximation factor Cand (iv) failure probability p. We would\nlike to minimize all the four parameters simultaneously. It turns out, however, that optimizing the failure\nprobabilitypcan lead to wildly different recovery schemes. Much of the sparse recovery or compressive\nsensing literature has focused on the case of either p= 0(which is called the “ forall ” model) or p= \n(1)\n(the “ foreach ” model). Cohen, Dahmen, and DeV ore [6] showed a lower bound of m= \n(N)for the\nnumber of measurements when p= 0, rendering a sparse recovery system useless as one must collect\n(asymptotically) as many measurements as the length of the original signal1. Thus, algorithmically there is\nnot much to do in this regime.\nThe case of p>\n(1) has resulted in much more algorithmic success. Cand ´es and Tao showed in [3]\nthatO(klog(N=k))random measurements with a polynomial time recovery algorithm are sufficient for\ncompressible vectors and Cohen, et al. [6] show that O(klog(N=k))measurements are sufficient for any\nvector (but the recovery algorithm given is not polynomial time). In a subsequent paper, Cohen, et al. [5]\ngive a polynomial time algorithm with O(klog(N=k))measurements. The next goal was to match the\nO(klog(N=k))measurements but with sub-linear time decoding. This was achieved by Gilbert, Li, Porat,\nand Strauss [9] who showed that there is a distribution on m\u0002Nmatrices with m=O(klog(N=k))and\na decoding algorithm Asuch that, for each x2RNthe`2=`2-error guarantee is satisfied with probability\np= \u0002(1) . The next natural goal was to nail down the correct dependence on C= 1 +\". Gilbert et\nal.’s result actually needs O(1\n\"klog(N=k))measurements. This was then shown to be tight by Price and\nWoodruff [23].\nAt this point, we completely understand the problem for the case of p= 0 orp= \n(1) . Somewhat\nsurprisingly, there is nowork that has explicitly considered the `2=`2sparse recovery problem when 0<\np6o(1). The main goal of this paper is to close this gap in our understanding.\nGiven the importance of the sparse recovery problem, we believe that it is important to close the gap. Similar\nstudies have been done extensively in a closely related field: coding theory. While the model of worst-case\n1For this reason, all of the forall sparse signal recovery results satisfy a different, weaker error guarantee. E.g. in the `1=`1\nforall sparse recovery we replace the condition (1) by kx\u0000A(\bx)k16Ckx\u0000xkk1.\n1errors pioneered by Hamming (which corresponds to the forall model) and the oblivious/stochastic error\nmodel pioneered by Shannon (which corresponds to the foreach model) are most well-known, there is a\nrich set of results in trying to understand the power of intermediate channels, including the arbitrarily vary-\ning channel [15]. Another way to consider intermediate channels is to consider computationally bounded\nadversaries [17]. Gilbert et al. [7] considered a computationally bounded adversarial model for the sparse\nrecovery problem in which signals are generated neither obliviously (as in the foreach model) nor adversar-\nially (in the forall model) in order to interpolate between the forall and foreach signal models. Our results\nin this paper imply new results for the `1=`1sparse recovery problem as well as the `2=`2sparse recovery\nproblem against bounded adversaries.\nOur main contributions are as follows.\n1. We prove that the number measurements has to be \n(klog(N=k) + log(1=p))for2\u0000\u0002(N)6p<1.\n2. We prove nearly matching upper bounds for sub-linear time decoding.\n3. We present applications of our result to obtain\n(i) the best known number of measurements for `1=`1forall sparse recovery with sublinear ( poly(k;logN))\ntime decoding, and\n(ii) nearly tight upper and lower bounds on the number of measurements needed to perform `2=`2-\nsparse recovery against information-theoretically bounded adversary.\nAs was motioned earlier, there are many parameters one could optimize. We will not pay very close attention\nto the approximation factor C, other than to stipulate that C6O(1). In most of our upper bounds, we can\nhandleC= 1 +\"for an arbitrary constant \", but optimizing the dependence on \"is beyond the scope of\nthis paper.\nLower Bound Result. We prove a lower bound of \n(log(1=p))on the number of measurements when\nthe failure probability satisfies 2\u0000\u0002(N)6p < 1. (Whenp62\u0000\n(N), our results imply a tight bound\nofm= \n(N).) The \n(log(1=p))lower bound along with the lower bound of \n(klog(N=k))from [23]\nimplies the final form of the lower bound claimed above. The obvious follow-up question is whether this\nbound is tight. Indeed, an upper bound result Cohen, Dahmen, and DeV ore [5] proves that this bound is\ntight if we only care about polynomial time decoding (see Theorem G.1). Thus, the interesting algorithmic\nquestion is how close we can get to this bound with sub-linear time decoding.\nUpper Bound Results. For the upper bounds, we provide several algorithms that span the trade-offs be-\ntween number of measurements and failure probability. For completeness, we include the running times and\nthe space requirements of the algorithms and measurement matrices in Table 1, which summarizes our main\nresults and compares them with existing results.\nWe begin by first considering the most natural way to boost the failure probability of a given `2=`2sparse\nrecovery problem: we repeat the scheme stimes with independent randomness and pick the “best” answer–\nsee Appendix A for more details. This boosts the decoding error probability from the original ptop\n(s)–\nthough the reduction does blow up the approximation factor by a multiplicative factor ofp\n3.\nThe above implies that if we can optimally solve the `2=`2sparse recovery problem for p>(N=k)\u0000k\n(i.e. withO(klog(N=k))measurements), then we can solve the problem optimally for smaller p. Hence,\nfor the rest of the description we focus on the case p>(N=k)\u0000O(k)(where the goal is to obtain m=\nO(klog(N=k))). Note that in this case, the amplification does not help as even for p= \n(1) , previous\n2Reference k m p Decoding time Space\n[5] Anykklog(N=k) + log(1=p) Anyp poly(N) poly(N)\n[9] Anyk klog(N=k) p>\n(1) k\u0001poly logNk\u0001poly logN\n[22] k>N\n(1)klog(N=k)p= (N=k)\u0000k=logckk1+\u000bpoly logNNk0:2\nAnyk klog(N=k) logc\nkNp=k\u0000k=logckk1+\u000bpoly logNNk0:2\n[9] Anyk klog(N=k) p>2\u0000k=logckk\u0001poly logNk\u0001poly logN\n+Lem 4.12\nThis paper k>N\n(1)klog(N=k)p= (N=k)\u0000k=logckk1+\u000bpoly logNk\u0001poly logN\nAnyk>log(N=k)klog(N=k) log\u000b\nkNp= (N=k)\u0000k=logckk2\u000b\u00001\npoly logNk\u0001poly logN\nTable 1: Summary of algorithmic results. The results in [22] are for `1=`1forall sparse recovery but their\nresults can be easily adapted to our setting with our proofs. cis some constant>8and\u000b>0is any arbitrary\nconstant and we ignore the constant factors in front of all the expressions.\nresults (e.g. [23]) imply that m>\n(klog(N=k)). Thus, if the original decoding error probability is pthen\nto obtain the (N=k)\u0000kdecoding error probability implies that the number of measurements will be larger\nthan the optimal value a factor of klog(N=k)=log(1=p). As we will see shortly the best know upper bound\ncan achievep= 2\u0000k, which implies that amplification will be larger than the optimal value of \n(klog(N=k)\nby a factor of log(N=k). In this work, we show how to achieve the same goal with an asymptotically smaller\nblow-up.\nForp>(N=k)\u0000k, there are two related works. The first is that of Porat and Strauss [22] who considered\nthe sparse recovery problem under the `1=`1forall guarantee. Despite the different error guarantee, our\nconstruction is closely related to that of [22] and our proofs imply the results for [22] listed in Table 1. Note\nthat the results for polynomially large kare pretty much the same except we have a better space complexity.\nFor general k, our result also has better number of measurements and failure probability guarantee. The\nsecond work is that of Gilbert et al. [9]. Even though the results in that paper are cited for p>\n(1) , it\ncan be shown that if one uses O(k)-wise independent random variables instead of the pair-wise independent\nrandom variables as used in [9], one can obtain a “weak system” with failure probability 2\u0000k. Then our\n“weak system to top level system conversion” (Lemma 4.12) leads to the result claimed in the second to last\nrow in Table 1. Our results have a better failure probability at the cost of larger number of measurements.\nIt is natural to ask whether decreasing the failure probability (the base changed from 2to(N=k)) is worth\ngiving up the optimality in the number of measurements (which is what [9] obtains). Note that achieving\na failure probability of (N=k)\u0000kis a very natural goal and our results are better than those in [9] when we\nanchor on the failure probability goal first. Interestingly, it turns out that this difference is also crucial to our\nresult for`1=`1sparse recovery which is discussed next.\n`1=`1forall sparse recovery. As was mentioned earlier, our construction is similar to that of Porat and\nStrauss. In fact, our techniques also imply some results for the `1=`1forall sparse recovery problem. (We\nonly need the “weak system” construction, after that results from [22] can be applied.) We highlight two\nresults here.\nFirst, fork>N\n(1)we can getO(klog(N=k))measurements with k1+\u000bpoly logNdecoding time and\nspace usage, for any \u000b > 0. This improves upon the `1=`1forall sparse recovery result from [22] with a\nbetter space usage and answers a question left open by [22].\nOur second result, which is for general k, achieves for any\u000b > 0,O(klog(N=k) log\u000b\nkN)number of\nmeasurements with decoding time k21=\u000bpoly logN. This greatly improves upon the extra factor in the num-\nber of measurements over the optimal O(klog(N=k)) from something around log8\nkNin [22] to log\u000b\nkN\nfor arbitrary \u000b > 0. Even though this “only” shaves of log factors, no prior work on sublinear time de-\ncodable`1=`1sparse recovery scheme has been able to breach the “log barrier.” There is a folklore con-\nstruction for a sub-linear time decodable matrix: using a “bit-tester matrix” along with a lossless bipartite\n3expander where each edge in its adjacency matrix is replaced with a random \u00061value. This scheme achieves\nm=O(klog(N=k) logN). If we consider a “weak system” only and small values of k, then the folklore\nconstruction achieves failure probability p= (N=k)\u0000kwhich matches that of our construction. This along\nwith a simple union bound (which we also use in our construction) and results in [22] implies an `1=`1\nsparse recovery system. However, this simple construction has an extra factor of logNin the number of\nmeasurements, which our result reduces to log\u000b\nkNfor any\u000b>0.2\nBounded Adversary Results. We also obtain some results for `2=`2-sparse recovery against information-\ntheoretically-bounded adversaries as considered by Gilbert et al. [7]. (See Section 2 for a formal definition\nof such bounded adversaries.) Gilbert et al. show that O(klog(N=k))measurements is sufficient for such\nadversaries with O(logN)bits of information. Our results allow us to prove results for a general number\nof information bits bound of s. In particular, we observe that for such adversaries O(klog(N=k) +s)\nmeasurements suffice. Further, if one desires sublinear time decoding then our results in Table 1 allows\nfor a similar conclusion but with extra poly logkfactors. We also observe that one needs ~\n(ps)many\nmeasurements against such an adversary (assuming the entries are polynomially large). In the final version\nof the paper, we will present a proof suggested to us by an anonymous reviewer that leads to the optimal\n\n(s)lower bound.\nLower Bound Techniques. Our lower bound technique is inspired by the geometric approach of Cohen et\nal. [6] for the p= 0case. Our bound holds for the entire range of failure probability p. Our technique also\nyields a simpler and more intuitive proof of Cohen et al. result. Both results hold even for sparsity k= 1.\nThe technical crux of the lower bound result in [6] for p= 0is to show that any measurement matrix \bwith\nO(N=C2)rows has a null space vector nthat is “non-flat,” – i.e. nhas one coordinate that has most of the\nmass of n. On the other hand since \bn=0, the decoding algorithm Ahas to output the same answer when\nx=nand when x=0. It is easy to see that then Adoes not satisfy (1) for at least one of these two cases\n(the output for 0has to be 0while the output for nhas to be non-flat and in particular not 0).\nTo briefly introduce our technique, consider the case of p= 2\u0000N(where we want a lower bound of m=\n\n(N)). The straightforward extension of Cohen, et al.’s argument is to define a distribution over, say,\nall the unit vectors in RN, argue that this gives a large measure of “bad vectors,” and then apply Yao’s\nminimax lemma to obtain our final result; i.e., that there are “a lot” of non-flat vectors in the null space of a\ngiven matrix \b. This argument fails because the distribution on the bad vectors must be independent of the\nmeasurement matrix \b(and algorithm A) in order to apply Yao’s lemma but null space vectors, of course,\ndepend on \b. On the other hand, if we define the “hard” distribution to be the uniform distribution, then the\nmeasure of null space vectors for any m>1is zero, and thus this obvious generalization does not work.\nWe overcome this obstacle with a simple idea. Our hard distribution is still the uniform distribution on the\nunit sphereSN\u00001. We first show that there is a region Ron this sphere with large measure ( >p) such that\nallvectors inRhave a positive “spike” (large mass) at one particular coordinate j\u00032[N]. (The region Ris\nsimply a small spherical cap above the unit vector ej\u0003.) In particular, to recover an input vector v2R, the\nalgorithm has to assign a large positive mass to the j\u0003th coordinate of A(\bv). Next, by applying a certain\ninvertible linear reflector to R, we can construct a region R0(which is also a region on the sphere, and is\njust a reflection of R) with the same measure satisfying the following: for each vector v2R, the reflection\nv0ofv(v02R0) has a negative spike at the same coordinate j\u0003; furthermore, \bv= \bv0, which forces the\nalgorithmAinto a dichotomy. The algorithm can not recover both vandv0well at once. Roughly speaking,\n2Interestingly one could use the weak system from [9] (with O(k)-wise independence) to obtain an `1=`1sparse recovery\nsystem with an extra factor of O(log(N=k)), which is not that much better than that obtained by the simple bit-tester construction.\n4the algorithm will be wrong with probability at least half the total measure of RandR0, which isp. Finally,\nYao’s lemma completes the lower bound proof. There are some additional technical obstacles that we need\nto overcome in this step—see Section F.2 for more details.\nIn the final version of the paper, we will present an alternate lower bound proof that was suggested to us by\nan anonymous reviewer.\nUpper Bound Techniques. We believe that our main algorithmic contributions are the new techniques\nthat we introduce in this paper, which should be useful in (similar) applications (e.g. in the `1=`1sparse\nrecovery problem as we have already pointed out).\nOur upper bounds follows the same outline used by Gilbert, Li, Porat and Strauss [9] and Porat and\nStrauss [22]. At a high level, the construction follows three steps. The first step is to design an “identi-\nfication scheme,” which in sub-linear time computes a set S\u0012[N]of size roughly kthat contains \n(k)\nof the “heavy hitters.” (Heavy hitters are the coordinates where if the output vector does not put in enough\nmass then (1) will not be satisfied.) In the second step, we develop a “weak level system” which essen-\ntially estimates the values of coordinates in S. Finally, using a loop invariant iterative scheme, we convert\nthe weak system into a “top level system,” which is the overall system that we want to design. (The way\nthis iterative procedure works is that it makes sure that after iteration i, one is missing only O(k=2i)heavy\nhitters– so after logksteps we would have recovered all of them.) The last two steps are designed to run in\ntimejSj\u0001poly logN, so if the first step runs in sub-linear time, then the overall procedure is sub-linear3.\nOur main contribution is in the first step, so we will focus on the identification part here. The second step\n(taking median of measurements like Count-Sketch [4]) is standard [13]. For a pictorial overview of our\nidentification scheme, see Figure 1.\nIn order to highlight and to summarize our technical contributions, we present an overview of the scheme\nin [22] (when adapted to the `2=`2sparse recovery problem). We focus only on the identification step.\nFor near-linear time identification, one uses a lossless bipartite expander where each edge in the adjacency\nmatrix is replaced by a random \u00061value. The intuition is that because of the expansion property most heavy\nhitters will not collide with another heavy hitter in most of the measurements it participates in. Further, the\nexpansion property implies that the `2\n2noise in most of the neighboring measurements will be low. (The\nrandom\u00061is a standard trick to convert this to an low `2noise.) Thus, if we define the value of an index\nto be median value of all the measurements, then we should get very good estimates for most of the heavy\nhitters (and in particular, we can identify them by outputting the top O(k)median values). Since this step\nimplies computing Nmedians overall we have a near linear time computation. However, note that if we had\naccess to a subset S0\u0012[N]that had most of the heavy hitters in it, we can get away with a run time nearly\nlinear injS0j(by just computing the medians in S0).\nThis seems like a chicken and egg problem as the set S0is what we were after to begin with! Porat and\nStrauss use recursion to compute S0in sub-linear time. (The scheme was also subsequently used by Ngo,\nPorat and Rudra to design near optimal sub-linear time decodable `1=`1forall sparse recovery schemes\nfor non-negative signals [21].) To give the main intuition, consider the scheme that results in ~O(p\nN)\nidentification time. We think of the domain [N]asL\u0002R, where both LandRare isomorphic to [p\nN].\n(Think ofLas the firstlogN\n2bits in the logN-bit representation of any index in [N]andRto be the\nremaining bits.) If one can obtain lists SL\u001aLandSR\u001aRthat contain the projections of the heavy hitters\ninLandR, respectively, then SL\u0002SRwill contain all the heavy hitters, i.e. S0\u0012SL\u0002SR. (We can use the\nnear linear time scheme to obtain SLandSRin~O(p\nN)time in the base case. One also has to make sure\n3We would also like to point out that Gilbert et al.’s construction has a failure probability of \n(1) in the very first iteration of the\nlast step (weak to top level system conversion) and it seems unlikely that this can be made smaller without significantly changing\ntheir scheme.\n5that when going from [N]to a domain of sizep\nN, not too many heavy hitters collide. This can be done\nby, say, randomly permuting [N]before applying the recursive scheme.) The simplest thing to do would be\nto setS0=SL\u0002SR. However, since both jSLjandjSRjcan be \n(k), this step itself will take \n(k2)time,\nwhich is too much if we are shooting for a decoding time of k1+\u000bpoly logNfor\u000b<1. The way Porat and\nStrauss solved this problem was to store the whole inversion map as a table. This allowed k\u0001poly logN\ndecoding time but the scheme ended up needing \n(N)space overall.\nTo get a running time of k1+\u000bpoly logNone needs to apply the recursive idea with more levels. One can\nthink of the whole procedure as a recursion tree with N=O(logkN)nodes. Unfortunately, this process\nintroduces another technical hurdle. At each node, the expander based scheme loses some, say \u0010, fraction\nof the heavy hitters. To bound the overall fraction of lost heavy hitters, Porat and Strauss use the naive\nunion bound of \u0010\u0001N. However, we need the overall fraction of lost heavy hitters to be O(1). This in\nturn introduces extra factors of logkNin the number of measurements (resulting in the ultimate number of\nmeasurements of klog(N=k) log8\nkNin [22]).\nWe are now ready to present the new ideas that improve upon Porat and Strauss’ solutions to solve the two\nissues raised above. Instead of dividing [N]into[p\nN]\u0002[p\nN], we first apply a code C: [N]![bp\nN]r.\n(Note that the Porat Strauss construction corresponds to the case when r=b= 2 andCjust “splits”\nthelogNbits into two equal parts.) Thus, in our recursive algorithm at the root we will get rsubsets\nS1;:::;Sr\u0012[bp\nN]with the guarantee that for (most) i2[r],SicontainsC(j)ifor most heavy hitters\nj. Thus, we need to recover the j’s for which the condition in the last sentence is true. This is exactly\nthe list recovery problem that has been studied in the coding theory literature. (See e.g. [24].) Thus, if we\ncan design a codeCthat solves the list recovery problem very efficiently, we would solve the first problem\nabove4. For the second problem, note that since we are using a code C, even if we only have C(j)i2Si\nfor sayr=2positionsi2[r], we can recover all such indices j. In other words, unlike in the Porat Strauss\nconstruction where we can lose a heavy hitter even if we lose it in any of the Nrecursive call, in our case\nwe only lose a heavy hitter if it is lost in multiple recursive calls. This fact allows us to do a better union\nbound than the naive one used in [22].\nThe question then is whether there exists code Cwith the desired properties. The most crucial part is that the\ncode needs to have a decoding algorithm whose running time is (near) linear in maxi2[r]jSij. Further, we\nneed such codes with r=O(1), i.e. of constant block length independent of maxi2[r]jSij. Unfortunately,\nthe known results on list recovery, be it for Reed-Solomon codes [12] or folded Reed-Solomon codes [10]\ndo not work well in this regime—these results need r>\n(maxi2[r]jSij), which is way too expensive. For\nour setting, the best we can do with Reed-Solomon list recovery is to do the naive thing of going through\nall possibilities in\u0002i2[r]Si. (These codes however can correct for optimal number of errors and lead to our\nresult in the last row of Table 1.) Fortunately, a recent result of Ngo, Porat, R ´e and Rudra [20] gave an\nalgorithmic proof of the Loomis-Whitney inequality [18]. The (combinatorial) Loomis-Whitney inequality\nhas found uses in theoretical computer science before [14, 16]. In this work, we present the first application\nof the algorithmic Loomis Whitney inequality of [20] and show that it naturally defines a code Cwith the\nrequired (algorithmic) list recoverability. This code leads to the result in the second to last row of Table 1.\nInterestingly, we get optimal weak level systems by this method. We lose in the final failure probability\nbecause of the weak level to top level system conversion.\nWe conclude the contribution overview by pointing out three technical aspects of our results.\n\u000fAs was mentioned earlier, we first randomly permute the columns of the matrix to make the recursion\nwork. To complete our identification algorithm, we need to perform the inverse operation on the\n4We would like to point out that [21] also uses list recoverable codes but those codes are used in a different context: they used\nit to replace expanders and further, the codes have the traditional parameters.\n6indices to be output. The naive way would be to use a table lookup, which will require O(NlogN)\nspace, but would still be an improvement over [22]. However, we are able to exploit the specific nature\nof the recursive tree and the fact that our main results use the Reed-Solomon code and the code based\non Loomis-Whitney inequality to have sub-linear space usage.\n\u000fIn the weak level to top level system, both Gilbert et al., and Porat and Strauss decrease the parameters\ngeometrically—however in our case, we need to use different decay functions to obtain our failure\nprobability.\n\u000fUnlike the argument in [22] we explicitly use an expander while Porat and Strauss used a random\ngraph. However, because of this, [22] need at least N-wise independence in their random variables\nto make their argument go through. Our use of expanders allows us to get away with using only\n~O(k)-wise independence, which among others leads to our better space usage.\n2 Preliminaries\nWe fix notations, terminology, and concepts that will be used throughout the paper.\nLet[N]denote the setf1;:::;Ng. LetG: [N]\u0002[`]![M]be an`-regular bipartite graph, and MGbe its\nadjacency matrix. We will often switch back and forth between the graph Gand the matrixMG.\nFor any subset S\u0012[N], let\u0000(S)\u0012[M]denote the set of neighbor vertices of SinG. Further, letE(S)\ndenote the set of edges incident on S. A bipartite graph G: [N]\u0002[`]![M]is a(t;\")-expander if for every\nsubsetS\u0012[N]ofjSj6t, we havej\u0000(S)j>jSj`(1\u0000\"). Several expander properties used in our proofs\nare listed in Appendix B. Appendix C has some probability basics.\nSparse Recovery Basics. For a vector x= (xi)N\ni=12RN, the set ofkhighest-magnitude coordinates\nofxis denoted by Hk(x). Such elements are called heavy hitters . Every element i2[N]nHk(x)such\nthatjxij>q\n\u00102\u0011\nk\u0001kzk2will be called a heavy tail element. Here, \u0010and\u0011are constants that will be clear\nfrom context. All the remaining indices will be called light tail elements; letLdenote the set of light tail\nelements. A vector w= (wi)N\ni=12RNis called a flat tail ifwi= 1=jSjfor every non-zero wi, where\nS= supp( w).\nDefinition 2.1. A probabilistic m\u0002NmatrixMis called an (k;C)-approximate sparse recovery system or\n(k;C)-top level system with failure probability pif there exists a decoding algorithm Asuch that for every\nx2RN, the following holds with probability at least 1\u0000p:\nkx\u0000A(Mx)k26C\u0001kx\u0000xHk(x)k2:\nThe parameter mis called the number of measurements of the system.\nWe will also consider (k;C)`1=`1top level systems, which are the same as above except we have p= 0\nand the`2norms are replaced by `1norms.\nDefinition 2.2. A probabilistic matrix MwithNcolumns is called a (k;\u0010;\u0011 )-weak identification matrix\nwith(`;p)-guarantee if there is an algorithm that, given Mxand a subset S\u0012[N], with probability at least\n1\u0000poutputs a subset I\u0012Ssuch that (i)jIj6`and (ii) at most \u0010kof the elements of Hk(x)are not\npresent inI. The time taken to compute Iwill be called identification time .\n7Definition 2.3. We will call a (random) m\u0002NmatrixMa(k;\u0010;\u0011 )weak`2=`2system if the following\nholds for any vector x=y+zsuch thatjsupp( y)j6k. GivenMxone can compute ^xsuch that there exist\n^y;^zthat satisfy the following properties: (1) x=^x+^y+^z; (2)jsupp( ^x)j6O(k=\u0011);5(3)jsupp( ^y)j6\u0010k;\n(4)k^zk26(1 +O(\u0011))\u0001kzk2\nWe will also consider (k;\u0010;\u0011 )weak`1=`1systems, which is the same as above except the `2norms are\nreplaced by `1norms (and the algorithm is deterministic).\nCoding Basics. In this section, we define and instantiate some (families) of codes that we will be interested\nin. We begin with some basic coding definitions. We will call a code C: [N]![q]rbe a(r;N)q-code.6\nVectors in the range of Care called codewords . Sometimes we will think of Cas a subset of [q]r, defined\nthe natural way. We will primarily be interested in list recoverable codes. In particular,\nDefinition 2.4. LetN;q;r;`;L>1be integers and 06\u001a61be a real number. Then an (r;N)q-\ncodeCis called a (\u001a;`;L )-list recoverable code if the following holds. Given any collection of subsets\nS1;:::;Sr\u0012[q]such thatjSij6`for everyi2[r], there exists at most Lcodewords (c1;:::;cr)2C\nsuch thatci2Sifor at least (1\u0000\u001a)nindicesi2[r]. Further, we will call such a code recoverable in time\nT(`;N;q )if all such codewords can be computed within this time upper bound.\nAn(r;N)q-code is said to be uniform if for everyi2[r]it is the case that C(x)iis uniformly distributed over\n[q]for uniformly random x2[N]. In our construction we will require codes that are both list recoverable\nand uniform. Neither of these concepts are new but our construction needs us to focus on parameter regimes\nthat are generally not the object of study in coding theory. In particular, as in coding theory, we focus on the\ncase whereNis increasing. Also we focus on the case when qgrows withN, which is also a well studied\nregime. However, we consider the case when ris afixed .\nWe now consider a code based on the Loomis-Whitney inequality [18]. Let d>2be an integer and assume\nthatNis a power of 2anddp\nNis an integer. Given x2[N]we think of it as x= (x1;:::;xd)2[dp\nN]d.\nFurther, for any i2[d]definex\u0000i= (x1;:::;xi\u00001;xi+1;:::;xd)2[dp\nN]d\u00001. Then define CLW(d)(x) =\n(x\u00001;:::;x\u0000d). The Loomis-Whitney inequality implies that CLW(d)is a(0;`;`d=(d\u00001))-list recoverable\ncode. Ngo, Porat, R ´e and Rudra [20] recently showed that the code is list-recoverable in time ~O(`d=(d\u00001)).\nThe result implies the following:\nLemma 2.5. The codeCLW(d)is a uniform code that is (0;`;`d=(d\u00001))-list recoverable code. Further, it is\nrecoverable in O(`d=(d\u00001)logN)time.\nFor the sake of completeness, we prove the above via Theorem E.1 (with a slightly different proof than the\none from [20]). Finally, we consider the well-known Reed-Solomon (RS) codes.\nLemma 2.6. Let\u001a<1=2(1\u0000b=r). Then the code CRSis a uniform code that is (\u001a;`;`r)-list recoverable\ncode. Further, it is recoverable in O(`rr2log2N)time.7\n5This part is different from the weak system in [22], where we have jsupp( ^x)j6O(k).\n6We depart from the standard convention and use the size of the code Ninstead of its dimension logqN: this makes expressions\nsimpler later on.\n7Ther2log2Nfactor follows from the fact that the Berlekamp Massey algorithm needs O(r2)operation over Fq, each of which\ntakesO(log2q)time.\n8Bounded Adversary Model. We summarize the relevant definitions of (computationally-)bounded adver-\nsaries from [7]. In this setting, Mallory is the name of the process that generates inputs xto the sparse\nrecovery problem. We recall two definitions for Mallory:\n\u000fOblivious: Mallory cannot see the matrix \band generates the signal xindependent from \b. For\nsparse signal recovery, this model is equivalent to the “foreach” signal model.\n\u000fInformation-Theoretic: Mallory’s output has bounded mutual information with the matrix. To cast\nthis in a computational light, we say that an algorithm Mis(s-)information-theoretically-bounded if\nM(x) =M2(M1(x)), where the output of M1consists of at most sbits. This model is similar to that\nof the “information bottleneck” [27].\nLemma 1 of [7] relates the information-theoretically bounded adversary to a bound on the success probabil-\nity of an oblivious adversary. We re-state the lemma for completeness:\nLemma 2.7. Pick`=`(N), and fix 0< \u000b < 1. LetAbe any randomized algorithm which takes input\nx2f0;1gN,r2f0;1gm, and “succeeds” with probability 1\u0000\f. Then for any information theoretically\nbounded algorithm Mwith space`,A(M(r);r)succeeds with probability at least\nminf1\u0000\u000b;1\u0000`=log(\u000b=\f)g\nover the choices of r.\n3 Lower bounds\n3.1 Lower bound for `2=`2-foreach sparse recovery with low risk\nThroughout this section, let \bdenote anm\u0002Nmeasurement matrix. Without loss of generality, we will\nassume that all our measurement matrices have full row rank: rank (\b) =m. LetNbe the (row) null-space\nof\b, and Pbe the matrix representing the orthoprojection onto N. Forj2[N], letejdenote thejth\nstandard basis vector. Note that P=P2=PTPbecause any orthogonal projection matrix is symmetric\nand idempotent. Hence, for any two vectors x;y2RN,\nhPx;yi=hx;Pyi=hx;PTPyi=hPx;Pyi:\nFor the sake of completeness we present a simplified version of the proof of the \n(N)lower bound from [6]\nfor the`2=`2forall sparse recovery in Appendix F.1.\nWe first prove an auxiliary lemma which generalizes Proposition F.1. The lemma implies that, for any\nmeasurement matrix \b, ifm=N is “small” then there will be “a lot” of unit vectors that are not “flat,” i.e. in\nthese vectors some coordinate j\u0003has relatively large magnitude compared to all other coordinates.\nLemma 3.1. Let\bbe an arbitrary real matrix of dimension m\u0002N. Then, there exists j\u00032[N]such that,\nfor any unit-length vector v2RNwe have\nhPv;ej\u0003i>hv;ej\u0003i\u0000q\n1\u0000hv;ej\u0003i\u0001p\n2m=N\u0000m=N: (2)\n9Proof. Letj\u0003be the coordinate for which kPej\u0003k2\n2>1\u0000m=N , guaranteed to exist by Proposition F.1. For\nany unit vector v, we have\nhPv;ej\u0003i=hv;Pej\u0003i\n=hv;ej\u0003i\u0000hv;ej\u0003\u0000Pej\u0003i\n=hv;ej\u0003i\u0000hv\u0000ej\u0003;ej\u0003\u0000Pej\u0003i\u0000hej\u0003;ej\u0003\u0000Pej\u0003i\n(by Cauchy-Schwarz) >hv;ej\u0003i\u0000kv\u0000ej\u0003k2\u0001kej\u0003\u0000Pej\u0003k2\u0000(1\u0000hej\u0003;Pej\u0003i)\n=hv;ej\u0003i\u0000q\n2\u00002hv;ej\u0003i\u0001q\n1\u0000kPej\u0003k2\n2\u0000(1\u0000kPej\u0003k2\n2)\n>hv;ej\u0003i\u0000q\n1\u0000hv;ej\u0003i\u0001p\n2m=N\u0000m=N:\nNext, we show that any unit vector vsufficiently close to ej\u0003can be paired up in a 1-to-1 manner (through\na reflection operator) with a unit vector v0such that the decoding algorithm cannot work well on both vand\nv0. Since the pairing is measure preserving, we can then infer that when m=N is small the algorithm Awill\nfail with “high” probability.\nLemma 3.2. Let(\b;A)be a (deterministic) pair of measurement matrix and decoding algorithm, where \b\nis anm\u0002Nmatrix. Define \u000e=m=N , and let\r>0;C>1be arbitrary constants such that\n1\u00002\r\u00002\u000e\u00002p\n2\r\u000e>Cp\n1 +C2: (3)\nLetj\u00032[N]be the index satisfying (2)guaranteed to exist by Lemma 3.1. Let vbe any unit vector such\nthathv;ej\u0003i>1\u0000\r, andv0= (I\u00002P)vwhere Iis the identity matrix. Then, the following two conditions\ncannot be true at the same time:\n(a)kv\u0000A(\bv)k26C\u0001kv\u0000vkk2\n(b)kv0\u0000A(\bv0)k26C\u0001kv0\u0000v0\nkk2\nProof. Since Pvis in the null space of \b, we have \bv0= \b(v\u00002Pv) = \b v:To simplify notation, define\nz=A(\bv0) =A(\bv):\nVector zis well-defined because Ais deterministic. Assumes (a)holds, we will show that (b)does not hold.\nLetj\u0003be the coordinate from Lemma 3.1. From (a), we have\n1\u0000\r\u0000hz;ej\u0003i6hv;ej\u0003i\u0000hz;ej\u0003i\n6jhv\u0000z;ej\u0003ij\n6kv\u0000zk2\n6C\u0001kv\u0000vkk2\n6C\u0001q\n1\u0000hv;ej\u0003i2\n6Cp\n1\u0000(1\u0000\r)2:\nHence,\nhz;ej\u0003i>1\u0000\r\u0000Cp\n1\u0000(1\u0000\r)2:\n10In particular, due to (3),\nhz;ej\u0003i>0:\nConsequently,\nkv0\u0000zk2>jhv0\u0000z;ej\u0003ij=jhv0;ej\u0003i\u0000hz;ej\u0003ij>hz;ej\u0003i\u0000hv0;ej\u0003i>\u0000hv0;ej\u0003i:\nWe next claim that\n\u0000hv0;ej\u0003i>Cq\n1\u0000hv0;ej\u0003i2: (4)\nBefore proving the claim, let us first see how it implies that (b)does not hold. To this end, observe that\nC\u0001kv0\u0000v0\nkk26Cq\n1\u0000hv0;ej\u0003i2<\u0000hv0;ej\u0003i6kv0\u0000zk2:\nFinally, we prove claim (4), which is equivalent to hv0;ej\u0003i<\u0000Cp\n1+C2:\nhv0;ej\u0003i=h(I\u00002P)v;ej\u0003i\n=hv;ej\u0003i\u00002hPv;ej\u0003i\n(Lemma 3.1 )61\u00002(1\u0000\r\u0000\u000e\u0000p\n2\r\u000e)\n=\u00001 + 2\r+ 2\u000e+ 2p\n2\r\u000e (5)\n(from (3) )<\u0000Cp\n1 +C2:\nLemma 3.3. Let(\b;A)be a fixed pair of (deterministic) measurement matrix and decoding algorithm,\nwhere \bis anm\u0002Nmatrix. Define \u000e=m=N , and let\r >0;C>1be arbitrary constants satisfying (3).\nSuppose we chose input vectors xuniformly on the unit sphere SN\u00001, then\nPr\nx[kx\u0000A(\bx)k2>Ckx\u0000xkk2]>p\n1=\r\u0001e\u0000N\n2ln(2=\r):\nProof. Letj\u00032[N]be the index satisfying (2) guaranteed to exist by Lemma 3.1. The set of vectors vfor\nwhichhv;ej\u0003i>1\u0000\ris called the (1\u0000\r)-cap about ej\u0003on the sphere SN\u00001. It is known (see, e.g., [1] –\nLemma 2.3) that the (1\u0000\r)-cap has measure at least\n(1=2)\u0001(p\n\r=2)N\u00001=p\n1=\r\u0001e\u0000N\n2ln(2=\r):\nThe mapping v!(I\u00002P)vfrom Lemma 3.2 is a reflection through the rowspace of \b, hence the image\nof the reflection is another cap on the sphere with exactly the same measure. From (5) and (3), the two caps\nare disjoint. Lemma 3.2 then completes the proof, because if algorithm Aworks well on a vector in one cap,\nthen it will not work well on the vector’s reflection.\nArmed with the lemma above we prove our final lower bound stated below (and proved in Appendix F.2).\nTo do this we need a continuous version of Yao’s lemma and a simple padding trick to reduce the case of\np= 2\u0000\u0002(N)to larger values of p.\nTheorem 3.4. LetC>1andpbe such thatp\n12 + 16C2\u0001e\u0000ln(6+8C2)\n2\u0001N6p < 1:Then, any`2=`2\nforeach sparse recovery scheme using m\u0002Nmeasurement matrices \bwith failure probability at most pand\napproximation factor Cmust havem>1\n(6+8C2) ln(6+8C2)ln\u0010p\n12+16C2\np\u0011\n= \n(log(1=p))measurements.\n113.2 Lower Bound for Bounded Adversary Model\nIn this section, we show the following result:\nTheorem 3.5. Any`2=`2sparse recovery scheme that uses at most bbits in each entry of \bneeds at\nleast \n\u0000ps\nb\u0001\nnumber of measurements to be successful against an s-information-theoretically-bounded\nadversary.\nThe result follows from the `2=`2forall lower bound argument from Corollary F.2 and the following simple\nobservation:\nLemma 3.6. LetAbe anm\u0002nmatrix withm6n. Consider the column sub-matrix A0which has the first\nn0(for somem6n06n). Ifn0is in the null space of A0, then n= (n0;0n\u0000n0)(i.e. vector n0followed by\nn\u0000n0zeroes) is in the null space of A.\nProof of Theorem 3.5. For the sake of contradiction assume that there exists an `2=`2-sparse recovery matrix\n\bagainst anys-information-theoretically-bounded adversary that achieves an approximation factor of Cand\nhasm<ps\nC2p\nbmeasurements. Next we present an s-information-theoretically-bounded adversary that can\nfoil such a system.\nLet\b0be the column sub-matrix of \bthat has only the first n0def=p\ns=bcolumns of \b. Then by Proposi-\ntion F.1, there is a unit vector n0that is in the null space of \b0and satisfieskn0k1>1\u00001\nC2. By Lemma 3.6,\nn= (n0;0)is in the null space of \b. Further, it is easy to verify that nis a unit vector and knk1>1\u00001\nC2.\nUsing the proof of Corollary F.2, one can then argue that any recovery algorithm will have to fail on either\nthe input 0orn.\nTo complete the proof, we need to argue that the adversary only needs to remember s-bits of information\nabout \bto compute n0. Indeed note that we only need at most m\u0001p\ns=b\u0001b6s=C2bits to describe the\nmatrix \b0, which is enough to compute n.\n4 Sublinear Decoding\nWe present known results with polynomial time decoding on `2=`2sparse recovery problem in Appendix G.1.\nOur strategy for designing sub-linear time decodable top level systems will be as follows: we will first\ndesign weak identification matrices that have sublinear identification time. Then we (in a black-box manner)\nconvert such matrices to sub-linear time decodable top level systems. We now present an outline of how we\nimplement our strategy.\nIn Section 4.4.1 we show how expanders can be used to construct various schemes that will be useful later. In\nSection 4.4.2, we show how to convert weak identification systems to top level systems. The rest of technical\ndevelopment (in Section 4.4.3 and 4.5) is in designing weak identification system with good parameters.\nOur first main result on sub-linear time decodable top levels systems will be:\nTheorem 4.1. For anyk>N\n(1)and\";\u000b> 0, there exists a (k;1+\")-top level system with O(\"\u000011klog(N=k))\nmeasurements, failure probability (N=k)\u0000k=log13+\u000bkand decoding time \"\u00004\u0001k1+\u000b\u0001logO(1)N.8This scheme\nusesO\"(k\u0001logO(1)N)bits of space.\n8TheO(\u0001)notation here hides the dependence on \u000b.\n12In fact, our results also work for k=No(1)but we then do not get the optimal number of measurements.\nHowever, an increase in the decoding time leads to our second main result, which has near-optimal number\nof measurements.\nTheorem 4.2. For any 16k6Nand\";\u000b> 0, there exists a (k;1 +\")-top level system with\nO(\"\u000011klog(N=k) log\u000b\nkN)measurements, failure probability (N=k)\u0000k=log13+\u000bkand decoding time (k=\")\u0002(2\u0000\u000b)\u0001\nlogO(1)N.9This scheme uses O\"(k\u0001logO(1)N)bits of space.\n4.1 Proof of Main Results\nLater in this paper, we will prove the following results, which we will use to prove Theorems 4.1 and 4.2.\nLemma 4.3. Let0<\u000b;\r;\u0011< 1be real numbers and 16k6k06Nbe integers. Then there exists an\nO\u0010\n\r\u00006\u0011\u00004\u0001k\u0001log(N=k)\u0001(logk0log(N=k0)N)6+log(1=\u000b)=log(1+\u000b)\u0011\n\u0002Nmatrix that is (k;\r;\u0011 )-weak iden-\ntification matrix with\u0010\nO(k=\u0011);(k0log(N=k 0))\u0000\n(\u000b\rk= logk0log(N=k0)N)\u0011\n-guarantee with identification time\ncomplexity of O\u0010\n\r\u00003\u0011\u00003\u0000\u000b\u0001k\u0001k\u000b\n0\u0001logO(1)N\u0011\n, and space complexity of O\u0010;\u0011(klogO(1)N).\nLemma 4.4. Let\u000bbe a small enough real number. Then for large enough n, there exists a (k;\u0010;\u0011 )-weak\nidentification matrix with (O(k=\u0011);(n=k)\u00002k)-guarantee with\nO\u0000\n\u0010\u00007\u0011\u00004\u0001k\u0001log(n=k)\u0001(logkn)\u000b\u0001\nmeasurements and a decoding time of\n\u0010\u00009\u0011\u00004\u0001kO(21=\u000b)\u0001poly(logn) +\u0010\u00003\u0011\u00002\u0001(k=\u0011)O(21=\u000b)\u0001poly(logn):\nLemmas 4.3 and 4.4 along our weak system to top level system conversion (Lemma 4.12 below) then prove\nTheorem 4.1 and 4.2 respectively.\nProof of Theorem 4.1. We first note that applying Lemma 4.3 with \r=\u0011=2andk0>N\n(1)implies\nthe existence of a (O(k=\u0011);\u0011=2;\u0011)weak identification matrix with O(O(k=\u0011);(N=K 0)\u0000\u000b\u0011k-guarantee\nwithO(\u0011\u000010klog(N=k))measurements. By Lemma 4.10 below, we can amplify the failure probability to\n(N=k 0)\u0000\n(sk)by increasing the number of measurements to O(\u000b\u00001\u0011\u000011klog(N=k)). This implies that the\nidentification algorithm will identify all but k=2elements of Hk+k=\u0011(x). This means like Lemma 4.11, we\ncan convert this into a weak system on which we can then applying the conversion technique of Lemma 4.12\nto get the claimed bounds. (When we’re applying the recursive procedure to k=2jwe use this value of kin\nthe weak systems and the “original” value of kask0in the weak system.) For the space requirement, we\nhave to add up the space requirement for each weak system, which can be done within the claimed bound.\n\u0003\nProof of Theorem 4.2. The proof is almost the same as that for Theorem 4.1 except we use Lemma 4.4\ninstead of Lemma 4.3. The rest of the proof is exactly the same. (The way Lemma 4.12 is stated it needs\nO(klog(N=k))measurements but it can be verified that the conversion also works with the extra O(log\u000b\nkN)\nin the number of measurements.) \u0003\n9TheO(\u0001)notation here hides the dependence on \u000b.\n134.2 Consequences for the Bounded Adversary Model\nOur first corollary is an upper bound for the information-theoretic bounded adversary and follows directly\nfrom Lemma 2.7 (by setting \f=\u000b2\u0000s=\u000b) and the result of [6] (see Theorem G.1).\nCorollary 4.5. Fix0< \u000b < 1. There is a randomized sparse signal recovery algorithm that with m=\nO(klog(N=k)+s=\u000b)measurements will foil an s-information-theoretically bounded adversary; that is, the\nalgorithm’s output will meet the `2=`2error guarantees with probability 1\u0000\u000b.\nThe algorithm in [6] does not have a sublinear running time. If the goal is to defeat such an adversary\nand to do so with a sublinear algorithm, we must adjust our measurements accordingly, using Table 1\nas our reference for the range of parameters pandm. We note that in [7], there was a single result for\nO(logN)-information-theoretically bounded adversaries ( O(klog(N=k)measurements are sufficient) and\nthis corollary provides an upper bound for the entire range of parameter s.\n4.3 Consequences for the `1=`1forall sparse recovery\nWe observe that our techniques also work for the `1=`1foreach sparse recovery. Actually the fact that an\n`2=`2-foreach sparse recovery system is also an `1=`1-foreach sparse recovery system follows easily from\nknown results. So in particular, Lemma 4.4 implies that one can get a similar system in the `1=`1sense\nwith the same parameters. Furthermore from Remark I.3, to convert this into a weak (k;\u0010;\u0011 )identification\nsystem with (O(k=\u0011);0)-guarantee (i.e. a deterministic system), we need to take union bound over\u0000N\nk0\u0001\n+Nx\nevents, where k0=O(\u0010\u00005\u0011\u00002k)andx=O(log(N=k)). Thus, for k>\n(log(N=k)with the existing\nmachinery from [22] to convert a weak system into a top-level system, we get the following result:\nTheorem 4.6. For any 16k6N(such thatk>\n(log(N=k))and\";\u000b > 0, there exists a (k;1 +\")-\n`1=`1top level system with O(\"\u000018klog(N=k) log\u000b\nkN)measurements and decoding time (k=\")\u0002(2\u0000\u000b)\u0001\nlogO(1)N.10This scheme uses O\"(k\u0001logO(1)N)bits of space.\nProof Sketch. The results in [22] imply that a weak (k;\u0011=2;\u0011)identification system (for any \u0011 >0) with\n(O(k=\u0011);0)-guarantee (i.e., it is deterministic ) can be converted into a (k;1+\")-`1=`1top level system with\nonly a constant blowup in the number of measurements (with the same dependence on \"as one has on \u0011).\nWe note that by Lemma 4.4 and Lemma 4.10 (and the observation above that an `2=`2guarantee implies\nan`1=`1guarantee), we obtain a weak (k;\u0010;\u0011 )identification system with (O(k=\u0011);(N=k)\u0000\n(\u0010\u00005\u0011\u00002k))-\nguarantee with O(\u0010\u000012\u0011\u00006klog(N=k) log\u000b\nkN)number of measurements. Thus, by Remark I.3, one can\nconvert such a system into a deterministic one, which by the discussion in the paragraph above completes\nthe proof.\n4.4 Basic Building Blocks\nIn this section, we lay out some basic building blocks that will help us prove Lemmas 4.3 and 4.4.\n4.4.1 Expander Based Sparse Recovery\nIn this section, we record three results that will be useful to prove our final result. The proofs modify those\nfrom [22] and use an expander instead of random graph. (The actual arguments are similar.) The proofs are\ndeferred to Appendix H. We will prove the following result:\n10TheO(\u0001)notation here hides th e dependence on \u000b.\n14Theorem 4.7. Lets>1be an integer and 0< \u0011;\r < 1be reals. Let G: [N]\u0002[`]![M]be a\n(4k;\")-expander with \"=O(\r3\u0011)and`>c\u0001log(N=k)(for some large enough constant c= \u0002(s)that\ncan depend on \"). LetMbe the random matrix obtained by multiplying each (non-zero) entry in MGby\nrandom (independent) \u00061. Then except with probability\u0000N\n\rk\u0001\u0000s,Mis a(k;2\r;p\u0011)weak`2=`2system.\nThe estimation algorithm in Appendix H with Corollary H.3 (where we substitute \u0011by\u00112) implies the\nfollowing:\nTheorem 4.8. Lets>1be an integer and 0< \u0011;\r < 1be reals. Let G: [N]\u0002[`]![M]be a\n(4k;\")-expander with \"=O(\r3\u00112)and`>c\u0001log(N=k)(for some large enough constant c= \u0002(s)that\ncan depend on \"). LetMbe the random matrix obtained by multiplying each (non-zero) entry in MGby\nrandom (independent) \u00061. Then except with probability\u0000N\n\rk\u0001\u0000s, the following is true.\nThere exists an algorithm that given as input S\u0012[N]in timeO(jSj\u0001`+k=\u0011\u0001log(jSj))outputsO(k=\u0011)\nitemsI\u0012Sthat contain all but \rkitemsi2Ssuch thatjxij>3p\n\u00112=kkzk2.\nFinally, the proof of of Theorem 4.7 also implies the following:\nTheorem 4.9. Lets>1be an integer and 0< \u0011;\r < 1be reals. Let G: [N]\u0002[`]![M]be a\n(4k;\")-expander with \"=O(\r3\u00112)and`>c\u0001log(N=k)(for some large enough constant c= \u0002(s)that\ncan depend on \"). LetMbe the random matrix obtained by multiplying each (non-zero) entry in MGby\nrandom (independent) \u00061. Then except with probability\u0000N\n\rk\u0001\u0000s, the following is true.\nMis(k;\u0010;\u0011 )weak`2=`2system. Further, there exists an algorithm that given as input S\u0012[N](that\ncontains all but \u0010k=2elements ofHk+k=\u0011(x)) in timeO(jSj\u0001`+k=\u0011\u0001log(jSj))outputs ^xwith the required\nproperties.\n4.4.2 Weak Identification to Top-Level system conversion\nWe begin with the following observation that follows by repeating the given weak identification matrix s\ntimes (proof in Appendix G.2):\nLemma 4.10. LetMbe a (k;\u0010;\u0011 )weak identification matrix with (`;p)guarantee. Then there exists a\n(k;3\u0010;\u0011)weak identification matrix M0with(2`;p\n(s))guarantee with stimes more measurements.\nBy combining Lemma 4.10 and Theorem 4.9, we get:\nLemma 4.11. LetMbe a(k;\u0010;\u0011 )weak identification matrix with (`;p)guarantee with mmeasurements\nand letGbe an expander as in Theorem 4.9. Then for any integer s>1, there exists a (k;3\u0010;\u0011)weak`2=`2\nsystem with O(m\u0001s+M)measurements with failure probability p\n(s) +\u0000N\n\rk\u0001\u0000s.\nNext we present a simple yet crucial modification of the weak level to top level system conversion from [22].\nLemma 4.12. Let the following be true for any 0< \u0010;\u0011 < 1and integers s;k>1. There is a (k;\u0010;\u0011 )\nweak`2=`2systemMwithO(s\u0001\u0010\u0000d\u0011\u0000c\u0001k\u0001log(N=k))measurements11, failure probability q\u0000\n(sk)and\ndecoding time T(k;\u0010;\u0011;N;s ). Then for any \";\u000b > 0andk>1, there is a (k;\")-top level system with\nfailure probability q\u0000\n(k=log(1+\u000b)c+2+\u000bk),O(\"\u0000c\u0001k\u0001log(N=k))measurements and O(logk\u0001T(k;\u0010;\u0011; 1))\ndecoding time.\n11canddare absolute constants.\n15Proof Sketch. We only sketch the differences from the corresponding argument in [22]. We will have logk\nstages. In stage i, we will pick a\u0000k\n2i;1\n2;\"\u00011\ni1+\u000b\u0001\nweak`2=`2system with s= 2i=i(1+\u000b)c+2+\u000band “stack”\nthem to obtain our final top level system. Using the “loop invariant” technique of Gilbert et al., and argu-\nments similar to [22], one can design a decoding algorithm that has an approximation factor of\n1 +O(\")\u00011X\ni=11\ni1+\u000b= 1 +O(\");\nas desired. (In the above we used the fact thatP1\ni=1i\u00001\u0000\u000b=O(1)for any constant \u000b.)\nWe briefly argue the claimed bounds on the number of measurements and the failure probability (the rest of\nthe bounds are immediate). Note that at stage i, the number of measurements (ignoring the constant in the\nO(\u0001)notation):\n1\ni(1+\u000b)c+2+\u000b\u00012i\u00012d\u0001\"\u0000c\u0001\u00121\ni1+\u000b\u0013\u0000c\n\u0001k\n2i\u0001log(N2i=k)61\ni1+\u000b\u00012d\u0001\"\u0000c\u0001klog(N=k):\nThe claimed bound on the final number of measurements follows from the fact thatP1\ni=11\ni1+\u000b=O(1).\nNext note that the failure probability at stage iisq\u0000\n(2ik=(2ii(1+\u000b)c+2+\u000b)=q\u0000\n\u0010\nk\ni(1+\u000b)c+2+\u000b\u0011\n. The final\nfailure probability is determined by the failure probability at i= logk, which implies that the final failure\nprobability is q\u0000\n(k=log(1+\u000b)c+2+\u000bk), as desired\n4.4.3 An Intermediate Result\nIn this section, we will present an intermediate result that will be the building block of our recursive con-\nstruction (in Section 4.5). Let N>M>1be integers and let h: [N]![M]be a random map that we will\ndefine shortly. Let G: [M]\u0002[`]![m]be a(4k;\")-expander. In this section, we will consider how good\nan identification matrix we can obtain from MG\u000eh(with random\u00061) entries.\nCall a maph: [N]![M]to be (k0;\u000b)-random if the following is true. For any subset S\u0012[N]of sizek0,\nthe probability that for any i62S,h(i) =h(j)for anyj2Sis upper bounded by O\u0010\njSj\nM1\u0000\u000b\u0011\n. Note that a\n(k0+ 1) -wise independent random map from [N]![M]is(k0;0)-random.\nWe will prove the following “identification” analogue of Theorem 4.7 (the proof appears in Appendix I):\nLemma 4.13. Let0<\u000b;\u0011;\r < 1be reals. Let f: [N]![M]be a(k(\u0010\u00005\u0011\u00002+ 1);\u000b)-random map with\nM>\n(\u0010\u00006\u0011\u00002\u0001k1\u0000\u000b\n1\u00002\u000b\u0001(log(N=k))2). LetG: [M]\u0002[`]![m]be a(4k;\")-expander with \"=O(\r3\u00112)\nand`>c\u0001log(M=k)(for some large enough constant c= \u0002(s)that can depend on \"). LetMbe the\nrandom matrix obtained by multiplying each (non-zero) entry in MG\u000efby random (independent) \u00061. Then\nexcept with probability\u0000M\nk\u0001\n(\u0000\u000b\rk), the following is true.\nThere exists an algorithm that given as input S\u0012[M]outputs in time O(jSj\u0001`)O(k=\u0011)itemsI\u0012Sthat\ncontainf(i)for all but\rkitemsi2Ssuch thatjxij>3p\n\u00112=kkzk2.\n4.5 Proof of Lemma 4.3\nSee Figure 1 for an overview of the proof of Lemma 4.3.\nWe note that Theorem 4.8 instantiated with an optimal (random) expander, implies the following:\n16Corollary 4.14. There exists a family of (k;\u0010;\u0011 )-weak identification matrices fMngn>\u0010\u00002with\n(O(k=\u0011);(n=k)\u0000\n(\rk))-guarantee,O(\u0010\u00006\u0011\u00004klog(n=k))measurements and O(\u0010\u00003\u0011\u00001jSj\u0001logn)identi-\nfication time.\nOur main result is that we can recursively construct a weak identification matrix that has efficient identifica-\ntion from the (family) of weak identification matrices guaranteed by Corollary 4.14 (that by itself does not\nhave a sub-linear identification algorithm). The main idea is as follows: using a tree structure, we map [N]\nto successively smaller sub-problems. At the “leaves” the domain size is ~O(k)and thus one can use any\nidentification matrix (including the identity matrix). The main insight is on how to break up the domains at\nan internal node. For illustration purposes, consider the root. We use a uniform and (good) list recoverable\ncode (r;N)bp\nN-codeC(for some parameter b>1). (At the end we will use one of the codes from Section 2\nasC.) More precisely the domain [N]is broken down to rcopies of the domain [bp\nN]andi2[N]gets\nmapped toC(i)jin thejth child/sub-problem. By induction, we will prove that each of the rchildren,\nthe weak identification problem can be solved efficiently. In other words, for each j2[r], we will have a\ncandidate set Sj\u0012[bp\nN], each of size O(k=\u0011). AsCis list recoverable we can recover a list Sthat contains\ni. However, this list could be bigger than the required O(k=\u0011)bound (though not much larger). To prune\ndown this to a list of size O(k=\u0011)we use a weak identification matrix Mto narrow down the list to Iin\nrunning time proportional to jSj.\nBelow we state our formal result.\nTheorem 4.15. Let0<\u0010;\u000b;\u0011< 1be real numbers. Let fMngn>\u0010\u00002be a family of m(n)\u0002nmatrices from\nCorollary 4.14 that are (k;\u0010;\u0011 )-weak identification matrices with\u0010\n`def=O(k=\u0011);p(n)def= (n=k)\u0000\n(\u0010k)\u0011\n-\nguarantee where m(n) =g(\u0010;\u0011)\u0001k\u0001log(n=k)for some function g(\u0010;\u0011). Finally, for any real numbers\nb>1,06\u001a<1and integerr>1, letfCngn>1be a family of codes, where Cnis a(r;n)bpncode that is\n(\u001a;`;L )-list recoverable in time T(`;n;b ).\nThen for large enough Nthere exists a m0(N)\u0002NmatrixM\u0003that is (k;\u00100;\u0011)-weak identification matrix\nwith(O(k=\u0011);p0(N))-guarantee and identification time complexity D(k;N)as follows. In what follows let\nA>\n(\u0010\u00006\u0011\u00002\u0001k1\u0000\u000b\n1\u00002\u000b\u0001(log(N=k))2)satisfy the lower bound in Lemma 4.13.\nThere exists two integers h6O(logrlogAN)andN= (logAN)such that the following hold:\n\u001006(\n\u0010\u0001NO\u0010log(1=\u001a)\nlogr\u0011\nif\u001a>0\n\u0010\u0001N otherwise; (6)\np0(N)6N\u0001\u0012A\nk\u0013\u0000\n(\u0010k)\n; (7)\nm0(N)6O\u0012\ng(\u0010;\u0011)\u0001k\u0001log(N=k)\u0001Nlogr\nb\nlogb\u0013\n; (8)\nand\nD(k;N) =O\u0000\n\u0010\u00003\u0011\u00002\u0001N\u0001A\u0001logA\u0001\n+h\u00001X\nj=0\u0010\nrjT(O(k=\u0011);bjp\nN;b) +O\u0010\n\u0010\u00003\u0011\u00002\u0001L\u0001logbjp\nN\u0011\u0011\n:\n(9)\nWe now instantiate Theorem 4.15 with Corollary 4.14 and the list recoverable code from Lemma 2.5 to\nprove Lemma 4.3.\n17Proof Sketch of Lemma 4.3. Letd= 1 + 1=\u000b. For the code from Lemma 2.5 we have \u001a= 0,r=d,\nb=d=(d\u00001),`=O(k=\u0011)andL=O(k1+\u000b=\u00111+\u000b). We apply Theorem 4.15 with the family from\nCorollary 4.14 with \u0010=\r\nNandA= \u0002(\u0010\u00006\u0011\u00002\u0001k\u0001k\u000b\n0(log(N=k)2). Note that this implies that N=\nO(logk0log(N=k0)N). The claimed bounds then follow from the bounds in Theorem 4.15 and Lemma 4.16.\nWe also need to take into account the randomness needed to for the random \u0006signs. However, it is easy\nto check that in each of the Nnodes, we need poly(\u0010;\u0011)\u0001k-wise independence (see Remark I.2). Storing\nthese random bits can be done within the claimed space usage.\nIn the rest of the section, we prove Theorem 4.15.\nProof of Theorem 4.15. The proof is essentially setting up the recursive structure (which we will state with\nthe help of a tree) and to define the overall identification algorithm in the natural recursive way. We then\nverify that all the claimed bounds on the parameters hold. The only non-trivial part is the bound on \u00100for\nthe case when \u001a>0, where we use a careful union bound to obtain the better bound.\nDefine\nh=dlogrlogANe;\nand\nN=rh+1\u00001\nr\u00001=O(logAN):\nTo prove (6), we will prove\n\u001006\u0010\u0001hX\nj=0 \nr\n\u001ar+1\n2!j\n=(\n\u0010\u0001NO\u0010log(1=\u001a)\nlogr\u0011\nif\u001a>0\n\u0010\u0001N otherwise; (10)\nto prove (7), we will prove\np0(N)6hX\nj=0rjp\u0010\nbjp\nN\u0011\n6N\u0001p(A)6N\u0001\u0012A\nk\u0013\u0000\n(\u0010k)\n; (11)\nand to prove (8), we will prove\nm0(N) =g(\u0010;\u0011)\u0001hX\nj=0rj\u0001k\u0001log\u0010\nbjp\nN=k\u0011\n6O\u0012\ng(\u0010;\u0011)\u0001k\u0001log(N=k)\u0001Nlogr\nb\nlogb\u0013\n: (12)\nThe construction. We will present the recursive construction via a tree. (See Figure 1 for an illustration.)\nIn particular, for ease of exposition, we assume that Nis a power of 2. Consider a complete r-ary tree with\nheighth. We will label the tree as follows: associate any node at level 06j6hwith the domain [bjp\nN].\nNote that the leaves are associated with the domain [A]and the tree hasNnodes.\nLetvbe an arbitrary node at level j. Order the routgoing edges in some (fixed) order and associate\ntheuth edge with the uth coordinate of the codewords in Cbjp\nN. In particular, this defines a function\n\u001ew: [N]![bj+1p\nN]for each node won the (j+ 1) th level. Let u1;:::;ujbe the order of the edges\nused from the root to vin the tree. Define \u001ev(i)to be the symbol obtained by successively applying the\ncorresponding code in each of the jlevels in the path from the root to v. In particular,\n\u001ev(i) =Cbj\u00001p\nN\u0010\n\u0001\u0001\u0001Cbp\nN(CN(i)u1)u2\u0001\u0001\u0001\u0011\nuj:\n18For every vertex vat levelj, we associate with it a m(bjp\nN)\u0002NmatrixM(v)as follows. For any i2[N],\ntheith column inM(v)is the same as the \u001ev(i)’th column inMbjp\nN.\nLetM0be the matrix that is the stacking of all the matrices M(v)for every node vin the tree. (The order\ndoes not matter as long as one can quickly figure out the rows of M(v)fromM0.) As the final step, we\nwill randomly rearrange the columns of M0. In particular, let f: [N]![N]be defined as follows: for\neveryi2[N]\u0011f0;1glogN, definef(i)to beiwith each of the logNbits (independently) flipped with\nprobability 1=2. Note that this implies that for every i2[N],f(i)is uniformly distributed in [N].\nFinally, defineM\u0003to be the matrix, where the ith column (for any i2[N]) is thef(i)’th column inM0.\nThe identification algorithm. The algorithm will be described recursively. First we assume that location\niis mapped to f(i)and hence we can now think of M\u0003asM0. We will start with a leaf `(i.e. a vertex\nat levelh). We claim that the matrix M(`)is of the form required in Lemma 4.13. Indeed, since each of\nthe codes used in the construction are uniform, \u001e`maps elements in [N]uniformly at random to [A]. Thus,\ngivenM(`)\u0001x, we use the algorithm in Lemma 4.13 (with S=S`def= [A]) and “send” I`to its parent node.\nNote thatjI`j6O(k=\u0011).\nNow consider a node vat level 06j <h . By induction for each of its rchildren, we will receive subsets\nI1;:::;Ir\u0012[bj+1p\nN]. Note that these form a valid input for list recovery of Cbjp\nN. We run the list recovery\nalgorithm on I1;:::;Irto obtain a subset Sv\u0012[bjp\nN]. Again using the arguments similar to the paragraph\nabove, we can assume we can run the algorithm from Lemma 4.13 to obtain a subset Iv\u0012[bjp\nN]and “send”\nit to its parent.\nIfwis the root of the tree then the final output is Iw. There is one catch: the output Iwis the collection of\nf(i)’s for the appropriate indices i. However, the way fis defined it is not a one to one function and thus,\nwe cannot just apply f\u00001on to the indices in Iw. We will return to this issue in Section 4.5.1.\nCorrectness of the construction. Consider an item i2[N]such thatjxij>3p\n\u0011=kkzk2. Now consider a\nnodevat leveljin the tree and let I1;:::;Ir\u0012[bj+1p\nN]be the sets passed up the rchildren ofvduring the\nidentification process above. Further, assume that for at least \u001arvaluesu2[r],Cbjp\nN(\u001ev(i))u2Iu. Then\nsinceCbjp\nNis(\u001a;O(k=\u0011);L)-list recoverable, we will have \u001ev(i)2Sv. Then when we run the algorithm\nfrom Lemma 4.13 on Sv. Now, two things can happen. Either, \u001ev(i)2Ivor not. In the latter case, we will\nloseibut we will account for such items iin the next part. In the former case, let vbe thej0th child of vertex\nu. Then note that \u001ev(i) =Cbj\u00001p\nN(\u001eu(i))j0and thus, we can apply the argument inductively to u. In other\nwords, ifiis never lost then we will have iin the final output Iw, assuming we can prove the base case. For\nthe base consider any leaf `. Since we picked S`= [A](and hence, trivially \u001e`(i)2S`), if the algorithm\nfrom Lemma 4.13 does not lose i, then\u001e`(i)2I`as desired.\nAnalyzing the number of lost heavy hitters. If\u001a= 0, then it is easy to see that the identification\nalgorithm above can lose at most N\u0001(\u0010k)of the “heavy enough” items. Thus, in this case the bound of\n\u001006N\u0001\u0010is obvious.\nNext, we consider the case when \u001a>0. Definee=\u001ar+ 1. We will argue soon that every heavy item ithat\nis not inIwit has to be the case that there exists a level jsuch that for at least (e=2)jnodesvsuch that\u001ev(i)\nis lost by the algorithm from Lemma 4.13 when it is run on Sv. In such a case we will assign item ito level\nj. Note that in the previous naive argument, we count each iat least (e=2)jtimes at level j. In total, level\n19jcan lose at most rj\u0001\u0010\u0001kitems, which implies that each level can lose at most\u0010\nr\ne\n2\u0011j\n\u0001\u0010\u0001kdistinct items\nat levelj. Summing over all the levels and noting that the total number of lost items is \u00100\u0001k, we obtain the\nfirst inequality in (10). It is easy to verify thatPh\nj=0(r=(e=2))j6(2r=e)O(h), from which the equality in\n(10) follows by substituting the values of hande.\nFinally, we argue that every lost item i62Iwis associated with a level j, where at least (e=2)jinvocations\nof the algorithm from Lemma 4.13 lose i. Towards this end, call a vertex vbad internal node foriif the\ninvocation of the algorithm from Lemma 4.13 does not lose \u001ev(i)but\u001ev(i)62Iv. Call the vertex vbad leaf\nforiif the invocation of the algorithm from Lemma 4.13 itself loses \u001ev(i). Note that is a node vis a bad\ninternal node for ithen at least eof its children are either a bad internal node of a bad leaf for i. In other\nwords, the set of bad internal nodes and bad leaves for iform a treeTiwith degree at least esuch the leaves\nofTiare exactly the bad leaves for i. Finally, note that we want to show for every lost i, there is a level\njwith at least (e=2)jleaves in level j. This now is a simple question on trees that we prove by a simple\ngreedy argument.\nW.l.o.g. assume that every internal node in Tihas degree exactly e. IfTihas one node then we are done. If\nnot, consider the echildren of the root. If at least e=2of these are leaves then we are done. If not, the root\nhas at leaste=2internal nodes. Thus, if we have not stopped for jlevels, we will have at least (e=2)j\u0001e\nchildren at level j+ 1. If at least half of them are leaves then we have at least (e=2)j+1leaves in level j+ 1\notherwise we have at least (e=2)j+1internal nodes. We can continue this argument in the worst-case till\nj=hbut all nodes at level hhave to be leaves, which means that the process will terminate with at least\ne(e=2)h\u00001leaves.\nAnalyzing the failure probability. The first inequality in (11) follows from the union bound (recall that\nthere arerjnodes at level j). The second inequality follows from the fact that p(n)is decreasing in nand\nthe final inequality follows from the definition of p(\u0001).\nAnalyzing the number of measurements. Note thatm0(M) =Ph\nj=0rj\u0001m(bjp\nN), which proves the\nequality in (12). The inequality follows by bounding the sumPh\nj=0(r=b)jas we did earlier.\nAnalyzing the identification time. Finally, in the identification algorithm for each node v(at levelj <h ),\nwe run two algorithms: one is the list recovery algorithm (which takes times T(O(k=\u0011);bjp\nN;b)) and the\nother is the algorithm from Lemma 4.13, which by Corollary 4.14 takes time O(\u0010\u00003\u0011\u00002jSvj\u0001logbjp\nN).\nThe sum in (9) comes from noting that jSj6L. Finally, at level j=h, we pickSv= [A]and hence\nLemma 4.13 and the fact that there are at most Nnodes at level hjustifies the first term in (9).\n4.5.1 Inverting the indices\nWe are done with the proof of Theorem 4.15 except for taking care of the issue that the random function\nf: [N]![N]chosen earlier might not be one to one.\nWe now mention two ways that we can modify the proof above to take care of this. The difference is in the\nfinal identification time we can achieve.\nFor simplicity we will assume that Nis a power of 2. Thus it makes sense to talk about fields FNand\nFN2. We will also go back and forth between the field representation and [N]and[N2]respectively. Let\nd=k(\u0010\u00005\u0011\u00002+ 1) .\n20With \n(N)storage one can solve this problem for anycode with the required properties. See Appendix J\nfor the details.\nNext, we present a specific but cheap solution. For this part, we will make a strong assumption on our list\nrecoverable codes C– we will assume that we are looking at the Loomis-Whitney code (from Lemma 2.5)\nor Reed-Solomon codes (from Lemma 2.6).\nWe pickg: [N]![N2]randomly by picking a random degree dpolynomial over FNand defineg(i)to\nbe the evaluation of this polynomial at i. It is well-known that such a random hash function is (d+ 1) -wise\nindependent (see e.g. [11]). Finally define f(i) = (i;g(i)).\nWe have to tackle three issues.\nFirst, we now have to first compute the matrix M0withN2columns (then we pick the columns of M0\nindexed by (i;g(i))fori2[N]) instead ofNcolumns as we have used so far. This however, is not an issue\nsince we have a family of matrices Mn. This change results in some constants in the parameters of M\u0003\nchanging but it does not affect the asymptotics.\nSecond, note that every f(i)is unique and recovering ifromf(i)can be done in O(1)time. Thus, we will\nnot need any additional space and will add an additive factor of O(k=\u0011)to the identification time.\nFinally, we have to show that this new fis fine with respect to applying Lemma 4.13. In particular, we\nhave to show the following: consider any node vin the recursion tree. Then we will show that for the (new)\n\u001ev: [N]![M]is(d;\u000b)-random (for some small \u000b). By our assumption on C, note that this implies that\nthelogMbits of\u001ev(i)for anyiare from the 2 logNbits from (i;g(i)). Unfortunately, for small M, all\nthese logMbits might all be from the deterministic part (i.e. these are some logMbits from the logNbit\nrepresentation of i), which implies that there will be lot of collisions (in fact N=M indices might collide).\nHowever, the main insight is that when we divide the domain at a node vinto itsrchildren, we have full\nfreedom in how we divide up the bits. (Recall that the code Cjust repeats some subset of message bits as a\ncodeword symbol.) The bad case mentioned above is only true if we do not do this division carefully. Thus,\nthe idea is to do this division carefully so that the “ g(i)” part off(i), which has full randomness, is passed\nalong.\nTo fully implement this, we will need to change ga bit. In particular, pick t=\u00061\n\u000b\u0007\n. Then pick gto be\nrandom degree dpolynomial over FNt\u00001. Now consider a code in the recursive tree structure. W.l.o.g., let\nus assume that Cis an(r;Nt)bp\nNtcode. Letj2[r], then we want to argue that C((i;g(i)))jis pretty much\nrandom.\nWe first consider the case that Cis the Loomis-Whitney code. First think of the message as coming form\n[N]\u0002[Nt\u00001](recall thatf(i) = (i;g(i))and hence is from this domain). Consider a message (i;`). We\npartition it up into dsymbols from [dp\nN]\u0002[dp\nNt\u00001]. I.e. each symbol has the same “proportion” of\nrandomness. In other words, a symbol (i0;`0)in the codeword for (i;`)will havei0to be deterministic and\nthe`0part to be completely random.\nNext we consider the case when Cis the (r;N)bp\nNReed-Solomon codes. First think of the message as\ncoming form [N]\u0002[Nt\u00001](recall thatf(i) = (i;g(i))and hence is from this domain). Consider a message\n(i;`). We partition it up into bsymbolsu0;:::;ub\u000012[bp\nN]\u0002[(bp\nN)t\u00001]. I.e. each symbol has the same\n“proportion” of randomness. Let us associate [bp\nN]with the field Fq, whereq=bp\nN. Recall that we need\nto chooserevaluation points for the code C. Since we have the full freedom in picking these elements, pick\ndistinct\f1;:::;\fr2Fq. (Recall that ris a constant, so this is always possible.) Now recall that the C(i;`)j\nwill be the element\nb\u00001X\ns=0\fs\nj\u0001us:\n21Let us think of usas an element of Ft\nqin the standard notation. I.e. us=Pt\u00001\na=0ua\ns\u0001\ri, where\ris a root of\nan irreducible polynomial of degree toverFq(and the one that defines the standard representation of Fqt).\nThen the above relation can be written as\nC(i;`)j=b\u00001X\ns=0\fs\nj\u0001 t\u00001X\na=0ua\ns\u0001\ra!\n=t\u00001X\na=0\ra\u0001 b\u00001X\ns=0\fs\nj\u0001ua\ns!\n:\nSince`=g(i)was completely random, the above implies that if one thinks of C(i;`)jas an element of\nFq\u0002Fqt\u00001, then the part in Fqt\u00001will be completely random.\nThe above implies that when Cis the Loomis -Whitney or Reed-Solomon code, for any \u001ev(i)has as its\nlast(1\u0000\u000b) logMof its bits to be uniformly random. We will now show that \u001evis(d;\u000b)-random, which\nwill be sufficient to apply Lemma 4.13. Towards that end, let us fix a subset S\u0012[N]of sized. Consider\ni2[N]nS. Consider the values f\u001ev(j)gj2S. Note that we do not have any control over the first \u000blogM\nbits in these values– so in the worst-case let us assume that they (as well as \u001ev(i)) have the same \u000blogM\nbit-vector as its prefix. However since, gisd-wise independent, we have that the (1\u0000\u000b) logMbit suffixes\noff\u001ev(j)gj2S[figare all uniformly random and independent . Thus, even conditioned on the values of \u001ev(j)\nfor everyj2S, the probability that \u001ev(i) =\u001ev(j)for somej2S, is upper bounded by\nd\n2(1\u0000\u000b) logM=d\nM1\u0000\u000b;\nas desired.\nWe will call this scheme above S CHEME -2. Note that we have show that S CHEME -2works and\nLemma 4.16. SCHEME -2addsO(k=\u0011)to the decoding time in Theorem 4.15 and needs O(\u0010\u00005\u0011\u00002\u0001k\u0001\nlog2N)bits of space.\nThe claim on the space comes from the amount of randomness needed to define g(which we need to store).\n4.6 Proof of Lemma 4.4\nIn this section, we do a better analysis of the earlier recursive construction (from the proof of Lemma 4.3)\nassuming the list recoverable code also leads to a limited-wise independent source, which lead us to the\nproof of Lemma 4.4. For our purposes, this code will be the RS code.\nWe begin with the following “inductive step” of the recursive construction.\nLemma 4.17. LetMbe anm\u0002bpnmatrix from Corollary 4.14 that is a (k;\u0010;\u0011 )-weak identification matrix\nwith(`;p)guarantee and identification time T2. LetCbe an (r;n)bpncode that is (\u001a;`;L )-list recoverable\nin timeT2. Assume that the following holds:\np<\u0010\u001a\n2e\u00112\n: (13)\nThen there exists a (rm)\u0002nmatrixM\u0003that is (k;\u00100;\u0011)-weak identification matrix with (L;p0)guarantee\nwith identification time r\u0001T1+T2with\n\u0010062\u0010\n\u001a(14)\nand\np06p\u001ar=4; (15)\nwhere the codewords in Cform ana-wise independent source for some a>\u001ar=2.\n22Proof. The basic idea is to first encode each index i2[n]withCand then use a copy of Mfor each of\ntherpositions in C(i). For identification, we first run the identification algorithm for Mon each of the r\npositions and then combine the result using the list recovery algorithm of C. The main insight in the analysis\nof the algorithm is that the \u0010fraction of the heavy hitters dropped in each of the rarea-wise independent.\nThus, by Chernoff the probability that a heavy hitter is dropped more than \u001artimes is exponentially small.\nNext we present the details.\nThe construction. Forj2[r], letMjdenote them\u0002nmatrix obtained as follows. For every i2[n],\nthei’th column inMjis theC(i)j’th column inM. LetM0be the stacking of the matrices M1;:::;Mr.\nLetf: [n]![n]be a completely random map. Then the final matrix M\u0003is obtained by putting as its i’th\ncolumn thef(i)’th column ofM0.\nIt is easy to verify that M\u0003hasrmrows, as desired.\nThe identification algorithm. We will first assume that the index ihas been mapped to f(i). (Ultimately,\nwe will need to do the inversion. We address this part separately in Section 4.5.1.) Thus, we can assume that\nwe are working with M0. Now if we consider the Mjpart ofM0(for anyj2[r]) and run the identification\nalgorithm forMj(from Corollary 4.14), then we will get a list Sjof the values C(h)jfor all but\u0010kheavy\nhittersh2[n]. We then run the list recovery algorithm for ConS1;:::;Srto obtain our final list of\ncandidate heavy hitters. Note that this algorithm takes time r\u0001T1+T2as desired. Next, we analyze the\ncorrectness of the algorithm.\nCorrectness of the algorithm. Consider a heavy hitter h2[n]. Note that it will not be output iff C(h)j62\nSjfor>\u001ar positionsj2[r]. Let\u00100be the fraction of heavy hitter we miss due to the above condition. Next\nwe bound\u00100and the failure probability p0.\nCall a codeword position j2[r]to be badif the identification algorithm for Mloses more than \u0010kheavy\nhitters when we run the algorithm for Mj. By Lemma C.2, the probability of having >\u001ar= 2bad positions\nis upper bounded by\n\u0012epr\n\u001ar=2\u0013\u001ar=2\n=\u00122ep\n\u001a\u0013\u001ar=2\n6p\u001ar=4;\nwhere we use the fact any a>\u001ar=2positions in a random codeword in Care independent and the inequality\nfollows from (13). We will show next that conditioned on there being at most \u001ar=2bad positions, at most\n\u00100kheavy hitters get lost, where \u00100satisfies (14). Note that this implies that the overall failure probability p0\nis upper bounded by the above probability, which then proves (15).\nTo bound\u00100, call a heavy hitter h2[n]corrupted ifC(h)j62Sjfor at least\u001ar=2of the good positions\nj. Note by the fact that Cis(\u001a;`;L )-list recoverable, if his not corrupted then, it is going to be output by\nour identification algorithm. Thus, we overestimate \u00100by upper bounding the number of corrupted heavy\nhitters. A simple counting argument show that number of such corrupted heavy hitters cannot be more than\n2\u0010k=\u001a , which proves (14).\nRecall that an (r=b=(1\u00002\u001a) + 1;n)bpnRS code by Lemma D.4, is an (\u001a;`;`r)-list recoverable. Further,\nrecall that such a code is b-wise independent. If \u001a < 2=5, then we will have Since b > \u001ar= 2, we have\nsufficient independent to use this code as Cin Lemma 4.17.\nUsing the RS code above with \u001a= 1=4in Lemma 4.17 one can prove Lemma 4.4.\n23Proof Sketch of Lemma 4.4. The proof is very similar to that of Theorem 4.15 and here we just sketch\nhow the current proof differs from the earlier one. As in Theorem 4.15, we start off with fMngn>\u0010\u00002—\na family of m(n)\u0002nmatrices from Corollary 4.14 that are (k;\u0010;\u0011 )-weak identification matrices with\u0010\n`def=O(k=\u0011);p(n)def= (n=k)\u0000\n(\u0010k)\u0011\n-guarantee where m(n) =g(\u0010;\u0011)\u0001k\u0001log(n=k)for some function\ng(\u0010;\u0011).\nHowever, unlike Theorem 4.15, we will pick a specific family of code. In particular, let fCngn>1be the\n(rdef= 2b+ 1;n)bpnReed-Solomon code: we will later pick bto beO(21=\"). Note that Lemma 2.6 implies\nthat this code will be (\u001adef= 1=4;`;`2b+1)-list recoverable in time O(`2b+1b2log2n).\nOther than the specific family of codes, the rest of the construction is exactly the same as in the proof of\nTheorem 4.15 to obtain for large enough Nam0(N)\u0002NmatrixM\u0003that is (k;\u00100;\u0011)-weak identifica-\ntion matrix with (O(k=\u0011);p0(N))-guarantee and identification time complexity D(k;N)as follows. (The\nanalysis is pretty much the same as in the proof of Theorem 4.15 except in the analysis of \u00100andp0in the\n“recursive” step instead of using the arguments in proof of Theorem 4.15, we use Lemma 4.17.)\nIn what follows let A>\n(\u0010\u00006\u0011\u00002\u0001k1\u0000\u000b\n1\u00002\u000b\u0001(log(N=k))2)satisfy the lower bound in Lemma 4.13. There\nexists two integers h6O(logrlogAN)andN= (logAN)such that the following hold:\n\u001006\u0010\u0001\u00122\n\u001a\u0013h\n(16)\np0(N)6\u0012A\nk\u0013\u0000\n(\u0010\u0001k\u0001(\u001a=4)h\u0001rh)\n; (17)\nm0(N)6O\u0012\ng(\u0010;\u0011)\u0001k\u0001log(N=k)\u0001Nlogr\nb\nlogb\u0013\n; (18)\nand\nD(k;N) =O\u0000\n\u0010\u00003\u0011\u00002\u0001N\u0001A\u0001logA\u0001\n+h\u00001X\nj=0\u0010\n(r=b)jO((k=\u0011)rlog2N) +O\u0010\n\u0010\u00003\u0011\u00002\u0001L\u0001logbjp\nN\u0011\u0011\n:\n(19)\nNoting thatrh= logANand that (O(1)=\u001a)h=NO(log(1=\u001a)=logr)along with the choice of A=\u0010\u00006\u0011\u00002\u0001\nkr\u0001log2(N=k)and some simple manipulation implies the following parameters:\n\u001006\u0010\u0001NO\u0010\nlog(1=\u001a)\nlogr\u0011\n(20)\np0(N)6\u0012N\nk\u0013\u0000\n(\u0010k=NO\u0012\nlog(1=\u001a)\nlogr\u0013\n)\n; (21)\nm0(N)6O\u0012\ng(\u0010;\u0011)\u0001k\u0001log(N=k)\u0001Nlogr\nb\nlogb\u0013\n; (22)\nand\nD(k;N) =O\u0000\n\u0010\u00003\u0011\u00002\u0001N\u0001A\u0001logA\u0001\n+\u0010\u00003\u0011\u00002(k=\u0011)r\u0001poly(logN): (23)\nTo obtain the claimed parameter p0one has to apply Lemma 4.10 with s=NO\u0010log(1=\u001a)\nlogr\u0011\n=\u0010. 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On general minimax theorems. Pacific Journal of Mathematics , 8:171–176, 1958.\n[27] N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. In The 37th annual\nAllerton Conference on Communication, Control, and Computing , pages 368–377, 1999.\nA Threshold for failure probability\nWe will prove the following result:\nLemma A.1. If there exists a (k;C)-top level system with mmeasurements and failure probability pthen\nfor every integer g>1, there exists a (k;p\n3\u0001C)-top level system with m\u0001smeasurements and p\n(s)failure\nprobability.\nThe above follows easily from the following result:\nLemma A.2 ([19], Lemma 5.3) .Lety1;:::;ys2Rnbe independent random vectors such that Pr[kyik2<\nD]>1\u0000pfor everyi2[s]. Lety2Rnbe obtained by taking the component-wise median of fy1;:::;ys.\nThen\nPr[kyk<p\n3\u0001D]>1\u0000(4ep)s=4:\n26In the rest of the section we prove Lemma A.1 using Lemma A.2 in the obvious way. Let M;Abe the given\ntop level system. The new top level system M0;A0is defined as follows. M0is justsindependent copies of\nM(call the copiesM1;:::;Ms). The new decoding algorithm is defined as follows: let z1;:::;zsbe the\noutput ofAon thescopies ofM, i.e.zi=A(Mix). ThenA0(M0x)is defined to be the component wise\nmedian of z1;:::;zs. Lemma A.1 follows by applying Lemma A.2 with yi=x\u0000ziandD=C\u0001kx\u0000xkk2,\nwhere xis the original signal.\nB Expander Basics\nWe will need the following three simple lemmas.\nLemma B.1. LetG: [N]\u0002[`]![M]be a(2t;\")-expander. Let S;T\u0012[N]such thatjTj=t,jSj>t\nandS\\T=;, then\nj\u0000(S)\\\u0000(T)j6jE(\u0000(S)\\\u0000(T))j64jSj`\":\nProof. The first inequality is obvious so we prove the second inequality. Divide Sintob=l\njSj\ntm\nmany\ndisjoint subsets S1;:::;Sbwhere allSi’s but possibly Sbhave size exactly t. (For what follows, by adding\nextra elements to Sif necessary, we will assume that jSbj=t.) Note thatb62jSj=tasjSj>t.\nFix ani2[b]and note that as Gis a(2t;\")expander, we have\nj\u0000(T[Si)j>2t`(1\u0000\"):\nAsj\u0000(Si)j6t`, the above implies that\nj\u0000(T)n\u0000(Si)j>t`(1\u00002\"):\nThe above in turn implies that\njE(\u0000(T)n\u0000(Si))j>j\u0000(T)n\u0000(Si)j>t`(1\u00002\"):\nSincejE(T)j=t`, the above implies that\njE(\u0000(T)\\\u0000(Si))j6t`\u0000t`(1\u00002\") = 2\"t`:\nFinally, asSis the disjoint union of S1;:::Sb, we have\njE(\u0000(T)\\\u0000(S))j62b\"t`64\"jSj`;\nas desired, where in the last inequality we have used the fact that bt62jSj.\nLemma B.2. Leta>2be an integer and let G: [N]\u0002[`]![M]be a(t;\" < 1=(2a))-expander. Let\nR\u0012[M]be any subset of size jRj6\rt`for some\r >0. Then there are <2a\rtelementsi2[N]such\nthatj\u0000(i)\\Rj>`=a.\nProof. For the sake of contradiction, assume that there exists a subset L\u0012[N]withjLj= 2a\rtsuch that\nfor everyi2L,j\u0000(i)\\Rj>`=a . Consider the neighbors \u0000(L)nR. Note that by assumption each i2L\nhas at most (1\u00001=a)\u0001`such neighbors. Thus, we have\nj\u0000(L)j62a(1\u00001=a)\rt`+jRj62a(1\u00001=a)\rt`+\rt`=\u0012\n1\u00001\n2a\u0013\n\u0001jLj\u0001`<(1\u0000\")\u0001jLj\u0001`;\nwhich contradicts the assumption that Gis a(t;\"< 1=(2a))-expander.\n27The following is a by-product of the well-known “unique neighborhood” argument (it also follows from the\nproof of Lemma B.1):\nLemma B.3. LetG: [N]\u0002[`]![M]be a(t;\")-expander. Then for any subset S\u0012[N]withjSj6t, we\nhave that at most 2\"jSj`verticesj2[M]each of which has more then one neighbor in S. (A vertex in [M]\nadjacent to exactly one vertex in Sis called a “unique neighbor” of S.)\nC Probability Basics\nGiven any two vectors x;y2Rn, we will denote its dot-product by hx;yi=Pn\ni=1xi\u0001yi.\nWe will use the following well-known result:\nLemma C.1. Letx= (x1;:::;xn)2Rnandr= (r1;:::;rn)2f\u0000 1;1gnbe a pair-wise independent\nrandom vector. Then\nPr\nr[jhx;rij>a\u0001kxk2]61\na2:\nWe will also use the following form of the Chernoff bound:\nLemma C.2. LetX1;:::;Xnbe random iid binary random variables where p= Pr[Xi= 1]. Then\nPr\"nX\ni=1Xi>a#\n6\u0010epn\na\u0011a\n;\nfor anya>pn . The above bound also holds if X1;:::;Xnarea-wise independent. Further, if the random\nvariables are k<a -wise independent, then the upper bound is (epn=a )k.12\nD Coding Basics\nIn this section, we define and instantiate some (families) of codes that we will be interested in. We begin\nwith some basic coding definitions.\nWe will call a code C: [N]![q]rbe a(r;N)q-code.13. Vectors in the range of Care called its codewords.\nSometimes we will think of C\u0012[q]r, defined the the natural way.\nWe will primarily be interested in list recoverable codes. In particular,\nDefinition D.1. LetN;q;r;`;L>1be integers and 06\u001a61be a real number. Then an (r;N)qcodeC\nis called a (\u001a;`;L )-list recoverable if the following holds. Given any collection of subsets S1;:::;Sr\u0012[q]\nsuch thatjSij6`for everyi2[r], there exists at most Lcodewords (c1;:::;cr)2Csuch thatci2Sifor\nat least (1\u0000\u001a)nindicesi2[r]. Further, we will call such a code recoverable in time T(`;N;q )if all such\ncodewords can be computed within this time upper bound.\nFurther, we will call an (r;N)qcode to be uniform , if for every i2[r]it is the case that C(x)iis uniformly\ndistributed over [q]for uniformly random x2[N]. In our construction we will require codes that are both\n12The statement of the Chernoff bound is pretty standard. The claim on the independence, follows e.g. from Lemma 3 and\nTheorem 2 in [25].\n13We depart from the standard convention and use the size of the code Ninstead of its dimension logqN: this makes expressions\nsimpler later on.\n28list recoverable and uniform. Neither of these concepts are new but our construction needs us to focus on\nparameter regime that is generally not the object of study in coding theory. In particular, as in coding theory,\nwe focus on the case where Nis increasing. Also we focus on the case when qgrows withN, which is\nalso a well studied regime. However, we consider the case when ris afixed . In particular, our “ideal” code\nshould have the following properties:\n1.qshould be as small as possible.\n2.rshould be as small as possible.\n3.Lshould be as close to `as possible.\n4.\u001ashould be as large as possible.\n5.T(`;N;q )should has poly-logarithmic dependence on both Nandq(and at the same time have as\nclose to linear dependence on `as possible).\nNext, we present three codes that we will use in our constructions (that achieve some of the properties above\nbut not all of the above).\nSplit code. In this case we assume that Nis both a power of 2and a perfect square. Given a x2[N],\nthink of it as a bit vector of length logN. Letx1(x2resp.) be the first logN=2(lastlogN=2resp.) bits\nofx. Then define Csplit(x) = (x1;x2). This is of course a trivial code and we record its properties below\nfor future use. The list recovery algorithm for this code is very simple: given S1andS2as the input, output\nS1\u0002S2.\nLemma D.2. The codeCsplitis a uniform code that is (0;`;`2)-list recoverable code. Further, it is recov-\nerable inO(`2logN)time.\nLoomis-Whitney code. We now consider a code based on the well-known Loomis-Whitney inequality.\nLetd>2be an integer and assume that Nis a power of 2anddp\nNis an integer. Given x2[N]think of it\nas(x1;:::;xd)2[dp\nN]d. Further, for any i2[d]definex\u0000i= (x1;:::;xi\u00001;xi+1;:::;xd)2[dp\nN]d\u00001.\nThen define CLW(d)(x) = (x\u00001;:::;x\u0000d). Note that for d= 2, we getCsplit. The Loomis-Whitney\ninequality shows that CLW(d)is a(0;`;`d=(d\u00001))-list recoverable code. We also show how to algorithmically\nachieve this bound in time ~O(`d=(d\u00001)). This implies the following:\nLemma D.3. The codeCLW(d)is a uniform code that is (0;`;`d=(d\u00001))-list recoverable code. Further, it is\nrecoverable in O(`d=(d\u00001)logN)time.\nThe above follows from Theorem E.1.\nReed-Solomon code. Finally, we consider the well-known Reed-Solomon (RS) codes. In particular, let\nb>1be an integer and let q=bp\nNbe a prime power and consider CRS: [N]![bp\nN]r. There are\nknown results on list recovery of Reed-Solomon codes but they need b=r=O(1=`), which is too weak for\nour purposes. Instead we first note that we can always first output S1\u0002S2\u0002\u0001\u0001\u0001\u0002Srand then for each\nvector check whether it is within Hamming distance of \u001anof some RS codeword. E.g., one can use the\nwell-known Berlekamp Massey algorithm that does unique decoding for \u001a <1=2(1\u0000b=r).14Further, it\n14One could potentially use list decoding to recover from even more errors but that does not seem to buy much for our application.\n29is well-known that RS codes are linear codes and it is well-known that linear codes are also uniform. This\nimplies the following:\nLemma D.4. Let\u001a<1=2(1\u0000b=r). Then the code CRSis a uniform code that is (\u001a;`;`r)-list recoverable\ncode. Further, it is recoverable in O(`rr2log2N)time.15\nE Constructive Proof of Loomis Whitney Inequality\nE.1 Notations\nWe begin with some notations. Let \u0006denote an arbitrary discrete “alphabet.” We will consider subsets\nS\u0012\u0006d. For any subset T\u0012[d], we will use STto denote the vectors in Sprojected down to T. Further,\nfor any integer 06i6d, we will use\u0000[d]\ni\u0001\nto denote the set of all subsets of dof size exactly i.\nNext, we define the “join” operators. Given any two subsets T1;T2\u0012[d], and two sets of vectors V1\u0012\u0006T1\nandV2\u0012\u0006T2and any subset G\u0012\u0006T1\\T2, we define the join ofV1andV2overG, denoted by V1o nGV2,\nto be the set of vectors (u1;a;u 2)in\u0006T1[T2\u0011\u0006T1nT2\u0002\u0006T1\\T2\u0002\u0006T2nT1, wherea2G,(u1;a)2V1and\n(a;u2)2V2. The join ofV1andV2(withoutGas an anchor), denoted by V1o nV2, is the set of vectors\n(u1;a;u 2)in\u0006T1[T2\u0011\u0006T1nT2\u0002\u0006T1\\T2\u0002\u0006T2nT1,(u1;a)2V1and(a;u2)2V2. In particular, when\nT1\\T2=;,V1o nV2is simplyV1\u0002V2whose coordinates are indexed by T1[T2.\nE.2 Projections of size d\u00001\nIn this subsection, we will consider the case when the projections are over [d]nfigfor everyi2[d]. We\nwill prove the following result:\nTheorem E.1. Letd>1be an integer. For each i2[d], let \u0016Si\u001a\u0006[d]nfigbe given finite sets where\nj\u0016Sij=kifor some integer ki. LetS\u0012\u0006[d]be the set of vectors such that S[d]nfig\u0012\u0016Sifor everyi2[d].\nThen,\njSj6(d\u00001)\u0001d\u00001sY\ni2[d]ki: (24)\nFurthermore, the “join” Scan be computed from the inputs fSigi2[d]in time ~O\u0010\nd\u0001d\u00001qQ\ni2[d]ki+P\ni2[d]ki\u0011\n.\nE.2.1 Facts about certain labeled trees\nLetTbe a binary tree. For every internal node v2T, let its left child be denoted by vLand its right child\nbe denoted by vR. Further, the subtree rooted at any node v2T will be denoted by T(v). Finally, letL(T)\ndenote the set of leaves in T.\nTo prove Theorem E.1, we will consider the following labeled trees. Given d>2, consider any binary tree\nTwithdleaves where each node v2T is labeled with a subset C(v)\u001a[d]. Without loss of generality,\nwe use the numbers f1;2;\u0001\u0001\u0001;dgto index the leaves L(T), i.e. each`2[d]is identified with a unique leaf\nnode ofT. The labeling is done as follows.\n\u000fEach leaf`2L(T) = [d]is labeled with the set C(`) = [d]nf`g.\n15Ther2log2Nfactor follows from the fact that the Berlekamp Massey algorithm needs O(r2)operation over Fq, each of which\ntakesO(log2q)time.\n30\u000fEach internal node vhas labelC(v) =C(vL)\\C(vR).\nWe record the following simple properties of such labeled trees:\nLemma E.2. Letd>2be an integer and let Tbe a binary tree with dleaves labeled as above. Then the\nfollowing are true:\n(i) For any internal node v,C(vL)[C(vR) = [d]; and\n(ii) For the root rofT,C(r) =;.\nProof. By induction, it is easy to see that for any node v,\nC(v) =\\\n`2L(T(v))C(`): (25)\nThe above immediately implies (ii)asT\n`2[d]([d]nf`g) =;. For any leaf `recall thatC(`) = [d]n`, which\nalong with (25) imply that C(v) = [d]nL(T(v)). This along with the fact that L(T(vL))\\L(T(vR)) =;\nimply (i).\nE.2.2 Proof of Theorem E.1\nFor notational convenience define\nP=d\u00001sY\ni2[d]ki:\nWe prove both parts by presenting an algorithm to compute Sfrom its potential projections \u0016Si,i2[d]. Let\nTbe an arbitrary labeled binary node with dleaves as described in the last subsection. Every node vwill be\nassociated with its labeled subset C(v)as well as two auxiliary sets \t(v)\u0012\u0006[d]and\u0005(v)\u0012\u0006C(v). The\nset\t(v)is supposed to contain the candidate vectors some of which will be members of the final output S.\nThe set \u0005(v)will be a superset of the projection (Sn\t(v))C(v).\nFor each leaf `2L(T), define \t(`) =;and\u0005(`) =\u0016S`. We next describe how to algorithmically compute\nthe sets \t(v)and\u0005(v)for internal nodes vrecursively from the leaves up to the root.\nLetvbe any internal node in Twhose left and right children’s auxiliary sets have already been com-\nputed. Without loss of generality, assume that \u0005(vL)C(v)= \u0005(vR)C(v). (If not, we can simply re-\nmove the elements from \u0005(vL)and\u0005(vR)whose projections on to C(v)lie in the symmetric difference\n\u0005(vL)C(v)4\u0005(vR)C(v).)\nWhenvisnotthe root, partition \u0005(vL)C(v)into two sets GandBsuch that for every vector uin the “good”\nsetG, the number of vectors in \u0005(vL)whose projection onto C(v)isuis upper bounded byl\nP\nj\u0005(vR)jm\n\u00001.\nThe “bad” set Bis\u0005(vL)C(v)nG. Finally, compute\n\t(v) = (\u0005(vR)o nG\u0005(vL))[\t(vL)[\t(vR)\n\u0005(v) =B:\nWhenvisthe root, we compute \t(v) = (\u0005(vR)o n\u0005(vL))[\t(vL)[\t(vR)and\u0005(v) =;. By induction\non each step of the algorithm, we will show that the following three properties hold for every node v2T:\n1.(Sn\t(v))C(v)\u0012\u0005(v);\n312.j\t(v)j6(jL(T(v))j\u00001)\u0001P; and\n3.j\u0005(v)j6min\u0010\nmin`2L(T(v))k`;Q\n`2L(T(v))k`\nPjL(T(v))j\u00001\u0011\n.\nAssuming the above are true, we first complete the proof of the theorem. Let rdenote the root of the tree T.\nBy property 1, (Sn\t(rL))C(rL)\u0012\u0005(rL)and(Sn\t(rR))C(rR)\u0012\u0005(rR). Also recall that by Lemma E.2\nC(rL)[C(rR) = [d]andC(rL)\\C(rR) =;. Hence,\nSn(\t(rL)[\t(rR))\u0012\u0005(rL)\u0002\u0005(rR) = \u0005(rL)o n\u0005(rR):\nThis implies S\u0012\t(r). Thus, from \t(r)we can compute Sby keeping only vectors in \t(r)whose\nprojection on any subset L2\u0000[d]\nd\u00001\u0001\nis contained in SL. In particular,jSj6\t(r)6(d\u00001)P, proving (24).\nFor the run time complexity of the above algorithm, we claim that for every node v, we need time ~O(j\t(v)j+\nj\u0005(v)j). To see this note that for each node v, we need to do the following:\n(i) Make sure \u0005(vL)C(v)= \u0005(vR)C(v),\n(ii) Compute Gfrom \u0005(vL),\n(iii) Compute \t(v) = \u0005(vR)o nG\u0005(vL)[\t(vL)[\t(vR)and\u0005(v) =B.\nIt can be verified that all of these steps can be computed in time near-linear in the size of the largest set\ninvolved (after sorting the sets all the required computation can be done with a linear scan of the input lists),\nwhich along with property 3 leads to a (loose) upper bound of ~O(P+ min`2L(T(v))k`)on the run time for\nnodev. Summing the run time over all the nodes in the tree gives the claimed run time.\nTo complete the proof we argue that properties 1-3 hold. For the base case, consider `2L(T). Recall that\nin this case \t(`) =;and\u0005(`) =\u0016S`. It can be verified that for this case, properties 1-3 hold.\nNow assume that properties 1-3 hold for all children of an internal node v. We first verify properties 2-3 for\nv. From the definition of G,\nj\u0005(vR)o nG\u0005(vL)j6\u0012\u0018P\nj\u0005(vR)j\u0019\n\u00001\u0013\n\u0001j\u0005(vR)j6P:\nFrom the inductive upper bounds on \t(vL)and\t(vR), property 2 holds at v. By definition of Gand an\naveraging argument, note that\njBj=j\u0005(v)j6j\u0005(vL)j\u00011\ndP=j\u0005(vR)je6j\u0005(vL)j\u0001j\u0005(vR)j\nP:\nFrom the induction hypotheses on vLandvR, we havej\u0005(vL)j6Q\n`2L(T(vL))k`\nPjL(T(vL))j\u00001andj\u0005(vR)j6Q\n`2L(T(vR))k`\nPjL(T(vR))j\u00001,\nwhich implies that j\u0005(v)j6Q\n`2L(T(v))k`\nPjL(T(v))j\u00001. Further, it is easy to see that j\u0005(v)j6min(j\u0005(vL)j;j\u0005(vR)j),\nwhich by induction implies that j\u0005(v)j6min`2L(T(v))k`. Property 3 is thus verified.\nFinally, we verify property 1. By induction, we have (Sn\t(vL))C(vL)\u0012\u0005(vL)and(Sn\t(vR))C(vR)\u0012\n\u0005(vR). This along with the fact that C(vL)\\C(vR) =C(v)implies that (Sn\t(vL)[\t(vR))C(v)\u0012\n\u0005(vL)C(v)\\\u0005(vR)C(v)=B]G. Further, every vector in (Sn\t(vL)[\t(vR))whose projection onto\nC(v)is inGalso belongs to \u0005(vR)o nG\u0005(vL). This implies that (Sn\t(v))C(v)=B= \u0005(v), as desired.\n32E.3 Error-Tolerant Constructive Loomis-Whitney Inequality\nIn this section, we prove the following version of the Loomis-Whitney inequality that can handle certain\n“errors,” i.e. we are interested in the set Ssuch that a few projections need not lie in the given input\nprojection sets. In particular, we will prove the following:\nTheorem E.3. Letd>2and06e6d\u00002be integers. Let \u0016Si\u0012\u0006[d]nfigbe sets of vectors such that for\neveryi2[d],j\u0016Sij=kifor some positive integer ki. LetS\u0012\u0006dbe the largest set such that for every vector\n(v1;:::;vd)2S, there are at least n\u0000evalues ofi2[d]for which (v1;:::;vi\u00001;vi+1;:::;vd)2\u0016Si.\nThen,\njSj6eX\ni=0X\nB2([d]\ni)(d\u0000i\u00001)\u00010\n@Y\nj2[d]nBkj1\nA1\nd\u0000i\u00001\n: (26)\nFurtherScan be computed from the projections f\u0016Sigi2[d]in time\n~O0\nB@eX\ni=0X\nB2([d]\ni)(d\u0000i\u00001)0\n@Y\nj2[d]nBkj1\nA1\nd\u0000i\u00001\n+eX\n`=1\u0012d\u00001\n`\u00001\u0013X\nj2[d]kj1\nCA:\nProof. For each potential “error set” B\u001a[d]with the number of errors jBj6e, we apply the algorithm\nfrom Theorem E.1 to join all \u0016S`, for`2[d]nB. The algorithm is identical to that of Theorem E.1, except\nfor two facts. First, the tree Tnow only has d\u0000jBjleaves, each identified by a member `2[d]nB. Each\nleaf`has labelC(`) = [d]nf`gas before. Second, the product Pis now defined to be\nP=0\n@Y\nj2[d]nBkj1\nA1\nd\u0000jBj\u00001\n:\nNote also that the label C(r)of the rootrofTis no longer the emptyset; however, this fact does not change\nthe analysis one bit.\nF Omitted Material from Section 3\nNote that P=P2=PTPbecause any orthogonal projection matrix is symmetric and idempotent. Hence,\nfor any two vectors x;y2RN,\nhPx;yi=hx;Pyi=hx;PTPyi=hPx;Pyi:\nIn particular,hPej;eji=kPejk2\n2for anyj2[N]. The following was proved in [6]. We provide here a\nvery short proof.\nProposition F.1. Let\bbe an arbitrary real matrix of dimension m\u0002N. Then, there exists j\u00032[N]such\nthat\nkPej\u0003k2\n2=hPej\u0003;ej\u0003i>1\u0000m=N:\nProof. SincehPej;ejiis precisely the jth diagonal entry of the matrix P, we have trace (P) =PN\nj=1hPej;eji.\nBut the trace of an orthoprojector is the dimension of the target space which is N\u0000min this case. Hence,\nN\u0000m=PN\nj=1hPej;eji, which completes the proof.\n33F.1 The forall case\nCorollary F.2 (Cohen-Dahmen-DeV ore [6]) .Let\bbe anm\u0002N `2=`2“forall” sparse recovery measure-\nment matrix with k>1, i.e. there exists a decoding algorithm Aand a constant C>1such that for any\ninput signal x2RN,\nkx\u0000A(\bx)k26C\u0001kx\u0000xkk2 (27)\nthen it must be the case that m>N=C2.\nProof. Lety=A(\b0) =A(0). Then, By applying (27) with x=0, it is easy to see that A(0) =A(\b0) =\n0. Next, from Proposition F.1 there exists j\u00032[N]such thathPej\u0003;ej\u0003i>1\u0000m=N . Letx=Pej\u0003\nkPej\u0003k2\nthenkxk2= 1andhx;ej\u0003i2>kPej\u0003k2\n2. Moreover, A(\bx) =A(0) =0because Pej\u0003is in the null space\nof\b. Consequently, from (27) we obtain\n1 =kxk2\n26C2\u0001kx\u0000xkk2\n26C2\u00010\n@X\nj6=j\u0003x2\nj1\nA=C2\u0001\u0000\n1\u0000hx;ej\u0003i2\u0001\n=C2\u0001\u0000\n1\u0000kPej\u0003k2\n2\u0001\n6C2\u0001m=N;\nwhich is the desired result.\nF.2 Proof of Theorem 3.4\nTo complete our lowerbound proof, we need a “continuous” version of Yao’s minimax principle.\nDefinition F.3. A sparse recovery system S= (\b;A)is defined to be a pair consisting of a measurement\nmatrix \b2Rm\u0002Nand a mapping A:Rm!RN. We assume that the mapping algorithm Ahas a finite\ndescription length, is deterministic, and is the best mapping for the matrix \b.\nWe consider sparse recovery systems (\b;A)in a compact set Y(that is, our matrices \bare in a compact\nset inRm\u0002N). LetRbe a probability measure on sparse recovery systems in Y, by which we mean R\nspecifies a distribution on measurement matrices \b(and the mapping Ais the best possible deterministic,\nfinite description length mapping for that distribution on \b). LetYbe a compact set of convex combinations\nof probability measures Ron sparse recovery systems in Y.\nLet us assume that our input vectors x2Xand thatXis a compact subset of RN. LetDbe a probability\nmeasure on input vectors in Xand letXbe a convex set of all such measures.\nDefinition F.4. We say that a sparse recovery system S= (\b;A)decodes xcorrectly if\nkx\u0000A(\bx)k26Ckx\u0000xkk2:\nWe define the cost of a sparse recovery system Son input xas\ncost(x;S) =(\n1ifAdoes not decode xcorrectly\n0otherwise:\nThus, the failure probability of a randomized sparse recovery system Son input xis\ncost(x;R) =ES\u0018R\u0010\ncost(x;S)\u0011\n:\n34Definition F.5. LetD2X andR2Y and define\nf(D;R) = cost(D;R) =Ex\u0018D \nES\u0018R\u0010\ncost(x;S)\u0011!\n=ES\u0018R \nEx\u0018D\u0010\ncost(x;S)\u0011!\n:\nObserve that fis bi-linear and always finite (hence, it’s a proper convex and concave function). Furthermore,\nwe can change the order of expectation by Fubini’s theorem.\nLemma F.6 (Continuous Yao’s Lemma) .With the compact, convex sets XandYand the function f:\nX\u0002Y! Rdefined above,\nmax\nD2Xmin\nS2Ycost(D;S) = min\nR2Ymax\nx2Xcost(x;R):\nProof. First, we observe that the hypotheses of the theorem match those of Sion’s Minimax Theorem [26];\nhence, we immediately have\nmax\nD2Xmin\nR2Ycost(D;R) = min\nR2Ymax\nD2Xcost(D;R):\n(in particular, we have used the fact that XandYare compact to ensure that the suprema and infima are\nattained.)\nTo finish the proof, we argue that for all distributions on recovery systems R02Y,\nmax\nDcost(D;R0) = max\nDEx\u0018DES\u0018R0\u0010\ncost(x;S)\u0011\n= max\nx2XES\u0018R0\u0010\ncost(x;S)\u0011\nand that for all distributions on inputs D02X,\nmin\nRcost(D0;R) = min\nRES\u0018REx\u0018D0\u0010\ncost(x;S)\u0011\n= min\nS2YEx\u0018D0\u0010\ncost(x;S)\u0011\nfrom the convexity of XandYand the compactness of XandY.\nSo, if we find some distribution D0on inputs for which the best sparse recovery system has failure probability\nat leastp(i.e. high cost), then we have established a lower bound on the failure probability for randomized\nrecovery systems:\np6min\nR2Ycost(D0;R)\n6max\nD2Xmin\nR2Ycost(D;R)\n= max\nD2Xmin\nS2Ycost(D;S)\n= min\nR2Ymax\nD2Xcost(D;R)\n= min\nR2Ymax\nx2Xcost(x;R):\nIn Lemma 3.3, we exhibited a hard distribution on input vectors xfor which the best sparse recovery system\nhas failure probability at leastp\n1=\r\u0001e\u0000N\n2ln(2=\r), given that\rand\u000e=m=N satisfy (3). It is not hard to\nsee that\r=\u000e=1\n12+16C2satisfy (3). Hence, any foreach sparse recovery system with failure probability at\nmostp=p\n12 + 16C2\u0001e\u0000ln(6+8C2)\n2\u0001Nmust have at least m>\u000eN=N\n12+16C2measurements. In particular,\nwe have shown that for failure probability 2\u0000\u0002(N), the number of measurements is \n(N).\n35We next give the simple reduction to handle the case of larger failure probability p >p\n12 + 16C2\u0001\ne\u0000ln(6+8C2)\n2\u0001N. DefineN0such thatp=p\n12 + 16C2\u0001e\u0000ln(6+8C2)\n2\u0001N0, i.e.\nN0=2\nln(6 + 8C2)ln p\n12 + 16C2\np!\n= \u0002(log(1=p)):\nIn this case for the hard distribution, we zero out the last N\u0000N0entries in the input vectors and then apply\nthe hard distribution on the first N0coordinates. This with the previous result implies for failure probability\nat mostp, we needm=\u000eN0= \n(log(1=p))measurements, as desired. We just proved Theorem 3.4.\nG Omitted Material from Section 4\nG.1 Known Results\nThe following result from Cohen, Dahmen, DeV ore [5] establishes a tight upper bound on the number of\nmeasurements with a polynomial time decoding algorithm for the foreach sparse recovery problem. The\nalgorithmAis Orthogonal Matching Pursuit (OMP) which runs in time O(MNk ).\nTheorem G.1. There is a distribution on M\u0002Nmatrices \bsuch that for each x2RN, the output of OMP\nafter 2kiterations satisfies\nkx\u0000A(\bx)k26Ckx\u0000xkk2\nwith probability larger than 1\u0000pprovided that M>C(klog(N=k) + log(1=p)).\nCohen, et al. provide three examples of such distributions on matrices: (i) iid Gaussian random entries\n\bi;jwith variance 1=M, (ii) iid Bernoulli random entries \bi;jwith values\u00061=p\nM, and (iii) columns of \b\ndrawn from a uniform distribution on SM\u00001.\nG.2 Proof of Lemma 4.10\nProof. We will use a simple repetition trick. M0is justscopies ofM, where each copy gets fresh random\nbits. The decoding algorithm is as follows: given the outputs I1;:::;Isfrom thescopies ofM, output an\ni2[s\nj=1Ijif it appears in >s=2Ij’s. Next we argue that the claimed bounds hold.\nFirst note that since j[s\nj=1Ijj6s`and eachithat is output appear in > s= 2intermediate outputs Ij, we\ncan only output <2`such indices. Next note that by Lemma C.2, except with probability p\n(s), at most\ns=4outputsIjmiss more than \u0010kelements from Hk(x). Call the remaining (at least 3s=4) intermediate\noutputs to be good . Note that we will not lose an iif it appears in >s=2goodIj’s. Then a simple counting\nargument implies that we can have at most 3\u0010kthat are missing form at least s=4goodIj’s.\nH Proof of Theorem 4.7\nLetu= (u1;:::;uM) =Mx. For anyi2[N], define it’s estimate \u0016xito the median of the values\nfMi;j\u0001ujgj2\u0000(i). We will show that\nLemma H.1. The following holds with probability 1\u0000\u0000N\n\rk\u0001\u0000s. Except for \rkpositionsi2[N], every other\nindex will have a good estimate: i.e.,\njxi\u0000\u0016xij6p\n\u0011=k\u0001kzk2:\n36In this section, we prove Theorem 4.7 using Lemma H.1. The proof is almost identical to the similar one\nin [22] (except ^xhas a larger support size).\nLet\u0016x= (\u0016x1;:::; \u0016xN)and define ^xto be \u0016xwith all but the top k0=k+k=p\u0011entries (by their absolute\nvalues) zeroed out. Further, let Tbe the set of items in [N]that do not have a good estimate (i.e. jxi\u0000\u0016xij>p\n\u0011=k\u0001kzk2). Note that by Lemma H.1, jTj6\rk.\nTo complete the proof of Theorem 4.7, we prove the following:\nLemma H.2. There exists a vector ^ywithjsupp( ^y)j62\rksuch that for ^z=x\u0000^x\u0000^y, we have\nk^zk2\n26(1 + 22p\u0011)\u0001kzk2\n2: (28)\nProof. We’ll prove the above by case analysis and adding up the contribution of different indices to k^zk2\n2\n(and also define the elements in ^yin the process):\n1. (i2supp( ^x)with a good estimate) In this case, each such item contributes\u0011\nk\u0001kzk2\n2tok^zk2\n2. Since\njsupp( ^x)j=k(1 + 1=p\u0011), items in this case contribute at most 2p\u0011\u0001kzk2\n2. (Set ^yi= 0.)\n2. (i2supp( ^x)with a bad estimate) These items do not contribute anything. (Set ^yi=xi\u0000^xi.)\n3. (i2supp( z)nsupp( ^x)) These contribute at most kzk2\n2. (Set ^yi= 0.)\n4. (i2Hk0(x)nsupp( ^x)with a good estimate is displaced by an item i02supp( ^x)nHk0(x)with a good\nestimate) In this case set ^yi= ^yi0= 0. Thus, in this case, we have ^xi= 0and^zi=xi;^zi0=xi0\u0000^xi0.\nNow note that by definitions of iandi0, we have\njxi0j>j^xi0j\u0000p\n\u0011=kkzk2>j^xij\u0000p\n\u0011=kkzk2>jxij\u00002p\n\u0011=kkzk2:\nNext, note that any item i00such thatjxi00j>p\n\u0011=kkzk2would satisfy i002Hk0(x). This in turn im-\nplies thatjxi0j6p\n\u0011=kkzk2. This along with the above inequalities implies that jxij63p\n\u0011=kkzk2.\nThis in turn implies that\n^z2\ni+ ^z2\ni069\u0011\nk\u0001kzk2\n2+\u0011\nk\u0001kzk2\n2=10\u0011\nk\u0001kzk2\n2:\nSince there are at most k+k=p\u0011such pairs (i;i0), the total contribution from this case is at most\n20p\u0011kzk2\n2.\n5. (i2Hk0(x)nsupp( ^x)with a bad estimate or is displaced by an item with a bad estimate) These do\nnot contribute anything. (Set ^yi=xi.)\nIt can be verified that the following three things holds: (i) x=^x+^y+^z. (ii) Since every item with bad\nestimate can contribute at most two non-zero items to ^y(once in item 2:and once in item 5:), we have that\njsupp( ^y)j62\rk. (iii) Finally, by items 1;3and4, (28) is satisfied.\nSome Remarks. We first note that it is possible to compute ^xefficiently in time near linear in N(as it\ninvolves computing Nmedian values and then outputting the top k+k=p\u0011values).\nThe argument in item 4 in the proof above also implies the following:\nCorollary H.3. supp( ^x)contains all but \rkitemsi2[N]that satisfyjxij>3p\n\u0011=k\u0001kzk2.\nIn particular, consider the following algorithm.\n370. The input is the vector Mxand a subset S\u0012[N].\n1. For each j2S, compute the estimate \u0016xjas the median of the values fj(Mx)bjgb2\u0000(j).\n2. Output in Ithe itemsj2Swith the top k+k=\u0011estimates \u0016xj.\nThe above along with Corollary H.3 (where we substitute \u0011by\u00112) implies Theorem 4.8.\nFinally, the proof of of Theorem 4.7 also implies Theorem 4.9.\nProof Sketch of Theorem 4.9. The proof is pretty much mimics the proof of Theorem 4.7 except the follow-\ning two small changes. First, we adjust the constants so that Lemma H.1 implies that the median estimate\nis correct for all but \u0010k=2elementsi\u0010[N], we have that \u0016xiis a good estimate of xi. Finally, in the argument\nof proof of Theorem 4.7, we also have to take into account the at most \u0010k=2elements of Hk+k=\u0011(x)that\nare missing from S. (Note that the algorithm to compute ^xis the one we used to prove Theorem 4.8– just\noutput the top k+k=\u0011median estimates for the elements in S.)\nH.1 Proof of Lemma H.1\nWe begin with some definitions and notation that will be useful in the proof. Let \u0010 >0be a real that we will\nfix later (to be \u0002(\r)) and let\"=\u00103\u0011.\nWe will call each element j2[M]abucket .\u0000(i)for somei2[N]will be the set of i’s buckets. Finally,\nwe will call a bucket j2\u0000(i)badfor indexi2[N]if at least one of the following conditions hold:\n\u000f(BAD-1)j2\u0000(h)for some heavy hitter h6=i.\n\u000f(BAD-2)j2\u0000(h)for some heavy tail element h6=i.\n\u000f(BAD-3) The`2\n2contribution of all light tail elements (other than i) tojis>\u0010\u0011\nk\u0001kzk2\n2.\n\u000f(BAD-4) Define ^xj=P\nb2Lnfigandj2\u0000(b)(Mb;j\u0001xb). Then\nk^xjk2>r\u0011\nk\u0001kzk2:\nIf a bucket satisfies (B AD-b) for someb2[4], then we will also call it B AD-b-bucket for item i. If a bucket\nis not B AD-bfor anyb2[4], then we will call it good fori. If we do not specify for which item a bucket\nis bad (or the more specific versions of bad as above) then, it’ll be assumed to be bad for some i2[N].\nFor anyi2[N]andb2[4], letBb\ni\u0012[M]denote the set of B AD-b-buckets for item iand for notational\nconvenience define Bi=B1\ni[B2\ni[B3\ni[B4\ni.\nNote that if for any i2[N], we havej\u0000(i)\\Bij<`\n2, thenjxi\u0000\u0016xij6p\n\u0011=k\u0001kzk2(this is because then\nin the majority of the buckets, xiis the only potential heavy element and the tail noise is low), as desired.16\nTo prove that at most \rkitems have bad estimates we will prove the following:\n16Actually we only need to consider buckets that are not B AD-1, not B AD-2 or not B AD-4 but not being B AD-3 makes the\nanalysis somewhat modular.\n38Lemma H.4. For any subset S\u0012[N]nHk(x)withjSj=k, the following are true\nj[i2Hk(x)[SB1\nij6\r\n12k`with probability 1: (29)\nj[i2Hk(x)[SB2\nij6\r\n12k`with probability 1: (30)\nj[i2Hk(x)[SB3\ni[B4\nij6\r\n12k`with probability at least 1\u0000exp(\u0000\n(\rk`)): (31)\nLetB=[i2Hk(x)[SBi. Then note that Lemma H.4 proves that except with probability exp(\u0000\n(\rk`)),\njBj6\r\n16\u0001(4k)`. This implies that by Lemma B.2 (with a= 2), there are at most 4(\r=16)(4k) =\rk\nelementsi2Hk[S, such that they have at least `=2bad buckets for item i. This shows that for every S,\nthe probability that there exists a subset T\u001aHk(x)[SwithjTj=\rksuch that every item in Thas a bad\nestimate is upper bounded by exp(\u0000\n(\rk`)). Taking the union bound over all the\u0000N\n\rk\u0001\nchoices forT, the\nprobability that there exists some set of \rkitems with a bad estimate is at most:\n\u0012N\n\rk\u0013\n\u0001e\u0000\n(\rk`)6\u0012N\n\rk\u0013\n\u0001\u0012\u0012N\n\rk\u0013\u0013\u0000s+1\n=\u0012N\n\rk\u0013\u0000s\n;\nwhere the inequality follows from the assumption that `>c\u0001log(N=k)for some large enough constant\nc>\n(slog(1=\r)). Thus, except with probability\u0000N\n\rk\u0001\u0000s, other than at most \rkitemsi2[N], every other\nitem has a good estimate, as desired.\nIn the rest of the section, we will prove Lemma H.4.\nH.2 Proof of Lemma H.4\nProof of (29). Fix anyi2Hk(x)[Sand consider a bucket j2\u0000(i). Ifjis B AD-1 forithen it means\nthatjis not a “unique neighbor” of Hk(x)[S. By Lemma B.3, we then get that\nj[i2Hk(x)[SB1\nij62\"(2k)`= 4\u00103\u0011k`6\r\n16k`<\r\n12k`;\nwhere the second inequality follows if we choose \u00106\r=4(as\r36\rand\u001161).\nProof of (30). We will make separate arguments for heavy tail elements that are in Sand those that are\nnot. To be more precise let Hin\u001aSbe the heavy tail elements in SandHout\u0012[N]nHinbe the heavy tail\nitems outside of S. We’ll make slightly different arguments depending on whether jHoutj<2kor not:\n\u000fCase 1 (jHoutj<2k) In this case we will prove the following stronger inequality:\nj[i2Hk(x)[S[HoutB2\nij6\r\n12k`:\nFix ani2Hk(x)[S[Houtand consider a bucket j2\u0000(i)that is B AD-2 fori. Then as in the previous\ncase, we get a non-unique neighbor in \u0000(Hk(x)[S[Hout). Note thatjHk(x)[S[Houtj<4kand\nthus by Lemma B.3, we have\nj[i2Hk(x)[S[HoutB2\nij<2\"(4k)`= 8\u00103\u0011k`6\r\n27k`<\r\n12k`;\nwhere the second inequality follows if we choose \u00106\r=6.\n39\u000fCase 2 (jHoutj>2k) In this case we will handle collisions with heavy tail elements from Hout\nseparately. Fix an element i2Hk(x)[Sand consider a bucket j2\u0000(i). Ifjis also in \u0000(h)for\nsome heavy tail element h2Hin, then call such a bucket light-B AD-2bucket for i. Otherwise if\nicollides with some heavy tail element h2Houtin bucketj, then we call jto be a heavy-B AD-\n2bucket for i. By the same argument as in the proof of (29), we can upper bound the number of\nlight-B AD-2buckets for any iby\n2\"(2k)`= 4\u00103\u0011k`64\r\n125k`<\r\n24k`; (32)\nwhere the first inequality follows if we choose \u00106\r=5.\nTo bound the number of heavy-B AD-2buckets we note that this would be upper bounded by j\u0000(Hk(x)[\nS)\\\u0000(Hout)j. By Lemma B.1, this is upper bounded by\n4\"jHoutj`64\"\u0001k\n\u00102\u0011\u0001`= 4\u0010k`6\r\n24k`; (33)\nwhere the first inequality follows by noting that the total number of heavy tail items (which upper\nboundsjHoutj) is at mostk=(\u00102\u0011)and the last inequality follows if \u00106\r=96.\nUpper bounds in (32) and (33) proves (30).\nProof of (31). We will prove (31) by proving the following two inequalities. We will show that the fol-\nlowing always holds:\nj[i2Hk(x)[SB3\nij6\r\n24k`; (34)\nand the following does not hold with probability at most exp(\u0000\n(\rk`)):\nj[i2Hk(x)[SB4\ninB3\nij6\r\n24k`: (35)\nProof of (34). We will prove (34) using an argument very similar to that in [22]. (The only difference is\nthat the argument for light tail with small support is less involved because of the use of expanders.) We will\nfirst compute the sum of the `2\n2contribution of the light tail elements to \u0000(Hk(x)[S). The main idea is\nto decompose this sum into (sub)-convex combination of flat tail contributions and contribution from a tail\nwith small support.\nFirst, we consider the contribution of the light tail elements from Sitself that contribute to the potential\nBAD-3 buckets. We take care of this contribution with an argument similar to the one we used to prove (29).\nIn particular, if bucket jis B AD-3bucket for i2Hk(x[S)due to a light tail element from Sthen bucket\njis not a unique neighbor. Thus, the number of such B AD-3buckets is upper bounded (due to Lemma B.3)\nby at most 4\u00103\u0011k`6\r\n54k`<\r\n48k`if we choose \u00106\r=6.\nNext, letL\u0012LnSbe the set of light tail elements with non-zero value outside of S. We will first bound\nthe sum of the `2\n2contribution of light tail elements from Lto\u0000(Hk(x)[S)– we denote the sum by \u0006L.\nDefine w= (x2\ni)i2L. (Note thatkwk16kzk2\n2.)\nWe first assume that jLj>2k. Letmbe the smallest wivalue among all i2Land consider the vector\nwmwhere supp( wm) =Land each non-zero value is m. (Note that wmis a flat tail scaled by mjLj).\nBy Lemma B.1, the contribution of wmto\u0006Lis upper bounded by 4\"`mjLj= 4\"`kwmk1. Update w \nw\u0000wmandL Lnfi2Ljw=mg. IfjLj>2kwe repeat the process above. We note that the total\ncontribution while jLj>2kin the process above is at most 4\"`kwk164\"`kzk2\n2. Finally, when we are left\n40withjLj62k, since each element in the residual wcan contribute at most \u00102\u0011=k\u0001kzk2\n2(repeated at most `\ntimes) to \u0006L, we conclude that the final contribution to \u0006Lis at most 2\u00102\u0011`kzk2\n2. This implies that\n\u0006L6(4\"+ 2\u00102\u0011)`kzk2\n266\u00102\u0011`kzk2\n2:\nIn the worst-case the sum \u0006Lcontributes to new B AD-3buckets. In particular, by the Markov argument\n(and the bound on \u0006Labove along with the fact that there are at most 2k`buckets), the number of new\nBAD-3buckets is upper bounded by 6\u0010k`6\r\n48k`, if we pick\r6\u0010=288.\nAdding the contributions of light tail elements from Sand outside of Sproves (34).\nProof of (35). Consider any bucket jthat is not a B AD-3-bucket. This implies that the `2contribution from\nthe light tail elements is at mostp\n\u0010\u0011=k\u0001kzk2. Then by Lemma C.1, we have that with probability at most\n\u0010, we havek^xjk2>p\n\u0011=k\u0001kzk2(where ^xjis as defined earlier). This implies that the expected number of\nBAD-4buckets is at most \u0010k`. Further, since every non-zero Mi;jis an independent\u00061value, we can apply\nChernoff bound (Lemma C.2) to bound the probability of more than 2\u0010k`BAD-4buckets by exp(\u0000\n(\u0010k`)).\nThus, except with probability exp(\u0000\n(\u0010k`))the number of B AD-4buckets is upper bounded by\n2\u0010k`6\r\n24;\nif we pick\u00106\r=48.\nWrapping up. Looking at all the conditions on \u0010, we note that the choice \u0010=\r\n288satisfies all the re-\nquired conditions. Note that this also implies that the probability of not satisfying (35) is upper bounded by\nexp(\u0000\n(\rk`)), as desired.\nH.3 Some observation on the use of randomness\nWe start off by observing that the only place that uses randomness in the proof of Theorem 4.7 is in\nLemma H.4. Further, the only place in the proof of Lemma H.4 that uses randomness is in the proof of\n(35).\nFirst note that by the dependence of the probability bound in Lemma C.2, we can argue the same probability\ndependence as in (35) even if the random \u00061values were O(k`)-wise independent. In other words,\nRemark H.5. Theorems 4.7, 4.8 and 4.9, hold even if the random \u00061entries areO(k`)-wise independent.\nNext we note that if we were proving the corresponding `1=`1result to Theorem 4.7, then we would not\nneed anyrandomness. In particular, for the `1=`1case, if we define B AD-3 event to be such that the total\n`1mass of all light tail elements other than iis>\u0010\u0011=kkzk1, then if a bucket is not B AD-3 then we do not\nneed to consider the B AD-4 event. In other words,\nRemark H.6. The versions of Theorems 4.7, 4.8 and 4.9 for `1=`1sparse recovery holds even without\nmultiplying the matrix MGwith random\u00061values. In other words, the results hold deterministically .\nI Proof of Lemma 4.13\nThe algorithm is the same as in the proof of Theorem 4.8.\n41The main idea in proving the correctness of the algorithm is to argue that under the map f, the heavy hitters\ndo not suffer much collisions or “acquire” heavy `2\n2noise. Then we apply Corollary H.3 (or more precisely\nthe proof of Lemma H.1) on the vector obtained by “applying” ftox. In what follows, we will be using\nnotation from the previous section.\nWe first consider the effect of fonx. Call a heavy hitter h2Hk(x)being corrupted if eitherf(h) =f(h0)\nfor some heavy hitter/heavy tail element h0or the`2\n2sum contribution of light tail elements to f(h)is\n>\u00102\u0011\nk\u0001kzk2\n2. We next argue that\nLemma I.1. Except with probability\u0000M\nk\u0001\n(\u0000\u000b\u0010k), at most 5\u0010kheavy hitters are corrupted.\nProof. The argument is very similar to that in [22].\nTo being with we upper bound the number of corruptions due to a collision. Just for this proof, we will refer\nto an element i2[N]as a heavy tail ifjxij2>\u00105\u00112kzk2\n2\nk.\nNote that even conditioned on the the maps of the heavy tail items (of which there are at most \u0010\u00005\u0011\u00002k)\nand the other heavy hitters (of which there are at most k\u00001), a given heavy hitter suffer a collision with\nprobability at most O(k(\u0010\u00005\u0011\u00002+1)=M1\u0000\u000b). Then using a similar argument used in [22], we can applying\nLemma C.2 to bound the probability of having more than \u0010kcorruptions due to collisions by\n\u0012ek2(1 +\u0010\u00005\u0011\u00002)\nM1\u0000\u000b\u0010k\u0013\n(\u0010k)\n6\u0012M\nk\u0013\n(\u0000\u000b\u0010k)\n; (36)\nwhere the inequality follows from the fact thatM1\u0000\u000b\nk\u0010\u00006\u0011\u00002>(M=k)\u000b(which in turn follows from the lower\nboundM>\n(\u0010\u00006\u0011\u00002\u0001k1\u0000\u000b\n1\u00002\u000b\u0001(log(N=k))2)).\nNext we upper bound the number of corruptions due to large `2\n2noise from light tail items. Define w2RN\nbe to zero on the locations of heavy hitters and heavy tail items. For a light tail element i, definewibex2\ni\nrounded up to the next highest number in fkzk2\n2=2igi>0. Note that all isuch thatjwij6\u00105\u00112kzk2\n2=Ncan\ncontribute at most \u00105\u00112kzk2\n2.\nThus, ignoring such elements, since the largest value of a light tail element is \u00105\u00112jzk2\n2=k, we are left with\nn0= log\u0000\n\u00105\u00112N=k) =O(log(N=k)\u0001\ndistinct values in w. In particular, we can decompose winton0\n(scaled) flat tails: call these tails w0;:::;wn0\u00001. (W.l.o.g. assume that kwi+1k16kwik1=2.) Call a tail\nwismall ifjsupp( wi)j6k=(\u00102\u0011). Otherwise, call a tail wilarge . Note that sincekw0k16\u00105\u00112kzk2\n2=k,\nthe total contribution of all small tails is at most\nn0\u00001X\ni=0k\n\u00102\u0011\u0001kw0k1\u00012\u0000i62k\n\u00102\u0011\u0001kw0k162\u00103\u0011kzk2\n2:\nNow consider a large tail wi. Each item in supp( wi)collides with a heavy hitter (independently) with prob-\nability at most O(k=M1\u0000\u000b)(even conditioned on the map of a heavy hitter under f). Thus, by Lemma C.2,\nexcept with probability\n\u0012ekjsupp( wi)j\n\u00103\u0011M\u00001\u000bjsupp( wi)j\u0013\n(\u00103\u0011jsupp( wi)j)\n6\u0012ek\n\u00103\u0011M1\u0000\u000b\u0013\n(\u0010k)\n; (37)\nwicontributes at most \u00103\u0011kwik2\n2`2\n2noise to the heavy hitters under f. Thus the total noise contribution of\nall the large tails is at most \u00103\u0011kzk2\n2except with probability n0\u0001\u0010\n\u00103\u0011M1\u0000\u000b\nk\u0011\u0000\u0010k\n6(M=k)\u0000\n(\u000b\u0010k), where\nthe last inequality follows from the lower bound on M.\n42Thus, the total `2\n2contribution of all the light tail elements to the heavy hitters under the map fis at most\n4\u00103\u0011kzk2\n2. Thus, by the Markov inequality at most 4\u0010kheavy hitters get corrupted by a light tail element.\nThus, except with probability\u0000M\nk\u0001\n(\u0000\u000b\u0010k), at most 5\u0010kheavy hitter get corrupted, as desired.\nThe rest of the proof is very similar to that of Theorem 4.7 so we will just sketch the differences below.\nThe main reason we can do this is because other than the analysis of the B AD-4 buckets, we essentially are\nworking with the `2\n2tail. (Also the analysis of the B AD-4 buckets on depends on whether the `2\n2noise in a\nbucket is large or not.)\nTo be more precise, define a vector related to xwhere all but the heavy hitters are squared: i.e. y=\n(y1;:::;yN), whereyi=xiifi2Hk(x)otherwiseyi=x2\ni. Now define f(y)vector to be the natural\n“partial sum” vector of yunderf: i.e,w= (w1;:::;wM) =f(y)such thatwj=P\ni:f(i)=jyi. For the\nproof, call an item j2[M]a heavy hitter if it contains an uncorrupted heavy hitter from xmapped under\nfto it. Note that by Lemma 4.13, we can assume that there are at least k(1\u00004\u0010)heavy hitters in w. As\nbefore, define an item j2[M]to be a heavy tail item if jwjj>\u00102\u0011kzk2\n2=k. Note that we can have at most\n\u0010\u00002\u0011\u00001k+ 4\u0010k(where the last term can arise from the corrupted heavy hitters from x). Now the rest of\nthe argument remains unchanged (by adjusting the constants), except for the following simple change– in\na bucket that is not B AD-1 B AD-2 or B AD-3, we still have to account for the `2\n2noise that an uncorrupted\nheavy hitter obtains due to the mapping f. However, this only increases the `2\n2noise by a constant factor\nthan what was handled earlier. Again by adjusting constants one can arrange for the claim in Lemma 4.13.\n(One also has to add up the failure probabilities from Lemma I.1 and the one gets from the expander part of\nMbut both are of the same order.)\nI.1 Some observations on the use of randomness\nIn this section, we make some observation on the use of randomness in the proof of Lemma 4.13 which were\nnot already covered in Section H.3.\nThe new place where one uses randomness in the proof of Lemma 4.13 is in Lemma I.1. In particular, we\nused the randomness of the map fin proving bounds (36) and (37). We note that both of these bounds would\nhold even if the map fwereO(k)-wise independent. Along with Remark H.5, this implies that\nRemark I.2. Lemma 4.13 hold even if the random \u00061entries areO(k`)-wise independent and the map f\nis onlyO(k)-wise independent.\nNext we note that if we were proving the corresponding `1=`1result to Lemma 4.13, then by Remark H.6\nwe do not need the random \u00061entries. There is another source of randomness– the map f. We will not\nget rid of the randomness used in the map f(at least not quite yet). However, for future use it would be\nbeneficial to note if one were to try and get rid of the randomness in proof of Lemma I.1, how many events\nwould one have to take a union bound against.\nLet us first consider (36). In this case we assume that the locations of the at most k0=\u0010\u00005\u0011\u00002k+k“heavy\nhitters” are fixed. Thus if we can do union bound over all\u0000N\nk0\u0001\nchoices of these items, then we would be able\nto prove that there are no more than \u0010kcollisions probabilistically. Now let us consider (37). We first note\nthat we only used randomness in trying to bound the corruption due to noise from light tail elements that\ncome fromO(log(N=k))large tails. In particular, we note that the argument assumes that the following are\nfixed: (i) The locations of the at most kheavy hitters and (ii) assignment of the light tail elements to one of\ntheO(log(N=k))large tails (or if it is not part of any large tail). The key observation is that we only care\nabout the locations of elements in (i) and (ii) but not the actual value at those locations. Note that then the\nnumber of choices are\u0000N\nk\u0001\n+NO(log(N=k)).\n43Remark I.3. The version of Lemma 4.13 for `1=`1sparse recovery holds even without multiplying the\nmatrixMGwith random\u00061values. Further, one can also get rid of the randomness in the map fif we are\nwilling to take union bound against\u0000N\nk0\u0001\n+Nxevents, where k0=O(\u0010\u00005\u0011\u00002k)andx=O(log(N=k)).\nJ Inverting a Random function\nWe pickf: [N]![N2]randomly by picking a random degree N\u00001polynomial over FN2and definef(i)\nto be the evaluation of this polynomial at i. It is well-known that such a random hash function is N-wise\nindependent (see e.g. [11]) or fully independent.\nWe have three issues to tackle.\nFirst, we now have to first compute the matrix M0withN2columns (then we pick the columns of M0\nindexed byf(i)2[N2]fori2[N]) instead ofNcolumns as we have used so far. This however, is not an\nissue since we have a family of matrices Mn. This change results in some constants in the parameters of\nM\u0003changing but it does not affect the asymptotics.\nSecond, we have to show that this new fis fine with respect to applying Lemma 4.13. In particular, we\nhave to show the following: consider any node vin the recursion tree. Then we will show that for the (new)\n\u001ev: [N]![M]is(d;\u000b)-random (for some small \u000b). We will in fact show that these functions are (d;0)-\nrandom. Indeed, since fisN > (d+ 1) -wise independent and the codes fCngare uniform, the new map\n\u001evis also (d+ 1) -wise independent. (This can be proven e.g. by induction on the level of v.) This in turn\nimplies that \u001evis(d;0)-random, which is sufficient for Lemma 4.13.\nFinally, we come to the inversion. Given j2[N], we would like to compute “ f\u00001(j).” (Recall we do this\nstep only once at the root of the recursion tree after the identification algorithm above has terminated with\nthe outputIw\u0012[N2]. Recall thatjIwj6O(k=\u0011).) Unfortunately, we cannot still guarantee that the inverse\nis unique. So we do the following: If we have j2Iwsuch thatjf\u00001(j)j>1, we just drop this index;\notherwise we output f\u00001(j). We first argue that this step does not drop too many indices and then consider\nhow quickly we can solve this step.\nRecall that we only care about whether we identify elements of Hk0(x), wherek0=O(\u0010\u00002\u0011\u00001k). Note that\nsincefis completely independent, for every i2Hk0(x)even conditioned on the value of f(j)for every\notherj2[N]nfig,f(i)is completely independent. Thus, the probability that f(i) =f(j)for one of these\njis at mostN=N2= 1=N. Thus, by the argument similar to the one in [22] (to care of the dependencies),\nwe can use Lemma C.2 to argue that the probability of more than \u0010k i’s colliding with another jis at\nmost (N=k)\u0000\n(\u0010k). Thus, the algorithm above loses an extra \u0010kindices while maintaining the same failure\nprobability. This extra \u0010kadditive factor only changes the constants and thus, can be absorbed into the\nanalysis without much trouble.\nWe now come to the part about computing f\u00001(j). To do this step, we just store the pairs (f(i);i)i2[N]in an\narray (sorted by the first entry) of size O(NlogN)bits. Note that given this array (by binary search), we can\nin timeO(logN), given aj2[N2], figure outf\u00001(j)(if it is unique). Since we have to do this inversion\nO(k=\u0011)times, we add an additional factor of O(k=\u0011logN)to the identification time. (This additive factor\nwill never be asymptotically significant in our final results.)\nWe will call this scheme above S CHEME -1. Note that we have show that S CHEME -1works and\nLemma J.1. SCHEME -1addsO(k=\u0011log(N=k))to the decoding time in Theorem 4.15 and needs O(NlogN)\nbits of space.\n44We also need O(\u0010\u00005\u0011\u00002\u0001k\u0001log2N)bits of space to store the randomness needed to define g(which we\nneed to store). However, this is subsumed by the O(NlogN)space to store the array above.\n45Level 1Identification algorithm Identification algorithmList Recovery AlgorithmIdentification algorithm\nIdentification algorithm000\n001\n010\n010\n100\n101\n110\n111\nLevel 0\nLevel 1 Level 1\nx0\nx1\nx2\nx3\nx4\nx5\nx6\nx7\nLemma 4.13 Lemma 4.13\n+\n+\n+\n++\n+\n+\n++\n+\n+\n+x1\nx3\nx2\nx2\nx5\nx4\nx7\nx6x1\nx3\nx4\nx2\nx5\nx4\nx7\nx6\nx1\nx3\nx4\nx2\nx5\nx4\nx7\nx6x1\nx3\nx4\nx2\nx5\nx4\nx7\nx6x1\nx3\nx4\nx2\nx5\nx4\nx7\nx6Lemma 2.5\nLemma 4.13Theorem 4.8f\nf\u00001\nSection 4.5.1\nFigure 1: Overview of the proof of Lemma 4.3 for the specific example of N= 8 and one level of recursion. The encoding\ncan be thought of as follows. The input vector (x0;:::;x 7)is moved around by a random map f(in this case fis a permutation).\nThen the shuffled vector is acted upon by one expander on level 0and three expanders on level 1. (For clarity the shuffled vector\nis copied next to the four expanders.) The green and red labels on the expander edges denote +1and\u00001weights respectively. At\nthe level 0expander, the eight values are routed on the edges and are collected in the output (gray) nodes by adding the incoming\nvalues (weighted by the corresponding \u00061label). At the level 1expander a similar process happens except the shuffled vector is\ncombined using the following code on the indices (which are represented in the vector next to the level 0expander for convenience)\nfrom Lemma 2.5: C(b0;b1;b2) = ((b0;b1);(b1;b2);(b0;b2)). The left most level 1expander corresponds to the first codeword\nposition, the middle one to the second position and the right one to the third position. Also the values corresponding to the indices\nthat map to the same symbol in f0;1g2are added up together. For convenience, we have used the same colored arrows for the same\nsymbol inf0;1g2for every level 1expander. E.g., in the middle expander, the top left vertex corresponds to (b1;b2) = (0;0),\nwhich means index 000(which has value x1) and index 100(which has value x5) get added up. The decoding process reverses the\nencoding logic. We first identify the heavy hitters for level 1expanders using the identification algorithm from Lemma 4.13. These\nlists are then combined for the f0;1g3domain by the list recovery algorithm for Cfrom Lemma 2.5. The output is then used as\nthe setSin the identification algorithm from Theorem 4.8 to get the locations of the heavy items in the shuffled vector. Finally, we\nused the inversion procedure as outlined in Section 4.5.1 to obtain our final indices, which are the 2nd and the 6th positions. (For\nclarity, the set of identified indices at every step are surrounded by a light blue curve.)46"    },    {        "title": "1304.7295v2.Landau_Lifshitz_theory_of_the_longitudinal_spin_Seebeck_effect.pdf",        "content": "Landau-Lifshitz theory of the longitudinal spin Seebeck e\u000bect\nSilas Ho\u000bman, Koji Sato, and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nThermal-bias-induced spin angular momentum transfer between a paramagnetic metal and ferro-\nmagnetic insulator is studied theoretically based on the stochastic Landau-Lifshitz-Gilbert (LLG)\nphenomenology. Magnons in the ferromagnet establish a nonequilibrium steady state by equilibrat-\ning with phonons via bulk Gilbert damping and electrons in the paramagnet via spin pumping,\naccording to the \ructuation-dissipation theorem. Subthermal magnons and the associated spin cur-\nrents are treated classically, while the appropriate quantum crossover is imposed on high-frequency\nmagnetic \ructuations. We identify several length scales in the ferromagnet, which govern qualitative\nchanges in the dependence of the thermally-induced spin current on the magnetic \flm thickness.\nPACS numbers: 85.75.-d,72.25.Mk,73.50.Lw,75.30.Ds\nI. INTRODUCTION\nOver the past three decades, spintronics has evolved\nfrom a focus on equilibrium phenomena in magnetic het-\nerostructures, such as giant magnetoresistance1and in-\nterlayer exchange interactions,2to dynamic processes,\nsuch as spin-transfer torque3,4and spin pumping,5,6and,\nmore recently, nonequilibrium thermodynamics, heralded\nby the spin Seebeck e\u000bect7{9and thermally-induced\nmotion of domain walls.10,11From a practical stand-\npoint, magnetic nanostructures are useful for \feld sens-\ning and nonvolatile information storage,12where magne-\ntoresistance is paramount for the readout, while current-\ninduced spin torques are useful for fast and scalable bit\nswitching.13\nOne rapidly-developing avenue of research concerns\nout-of-equilibrium spin phenomena in insulating systems,\nwhere spin is carried by collective excitations, such as\nspin waves (magnons), rather than electronic quasipar-\nticles. To this end, spin waves in the ferrimagnetic in-\nsulator yttrium iron garnet (YIG) appear particularly\npromising as they su\u000ber from a remarkably low Gilbert\ndamping (at microwave frequencies), \u000b\u001810\u00004, and\nthe host material has Curie temperature of \u0018500 K,\nthus remaining magnetic at room temperature.14Spin\nwaves in YIG have recently been shown to undergo room-\ntemperature Bose-Einstein condensation under nonlinear\nmicrowave pumping,15exhibit large spin pumping into\nadjacent conductors,16{18manifest the longitudinal spin\nSeebeck e\u000bect,19and e\u000eciently move domain walls un-\nder small thermal gradients.11These phenomena hold\npromise for integrated circuits based on nonvolatile mag-\nnetic elements20with essentially no Ohmic losses and\nthus very low dissipation.\nFurthermore, thermal control of magnetic dynamics\nand spin currents21provides an attractive alternative\nto voltage control, especially since magnons, which are\nneutral objects, can respond more directly to tempera-\nture gradients. The spin Seebeck e\u000bect, i.e., the gen-\neration of thermal spin current between magnetic insu-\nlators and normal metals, is the basic phenomenon of\ncentral interest in this context. The purpose of this pa-per is to develop a systematic semi-phenomenological ap-\nproach to this problem, based on the Landau-Lifshitz-\nGilbert (LLG) theory of ferromagnetic dynamics,22de-\nparting from the spin-pumping6perspective on the inter-\naction between electrons and magnons at ferromagnetic-\ninsulatorjnormal-metal interfaces put forward in Ref. 8.\nIn Sec. VIII, we comment on how the theory could be ex-\npanded to account for magnon and phonon kinetics when\nthe standard LLG phenomenology fails.\nII. FERROMAGNETIC BULK DYNAMICS\nIn the ferromagnetic bulk, away from the Curie\ntemperature, magnetic dynamics are described by the\nstochastic LLG equation22\n@tm=\u0000\rm\u0002(He\u000b+hl) +\u000bm\u0002@tm; (1)\nwhere m=M=Msis the unit-vector magnetization di-\nrection (Ms=jMjbeing the saturated magnetization\nmagnitude), \r(minus) the gyromagnetic ratio ( \r > 0\nfor free electrons), \u000bdimensionless Gilbert damping con-\nstant,\nHe\u000b\u0011\u0000\u000eMF=Haz+Axr2m+Hr (2)\nthe e\u000bective \feld (consisting of applied \feld Hain the\nzdirection, exchange \feld /Ax, and relativistic cor-\nrections Hrthat include dipolar interactions and crys-\ntalline anisotropies), and hlrandom Langevin \feld with\ncorrelator23\nhhl;i(r;t)hl;j(r0;t0)i=2\u000b\n\rMskBT(r)\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);\n(3)\nin accordance with the \ructuation-dissipation theorem.\nWe are interested at intermediate temperatures: much\nlower than the Curie temperature, such that the Landau-\nLifshitz phenomenology based on the directional magne-\ntization dynamics [SO(3) nonlinear \u001bmodel] is appro-\npriate, while not too low such that the classical theory\ncan be used as a starting point. We will, furthermore, ne-\nglectHrin Eq. (2), for simplicity, which is justi\fed whenarXiv:1304.7295v2  [cond-mat.mes-hall]  12 Aug 20132\nxyz0dT1T2With a little effort the layout of this diagram can actually be improvedby enlarging the inner box, see page 29 below.Here is the resonants-channel contribution toe+e−→4f. (From nowon, we do no longer display the\\begin{fmfgraph}(40,25)\\fmfpen{thick}...\\end{fmfgraph}environment surrounding all pictures.)\\fmfleftn{i}{2}\\fmfrightn{o}{4}\\fmf{fermion}{i1,v1,i2}\\fmf{photon}{v1,v2}\\fmfblob{.15w}{v2}\\fmf{photon}{v2,v3}\\fmf{fermion}{o1,v3,o2}\\fmf{photon}{v2,v4}\\fmf{fermion}{o4,v4,o3}\ne−e+µ+νµs¯cAnd the resonantt-channel contribution:\\fmfleftn{i}{2}\\fmfrightn{o}{4}\\fmf{fermion}{i1,v1,v2,i2}\\fmf{photon}{v1,v3}\\fmf{fermion}{o1,v3,o2}\\fmf{photon}{v2,v4}\\fmf{fermion}{o4,v4,o3}\ne−e+µ+νµs¯cTwo point loop diagrams pose another set of problems. We must havea way of specifying that one or more of the lines connecting the twovertices arenotconnected by a straight line. The optionsleft,rightandstraightoffer the possibility to connect two vertices by a semicircledetour, either on the left or on the right. Since by default all lines con-tribute to the tension between two vertices, thetensionoption allows usto reduce this tension. The next examples shows both options in action.The lower fermion line is given an tension of 1/3 to make is symmetricalwith the upper line with consists of three parts. The loop photon is usinga detour on the right and does not contribute any tension.\\fmfleft{i1,i2}\\fmfright{o1}\\fmf{fermion,tension=1/3}{i1,v1}\\fmf{plain}{v1,v2}\\fmf{fermion}{v2,v3}\\fmf{photon,right,tension=0}{v2,v3}21nHa⊗N1N2Fjs1js2\nFIG. 1. Schematic of an N 1jFjN2sandwich structure studied\nin this paper. The normal-metal layer N 1is treated as a poor\nspin sink, which blocks spin current, js1\u00190. The normal-\nmetal layer N 2, on the other hand, is a perfect spin sink, thus\nestablishing a thermal contact between its itinerant electrons\nand magnons in the ferromagnetic insulator (F), which results\nin spin current js2\u0011js. We assume the phonons in the F layer\nfollow a linear temperature pro\fle from T1atx= 0 (N 1jF\ninterface) to T2atx=d(FjN2interface), corresponding to\nelectron temperatures in N 1and N 2, respectively.\nkBT\u001d~\rMs. The Langevin correlator (3) is white at\nfrequencies !\u001ckBT=~, corresponding to classical be-\nhavior. In Sec. VI, we will adapt our theory to account\nfor quantum \ructuations at !&kBT=~, by matching\nwith the fully quantum treatment of Ref. 24.\nIn order to streamline discussion of the spin transfer,\nlet us switch from the magnetization to the spin density:\ns\u0011sn=\u0000Ms\n\rm; (4)\nwheres=Ms=\ris the saturated spin density and n=\n\u0000mits direction. The LLG equation then becomes\ns(1 +\u000bn\u0002)@tn+n\u0002(Hz+h) +@ijs;i= 0;(5)\nwhere\njs;i=\u0000An\u0002@in (6)\nis identi\fed as the magnetic spin current and\nhhi(r;t)hj(r0;t0)i= 2\u000bskBT(r)\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);(7)\nwhereiandjstand for the Cartesian coordinates. Here,\nH\u0011MsHaandA\u0011MsAx. In equilibrium, n=\u0000z,\nassuming the applied \feld Ha>0.\nIn this paper, we focus on the trilayer heterostructure\ndepicted in Fig. 1. The temperature T(r) entering Eq. (3)\nis taken to correspond to the x-dependent phonon tem-\nperature inside of the ferromagnetic \flm,\nT(x) =T1+x\nd(T2\u0000T1); (8)\nassuming Gilbert damping stems from the local magnon-\nphonon scattering. (We will revisit this assumption in\nSec. VIII.)III. BOUNDARY CONDITIONS\nThe boundary conditions for a ferromagnet sand-\nwiched between two normal metals need to be simi-\nlarly constructed to account both for deterministic6and\nstochastic25spin-transfer torques. We will start with\nthe former and then include the latter according to the\n\ructuation-dissipation theorem.\nWe assume the spin current is blocked by the N 1layer\natx= 0, due to its weak spin-relaxation rate:\njs;x= 0 (x= 0): (9)\nIn other words, the spin current pumped across the F jN1\ninterface is balanced by an equal back\row.6For our pur-\nposes, N 1can thus be replaced by an insulator, as long\nas it makes a good thermal contact with phonons in the\nferromagnet. A net spin current across the F jN2inter-\nface, on the other hand, is allowed, if we treat N 2as a\nperfect spin sink:6\njs;x=~g\"#\n4\u0019n\u0002dn\ndt(x=d); (10)\nwhereg\"#is the real part of the dimensionless interfa-\ncial spin-mixing conductance (per unit area).26We disre-\ngard the imaginary part of the spin-mixing conductance,\nsince it governs the typically smaller6nondissipative spin-\ncurrent component /dn=dt, which vanishes over a cycle\nof precession. A speci\fc realization for such an N 1jFjN2\ntrilayer could be provided by the Cu jYIGjPt combina-\ntion, where Cu (Pt) is a light (heavy) element with weak\n(strong) spin-orbit interaction. (We will generalize our\n\fndings to arbitrary N 1jFjN2trilayers, such as symmetric\nPtjYIGjPt type structures or general asymmetric struc-\ntures, in Sec. VII.)\nFor the two N 1jFjN2interfaces, we correspondingly\nhave the following (deterministic) boundary conditions\nfor magnetic dynamics (as T!0):\n8\n<\n:@xn= 0; x = 0\nA@xn+~g\"#\n4\u0019@tn= 0; x=d; (11)\nre\recting continuity of spin current, which is given by\nEq. (6) inside the ferromagnet and Eqs. (9) and (10) in\nN1and N 2, respectively, across the corresponding inter-\nfaces. Since spin pumping (10) a\u000bects magnetic dynamics\nsimilarly to Gilbert damping,6it is accompanied with a\nsimilar stochastic term.25The latter can be accounted\nfor by modifying the boundary condition at x=d:\nA@xn+~g\"#\n4\u0019@tn+h0= 0; (12)\nwhere\n\nh0\ni(\u001a;t)h0\nj(\u001a0;t0)\u000b\n=~g\"#\n2\u0019kBT2\u000eij\u000e(\u001a\u0000\u001a0)\u000e(t\u0000t0) (13)3\nand\u001a= (y;z) is the two-dimensional position along the\ninterface at x=d. The Langevin correlator strength is\nproportional to the electron temperature T2at the FjN2\ninterface, since the noise originates in the thermal \ruc-\ntuations of electronic spin currents in N 2.\nThe spin Seebeck e\u000bect is embodied in the thermal-\naveraged spin current \rowing through the F jN2\ninterface:8\njs;x=\u0000An\u0002@xn=n\u0002\u0012~g\"#\n4\u0019@tn+h0\u0013\n: (14)\nSince our system is axially symmetric with respect to the\nzaxis, it is convenient to switch to complex notation:\nn\u0011nx\u0000iny. Thermal spin-current density, hjs;xi=jsz,\ncan thus be written for small-angle dynamics (relevant\nat temperatures well below the Curie temperature) as\njs=AImhn\u0003@xnijx=d: (15)\nExploiting, furthermore, translational invariance in the\nyzplane, we \fnd in the steady state:\njs=AImZd2qd!\n(2\u0019)3hn(q;!)\u0003@xn(q0;!0)i\n(2\u0019)3\u000e(q\u0000q0)\u000e(!\u0000!0);(16)\nwhere\nn(q;!) =Z\nd2\u001adtei(!t\u0000q\u0001\u001a)n(\u001a;d;t) (17)\nis the Fourier transform over \u001aand timet. The delta\nfunctions in the denominator of Eq. (16) cancel delta\nfunctions that factor out of the numerator when evaluat-\ning the averageh:::i(with the remaining integrand inde-\npendent of q0and!0). Similarly transforming Langevin\ncorrelators, Eqs. (7) and (13), we have:\nhh(x;q;!)\u0003h(x0;q0;!0)i=4(2\u0019)3\u000bskBT(x)\n\u0002\u000e(x\u0000x0)\u000e(q\u0000q0)\u000e(!\u0000!0);(18)\nfor the bulk and\nhh0(q;!)\u0003h0(q0;!0)i= 4(2\u0019)3\u000b0skBT2\u000e(q\u0000q0)\u000e(!\u0000!0)\n(19)\nfor the FjN2interface, de\fning\n\u000b0\u0011~g\"#\n4\u0019s; (20)\nwhich has dimensions of length. \u000b0=dis the enhanced\nGilbert damping for a monodomain precession of the fer-\nromagnetic \flm.6\nIV. SPIN SEEBECK COEFFICIENT\nWe now have all the necessary ingredients in order\nto evaluate the (longitudinal) spin Seebeck coe\u000ecient\n(which has units of inverse length squared)27\nS\u0011js\nkB(T1\u0000T2)(21)of the N 1jFjN2structure shown in Fig. 1. To simplify\nour subsequent analysis, let us optimize the notation, as\nfollows. The stochastic LLG equation (5) in the \flm bulk\nis written as\nA(@2\nx\u0000\u00142)n(x;q;!) =h(x;q;!); (22)\nafter linearizing transverse dynamics and Fourier trans-\nforming it in the yzplane and time. Here,\n\u00142\u0011q2+H\u0000(1 +i\u000b)s!\nA: (23)\nThe stochastic boundary condition at x=d, Eq. (12), in\nthis notation is\nA(@x\u0000\u00140)n(x;q;!) =\u0000h0(q;!) (x=d); (24)\nwhile@xn= 0 atx= 0. Here,\n\u00140\u0011i\u000b0s!\nA: (25)\nEqs. (22)-(25) now form a closed system of inhomo-\ngeneous linear di\u000berential equations, with source terms\ngiven by stochastic \felds handh0. These are straight-\nforward to solve for nusing Green's functions. Substitut-\ning the solution for ninto Eq. (16), we \fnd, after some\nalgebra, the spin Seebeck coe\u000ecient (21):\nS=\u000b\u000b0s2\n2\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\n\u0002Zd\n0dxx\f\f\f\fcosh[\u0014(x\u0000d)]\n\u0014sinh(\u0014d)\u0000\u00140cosh(\u0014d)\f\f\f\f2\n:(26)\nIntegrating over the longitudinal coordinate x, this \fnally\nbecomes\nS=\u000b\u000b0s2\n8\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\nj\u0014sinh(\u0014d)\u0000\u00140cosh(\u0014d)j2\n\u0002\u0014sin2(\u0014id)\n\u00142\ni+sinh2(\u0014rd)\n\u00142r\u0015\n; (27)\nwhere\u0014r(\u0014i) is the real (imaginary) part of \u0014. Eq. (27)\nis our central results and the main departure point for\nthe subsequent analysis.\nLet us, for convenience, de\fne the following length\nscales:\n\u0018\u0011r\nA\nH(28)\nis the magnetic exchange length,\nl0\u0011\u000b0\n\u000b(29)\nis the spin-pumping length (i.e., the F thickness at which\nthe monodomain Gilbert damping enhancement due to\nspin pumping6equals the intrinsic damping),\n\u0015\u0011r\n~A\nskBT(30)4\nis the thermal de Broglie wavelength (in the absence of\napplied \feld), where Tis the ambient temperature, and\nl\u0011\u0015\n\u000b(31)\nis the decay length for thermal magnons in the bulk. In\nthis notation,\n\u00142=q2+1\n\u00182\u00001 +i\u000b\n\u00152~!\nkBT: (32)\nV. QUASIPARTICLE APPROXIMATION\nIn the following, we are primarily interested in the\nthickness,d, dependence of the spin Seebeck coe\u000ecient,\nS, assuming the the length-scale hierarchy \u0015\u001cl0\u001cl.\nIn YIG, for example, taking144\u0019Ms\u00182 kG,A\u0018\n1=2\u000210\u00006erg/cm, and \u000b\u001810\u00004, we \fnd the follow-\ning lengths: (1) \u0015.1 nm, at room temperature, (2)\nl0\u0018100 nm, taking g\"#\u00181014cm\u00002from Ref. 17 and\nproportionately larger l0withg\"#\u00185\u00021014cm\u00002from\nRef. 18 (g\"#is very sensitive to the preparation and qual-\nity of the YIGjmetal interfaces), (3) l.10\u0016m, and (4)\n\u0018\u001810 nm at 1 kG (corresponding to typical magneto-\nstatic \felds).\nWe start by performing integration over frequency !\nin Eq. (27) in the limit of low damping (both intrinsic\nand spin pumping). In this case, the integrand is peaked\nat sinh(\u0014d)\u00190, corresponding to\n!n(q) =A\ns\u0012\nq2+n2\u00192\nd2+1\n\u00182\u0013\n; (33)\nwheren= 0;1;2;3;::: labels magnon subbands [not to\nbe confused with unit vector nintroduced in Eq. (4)].\nThese resonances are well separated when their width is\nmuch smaller than their spacing, allowing for a quasipar-\nticle treatment of the energy integral. For the bulk damp-\ning, this condition is \u000b\u001c1=dp\n1=\u00152\u00001=\u00182, for thermal\nmagnons. Additionally, the occupation of magnons is ex-\nponentially suppressed when the temperature is smaller\nthan the gap, i.e., kBT=~.!0\u0011H=s. Therefore, we are\ninterested in the opposite regime, when the thermal de\nBroglie wavelength is smaller than the magnetic exchange\nlength,\u0015 < \u0018 . Thus in the regime d\u001cl, these reso-\nnances are well-de\fned quasiparticle peaks corresponding\nto monodomain precession ( n= 0) and standing waves\n(n>0) along the longitudinal direction. See Fig. 2. This\nallows us to evaluate the spin Seebeck coe\u000ecient by sum-\nming the contributions from individual magnon modes,\nwhend\u001cl.\nExpanding around !0, the contribution from the lowest\nenergy resonance is\nS0=\u000b\u000b0\n4\u00193dZ1\n\u00001d2qZ1\n\u00001d!!\n(!\u0000!0)2+ (\u000b+\u000b0=d)2!2;\n(34)\n1.01.21.41.61.82.011.52!/!0(q)FIG. 2. Integrand of the spin Seebeck coe\u000ecient, Eq. (27),\nin arbitrary units, illustrating the \frst four quasiparticle res-\nonances,n= 0;1;2;3, according to Eq. (33), for a \fxed q.\nWe set\u000b= 10\u00004,\u000b0= 10\u00002nm,q2+\u0018\u00002= 1 nm\u00002, and\nd= 10 nm in the main plot. The inset shows the essentially\ncontinuum spectrum when d= 100\u0016m, keeping other param-\neters unmodi\fed.\nwhich can be readily integrated over frequency:\nS0=\u000b\u000b0\n4\u00192dZ1\n\u00001d2q\n(\u000b+\u000b0=d) [1 + (\u000b+\u000b0=d)2]\n\u0019\u000b0=d\n1 +l0=dZ1\n\u00001d2q\n(2\u0019)2; (35)\nwhere we have assumed small damping, \u000b+\u000b0=d\u001c1.\nSimilarly, we \fnd for the n>0 subbands:28\nSn\u00192\u000b0=d\n1 + 2l0=dZ1\n\u00001d2q\n(2\u0019)2: (36)\nThe total Seebeck coe\u000ecient in the quasiparticle approx-\nimation is thus\nS=S0+X\nn>0Sn: (37)\nThe wave vector qin integrals (35) and (36) should be\nbounded by requiring that ~!n(q)<kBT, at which point\nour classical treatment breaks down, as discussed in the\nnext section. The spurious ultraviolet divergence is elim-\ninated by cutting o\u000b at a total three-dimensional wave\nnumber,qc=p\n1=\u00152\u00001=\u00182, corresponding to energy\nkBT. While a more careful quantum treatment for large\nwave numbers is constructed in the next section, we ex-\npect a crude cuto\u000b to adequately capture the behavior of\nthe Seebeck coe\u000ecient, up to an overall factor of order\none.\nAllow us to momentarily focus on the regime when d\u001d\n\u0015, where the quasiparticle peaks are dense (recovering\nthree-dimensional behavior), so that\nS\u0019X\nn>0Sn\u00192\u000b0\n1 + 2l0=dZ\nqcd3q\n(2\u0019)3=\u000b0\n1 + 2l0=dq3\nc\n3\u00192(38)5\nand, therefore,\nS=\u00121\n\u00152\u00001\n\u00182\u00133=2\n\u00028\n>><\n>>:\u000bd\n6\u00192; d\u001cl0\n\u000b0\n3\u00192; d\u001dl0: (39)\nThat is, when the thickness of the ferromagnet is much\nsmaller or larger than spin-pumping length, the spin cur-\nrent scales linearly with dor isdindependent, respec-\ntively.\nIf the quasiparticle peaks are not dense, d\u0018\u0015, \fnite-\nsize e\u000bects are important, re\recting individual magnon\nsubbands. In the extreme low-temperature case when\nd\u001c\u0015, only monodomain precession along the longi-\ntudinal direction contributes to the Seebeck coe\u000ecient:\nS\u0019S0. If the transverse dimensions are also much\nsmaller than \u0015, the full volume of the (nano)magnet un-\ndergoes stochastic monodomain precession, and\nS=\u000b0\nV1\n1 +l0=d; (40)\nwhere we have retained only one mode associated with\nthe transverse momentum in Eq. (35). Vhere is the vol-\nume of the F layer. This coincides with the spin Seebeck\ncoe\u000ecient for a monodomain obtained in Ref. 8 [de\fning\n(T1+T2)=2!TF,T2!TN, and\u000b0=d!\u000b0, to match\ntheir notation].\nFinally, for largest thicknesses d\u001dl, the quasiparticles\nare no longer well-de\fned (see inset of Fig. 2) and the\nabove analysis cannot be applied. Because the thickness\nis beyond the magnon propagation length, only magnons\nwithin a distance lfrom the FjN2interface contribute to\nthe spin current, which should, therefore, be independent\nof thickness, d, for a \fxed thermal gradient, ( T1\u0000T2)=d.\nSince, in this regime, the magnon propagation length is\nthe largest length scale in the problem, we can send d!\n1in Eq. (27), which gives\nS=\u000b\u000b0s2\n8\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\n\u00142rj\u0014\u0000\u00140j2: (41)\nThe integrand, which can be evaluated numerically, is\nindependent of thickness, and, therefore, S/1=d, as\nexpected.\nVI. QUANTUM CROSSOVER\nOur classical Langevin theory needs to be appropri-\nately modi\fed when approaching magnon frequencies of\n~!\u0018kBT. On the one hand, this is an important limit,\nas the spin transport is dominated by thermal magnons\nin our model. On the other hand, the classical theory\nis inadequate for the treatment of quantum \ructuations\nthat dominate at high (on the scale of the ambient tem-\nperature) frequencies.To this end, we use a quantum-mechanical result24for\nthe thermal spin current, which is exact for a tunneling\nspin-exchange Hamiltonian at an F jN interface:29\njs=\u00004kB\u000eT\u000b0Z1\n\u000f0d\u000f\u000fD (\u000f)\f2@nBE(\f\u000f)\n@\f; (42)\nwhereD(\u000f) is the magnon density of states, nBE(x)\u0011\n(ex\u00001)\u00001is the Bose-Einstein distribution, \f\u00111=kBT,\nand\u000eTis the temperature drop across the F jN inter-\nface (assuming magnons are equilibrated to a uniform\ntemperature T, such that \u0015\u001cd). This limit can be di-\nrectly compared to our Eqs. (21) and (27), by \frst send-\ning\u000b0!0 in the integrand (thus reproducing the weak\nFjN contact and allowing for the magnons in F to equili-\nbrate with phonons) and then sending \u000b!0 [such that\nthe magnon spectral properties are una\u000bected by Gilbert\ndamping, as assumed in the derivation of Eq. (42)]. The\nmagnons are correspondingly equilibrated to the average\nphonon temperature ( T1+T2)=2, such that we identify\n\u000eT= (T1\u0000T2)=2, in the present notation. According to\nEq. (38), our semiclassical spin current becomes in this\nlimit\njs!2kB(T1\u0000T2)\u000b0Zd3q\n(2\u0019)3: (43)\nThis agrees with Eq. (42) in the limit \u000f\u001ckBT, where\n\u0000Z1\n\u000f0d\u000f\u000fD (\u000f)\f2@nBE(\f\u000f)\n@\f!Z1\n\u000f0d\u000fD(\u000f)\u0011Zd3q\n(2\u0019)3:\n(44)\nWe conclude that the classical-to-quantum crossover\ncan be accounted for by inserting the factor\n\u0000\u000f\f2@nBE(\f\u000f)\n@\f=\u0014\f~!=2\nsinh(\f~!=2)\u00152\n\u0011F(\f~!) (45)\nin the energy integrand of Eq. (27), which e\u000bectively cuts\no\u000b the contribution from magnons with energy \u000f\u0011~!\u001d\nkBT.\nVII. RESULTS\nWe summarize the spin Seebeck coe\u000ecient dependence\non the ferromagnetic layer thickness, when \u0015\u001cl0\u001cl:\nS(d)\u00181\n2\u0019\u001528\n><\n>:\u000b; d\u001c\u0015\nd=l; \u0015\u001cd\u001cl0\nl0=l; l0\u001cd\u001cl\nl0=d; l\u001cd; (46)\nassuming\u0015\u001c\u0018(or else the magnon transport is ex-\nponentially frozen out). The four regimes correspond\nrespectively to the following physical situations: (1)\nOnly the lowest magnon subband is thermally active\n(d\u001c\u0015), (2) quasi-3D subband structure is activated, but\ndamping is still dominated by interfacial spin pumping6\n0.01110010410610-510-410-310-210-1\u0000l0l\nd/\u0000S\u00002\nFIG. 3. Plot of the spin Seebeck coe\u000ecient, Eq. (27), as a\nfunction of ferromagnet thickness dfor\u0015= 1 nm,\u0018= 10 nm,\nl0= 100 nm, and l= 10\u0016m. We use Eq. (37) (solid curve\non the left) and Eq. (41) (solid line on the right), which are\nvalid when d.landd&l, respectively, when N 1is a poor\nspin sink and N 2a perfect spin sink. To account for the\nclassical-to-quantum crossover, we have inserted factor (45)\nin the integrands of Eqs. (35) and (36) [with !!!n(q)] and\nEq. (41). The dotted curve shows the enhanced spin Seebeck\ncoe\u000ecient when also N 1is a perfect spin sink (which increases\nthe e\u000bective thermal bias between magnons and electrons at\nthe FjN2interface).\n(\u0015\u001cd\u001cl0), (3) bulk damping overtakes spin pumping,\nbut the magnetic \flm is still thinner than the thermal\nmagnon decay length, such that magnons probe the full\n\flm width ( l0\u001cd\u001cl), and (4) bulk regime is \fnally\nestablished when the \flm is thicker than the magnon de-\ncay length ( l\u001cd). To illustrate these crossovers, we plot\nthe spin Seebeck coe\u000ecient as a function of din Fig. 3,\nusing lengths characteristic of YIG, which are consistent\nwith the above length-scale hierarchy. Notice that even\nthough\u0018determines the magnon energy gap and the as-\nsociated Ginzburg-Landau correlation length in the clas-\nsical theory, it does not govern any prominent crossover\nin the function S(d).\nWe conclude that S(d) has nonmonotonic thickness de-\npendence, with the maximum value\nS(max)\u0018\u000b0\n2\u0019\u00153; (47)\nattained at l0.d.l, i.e., below the magnon decay\nlength but above l0, such that magnons equilibrate fully\nto the average phonon temperature \u0016T= (T1+T2)=2\n(d&l0) but still remain coherent on the scale of d\n(d.l). This agrees with the result obtained in Ref. 30.\nS(max)is proportional to the spin-mixing conductance\n[see Eq. (20)] but is independent of the bulk Gilbert\ndamping (as the magnon quasiparticle structure is still\nwell resolved). According to Eq. (21), S(max)determines\nthe largest spin current emitted thermally by a \flm of\nmagnetic insulator, as a function of d, when subjectedto a certain temperature di\u000berence (for example, in a\nwedged magnetic insulator coated by metallic contacts).\nIf, on the other hand, a well-de\fned temperature gradi-\nentis supplied (corresponding, for example, to a certain\nphonon-dominated heat-\rux density), while thickness d\nis varied, the spin-current density js/Sdincreases with\ndsaturating at d&l(the magnon decay length):\nj(max)\ns\f\f\f\n\fxed@xT\u0018l0\n2\u0019\u00152kB@xT; (48)\nwhich corresponds to the bulk regime. j(max)\ns vanishes\nwhen\u000b0!0 (no spin pumping) or \u000b!1 (no magnetic\ndynamics).\nIt is interesting to ask how the above results would\nmodify if both N 1and N 2in our model (see Fig. 1) were\nperfect spin sinks. For an inversion-symmetric struc-\nture (e.g., PtjYIGjPt), spin currents at the two inter-\nfaces must be equal, js1=js2\u0011js. Whend\u001dl, we\nshould recover the bulk limit (41), since the magnons de-\ncay before traversing the full width of the \flm (and thus\nthe spin current at one interface should not be sensitive\nto the boundary condition at the other). When d\u001cl,\nhowever,S0andSnentering Eq. (37) need to be modi-\n\fed. To that end, we notice that the factor (1+ l0=d)\u00001in\nEq. (35) re\rects the di\u000berence between Tm;0, the e\u000bective\ntemperature of the magnons, and T2, the temperature of\nthe electrons in N 2:8\nTm;0\u0000T2=\u000b\u0016T+ (\u000b0=d)T2\n\u000b+\u000b0=d=T1\u0000T2\n21\n1 +l0=d:(49)\nSimilarly for the n>0 subbands, the factor (1+2 l0=d)\u00001\nin Eq. (36) stems from the e\u000bective magnon-electron tem-\nperature di\u000berence across the F jN2interface of\nTm;n\u0000T2=T1\u0000T2\n21\n1 + 2l0=d: (50)\nWhen the bulk damping dominates over the interfa-\ncial spin pumping, i.e., l0\u001cd(while still d\u001cl),\nTm;n!(T1+T2)=2, while in the opposite limit, i.e.,\nd\u001cl0,Tm;n!T2. The magnon temperature thus\nbecomes strongly skewed toward the F jN2interface for\nthinner ferromagnetic \flms, when N 1is a poor spin sink\n(with electrons and magnons, therefore, being essentially\ndecoupled at the N 1jF interface, for our purposes). In\nthe case when both N 1and N 2are perfect spin sinks, on\nthe other hand, the e\u000bective magnon-electron tempera-\nture di\u000berence driving spin current is given simply by\n\u0016T\u0000T2= (T1\u0000T2)=2 for all subbands (when d\u001cl). We\naccount for this increased thermal gradient by dropping\nthe factor (1 + l0=d)\u00001on the right-hand side of Eq. (35)\nand likewise (1+2 l0=d)\u00001in Eq. (36). The corresponding\nenhancement of the spin Seebeck coe\u000ecient reverses the\ntrend in Eq. (46) at d.l0to give\nS(d)\u00181\n2\u0019\u001528\n<\n:\u000b0=d; d\u001c\u0015\nl0=l; \u0015\u001cd\u001cl\nl0=d; l\u001cd; (51)7\n0.01110010410610-510-410-310-210-1\u0000l0l\nd/\u0000S\u00002\nFIG. 4. Plot of the spin Seebeck coe\u000ecient using magnetic\nlength scales as in Fig. (3) but calculated for a non-inversion-\nsymmetric N 1jFjN2structure with spin-pumping parameters\n\u000b0\n1=\u000b0=10,\u000b0\n2=\u000b0, respectively, at the two interfaces (upper\ntrace) and\u000b0\n1=\u000b0,\u000b0\n2=\u000b0=10 (lower trace). Note that the\nformer case is intermediate between the two curves plotted in\nFig. 3 (where the solid curve corresponds to \u000b0\n1= 0,\u000b0\n2=\u000b0\nand the dotted curve to \u000b0\n1=\u000b0\n2=\u000b0).\nwhich is now monotonically decreasing with d, as plotted\nby the dotted line in Fig. 3.\nWhen the structure N 1jFjN2is not mirror symmetric\n(either because the spin-mixing conductances or the spin-\nsink characteristics are di\u000berent), which we characterize\nby di\u000berent \u000b0\n1and\u000b0\n2spin-pumping parameters at the\ntwo interfaces, we can repeat the above analysis for d\u001cl,\n\fnding\nTm;0\u0000T2=\u000b\u0016T+ (\u000b0\n1=d)T1+ (\u000b0\n2=d)T2\n\u000b+\u000b0\n1=d+\u000b0\n2=d\n=T1\u0000T2\n21 + 2l0\n1=d\n1 + (l0\n1+l0\n2)=d(52)\nand\nTm;n\u0000T2=T1\u0000T2\n21 + 4l0\n1=d\n1 + 2(l0\n1+l0\n2)=d; (53)\nwhere we de\fned the spin-pumping lengths l0\ni\u0011\u000b0\ni=\u000bas-\nsociated with the left, i= 1, and right, i= 2, interfaces.\nWe thus generalize the spin Seebeck contributions (at the\nFjN2interface) from di\u000berent magnon subbands to\nS0\u0019\u000b0\n2\nd1 + 2l0\n1=d\n1 + (l0\n1+l0\n2)=dZ1\n\u00001d2q\n(2\u0019)2F[\f~!0(q)] (54)\nand\nSn>0\u00192\u000b0\n2\nd1 + 4l0\n1=d\n1 + 2(l0\n1+l0\n2)=dZ1\n\u00001d2q\n(2\u0019)2F[\f~!n(q)]\n(55)\nin lieu of Eqs. (35) and (36), respectively. The asymptotic\nand crossover trends are now given by (assuming \u0015\u001cl0\n1;l0\n2\u001cl):\nS(d)\u00181\n2\u0019\u001528\n>><\n>>:2=(\u000b0\n1\u00001+\u000b0\n2\u00001)d; d\u001c\u0015\n2=(l0\n1\u00001+l0\n2\u00001)l; \u0015\u001cd\u001cl0\n1\nl0\n2=l; l0\n1;l0\n2\u001cd\u001cl\nl0\n2=d; l \u001cd;\n(56)\nwhich coincides with Eq. (51) when \u000b0\n1=\u000b0\n2but has an\nadditional shoulder-like feature at d\u0018(l0\n1+l0\n2)=2 . This\nfeature makes S(d) nonmonotonic when \u000b0\n1< \u000b0\n2. In\nFig. 4, we plot the spin Seebeck coe\u000ecient for two asym-\nmetric cases: (1) \u000b0\n1=\u000b0=10,\u000b0\n2=\u000b0and (2)\u000b0\n1=\u000b0,\n\u000b0\n2=\u000b0=10 (physically corresponding to spin currents on\ntwo sides of a non-inversion-symmetric N 1jFjN2struc-\nture, such as, PdjYIGjPt).\nVIII. DISCUSSION\nOur theory provides a minimalistic application of the\nLLG phenomenology to the problem of the spin See-\nbeck e\u000bect, yet disregards magnon-magnon interactions.\nThese become important at high temperatures, especially\napproaching the Curie temperature. Magnon-phonon in-\nteractions are included only insofar as a contribution to\nthe total Gilbert damping. Elastic magnon scattering on\nimpurities, which would manifest as an inhomogeneous\nbroadening of ferromagnetic-resonance linewidth, may\nbe an important impediment to the thermally-induced\nspin currents in disordered \flms. The bulk limit of spin\ncurrent, Eq. (48), is reduced by disorder, as well as the\nmagnon decay length describing the crossover to the bulk\nregime. When the magnon mean free path l\u0003is shorter\nthan our Gilbert damping decay length l, in particular,\nwe expect the e\u000bective decay length to be le\u000b\u0018p\nll\u0003(the\nspin di\u000busion length) and j(max)\ns in Eq. (48) to be reduced\nby a factor of l=le\u000b\u0018p\nl=l\u0003(assuming that l0\u001cle\u000b,\nsuch that our length-scale hierarchy is unchanged). The\nSeebeck coe\u000ecient behavior (46) [as well as Eq. (47)],\nhowever, remain essentially intact up to the thickness\nd\u0018le\u000b.\nFinally, we want to comment on a possibility of nonlo-\ncal magnetic relaxation. In this paper, we have assumed\nthat Gilbert damping is a local and isotropic tensor.\nThe locality would be a reasonable approximation if the\ndamping bottleneck was due to some local dynamic de-\nfects. In the case of YIG,31which is known for its highly\ncoherent elastic properties, nonlocality of the bulk mag-\nnetic relaxation could signi\fcantly modify our \fndings.\nFirst of all, this could introduce new phonon-dependent\nlength scales into the problem, which would show in the\nS(d) dependence. Standard long-wavelength ferromag-\nnetic resonance on thick \flms would reveal damping \u000b\nthat could be very di\u000berent from that of short-wavelength\nthermal magnons relevant here. An e\u000bective damping\nparameter ~ \u000bof thermal magnons (which may itself be\nthickness dependent) would result in the bulk crossover\nthickness of ~l\u0018\u0015=~\u000b6=l. Whend.~l, we may still invoke8\nthe \frst three regimes of our \fndings, Eq. (46) (with \u000b\nandlcorresponding to the thermal-magnon inverse qual-\nity factor and decay length, respectively, due to magnon-\nphonon scattering), which should, furthermore, be indif-\nferent to the fact that the local temperature, Eq. (8), is\nnot well de\fned for highly-coherent phonons. The rea-\nson for this is that, in these regimes, when dis below the\nmagnon decay length ~l, only the average phonon tem-\nperature \u0016Tis relevant for our theory. 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W. Bauer, K. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 82, 099904(E) (2010).\n31V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep.\n229, 81 (1993)."    },    {        "title": "1305.4961v1.Characterization_and_Synthesis_of_Rayleigh_Damped_Elastodynamic_Networks.pdf",        "content": "Manuscript submitted to Website: http://AIMsciences.org\nAIMS' Journals\nVolume X, Number 0X, XX200X pp.X{XX\nAlessandro Gondolo and Fernando Guevara Vasquez\nMathematics Department, University of Utah\n155 S 1400 E RM 233\nSalt Lake City, UT 84112-0090, USA\nCHARACTERIZATION AND SYNTHESIS OF\nRAYLEIGH DAMPED ELASTODYNAMIC NETWORKS\nAbstract. We consider damped elastodynamic networks where the damping\nmatrix is assumed to be a non-negative linear combination of the sti\u000bness\nand mass matrices (also known as Rayleigh or proportional damping). We\ngive here a characterization of the frequency response of such networks. We\nalso answer the synthesis question for such networks, i.e., how to construct\na Rayleigh damped elastodynamic network with a given frequency response.\nOur analysis shows that not all damped elastodynamic networks can be realized\nwhen the proportionality constants between the damping matrix and the mass\nand sti\u000bness matrices are \fxed.\n1.Introduction. The second Newton's law applied to a network of springs, masses,\nand dampers gives\nKu+C_u+Mu=f; (1)\nwhere uis the vector of displacements of the network nodes and fis the vector\nof external forces. The sti\u000bness, damping, and mass matrices are K,C, and M,\nrespectively. These matrices are de\fned in section 2. We are mainly concerned\nwith networks where the damping is proportional or of Rayleigh type , which means\nthat there are some known proportionality constants \u000b;\f\u00150 linking the damping\nmatrix to the sti\u000bness and mass matrices:\nC=\u000bK+\fM: (2)\nThe Rayleigh damping assumption means that:\n(a) The springs have a dashpot in parallel with damping constant proportional to\nthe spring constant. One way of achieving this is to construct springs from\nKelvin-Voigt solids (see e.g., [5, x7.3]), where the damping constant is propor-\ntional to the sti\u000bness constant.\n(b) Each mass lies in a \fxed cavity \flled with a viscous liquid, where the viscosity\nconstant is proportional to the mass, or the cavity size is adjusted to obtain the\nsame e\u000bect (i.e., inversely proportional to the mass).\nThe proportionality constants \u000b(linking the damping to the sti\u000bness) and \f(linking\nthe damping to the mass) are assumed to be the same for all the springs and nodes\nin the network. An example of such a network is given in \fgure 1.\n2000 Mathematics Subject Classi\fcation. Primary: 74B05, 35R02.\nKey words and phrases. Elastodynamic networks, Response function, Damping, Network syn-\nthesis, Proportional damping.\n1arXiv:1305.4961v1  [math-ph]  21 May 20132 A. GONDOLO AND F. GUEVARA VASQUEZ\n1\nFigure 1. An example of a Rayleigh damped network. For the\nterminal nodes the masses are colored in white and for the in-\nternal nodes in black. Each mass is surrounded by a cavity con-\ntaining a viscous liquid. The damping coe\u000ecient of each of the\nlinear dampers is \u000btimes the sti\u000bness constant of the correspond-\ning spring, and the viscous damping coe\u000ecient of each mass is \f\ntimes the corresponding mass.\nWe answer two questions about the (frequency) response of networks with this\nclass of damping. The response is the frequency-dependent linear relationship be-\ntween the displacements and forces at a few terminal, or accessible, nodes. The\n\frst question we answer is the characterization question, i.e., we give the form of\nall possible responses for this particular class of networks. The second question is\nthe synthesis: can we build a network from this class that mimics any admissible\nresponse?\nFor static networks (zero frequency), the characterization and synthesis questions\nwere established by Camar-Eddine and Seppecher [3], as part of a characterization\nof the possible macroscopic behaviors of a static elastic material under a single\ndisplacement \feld. Then, Milton and Seppecher [11] characterized the response of\ndamped elastic networks at a single, non-resonant frequency, and found it is possible\nto build a damped elastic network that mimics a prescribed response at one single\nfrequency . The complete characterization and synthesis for the undamped case is\ndone in [10]: it is shown that the response is a matrix with rational function (of\nfrequency) entries, and that given the response for an undamped elastic network,\nit is possible to build a network that mimics this response for all non-resonant\nfrequencies . The new characterization and synthesis results that we present here\nare a generalization of the results in [10] to damping of Rayleigh type. In particular,\nour results show that the Rayleigh damping model is incomplete, in the sense that\nit cannot describe, by itself, the responses of all possible elastodynamic networks\nwith damping.\nThe characterization and synthesis for elastic networks have been used by Camar-\nEddine and Seppecher [3] to show how to design a (linear) elastic material that\nhas a prescribed response under a single displacement \feld. A similar technique\nexploiting the characterization of networks of resistors was used by Camar-Eddine\nand Seppecher [2] to show how to design a conductor that has a prescribed response\nunder a single voltage \feld. Synthesis results are also known for electrical networks\nwith resistors (Kirchho\u000b's Y\u0000\u0001 theorem, Curtis, Ingerman, and Morrow [6] for\nplanar resistor networks); networks of resistors, capacitors and inductors (FosterRAYLEIGH DAMPED NETWORKS 3\n[7, 8], Bott and Du\u000en [1], Milton and Seppecher [11]); acoustic networks (Milton\nand Seppecher [11]); and an electromagnetic version of elastodynamic networks\n(Milton and Seppecher [12, 13]).\nWe start in section 2 by de\fning the sti\u000bness, mass, and damping matrices for\na damped elastodynamic network and, given only access to a few terminal nodes,\nits frequency response. Then in section 3, we show how the characterization and\nsynthesis results for static elastic networks in [3, 10] can be generalized to massless\nelastic networks with Rayleigh damping. For general Rayleigh damped networks,\nthe characterization result appears in section 4 and the synthesis result in section 5.\nWe \fnish by giving in section 6 the loci of the resonances of Rayleigh damped\nnetwork.\n2.The frequency response of an elastodynamic network with damping.\nThe linearized Hooke's Law for a single spring located between nodes x1andx2\nrelates the forces ai(supported at node xi) to the displacements ui(assuming these\nare small) through the relation\na2=\u0000k1;2(x2\u0000x1)(x2\u0000x1)T\nkx2\u0000x1k2(u2\u0000u1) =\u0000a1:\nLetKbe the sti\u000bness matrix, Cbe the damping matrix, and Mbe a diagonal\nmatrix with the masses of the nodes in the diagonal. In the case of a single spring,\nthe sti\u000bness and mass matrices are\nK=k1;2\u0014n1;2nT\n1;2\u0000n1;2nT\n1;2\n\u0000n1;2nT\n1;2n1;2nT\n1;2\u0015\n;M= diag (m1e;m2e);\nn1;2=x2\u0000x1\nkx2\u0000x1k;\nwhere e= (1;:::; 1)T2Rd,d= 2;3. For a network with Nnodes, the mass\nmatrix is an Nd\u0002Nddiagonal matrix that is de\fned similarly with the nodal\nmassesm1;:::;mN. The sti\u000bness and damping matrices of a network are Nd\u0002Nd\nmatrices obtained by adding the contributions from individual springs or dashpots:\nK=X\nspringijkijNijandC=X\nspringijcijNij (3)\nwherekij(resp.cij) is the spring (resp. damping) constant for the spring (resp.\ndamper) between nodes iandjand\nNij=\u0012\u0002eiej\u0003\u0014\n1\u00001\n\u00001 1\u0015\u0014eT\ni\neT\nj\u0015\u0013\n\n(nijnT\nij): (4)\nHere we used the Kronecker product \nand thei\u0000th canonical basis vector ei2RN.\nAs in the case of a single spring, the vector nij2Rdis a unit length vector with\ndirection xi\u0000xj.\nFor a network of springs, masses, and dashpots, the balance of forces (Newton's\nsecond law) gives a system of ODEs, which is given in (1). Now, recall the Laplace\ntransform:\neu(\u0015) =L[u(t)](\u0015) =Z1\n0e\u0000\u0015tu(t)dt:\nApplying the Laplace transform to (1), the ODE becomes\n(K+\u0015C+\u00152M)eu=ef: (5)4 A. GONDOLO AND F. GUEVARA VASQUEZ\nIn the following, we only work in the Laplace domain; the tilde notation is dropped\nfor simplicity. Our results can also be formulated in the frequency domain by using\nthe Fourier transform instead of the Laplace transform on (1), or simply by setting\n\u0015={!.\nFor the purpose of introducing the problem, we \frst consider the case where all\nnodes have a non-zero mass. Let us partition the nodes of the network into terminal\nnodes (B) and interior nodes ( I). At the interior nodes, the external forces are zero;\nthe displacement of the interior nodes is determined by the displacement of the\nboundary nodes, except at the few frequencies corresponding to the resonances of\nthe system. If subscripts BandIdenote the quantities referring to their respective\nnodes, then the displacements at the boundary nodes are related to the forces at\nthe boundary nodes by\nfB=W(\u0015)uB;\nwhere the displacement to forces map is\nW(\u0015) =KBB+\u0015CBB+\u00152MBB\n\u0000(KBI+\u0015CBI)(KII+\u0015CII+\u00152MII)\u00001(KIB+\u0015CIB);(6)\nwhich is the Schur complement of the IIblock in the matrix K+\u0015C+\u00152M(see\nAppendix A for the de\fnition of the Schur complement and properties).\nRemark 1. We assume throughout this paper that each spring-damper pair oc-\ncupies an arbitrarily small volume surrounding the segment connecting the corre-\nsponding nodes. We also assume that the cavities with viscous \ruid (and hence the\nmasses) can be made arbitrarily small. We need these assumptions to constrain the\ninternal nodes in our synthesis result to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes.\n3.Massless Rayleigh damped networks. First, we characterize the networks\nwith Rayleigh damping and no mass. Later, in section 4, we use such networks to\nsimplify the work for general networks. The theorems in this section are a natural\nextension of the characterization and synthesis in [3, 10] for static networks, i.e.,\nnetworks with springs only. In the static case the response matrix is given by (6)\nwith\u0015= 0. The forces f= (fT\n1;:::;fT\nn)Tat the nodes x1;:::;xnform a balanced\nsystem of forces when\nnX\ni=1fi=0(equilibrium of forces) andnX\ni=1xi\u0002fi=0(equilibrium of torques) ;(7)\nwhere u\u0002v= (u2v3\u0000u3v2;u3v1\u0000u1v3;u1v2\u0000u2v1) ifd= 3 and u\u0002v=u1v2\u0000u2v1\nifd= 2, is the usual cross product. The characterization and synthesis theorems in\n[3, 11, 10] are summarized in the next theorem.\nTheorem 3.1. The response matrix Wof any network of springs is symmetric\npositive semide\fnite where each column is a balanced system of forces at the terminal\nnodes. (Characterization)\nConversely, any matrix Wthat is symmetric positive semide\fnite where each\ncolumn is a balanced system of forces at the terminals can be realized by a network\nof only springs. Moreover the internal nodes of such network can be chosen so as\nto avoid a \fnite number of points and to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes. (Synthesis)RAYLEIGH DAMPED NETWORKS 5\nProof. See e.g. [10, Lemma 2] for the characterization and [10, Theorem 1] for the\nsynthesis.\nTheorem 3.1 can be readily extended to massless Rayleigh damped networks.\nTheorem 3.2. The response of a massless network with Rayleigh damping is of\nthe form\n(1 +\u000b\u0015)W; (8)\nwhere\u000b\u00150,\u0015is the Laplace parameter, and Wis the response of a static network,\ni.e.,Wis symmetric positive semide\fnite with each column being a balanced system\nof forces at the terminals x1;:::;xn(as in theorem 3.1).\nConversely, given any matrix-valued function of \u0015of the form (8)where Wis\nthe response of a static network at the nodes x1;:::;xn, there is a massless Rayleigh\ndamped network with terminals at the nodes x1;:::;xnrealizing it. Moreover the\ninternal nodes of such network can be chosen so as to avoid a \fnite number of points\nand to be within an \u000f\u0000neighborhood of the convex hull of the terminal nodes.\nProof. The forward direction is due to the homogeneity of degree 1 of the Schur\ncomplement (see Appendix A). Let Kbe the sti\u000bness matrix for a static network\nandWthe response matrix of the network at some terminal nodes B. Then the\nfrequency response of the massless network with Rayleigh damping when all the\nnodes are terminals is (1 + \u000b\u0015)K. Then by taking the Schur complement, the\nresponse at the nodes Bis (1 +\u000b\u0015)W.\nTo prove the converse, assume we are given a frequency response of the form\n(1 +\u000b\u0015)W, where Wis the response of a static network. By the second part of\ntheorem 3.1, there is a network of springs with sti\u000bness matrix Kwith response\nWat the nodes B. The internal nodes of this network can be chosen so as to\navoid a \fnite number of points and to be in an arbitrarily small neighborhood\nof the convex hull of the terminals. Then a network that has (1 + \u000b\u0015)Was its\nresponse is the network with response at all the nodes (1 + \u000b\u0015)K, i.e., the network\nobtained from the second part of theorem 3.1 with dashpots in parallel with each\nspring, each dashpot having a damping constant being \u000btimes the sti\u000bness of the\nassociated spring. Here we have used the assumption that each spring and damper\npair occupies an arbitrarily thin segment between the corresponding nodes, so the\nnetwork transformations that do not change the response in [10, x2.3] are still\nvalid.\nRemark 2. Theorem 3.2 is valid for planar networks, i.e., networks for which all\nthe springs lie in a plane. This is because theorem 3.1 is valid for planar networks:\none can always realize the response of a planar network with a planar network.\n4.Characterization of general Rayleigh damped networks. In this section,\nwe give conditions that the response of Rayleigh damped networks needs to satisfy.\nIn general, we may have massless nodes in the network; these are dealt with in\nlemma 4.1, where we use the characterization for the massless case from section 3\nto eliminate any massless interior nodes. Theorem 4.2 is the characterization result\nfor general Rayleigh networks. Then Proposition 1 shows that the conditions of\ntheorem 4.2 are consistent with the characterization at a single frequency found\nby Milton and Seppecher [11] (a condition which means, in physical terms, that\ndamping can only consume energy).6 A. GONDOLO AND F. GUEVARA VASQUEZ\nLet us partition the interior nodes Iinto the set of interior nodes Jwith positive\nmass and the set of massless nodes L. Clearly, MJJis positive de\fnite while\nMLL=0. The following lemma is similar to [10, Lemma 3] and reduces the\ncharacterization problem for networks with massless nodes to networks where each\nnode has mass.\nLemma 4.1. LetKbe aNd\u0002Nd sti\u000bness matrix and let Mbe aNd\u0002Nd\n(diagonal) mass matrix, where N=jB[Ij. The response at the terminals is\nW(\u0015) =eKBB+\u0015eCBB+\u00152MBB\n\u0000(eKBJ+\u0015eCBJ)(eKJJ+\u0015eCJJ+\u00152MJJ)\u00001(eKJB+\u0015eCJB);\nfor all frequencies \u0015for which det(eKJJ+\u0015eCJJ+\u00152MJJ)6= 0. Here the tilde\nmatrices are submatrices of the matrices\neK=\"\neKBBeKBJ\neKJBeKJJ#\n=\u0014KBBKBJ\nKJBKJJ\u0015\n\u0000\u0014KBL\nKJL\u0015\nKy\nLL\u0002KLBKLJ\u0003\nand\neC=\"\neCBBeCBJ\neCJBeCJJ#\n=\u0014CBBCBJ\nCJBCJJ\u0015\n\u0000\u0014CBL\nCJL\u0015\nCy\nLL\u0002CLBCLJ\u0003\n;(9)\nwhere the symbol ydenotes the Moore-Penrose pseudoinverse. Also, eKJJandeCJJ\nare positive semide\fnite.\nProof. The response of the network with terminals B[Jand internal nodes Lis\neK+\u0015eC+\u00152diag ( MBB;MJJ). Equilibrating forces at the nodes J(i.e., taking\nthe Schur complement of the JJblock) gives the desired result. Note that eKJJ\nandeCJJare positive semide\fnite as they are the Schur complement of positive\nsemide\fnite matrices (see e.g., (14)).\nRemark 3. For Rayleigh networks, the elimination of massless nodes made in\nlemma 4.1 preserves the Rayleigh damping structure. If we have Rayleigh damping\nthenC=\u000bK+\fM(at all nodes) and\neC=\u000beK+\fdiag ( MBB;MJJ);\nwhere the tilde matrices are de\fned as in lemma 4.1. Indeed a simple calculation\ngives\neC=\u000b\u0014KBBKBJ\nKJBKJJ\u0015\n\u0000\u000b2\u0014KBL\nKJL\u0015\n(\u000bKLL)y\u0002KLBKLJ\u0003\n+\f\u0014MBB\nMJJ\u0015\n:\nWe can now formulate a characterization of the response at the terminal nodes\nof a Rayleigh damped network.\nTheorem 4.2. Consider a damped mass-spring network with terminals x1;:::;xn\nwith Rayleigh damping, i.e., the damping matrix is of the form C=\u000bK+\fM,\nwhere\u000b;\f\u00150are given. The displacement-to-forces map of any such network is\nof the form:\nW(\u0015) = (1 +\u000b\u0015)A+ (\f\u0015+\u00152)M\u0000pX\nj=1(1 +\u000b\u0015)2R(j)\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152; (10)\nwhereRAYLEIGH DAMPED NETWORKS 7\ni.R(j)is real symmetric positive semide\fnite and \u001bj>0.\nii.Mis real diagonal positive semide\fnite.\niii.Ais real symmetric positive semide\fnite.\niv. There are at most 2pdistinct poles: namely the roots of the polynomials\nqj(\u0015) =\u001bj+\u0015(\u000b\u001bj+\f) +\u00152;forj= 1;:::;p:\nMoreover, for all roots \u0015\u0003ofqj(\u0015), we have Re(\u0015\u0003)\u00140. This is the second law\nof thermodynamics: damping consumes energy.\nv. The response for \u0015= 0, i.e.,\nW(0) = A\u0000pX\nj=1R(j)\n\u001bj;\nis the response of a static network. The characterization for static elastic\nnetworks [3] states that W(0)must be real symmetric positive semide\fnite\nwith every column being a balanced system of forces (see (7)) at the terminals\nx1;\u0001\u0001\u0001;xn.\nProof. We begin by using lemma 4.1 to remove the massless interior nodes. Let\nI=J[LwhereJare the positive mass nodes and Lare the massless nodes. Let\neKandeCbe as in (9). Then lemma 4.1 gives\nW(\u0015) =eKBB+\u0015eCBB+\u00152MBB\n\u0000(eKBJ+\u0015eCBJ)(eKJJ+\u0015eCJJ+\u00152MJJ)\u00001(eKJB+\u0015eCJB):(11)\nSince MJJis real diagonal positive de\fnite, it has a square root M1=2\nJJ. Now\neKJJis real symmetric positive semide\fnite and so is M\u00001=2\nJJeKJJM\u00001=2\nJJ. Then by\nthe spectral theorem, there exists Uunitary and \u0006real diagonal with nonnegative\nentries so that\nM\u00001=2\nJJeKJJM\u00001=2\nJJ =U\u0006UT:\nConsider the matrix X=M\u00001=2\nJJU, which clearly is invertible and is so that\nXTeKJJX=\u0006andXTMJJX=I;\nwhere Iis the identity matrix (in this case jJj\u0002jJj). The resolvent part of (11)\nbecomes\nQ(\u0015) =eKJJ+\u0015eCJJ+\u00152MJJ\n=X\u0000T(XTeKJJX+\u0015XTeCJJX+\u00152XTMJJX)X\u00001\n=X\u0000T(\u0006+\u0015(\u000b\u0006+\fI) +\u00152I)X\u00001\n=X\u0000TD(\u0015)X\u00001:\nHere we used the result from remark 3, i.e., that eCJJ=\u000beKJJ+\fMJJ. The matrix\nD(\u0015) is diagonal and therefore easily inverted. Then the resolvent [ Q(\u0015)]\u00001can be\nwritten as\n[Q(\u0015)]\u00001=XD(\u0015)\u00001XT\n=Xdiag\u00121\n\u001bi+\u0015(\u000b\u001bi+\f) +\u00152\u0013\nXT;8 A. GONDOLO AND F. GUEVARA VASQUEZ\nwhere\u001biare the diagonal elements of \u0006. Let KBJX= [v1\u0001\u0001\u0001vjJj]. By remark 3\nwe haveeCBB=\u000beKBB+\fMBBandeCBJ=\u000beKBJ. Therefore the response (11)\ncan be written in the form\nW(\u0015) = (1 +\u000b\u0015)eKBB+ (\f\u0015+\u00152)MBB\u0000jJjX\ni=1vivT\ni(1 +\u000b\u0015)2\n\u001bi+\u0015(\u000b\u001bi+\f) +\u00152:\nNotice that the \u001bimay be zero. This is the case when the network has a so\ncalled \\\roppy mode\" (non-zero displacements with zero force required). The same\nargument in [10, Lemma 12] can be used to deal with \roppy modes. Indeed Lemma\n1 in [10] can be used to show that \u001bi= 0 implies vi= 0. By adding up the non-\nzero residues vivT\nithat share the same denominator, we get the symmetric positive\nsemide\fnite residues R(j)of the statement of the theorem. The result follows by\nnoticing that A=eKBBis symmetric positive semide\fnite, MBBis diagonal with\nnonnegative entries and at \u0015= 0, the response\nW(0) = A\u0000pX\nj=1R(j)\n\u001bj;\nis the response of a static elastic network.\nFinally for the roots \u0015(j)\n\u0006of\u001bj+ (\u000b\u001bj+\f)\u0015+\u00152we have\n2 Re\u0015(j)\n+=\u0000(\u000b\u001bj+\f)\u00140:\nHence the resonances lie in the left half plane.\nRemark 4. Setting\u0015= 0 in Theorem 4.2 gives the static or zero frequency result\nin [3]. Setting \u000b=\f= 0 (i.e., no damping) gives the result in [10] for elastodynamic\nnetworks with no damping.\nFor the Laplace parameter \u0015in the imaginary axis (i.e., real frequencies), the\neigenvalues of the imaginary part of the response should have a sign consistent\nwith the energy losses due to damping (again a manifestation of the second law of\nthermodynamics). These inequalities are essential to the single-frequency charac-\nterization and synthesis of Milton and Seppecher [11], and hence these inequalities\nshould also hold in our case. The next proposition shows that these inequalities hold\nautomatically for matrix functions of \u0015satisfying the hypothesis of theorem 4.2.\nProposition 1. Any matrix function of \u0015of the form (10) and with the properties\nof theorem 4.2 is such that\n!ImW({!)\u00150for any!2R;\nwhere the inequality A\u00150for a symmetric matrix A, means Ais positive semi-\nde\fnite.\nProof. Let us rewrite the matrix-valued function W(\u0015) in (10) as\nW(\u0015) =W1(\u0015) +W2(\u0015) +W3(\u0015);RAYLEIGH DAMPED NETWORKS 9\nwhere\nW1(\u0015) = (1 +\u000b\u0015)0\n@A\u0000pX\nj=1R(j)\n\u001bj1\nA= (1 +\u000b\u0015)W(0);\nW2(\u0015) = (\f\u0015+\u00152)M;and\nW3(\u0015) =pX\nj=1\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001bj\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152\u0015R(j)\n\u001bj:\nTo prove the \fnal result, it is enough to show that !ImWk({!)\u00150,k= 1;2;3. The\n\frst two cases are clear since: !ImW1({!) =\u000b!2W(0)\u00150 because W(0)\u00150;\nand!ImW2({!) =\f!2M\u00150 because Mis diagonal with nonnegative entries.\nSince R(j)=\u001bj\u00150, we have proved the inequality for the third case if for all\n\u001b>0 the function\nf(\u0015) = (1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+\u0015(\u000b\u001b+\f) +\u00152\nis such that\n!Imf({!)\u00150;for!2R.\nTo show this inequality, consider the 2 \u00022 complex symmetric matrix\nB(\u0015) =\"\n1 +\u000b\u0015 1 +\u000b\u0015\n1 +\u000b\u0015\u001b+\u0015(\u000b\u001b+\f)+\u00152\n\u001b#\n:\nWe start with !\u00150. Clearly Im B({!)\u00150 since det(Im B({!)) =!\u000b\f=\u001b\u00150\nand all the entries of Im B({!) are non-negative. Then noticing that f(\u0015) is the\nSchur complement of the 2 ;2 entry in B(\u0015) and using the property of the Schur\ncomplement (14), we have that Im B({!)\u00150)Imf({!)\u00150. So we get the result\n!f({!)\u00150 when!\u00150.\nFor!\u00140, the same reasoning holds. We only need to check that Im B({!)\u00140.\nThis is true because all the entries of Im B({!) are non-positive and det(Im B({!)) =\n!\u000b\f=\u001b\u00140. Therefore !f({!)\u00150 when!\u00140, which completes the proof.\n5.Synthesis of general Rayleigh damped networks. The basic building block\nfor the synthesis is a Rayleigh damped network with: a rank one response; a complex\nconjugate pair or a real pair of prescribed resonances; and zero static response. The\nexistence of such a network is stated in lemma 5.1 and proved at the end of this\nsection (it is a similar construction to that in [10, Lemma 12]). The synthesis of a\ngeneral Rayleigh damped network is done in theorem 5.2.\nLemma 5.1. Letx1;:::;xnbe some terminal nodes and let f1;:::;fnbe an ar-\nbitrary system of forces at the corresponding terminals (the forces do not need to\nbe balanced). Then for any \u000b;\f\u00150and\u001b > 0, it is possible to build a Rayleigh\ndamped network with zero static response and with\nW(\u0015) =\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+ (\u000b\u001b+\f)\u0015+\u00152\u0015\n\u000bT; (12)\nwhere f= [fT\n1;:::;fT\nn]T. The internal nodes of such a network can be chosen to\navoid a \fnite number of points and to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes.10 A. GONDOLO AND F. GUEVARA VASQUEZ\nTheorem 5.2. Given any matrix-valued function W(\u0015)of the form (10) and sat-\nisfying the conditions of theorem 4.2 with \u000band\f\fxed, there exists a Rayleigh\ndamped network with proportionality constants \u000band\fand response W(\u0015). The\ninternal nodes of such a network can be chosen to avoid a \fnite number of points\nand to be within an \u000f\u0000neighborhood of the convex hull of the terminal nodes.\nProof. The proof relies on the superposition of networks, i.e., using the fact that\nthe response of two networks that share terminal nodes and only terminal nodes,\nis the sum of the responses of each network (see e.g., [10, x2.4]). To specify the\nbuilding blocks needed to realize the matrix-valued function W(\u0015) in (10), we \frst\nrewrite it as:\nW(\u0015) =W1(\u0015) +W2(\u0015) +W3(\u0015);\nwhere\nW1(\u0015) = (1 +\u000b\u0015)0\n@A\u0000pX\nj=1R(j)\n\u001bj1\nA= (1 +\u000b\u0015)W(0);\nW2(\u0015) = (\f\u0015+\u00152)M;and\nW3(\u0015) =pX\nj=1\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001bj\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152\u0015R(j)\n\u001bj:\nWe now show how to realize each term W1,W2andW3separately by a network.\nThe superposition principle can then be used to \fnd a network realizing W.\nWe \frst use theorem 3.2 to see that a network realizing W1(\u0015) is the network\nof springs realizing the static response W(0) (existence guaranteed by theorem 3.1\nor [3]), with a damper with damping constant \u000btimes the spring constant added\nin parallel to each spring. To realize W2it su\u000eces to endow each terminal node\nby the mass dictated by Mand surrounding it by a cavity such that the resulting\ndamping is \ftimes the mass.\nWe now show that each term in the sum in the expression of W3is realizable.\nThen the realizability of W3follows from the superposition principle. Let us drop\nthe indexjfor the sake of simplicity and show that there is a Rayleigh damped\nnetwork with response:\n\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+\u0015(\u000b\u001b+\f) +\u00152\u0015R\n\u001b;\nwhen Ris real symmetric positive semide\fnite. By the spectral theorem it is\npossible to \fnd real vectors vksuch that\nR\n\u001b=rX\nk=1vkvT\nk;\nwherer >0 is the rank of R. (The case r= 0 is the trivial R=0case). Hence\nwe have reduced the problem to that of \fnding a network with zero static response,\na rank one response and as resonances the roots of p(\u0015) =\u001b+\u0015(\u000b\u001b+\f) +\u00152.\nLemma 5.1 shows how to build such a network. This completes the construction.\nWe now prove Lemma 5.1.\nProof. The main idea here is to \fnd a massless Rayleigh damped network and add\nappropriate masses to the internal nodes to get the desired resonances. According\nto [10, Lemma 12], it is possible to choose two distinct nodes xn+1andxn+2andRAYLEIGH DAMPED NETWORKS 11\ntwo forces fn+1andfn+2such that the system of forces f1;:::;fn+2is balanced\nwhen supported at the nodes x1;:::;xn+2, regardless of the choice of f. Then\nby theorem 3.2, there exists a massless Rayleigh damped network with rank one\nresponse (1 + \u000b\u0015)[fT;gT]T[fT;gT], where g= [fT\nn+1;fT\nn+2]T. We now endow the\ntwo internal nodes xn+1andxn+2with the same mass m, that is determined later\nto match the desired resonances. We also surround the nodes xn+1andxn+2by\na cavity with a viscous \ruid, designed so that the damping term is \fm. Then\nNewton's second law becomes\n\u0012\u0014f\ng\u0015\u0002\nfTgT\u0003\n(1 +\u000b\u0015) +\u00140\nmI\u0015\n(\f\u0015+\u00152)\u0013\u0014uB\nuI\u0015\n=\u0014W(\u0015)uB\n0\u0015\n;\nwhere uBanduIare the displacements of the boundary nodes x1;:::;xnand\ninterior nodes xn+1andxn+2, respectively, and W(\u0015) is the frequency response of\nthis network.\nNext, we take the Schur complement of the IIblock to \fnd the response W(\u0015):\nW(\u0015) = (1 +\u000b\u0015)\u000bT+ (1 +\u000b\u0015)\u0012\u0000(1 +\u000b\u0015)jgj2\n(1 +\u000b\u0015)jgj2+ (\f\u0015+\u00152)m\u0013\n\u000bT\n= (1 +\u000b\u0015)\u000bT\u0000\u0012(1 +\u000b\u0015)2(jgj2=m)\n\u00152+ (\u000b(jgj2=m) +\f)\u0015+ (jgj2=m)\u0013\n\u000bT:\nLetm=jgj2=\u001b. Then the response is\nW(\u0015) = (1 +\u000b\u0015)\u000bT\u0000\u0012(1 +\u000b\u0015)2\u001b\n\u00152+ (\u000b\u001b+\f)\u0015+\u001b\u0013\n\u000bT;\nwhich is the desired result since W(0) = 0.\n6.Resonances of Rayleigh damped networks. A natural question to ask is\nwhether it is possible to \fnd a Rayleigh network with \u000band\f\fxed with a resonance\nthat is located anywhere in the left half complex plane. Since the conditions of\ntheorem 4.2 are necessary and su\u000ecient for a matrix valued function of frequency\nto be the response of a Rayleigh damped network, this question is equivalent to\n\fnding the set of all possible resonances for a Rayleigh damped network with \u000band\n\f\fxed. Our analysis shows that this is set is not the left hand plane. This means\nthat the Rayleigh damping model is incomplete, since a general damped network\ncan have resonances anywhere in the left half complex plane.\nWe state here the loci of the resonances that can be realized with Rayleigh\ndamped networks. The derivation of these loci is standard and is not included\nhere. According to theorem 4.2, the resonances that can be realized with Rayleigh\ndamped networks are the roots of the quadratic \u00152+ (\u000b\u001b+\f)\u0015+\u001b, i.e.,\n\u0015\u0006=\u0000(\u000b\u001b+\f)\u0006p\n(\u000b\u001b+\f)2\u00004\u001b\n2:\nBecause we only can choose \u001b, the resonances lie in curves in the complex plane.\nThe speci\fc curves depend on \u000band\fand are as follows.\nDamping at the nodes only ( \u000b= 0):Union of the segment Im \u0015= 0;\n\u0000\f\u0014Re\u0015<0 and the line Re \u0015=\u0000\f=2.12 A. GONDOLO AND F. GUEVARA VASQUEZ\nRe\u0015Im\u0015\u0000\u0000\n\u0000\f\u0000\f=2\nDamping with dashpots between the nodes only ( \f= 0):Union of the\nnegative real axis Im \u0015= 0;Re\u0015 < 0 and the circlej\u0015+\u000b\u00001j=\u000b\u00001(the origin\nexcepted).\nRe\u0015Im\u0015\u0000\u0000\u000b\u00001\n\u00001=\u000b\nCase\u000b;\f6= 0and\u000b\f < 1:Union of the negative real axis Im \u0015= 0;Re\u0015<0\nand the circlej\u0015+\u000b\u00001j=\u000b\u00001p1\u0000\u000b\f.\nRe\u0015Im\u0015\u0000\n\u00001=\u000bp1\u0000\u000b\f\n\u000b\nCase\u000b;\f6= 0and\u000b\f\u00151:the negative real axis Im \u0015= 0;Re\u0015<0.\nRe\u0015Im\u0015\n7.Discussion and future work. We have established the characterization and\nsynthesis of the response for mass-spring networks with Rayleigh damping. In\nparticular, our result shows that, for each pair of \u000band\f, there is a class of\nresonances that can be realized when the damping matrix is C=\u000bK+\fM. Clearly,\nfrom section 6, when choosing di\u000berent values of \u000band\f, it is possible to have any\nresonance with negative real part. Hence, if we superimpose Rayleigh damped\nnetworks with di\u000berent values of \u000band\f, it is possible to construct a network\nwith any \fnite number of resonances in the left half complex plane (provided they\nare real or come in complex conjugate pairs). However, it is not clear whether\nthe response of a general damped network can be realized by superposing several\nRayleigh networks with di\u000berent \u000band\f. This is a question that we plan to explore.REFERENCES 13\nAnother extension of this work would be to consider general damping by using\nthe quadratic eigenvalue problem [14, 9]. Speci\fcally, we believe it is possible to use\nthe spectral decomposition for real symmetric quadratic pencils by Chu and Xu [4]\nto characterize the response of general damped networks. Then for the synthesis,\nwe would need to construct networks that realize the form of the response, but this\nis left for future work.\nAppendix A.Schur complement properties. Consider the partition of a ma-\ntrixA2Cn\u0002ninduced by the partition of f1;:::;nginto two sets IandB:\nA=\u0014ABBABI\nAIBAII\u0015\n:\nThe Schur complement of the IIblock in Ais de\fned as\nS=ABB\u0000ABIA\u00001\nIIAIB;\nprovided AIIis invertible. The Schur complement is homogeneous of degree 1, since\nfor nonzero \u00152C,\n\u0015S= (\u0015ABB)\u0000(\u0015ABI)(\u0015AII)\u00001(\u0015AIB):\nA quadratic form of the Schur complement is equivalent to a quadratic form of\nthe original matrix A, indeed by simple manipulations we have\nv\u0003\nBSvB=\u0014\nvB\nvI\u0015\u0003\nA\u0014\nvB\nvI\u0015\n;where vI=\u0000A\u00001\nIIAIBvB: (13)\nA consequence of (13) is that if A2Rn\u0002n,A\u00150 implies S\u00150 (and similarly\nfor the reverse inequality). Here the inequality A\u00150 is understood as Apositive\nsemi-de\fnite.\nThe quadratic form v\u0003AvforA2Cn\u0002nwithAT=A(i.e., complex symmetric),\ncan be written as\nRe(v\u0003Av) = (Re v)T(ReA)(Rev) + (Im v)T(ReA)(Imv);\nIm(v\u0003Av) = (Re v)T(ImA)(Rev) + (Im v)T(ImA)(Imv):\nBy combining this fact with (13) we have that for complex symmetric Aand its\nSchur complement S:\nReA\u00150)ReS\u00150 and Im A\u00150)ImS\u00150; (14)\nand similarly for the reverse inequalities.\nAcknowledgments. The authors are thankful to Graeme W. Milton, Daniel Onofrei\nand Pierre Seppecher for insightful conversations on this subject. The work of AG\nwas supported through the University of Utah Mathematics Department VIGRE-\nREU program. The work of FGV was partially supported by the National Science\nFoundation DMS-0934664.\nREFERENCES.\n[1] R. Bott and R. J. Du\u000en. Impedance synthesis without use of transformers.\nJournal of Applied Physics , 20:804, 1949. doi: 10.1063/1.1698532.\n[2] M. Camar-Eddine and P. Seppecher. Closure of the set of di\u000busion functionals\nwith respect to the Mosco-convergence. Math. Models Methods Appl. Sci. , 12\n(8):1153{1176, 2002. ISSN 0218-2025. doi: 10.1142/S0218202502002069.14 REFERENCES\n[3] M. Camar-Eddine and P. Seppecher. Determination of the closure of the set of\nelasticity functionals. Arch. Ration. Mech. Anal. , 170(3):211{245, 2003. ISSN\n0003-9527. doi: 10.1007/s00205-003-0272-7.\n[4] M. T. Chu and S.-F. Xu. Spectral decomposition of real symmetric quadratic\n\u0015-matrices and its applications. Math. Comp. , 78(265):293{313, 2009. ISSN\n0025-5718. doi: 10.1090/S0025-5718-08-02128-5.\n[5] T. J. Chung. General continuum mechanics . Cambridge University Press,\nCambridge, 2007. ISBN 978-0-521-87406-9; 0-521-87406-8.\n[6] E. B. Curtis, D. Ingerman, and J. A. Morrow. Circular planar graphs and\nresistor networks. Linear Algebra Appl. , 283(1-3):115{150, 1998. ISSN 0024-\n3795. doi: 10.1016/S0024-3795(98)10087-3.\n[7] R. M. Foster. A reactance theorem. The Bell System Technical Journal , 3:\n259{267, 1924. ISSN 0005-8580.\n[8] R. M. Foster. Theorems regarding the driving-point impedance of two-mesh\ncircuits. The Bell System Technical Journal , 3:651{685, 1924. ISSN 0005-8580.\n[9] I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials . Academic\nPress Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. ISBN\n0-12-287160-X. Computer Science and Applied Mathematics.\n[10] F. Guevara Vasquez, G. W. Milton, and D. Onofrei. Complete characterization\nand synthesis of the response function of elastodynamic networks. J. Elasticity ,\n102(1):31{54, 2011. ISSN 0374-3535. doi: 10.1007/s10659-010-9260-y.\n[11] G. W. Milton and P. Seppecher. Realizable response matrices of multi-terminal\nelectrical, acoustic and elastodynamic networks at a given frequency. Proc. R.\nSoc. Lond. Ser. A Math. Phys. Eng. Sci. , 464(2092):967{986, 2008. ISSN 1364-\n5021. doi: 10.1098/rspa.2007.0345.\n[12] G. W. Milton and P. Seppecher. Electromagnetic circuits. Netw. Heterog.\nMedia , 5(2):335{360, 2010. ISSN 1556-1801. doi: 10.3934/nhm.2010.5.335.\n[13] G. W. Milton and P. Seppecher. Hybrid electromagnetic circuits. Phys-\nica B: Condensed Matter , 405(14):2935 { 2937, 2010. ISSN 0921-4526. doi:\n10.1016/j.physb.2010.01.007. Proceedings of the Eighth International Confer-\nence on Electrical Transport and Optical Properties of Inhomogeneous Media,\nETOPIM-8.\n[14] F. Tisseur and K. Meerbergen. The quadratic eigenvalue problem. SIAM Rev. ,\n43(2):235{286, 2001. ISSN 0036-1445. doi: 10.1137/S0036144500381988.\nE-mail address :agondolo@math.utah.edu\nE-mail address :fguevara@math.utah.edu"    },    {        "title": "1305.5945v2.Thermally_Assisted_Spin_Transfer_Torque_Dynamics_in_Energy_Space.pdf",        "content": "Thermally-Assisted Spin-Transfer Torque Dynamics in Energy\nSpace\nD. Pinna,1,\u0003A. D. Kent,1and D. L. Stein1, 2\n1Department of Physics, New York University, New York, NY 10003, USA\n2Courant Institute of Mathematical Sciences,\nNew York University, New York, NY 10012, USA\nAbstract\nWe consider the general Landau-Lifshitz-Gilbert theory underlying the magnetization dynamics\nof a macrospin magnet subject to spin-torque e\u000bects and thermal \ructuations. Thermally activated\ndynamical properties are analyzed by averaging the full magnetization equations over constant-\nenergy orbits. After averaging, all the relevant dynamical scenarios are a function of the ratio\nbetween hard and easy axis anisotropies. We derive analytically the range of currents for which\nlimit cycles exist and discuss the regimes in which the constant energy orbit averaging technique\nis applicable.\n1arXiv:1305.5945v2  [cond-mat.mes-hall]  15 Jul 2013I. INTRODUCTION\nMore than a decade has passed since the switching e\u000bects of spin-polarized currents\ntraversing nanomagnets were \frst explored experimentally1{4. The quest to characterize\nhow a current may be used to transfer spin-angular momentum into a magnetic system has\nled to sweeping advances in the \feld of spintronics. The study and development of spin-valves\nand magnetic tunnel junctions (see, for example,5) with the aim of constructing denser and\nmore e\u000ecient memory technologies is one important result of spin-transfer-torque related\nresearch. Theoretical treatments usually begin by treating the magnetization within the\nferromagnetic \flm as a single macrospin6. Spin-torque e\u000bects are then taken into account\nphenomenologically by modifying the macrospin's dynamical Landau-Lifshitz-Gilbert (LLG)\nequation1.\nAt the nanoscale size, however, one must extend the deterministic treatment described\nabove to include random forces due to thermal noise. The e\u000bects of noise become even more\npronounced at low currents. A comprehensive theoretical treatment therefore requires taking\ninto account the interplay between noise and the deterministic dynamics. The analysis of\nsuch e\u000bects is complicated by the non-conservative nature of the spin-torque interaction.\nThe energetics underlying the thermally activated behavior of uniaxial macrospins have\nbeen studied in simple situations both analytically and numerically7{10, the latter is possible\nbecause of signi\fcant computational improvements in graphics processing unit (GPU) tech-\nnology. The advantage of using a uniaxial macrospin model is that its thermally activated\nswitching dynamics can be modeled using a 1 Dstochastic di\u000berential equation (SDE), which\nis solvable via standard statistical methods. In practical applications, though, magnets pos-\nsess a hard axis away from which magnetization deviations have an energy cost. This is\nparticularly true in thin \flm nanomagnets where, due to the shape of the sample, out of\nplane magnetizations are energetically suppressed.\nUpon adding a hard axis anisotropy to the macrospin model, one is forced to deal with\nthree coupled SDEs. A complete solution is then obtainable only numerically. Nonetheless,\na useful approximation can be employed whenever the conservative precessional timescale is\nsmall compared to the di\u000busion timescale, the latter determined by the interplay between\nnoise and spin transfer dynamics. This approximation allows for the construction of a\nconstant-energy orbit averaged (CEOA) LLG equation, which characterizes the macrospin\n2dynamics by focusing on its slow di\u000busion in energy space.\nAn early attempt that used the above approach consisted in writing down a Fokker-Planck\nequation describing the energy di\u000busion of the biaxial monodomain12,13. Unfortunately, an\nenergy di\u000busion equation of this type remains almost as intractable as the original set of\ndynamical equations. Nonetheless, Apalkov and Visscher12estimated a linear exponential\nscaling (log \u001c/(1\u0000I)) between current and mean switching time for thermal switching.\nHowever, research work done on uniaxial monodomains has found that, in contrast to the\nscaling form above, the dependence of mean switching time on current | written generally\nas log\u001c/(1\u0000I)\f| depends on whether one is studying a uniaxial or a biaxial macrospin\nmodel. For the former, the exponent \fwas found to be 2 rather than 18,10,15.\nAlternatively, an e\u000bective Langevin equation for the energy dynamics of a uniaxial\nmacrospin can also be contructed by averaging the magnetization dynamics over constant\nenergy trajectories15. Newhall and Vanden-Eijnden11numerically constructed an analogous\nLangevin equation for the energy dynamics of a biaxial macrospin and found numerical evi-\ndence for the appearance of stable limit cycles in the macrospin dynamics for certain current\nregimes. Their analysis supported the Apalkov-Visscher linear scaling result ( \f= 1)12, and\ndemonstrated a current dependence of \ffor thermal switching starting from a stable limit\ncycle state. These di\u000bering results implied that the transition between the uniaxial and\nbiaxial macrospin models had to be either continuous or at most display a \fnite disconti-\nnuity. Taniguchi14, by also approximating the Fokker-Planck equation over constant energy\ntrajectories, showed how the exponent \fdi\u000bers from the result of Apalkov and Visscher in\nthe thermally activated regime, as well as from the results of Newhall and Vanden-Eijnden\nfor applied currents greater than a certain critical value. Above this value, \fceases to be\nconstant and instead varies nonlinearly as a function of applied current.\nIn this paper we extend the energy Langevin approach15in order to study the biaxial\nmacrospin model. We obtain analytical expressions for three critical current values across\nwhich transitions between di\u000berent dynamical regimes occur. Two of these are found to\nbe identical to those independently derived by Taniguchi via di\u000berent means14. The third\ncritical current sets a limit on the current regimes for which CEOA is a valid approximation.\nOur analysis explains the transition between the biaxial and uniaxial macrospin regimes,\nthus shedding light on the contradictory conclusions obtained in the previous literature.\nWe further apply our analytical approach to thermally activated switching and obtain an\n3analytical expression for the non-linear exponential scaling discussed by Taniguchi14.\nII. GENERAL FORMALISM\nA simple model of a ferromagnet uses a Stoner-Wohlfarth monodomain with magnetiza-\ntionMwhose properties are \fxed by its shape anisotropy. The energy landscape experienced\nbyMis generally described by three terms: an applied \feld H, an easy-axis anisotropy en-\nergyUKwith axis along ^ nK(Fig. 1) in the ex\u0000ezplane making an angle \u0012with the ez\naxis, and a hard-axis anisotropy energy Upwith normal direction ^ nDperpendicular to ^ nK.\nThe magnetization Mis assumed to be constant in magnitude and for simplicity normalized\ninto a unit direction vector m=M=jMj. A spin-polarized current Jenters the magnetic\nbody in the\u0000eydirection, with spin polarization \u0011, and spin direction along the ezaxis.\nThe current exits in the same direction, but with its average spin direction aligned along M.\nThe self-induced magnetic \feld of the current is ignored here because the dimension ais\nconsidered to be smaller than 100 nm, where spin-current e\u000bects are expected to become\ndominant over those due to the current-induced magnetic \feld. The dynamics are described\nby the standard LLG equation:\n_ m=\u0000\r0m\u0002He\u000b\u0000\u000b\r0m\u0002(m\u0002He\u000b)\n\u0000\r0jm\u0002(m\u0002^ np) +\r0\u000bjm\u0002^ np; (1)\nwhere\r0=\r=(1+\u000b2) is the Gilbert ratio, \ris the usual gyromagnetic ratio and j= (\u0016h=2e)\u0011J\nis the spin-angular momentum deposited per unit time with \u0011= (J\"\u0000J#)=(J\"+J#) the\nspin-polarization of incident current J. The last two terms describe a vector torque\ngenerated by current polarized in the direction ^ np. These are obtained by assuming that\nthe macrospin absorbs angular momentum from the spin-polarized current only in the\ndirection perpendicular to m.1\nTo write He\u000bexplicitly, we must construct an energy landscape for the magnetic body.\nThere are three main components that need to be considered: a uniaxial anisotropy energy\nUK, a hard-axis anisotropy UPand a pure external \feld interaction UH. These are written\nas follows:\n4UK=\u0000K(^ nK\u0001m)2(2)\nUP=KP(^ nD\u0001m)2(3)\nUH=\u0000MSVm\u0001Hext (4)\nIn these equations, MSis the saturation magnetization, KPis the hard-axis anisotropy,\nK= (1=2)MSVHK,HKis the Stoner-Wohlfarth switching \feld (in units of Teslas) and\nHextis the externally applied magnetic \feld. The full energy landscape then becomes\nU(m) =UK+UP+UHand reads:\nU(m) =K\u0002\nD(^ nD\u0001m)2\u0000(^ nK\u0001m)2\u00002h\u0001m\u0003\n; (5)\nwhereD\u0011(KP=K),h=Hext=HKand^ nKis the unit vector pointing in the orientation of\nthe uniaxial anisotropy axis. Such an energy landscape has stable magnetic con\fgurations\nparallel and anti-parallel to ^ nK. The e\u000bective interaction \feld He\u000bis then given by\nHe\u000b=\u00001\nMSVrmU(m) =\u0000HK[D(^ nD\u0001m)^ nD\u0000(^ nK\u0001m)^ nK\u0000h]; (6)\nand the deterministic LLG dynamics can then be expressed as:\n_ m=\u0000m\u0002[(^ nK\u0001m)^ nK\u0000D(^ nD\u0001m)^ nD+h]\n\u0000\u000bm\u0002[m\u0002((^ nK\u0001m)^ nK\u0000D(^ nD\u0001m)^ nD+h)]\n\u0000\u000bIm\u0002\u0010\nm\u0002^k\u0011\n+\u000b2Im\u0002^k; (7)\nwhere we have de\fned I=j=(\u000bHK), introduced the natural timescale \u001c=\r0HKt, and have\nchosen our z-axis to be aligned with the spin polarization axis ( ^k\u0011^np).\nIn practice, the uniaxial and hard-axis axes cannot simply be oriented in any given\ndirection but are, in fact, perpendicular (see Fig. 1). We can write the equations of motion\nfor the magnetization in a rotated coordinate system, where ^ nK,^ nDand^ nOde\fne our\nnew coordinate system. Let ^ nK= (nx;0;nz),^ nD= (nzp\n1\u0000n2;n;\u0000nxp\n1\u0000n2) and ^ nO=\n(\u0000nzn;p\n1\u0000n2;nxn) be the orientation of the uniaxial and hard-axis anisotropy vectors with\nrespect to our original coordinate system (where nz= cos\u0012,nx= sin\u0012andn= sin\u001e). We\ncan then write the dynamical equations in terms of the projections q\u0011^ nK\u0001m,s\u0011^ nD\u0001mand\n5FIG. 1: Uniaxial and hard-axis anisotropy axes tilted with respect to polarizer by angle \u0012.\np\u0011^ nO\u0001mof the magnetization vector along this new set of axes. Written componentwise,\nthey are:\n_s=qp\u0000\u000bh\n(Inxp\n1\u0000n2+Ds)(1\u0000s2) +qs(Inz+q) +Inxnspi\n(8)\n_q=Dsp+\u000bh\n(Inz+q)(1\u0000q2) +Iqnx(sp\n1\u0000n2\u0000np) +Ds2qi\n(9)\n_p=\u0000(D+ 1)qs+\u000bh\nsp(Inxp\n1\u0000n2+Ds)\u0000qp(Inz+q) +Inxn(1\u0000p2)i\n:(10)\nIII. THERMAL EFFECTS\nThermal e\u000bects are included by considering uncorrelated \ructuations in the e\u000bective\ninteraction \feld: He\u000b!He\u000b+Hth. These transform the LLG equation into its Langevin\nform. We model the stochastic contribution Hthby specifying its correlation properties,\nnamely:\n6hHthi= 0 (11)\nhHth;i(t)Hth;k(t0)i= 2C\u000ei;k\u000e(t\u0000t0) (12)\nThe e\u000bect of the random torque Hthis to produce a di\u000busive random walk on the sur-\nface of the M-sphere. An associated Fokker-Planck equation describing such dynamics was\nconstructed by Brown6. At long times, the system attains thermal equilibrium; by the\n\ructuation-dissipation theorem we have\nC=\u000bkBT\n2K(1 +\u000b2)=\u000b\n2(1 +\u000b2)\u0018; (13)\nwhere\u0018=K=kBTis the barrier height due to the uniaxial anisotropy energy divided by the\nthermal energy.\nSetting the external magnetic \feld to zero and considering only thermal \ructuations, the\nstochastic LLG equation reads:\n_mi=Ai(m) +Bik(m)\u000eHth;k (14)\nwhere\nA(m) =\u000bIh\n\u000bm\u0002^k\u0000m\u0002\u0010\nm\u0002^k\u0011i\n\u0000(^ nK\u0001m) [m\u0002^ nK\u0000\u000b(^ nK\u0000(^ nK\u0001m)m)]\n+D(^ nD\u0001m) [m\u0002^ nD\u0000\u000b(^ nD\u0000(^ nD\u0001m)m)]; (15)\nBik(m) =\u0000\u000fijkmj\u0000\u000b(mimk\u0000\u000eik): (16)\nand `\u000eHth;k' means that the multiplicative noise term in the dynamical equation is to be\ninterpreted in the sense of Stratonovich calculus2516.\nWe numerically solve the above Langevin equations by using a standard second-order\nHeun scheme to ensure proper convergence. At each time step, the strength of the random\nkicks is given by the \ructuation-dissipation theorem. Statistics were gathered from an\nensemble of 5000 events with a natural integration stepsize of 0 :01. For concreteness, we set\nthe damping constant \u000b= 0:04 and the barrier height to \u0018= 80. To explore the simulations\nfor long time regimes, the necessary events were simulated in parallel on an NVidia Tesla\n7C2050 graphics card. To generate the large number of necessary random numbers, we chose\na proven combination17of the three-component combined Tausworthe \\taus88\"18and the\n32-bit \\Quick and Dirty\" LCG19. The hybrid generator provides an overall period of around\n2121.\nIV. SWITCHING DYNAMICS\nIn experiments, one is generally interested in understanding how the interplay between\nthermal noise and spin torque e\u000bects switch an initial magnetic orientation from parallel\nto antiparallel. The role of spin torque from an energy landscape point of view can be\nelucidated by considering how energy is pumped into the system. As in the previous section,\nthe monodomain's magnetic energy (for h= 0) can be written in dimensionless form:\n\u000f(m)\u0011U(m)=K=D(^ nD\u0001m)2\u0000(^ nK\u0001m)2=Ds2\u0000q2: (17)\nStable states with minimum energy have q=\u00061 and thus \u000f=\u00001. The change in energy\nover time can then be written as:\n_\u000f= 2 [Ds_s\u0000q_q]: (18)\nThe dynamical equations (14) for _ sand _qcan be plugged back in to (17) to obtain a\ndynamical equation for the energy:\n_\u000f=f(s;q) +g(s;q)\u000e_W: (19)\nNanomagnets typically operate in a regime where the nonconservative parts of the dynamics,\nincluding the e\u000bects of damping, STT and thermal noise, act on timescales that are long\ncompared to the energy-conserving piece of the Hamiltonian. Trajectories are therefore\nexpected to remain close to periodic Hamiltonian orbits for a long time, slowly drifting from\none orbit to another. Using this information, we average the energy evolution equation (19)\nover constant energy trajectories, thereby obtaining a di\u000busion equation in energy space11{13.\nFrom (7) we have\n8_s0=\u0000q0p0(20)\n_q0=\u0000Ds0p0(21)\n_p0= (D+ 1)q0s0; (22)\nthe solutions of which can be expressed in terms of Jacobi elliptic functions:\ns0(t) =\u0007r\nD1 +\u000f\nD+ 1cnhp\nD\u0000\u000ft;k2i\n(23)\nq0(t) =\u0006r\nD\u0000\u000f\nD+ 1dnhp\nD\u0000\u000ft;k2i\n(24)\np0(t) =\u0000p\n1 +\u000fsnhp\nD\u0000\u000ft;k2i\n; (25)\nwherek2\u0011D1+\u000f\nD\u0000\u000f. The period of these trajectories as a function of energy can be expressed\nas a complete elliptic integral of the \frst kind:\nT(\u000f) =4p\nD\u0000\u000fZ1\n0dxp\n(1\u0000x2)(1\u0000k2x2)=4p\nD\u0000\u000fK(k2): (26)\nA sample of these trajectories for energies greater and less than 0 are shown in Fig. 2.\nThe\u000f<0 dynamics consist of the magnetization precessing around the uniaxial axis ( ^ nK),\nthus remaining in the energy minimum's basin of attraction. On the other hand, for \u000f>0\nthe magnetization precesses around the hard axis ( ^ nD). Magnetic switching, de\fned as a\ntransition from one \u000f <0 basin to another, must proceed through an intermediate \u000f >0.\nAs a result, we will consider \u000f= 0 to be the critical threshold energy that a monodomain\nmust surpass to achieve switching.\nThe qualitative properties of such trajectories are better understood by parametrizing\nthem geometrically as follows. The energy landscape (17) determines the geometrical con-\nditions these trajectories must satisfy. For \u000f>0 we have\nD\n\u000fs2\u00001\n\u000fq2= 1 (27)\ns2+q2+p2= 1: (28)\n9FIG. 2: Constant enegrgy trajectories. \u000f<0 trajectories are shown in red whereas \u000f>0 trajectories\nare shown in green.\nThese are satis\fed by the parametrization:\ns0(w) =\u0006r\nj\u000fj\nDcosh(w) (29)\nq0(w) =p\nj\u000fjsinh(w) (30)\np0(w) =\u0006p\n1 +j\u000fjq\n1\u0000\r2\n+cosh2(w); (31)\n\r2\n+=\u000f(D+ 1)\nD(\u000f+ 1): (32)\nwhere the limits of validity of \u0000acosh(1=\r+)<w< acosh(1=\r+) are found by imposing the\nextra condition that for p= 0,s2+q2= 1. The identical reasoning carried out for \u000f <0\n10leads to the parametrization:\ns0(w) =\u0006r\nj\u000fj\nDsinh(w) (33)\nq0(w) =\u0006p\nj\u000fjcosh(w) (34)\np0(w) =\u0006r\nD+j\u000fj\nDq\n1\u0000\r2\n\u0000cosh2(w); (35)\n\r2\n\u0000=j\u000fj(D+ 1)\nD+j\u000fj; (36)\nwith limits\u0000acosh(1=\r\u0000)<w< acosh(1=\r\u0000).\nWe are now ready to average (18) over constant energy orbits. The averaging procedure\nis simpli\fed by noting that we are interested in switching behavior starting from the \u000f<0\nbasin and that in that basin the constant energy trajectories are symmetric with respect to\nthe^ nKaxis. As such, the majority of terms obtained by inserting (8) into (18) will average\nto zero. The remaining nonzero terms lead to the averaged energy equation:\nh@t\u000fi=\u00002\u000b\u0002\nInz(1 +\u000f)hqi+ (1 +\u000f)hq2i+D(D\u0000\u000f)hs2i\u0003\n+ 2s\n\u000b(D+ 1)\n\u0018r\nhq2i+D\u0000\u000f\nD+ 1\u000f\u000e_W (37)\nwhereh\u0001iimplies averaging over constant energy orbits as described. Construction of the\nstochastic term requires averaging the variance of (18) following the rules of additivity for\nGaussian random variables. It is seen from (37) that the applied current factors into this\nequation only in the form Inz, wherenzis the cosine of the angular tilt between uniaxial\nanisotropy and polarizer axes. This leads to a more general form of a result obtained in our\nprevious work8, namely that scalings between switching times and applied current will be\nfunctionally identical independent of the incoming angle of the spin-polarized current. The\nonly caveat here is that every current must be rescaled by multiplying it by nz. Analogously,\nthe azimuthal tilt parameter ndoes not appear at all in the energy-averaged equation, im-\nplying that only coplanar setups between polarizer, easy and hard axes need to be considered\nin what follows.\nWith this understanding, we can now omit the notation nzand consider macrospin setups\nwith no angular tilt between polarizer and uniaxial anisotropy axes. The next step is to\nshow how the necessary averages can be computed explicitly. As an example, consider hqi;\nits average is given by:\n11hqi=1\nT(\u000f)ZT\n0dtq0(t); (38)\nwhere the integration is over time and one uses the expression for q0in terms of its Jacobi\nelliptic function. We can express the same integral in terms of the geometrical parametriza-\ntion (33). In fact\nhqi=1\nT(\u000f)ZT\n0dtq0(t) =1\nT(\u000f)I\ndwds0=dw\n_s0q0(w): (39)\nPutting together the expression for _ s0from (20) and, again, employing the geometrical\nparametrizations from (33), we obtain\nhqi=\u00004\nT(\u000f)p\nD+ 1Zacosh(1=\r)\n0dwcosh(w)q\n1\u0000\r2cosh2(w)=\u00002\u0019\nT(\u000f)p\nD+ 1; (40)\nwhere, for notational convenience, we rede\fne \r=\r\u0000.\nRepeating the procedure for the other terms in (8), we write\nhq2i(\u000f) =4\nT(\u000f)p\nD+ 1s\nD\nD+ 1\u0000\r2\u00111 (41)\nhs2i(\u000f) =4\nT(\u000f)p\nD+ 1s\nD\nD+ 1\u0000\r21\nD\u0000\n\u00111\u0000\r2\u00110\u0001\n; (42)\nwith\n\u00110(\u000f) =Z1\n0dxp\n(\r2+ (1\u0000\r2)x2)(1\u0000x2)=K(1\u00001\n\r2)\n\r= K(1\u0000\r2) (43)\n\u00111(\u000f) =Z1\n0dxr\n\r2+ (1\u0000\r2)x2\n1\u0000x2=\rE(1\u00001\n\r2) = E(1\u0000\r2) (44)\nT(\u000f) = 4s\nD+ 1\u0000\r2\nD(D+ 1)K(1\u0000\r2) = 4s\nD+ 1\u0000\r2\nD(D+ 1)\u00110(\r) (45)\n\r(\u000f) =j\u000fj(D+ 1)\nD+j\u000fj; (46)\nwhere K(x) and E(x) are elliptic functions of the \frst and second kind respectively (Ap-\npendix A). The energy equation for the dynamics starting in the negative energy well\n(0<j\u000fj<1) then reads (expressed in terms of \r26):\n@tj\u000fj(\r) =8\u000b\nT(\r)\u0012D(D+ 1)1=3\nD+ 1\u0000\r2\u00133=2\"\n\u00111(\r)\u0000\r2\u00110(\r) +1\u0000\r2\nD \n\u00111(\r)\u0000\u0019I\n2r\nD+ 1\u0000\r2\nD!#\n+ 4r\u000b\n\u0018T(\r)\u0012D(D+ 1)\nD+ 1\u0000\r2\u00131=4s\n\u00111(\r)\u0000D\r2\nD+ 1\u0000\r2\u00110(\r)\u000e_W: (47)\n12In outlining the steps above, we have succeded in reducing the multidimensional complex-\nity of the full magnetization dynamics to a one-dimensional stochastic di\u000berential equation\nwhose properties we now proceed to study. The relation to previous studies8,9on uniaxial\nmacrospins can be rederived by considering equations (20) and (47) in the limit D!0.\nDoing so leads to the much simpli\fed energy di\u000busion equation15\n@tj\u000fj= 2\u000bp\nj\u000fj(1\u0000j\u000fj)(p\nj\u000fj\u0000I) + 2r\u000b\n\u0018j\u000fj(1\u0000j\u000fj)\u000e_W; (48)\nwith a stable energy minimum at \u000f=\u00001 and saddle point at \u000f=\u0000I2. This equation also\nshows that, for currents I > Ic= maxj\u000fj2[0;1]p\nj\u000fj= 1, the \row becomes negative for all\nvalues ofj\u000fj, so that in this regime all states will deterministically switch.\nV. STABILITY ANALYSIS\nIn the absence of applied currents, @tj\u000fj>0 as determined by (47) and \u000f\rows toward its\nminimum value of \u00001 (the stable \fxed point of its dynamics). To understand switching one\nmust therefore investigate under what circumstances \u000fmay become greater than zero, thus\nimplying a transition from the red to the green trajectories discussed in Fig. 2. It su\u000eces\nto understand the behavior of the energy \row at the stable \fxed point of the zero current\ndynamics (j\u000fj=\r= 1) and at the energy threshold for switching ( j\u000fj=\r= 0). These\nlimiting \rows are found by computing the integrals \u00110(\r) and\u00111(\r). One \fnds:\nlim\n\r!1\u00110(\r)\u0018\u0019\n2+\u0019\n8D\nD+1\u0000\r2(1\u0000\r2) +o((1\u0000\r2)2) (49)\nlim\n\r!1\u00111(\r)\u0018\u0019\n2\u0000\u0019\n8D\nD+1\u0000\r2(1\u0000\r2) +o((1\u0000\r2)2); (50)\nand\nlim\n\r!0\u00110(\r)\u0018log(\r) +o(log(\r)\r2) (51)\nlim\n\r!0\u00111(\r)\u00181 +o(\r): (52)\n13Using these results in (47), one \fnds that in the absence of thermal noise (i.e., the zero\ntemperature limit), the limiting \rows are:\nlim\n\r!0@tj\u000fj=8\u000b\nTp\nD\nD+1h\nD+ 1\u0000\u0019I\n2q\nD+1\nDi\n(53)\nlim\n\r!1@tj\u000fj=4\u0019\u000b\nTpD+1\nD(1\u0000\r2) [D+ 2\u00002I]: (54)\nAs expected, both eqs. (53) and (54) show how the qualitative behavior of the deter-\nministic energy \row can be tuned by the value of applied current. The stability of the zero\ncurrent stable \fxed point ( j\u000fj=\r= 1) can in fact be be rendered unstable once (54) changes\nsign at the applied current value:\nIj\u000fj=1\nc\u0011I1\nC=D+ 2\n2: (55)\nAnalogously, atj\u000fj=\r= 0 the \row is either positive or negative depending on the applied\ncurrent. Using (53), we \fnd that the sign of the \row switches from positive to negative at\nIj\u000fj=0\nc\u0011I0\nC=2\n\u0019p\nD(D+ 1): (56)\nIdentical expressions for the critical currents have recently been derived by Taniguchi14via\ndi\u000berent means. These two critical stability thresholds are identical at a critical value of D:\nD0= 2\u00192\u00004\n16\u0000\u00192\"\n1 + 2p\n4 + 2\u00192\n\u00192\u00004#\n\u00185:09: (57)\nThe value of Drelative to D0, as well as the applied current, will select between two\nqualitatively di\u000berent dynamical regimes. We now proceed to explore both.\nA.D>D 0\nWhen the ratio between hard and easy axis anisotropies is greater than the threshold\nvalueD0,I0\nC>I1\nC(see Fig. 3). Unless the applied current is greater than I0\nC, deterministic\nswitching of the monodomain cannot be achieved. This di\u000bers from conclusions drawn by\nprevious work20where the critical current for deterministic switching was assumed to be\nthat which renders the stable \fxed point ( j\u000fj=\r= 1) unstable.\nFor small currents such that I < I1\nC, on the other hand, the deterministic \row pushes\nthe energy toward the zero current stable \fxed point. In such a scenario, switching can only\noccur via thermal activation over the j\u000fj=\r= 0 e\u000bective barrier.\n14FIG. 3: Critical currents versus the ratio of the hard and easy axis anisotropies D. The blue curve\nisI1\nCand the red curve is I0\nC. ForD<D 0, currents greater than I1\nClead to deterministic switching\n(labeled DS). For D>D 0currents between I0\nCandI1\nClead to limit cycles (LC). Limit cycles can\nalso occur for currents just below and approximately equal to I1\nCforD<D 0, as shown in Fig. 7\nFinally, for currents I0\nC> I > I1\nC, the zero current stable \fxed point has been rendered\nunstable while the \row at the switching energy threshold remains positive. The monodomain\nwill move (deterministically) from the j\u000fj=\r= 1 \fxed point, but switching over the\nj\u000fj=\r= 0 must still take place via thermal activation. This implies that a new stable\nenergy equilibrium exists for some value 0 <~\rS<1. The precessional macrospin dynamics\nwill then manifest itself in the form of stable limit cycles. This has been observed both\nexperimentally21and numerically11. These three dynamical scenarios are displayed in Fig. 4\nwhere the deterministic energy \row is plotted as a function of energy.\n15FIG. 4: Three sample regimes of deterministic energy \row _ \u000fas a function of energy for D >D 0.\nColoring (online) is included to better distinguish the various curves. a) I <I1\nC: Subcritical regime,\nthermal noise must oppose a positive energy \row to achieve switching; b) I1\nC<I <I0\nC: Limit cycle\nregime; and c) I >I0\nC: Supercritical regime, negative \row leads to determinist switching.\nB.D<D 0\nWhen the ratio between hard and easy axis anisotropies is smaller than the threshold\nvalueD0, we haveI1\nC> I0\nC(see Fig 3). The threshold critical current that switches all\nmagnetic states deterministically is now I1\nC20. Above this value of the applied current,\nswitching will occur independently of thermal noise e\u000bects.\nFor applied currents such that I0\nC< I < I1\nC, a saddle point emerges for some value\n0<~\rU<1. For switching to occur, thermal activation must move the monodomain energy\npast this saddle point. As the current is lowered, ~ \rUbecomes progressively smaller until\nthe limiting value ~ \rU= 0. At this point, switching requires thermal activation throughout\nthe whole energy range. These three dynamical scenarios are displayed in Fig. 5 where the\n16FIG. 5: Three sample regimes of deterministic energy \row _ \u000fas a function of energy for D <D 0.\nColoring (online) is included to better distinguish the various curves. a) I < I0\nC: Subcritical\nregime, thermal noise must oppose a positive energy \row to achieve switching; b) I0\nC< I < I1\nC:\nCrossover regime, switching is still achieved via thermal activation but the unstable equlibirum\nhas now shifted; and c) I >I1\nC: Supercritical regime, negative \row leads to determinist switching.\ndeterministic energy \row is plotted as a function of energy.\nIt is interesting to note that the condition D < D 0can also result in the appearance of\nlimit cycles. Limit cycle regimes in fact can be seen for very small applied ranges of current\nless thanI0\nC(Fig. 6).\nThe uniaxial macropin model is a particular case of a D<D 0model (D= 0). By taking\nthe limitD!0, we \fndI0\nC!0 andI1\nC!1, as found in previous work.8For all values of\nthe applied current strictly between 0 and 1, uniaxial macrospin switching must take place\nvia thermal activation over an e\u000bective energy barrier.\n17FIG. 6: Energy \row for a D= 4 macrospin and applied current I= 2:82<I0\nC. Circles and squares\nrespectively represent stable and unstable equilibria. For these parameters ( D= 4 andI= 2:82) ,\ntwo stable equilibria of the zero temperature dynamics coexist.\nVI. LIMITS OF THE CEOA APPROACH\nIn this section we will determine under what conditions the CEOA approach is valid.\nThe fundamental assumption is that the deterministic precession timescale of the constant\nenergy orbits be small compared to the timescale over which signi\fcant energy di\u000busion due\nto noise and spin-torque transfer takes place. We now quantify this condition and obtain\nprecise limits in terms of the applied current intensity Iand the anisotropy ratio D. For our\napproximations to be valid, the averaged energy \row ( T(\u000f)@tj\u000fj) over any given orbit must\nbe small compared to the maximum allowable energy (0 <j\u000fj<1):\n\u000fmaxT(\u000f)j@tj\u000fjj=\rmaxT(\r)j@tj\u000fjj\u001c 1; (58)\nwhere we continue to use the variable \r(\u000f) =j\u000fj(D+ 1)=(D+j\u000fj). To proceed, we consider\nthe deterministic drift and random components of the energy \row separately. Noise averag-\n18ing (47) will give the contribution of the drift to the energy \row. The intensity of the orbit\naveraged drift then reads:\nT(\r)jh@tj\u000fjij= 8\u000b\u0012D(D+ 1)1=3\nD+ 1\u0000\r2\u00133=2\f\f\f\f\f\u00111(\r)\u0000\r2\u00110(\r) +1\u0000\r2\nD \n\u00111(\r)\u0000\u0019I\n2r\nD+ 1\u0000\r2\nD!\f\f\f\f\f;\n(59)\nwhere angular brackets denote noise averaging. This expression can be shown to be \fnite\nfor all values of \r,IandD. Furthermore, for applied currents greater than I0\nC, (59) has\na maximum at \r= 0; its value in this limit is given by (53). Enforcing the conditions for\nvalidity discussed above results in an upper bound for the current:\nI\u001c(1 +1\n8\u000bp\nD)I0\nC\u0011IM: (60)\nIn the limit D!0, this expression converges to the correct uniaxial limit inequality I\u001c\n(4\u0019\u000b)\u00001, discussed previously in15.\nWhen switching occurs deterministically (and I >I0\nC), this bound is the sole limit to the\nvalidity of the CEOA approximation. This is because random contributions to the energy\n\row dynamics have zero mean. However, in scenarios where switching occurs due to thermal\nactivation, one must consider the standard deviation of random contributions to the energy\n\row. These determine the rate of the rare events that drive the system over its con\fning\nenergy barrier. The energy \row's standard deviation isp\nh[@tj\u000fj\u0000h@tj\u000fji]2i. The orbit-\naveraged rate of random events driving noise-induced energy di\u000busion can then be written\nas:\nT(\r)p\nh[@tj\u000fj\u0000h@tj\u000fji]2i= 8s\n\u000b\n\u0018\u00110(\r)\u0012\n\u00111(\r)\u0000D\r2\nD+ 1\u0000\r2\u00110(\r)\u0013\n; (61)\nwhich can be shown to be both a monotonically decreasing function of \rand logarithmically\ndivergent for \r!0. The latter is due to the limiting behavior of \u00110(\r) (see (51)) for all\nD > 0. The physical origin of this behavior lies in the divergence of the constant energy\nprecessional period as the zero energy switching threshold is reached (a fact also pointed\nout by Taniguchi et al.14).\nThe relevant maxima of (61) will be located at the energy \row's saddle point \rU. Unfor-\ntunately, solving for \rUwhile imposing that the RHS of (61) be much less than 1 results in\na set of transcendental equations which can only be solved numerically. However, we can set\na precise bound by requiring that the saddle point never be at \rU= 0 (where (61) diverges\n19for allD > 0). The applied current at which such a saddle point occurs has already been\nderived in (56), leading us to the lower bound inequality\nI\u001dI0\nC: (62)\nas the validity condition for CEOA in thermally activated scenarios. The reason for limiting\nourselves to applied currents greater than I0\nCin deriving the upper bound condition (60) is\nnow justi\fed.\nAs a result of (62), there are only two scenarios in which thermally activated switching\ncan be analyzed and understood using CEOA dynamics. The \frst is in the limit D!0\n(the uniaxial macrospin limit) where one has \r(\u000f)!1 indpendently of \u000f. Divergences in\nthe intensity of random contributions are then avoided for all \u000fand (61) takes the simple\nform 4\u0019p\n(\u000b=\u0018)(1\u0000\u000f). At\u000f= 0 it converges to the \fnite value 4 \u0019p\n\u000b=\u0018(which may still\nbe large, however, depending on the ratio \u000b=\u0018). The second scenario is for models with\nD < D 0and applied current values I > I0\nC. In these, the saddle point to be reached via\nthermal activation shifts to nonzero values \rU6= 0 where (61) does not diverge and the\nvalidity condition may be satis\fed.\nThe above discussion demonstrates that thermally activated switching between dynamical\nlimit cycle equilibria cannot be modeled with our technique. Thermal switching from a\nlimit cycle, in fact, must proceed via noise di\u000busion up to a saddle point of the type \rU=\n0 where noise contributions diverge. Nonetheless, it should be noted that this does not\nnecessarily invalidate the conditions for their existence discussed in the previous section. In\nfact, relaxation to a limit cycle is a drift-driven process for which noise related contributions\naverage to zero. Our analysis merely shows that, once the macrospin has relaxed to its limit\ncycle state, further dynamics must proceed via thermal activation for which CEOA does not\nprovide a suitable description.\nIn Fig. 7, the dynamical behavior for D= 4 is displayed. Here D < D 0andI0\nC< I1\nC.\nFig. 7a shows the dependence between rescaled current and mean switching time for di\u000berent\npolar angular tilts \u0012and \fxed azimuthal angle \u001e. For reference, the critical currents I0\nC,I1\nC\nandIM\nCare shown. For large applied currents, deviations between the di\u000berent sets of data\nare clearly visible. This discrepancy decreases as the current is lowered.\nHowever, for I < I0\nC, the procedure is expected to fail due to the previously discussed\ndivergence of the orbital period. To show this, we calculate the maximum deviation of\n20FIG. 7: a)Mean switching time versus current for D= 4,\u000b= 0:04 and\u0018= 80 at di\u000berent \u0012\nangular tilts with \u001e= 0 kept \fxed. All currents have been rescaled by 1 =cos\u0012. Times are shown\nin units of ( s\u0001T) whereTstands for Tesla: real time is obtained upon division by HK. For\nvisual guidance, the critical currents I0\nCandI1\nChave been included. In a regime where the CEOA\ntechnique is applicable, the switching data from the various angular con\fgurations should all fall\non top of each other. b)Double axis plot of max[ T(\u000f)j@tj\u000fjj] and the percent deviation of data\nfrom ( a)) as a function of normalized current. In the current range where the deterministic \row\nachieves its minimum, the deviation of the data does also. As the critical current I0\nCis approached,\ndeviation spikes are observed analogously to what can be inferred by the theory.\nthe mean switching time between all the angular data points at each value of the rescaled\ncurrent (see Fig. 7b). As a further test, we have also numerically computed the LHS of (58)\nand compared it to the angular deviations. The critical currents I0\nC,I1\nCandIM\nCare again\nshown for reference. The theory is in satisfactory agreement with numerical simulations as\ncan be seen from both the good alignment of the deviation minimum with the minimum of\nthe maximum \row and the rapid spike in deviation as I0\nCis approached.\nThis discussion implies that the CEOA tecnique applies best to small Dmacrospin mod-\nels, and becomes increasingly inaccurate for larger D. Figs. 8 and 9 show the same analysis\nof a macrospin with D= 50 and one with D= 7. Fig. 8a indicates that the technique is not\napplicable for su\u000eciently high values of D. Nonetheless, theory and experiment approach\neach other near the minimum of the deterministic \row, as seen in Fig. 8b. As in Fig. 7, as\ncurrents drop below I0\nC, deviations in the data rapidly spike.\n21FIG. 8: a)Mean switching time versus current for D= 50,\u000b= 0:04 and\u0018= 80 at di\u000berent\n\u0012angular tilts with \u001e= 0 kept \fxed. All currents have been rescaled by 1 =cos\u0012. Times are\nshown in units of ( s\u0001T) whereTstands for Tesla: real time is obtained upon division by HK.\nFor visual guidance, the critical currents I0\nCandI1\nChave been included. b)Double axis plot of\nmax[T(\u000f)j@tj\u000fjj] and the percent deviation of data from ( a)) as a function of normalized current.\nIn the current range where the deterministic \row achieves its minimum, the deviation of the data\ndoes also. As the critical current I0\nCis approached, deviation spikes are observed analogously to\nwhat can be inferred by the theory.\nThe analysis and its comparison to numerical data indicate that the CEOA tecnique is\nbest suited for studying macrospin dynamics in the crossover regime separating ballistic\nfrom fully activated thermal switching. Furthermore, the stability analysis of the energy\ndynamics (47) manages to capture the emergence of limit cycles even though it fails to\ndescribe thermally activated processes proceeding from them.\nVII. THERMALLY ACTIVATED SWITCHING\nScenarios in which thermally activated switching can be studied using the approach de-\nscribed above are generally rather limited. Nonetheless, in the previous section, the tech-\nnique was shown to be applicable in models with D<D 0for currents I0\nC<I <I1\nC. Starting\nfrom (47) one could in principle construct a Fokker-Planck equation describing energy di\u000bu-\nsion and then attempt to solve an appropriate mean \frst passage time problem numerically.\n22FIG. 9: a)Mean switching time versus current for D= 7,\u000b= 0:04 and\u0018= 80 at di\u000berent \u0012\nangular tilts with \u001e= 0 kept \fxed. All currents have been rescaled by nz= 1=cos\u0012. Times are\nshown in units of ( s\u0001T) whereTstands for Tesla: real time is obtained upon division by HK.\nFor visual guidance, the critical currents I0\nCandI1\nChave been included. b)Double axis plot of\nmax[T(\u000f)j@tj\u000fjj] and the percent deviation of data from ( a)) as a function of normalized current.\nIn the current range where the deterministic \row achieves its minimum, the deviation of the data\ndoes also. As the critical current I0\nCis approached, deviation spikes are observed analogously to\nwhat can be inferred by the theory.\nInstead, we will simplify the thermal activation problem by deriving the exponential scaling\ndependence between mean switching time and current, using an analytical tool described\nin15. In that paper, it was shown how, starting from energy di\u000busion dynamics analogous\nto (47), the exponential scaling dependence can be written in terms of an integral from the\ninitial stable state to the saddle using a Friedlin-Wentzell22type formulation:\nlog(h\u001ci)/2Z1\nj\u000fUjf(\u000f0)\ng2(\u000f0)d\u000f0(63)\nwheref(\u000f) andg(\u000f) are, respectively, the drift and noise terms of (47), now expressed in\nterms of the energy \u000f. This integral can be computed numerically once the saddle point of\nthe energy \row has been identi\fed (e.g. via a Newton algorithm). The integrand can be\nfurther expanded for D>> 1 and approximated around the stable point j\u000fj= 1. This gives,\n23to \frst order in 1 \u0000j\u000fj,\nlog(h\u001ci)/\u0018Z1\nj\u000fUj \n1\u0000(I=I0\nC)r\nD\nD+\u000f0\u00121\u0000\u000f0\n\u00111(\u000f0)\u0000\u000f0\u00110(\u000f0)\u0013!\nd\u000f0(64)\n'\u0018Z1\nj\u000fUj\u0012\n1\u0000(I=I0\nC)1\u0000\u000f0\n1\u0000\u000f0\u0010(D)\u0013\nd\u000f0(65)\nwhere\u0010(D) is given by\n\u0010(D) =I1\nC\nI0\nC=\u0019\n4 \nD+ 2p\nD(D+ 1)!\n: (66)\nThe approximated integral can now be written in closed form in terms of hypergeometric\nfunctions. The resulting expression is accurate to within 2% for values D> 0:1.\nIn Fig. 10 we plot the dependence of the mean switching time as a function of applied\ncurrent, and compare it to log( \u001c)/\u0018(1\u0000I=IC)\f. In the limit D!0, the uniaxial switching\nexponent\f= 2 is recovered. For larger values of D, the exponent \fdepends non-linearly on\nthe applied current I. In the limit I!0 the limiting value of \fcan be derived analytically:\nlim\nI!0\f(I) = lim\nI!0log(1\u0000q(D)\nI0\nCI)\nlog(1\u0000I=I1\nC)=q(D)I1\nC\nI0\nC; (67)\nwith\nq(D)\u0011Z1\n0r\nD\nD+x\u00121\u0000x\n\u00111(x)\u0000x\u00110(x)\u0013\ndx: (68)\nThe same calculation can be performed in the limit I!I1\nCand used to show that the\nexponent\fdiverges for all non-zero values of D. The mean switching time is nonetheless well\nbehaved, as can be seen in Fig. 10a. This di\u000bers from the results obtained by Taniguchi14, in\nwhich the limiting value of the exponent \fis roughly 2 :2, as the applied current approaches\nthe critical threshold for determinisic switching.\nVIII. CONCLUSION\nThis paper extends the analytical approach introduced by the authors in15and applies it\nto the biaxial macrospin model. We have described a technique that reduces the complexity\nof the 3Dmacrospin reversal dynamics, under the action of both spin-torque and thermal\nnoise, to a 1 Dstochastic equation in the energy space of the macrospin. To do so, the\ncomplete macrospin dynamics were approximated over precessional constant energy orbits.\n24FIG. 10: a)Scaling dependence of mean switching time as a function of applied current Ifor\nmodels with varying D<D 0.\u0018is the energy barrier height and I1\nCthe critical current threshold for\ndeterministic switching. b)Fit of (64) to the form (1 \u0000I=I1\nC)\f. Dashed lines represent continuation\nof analytical results outside the tecnique's regime of validity. Fitting exponent \fis plotted as a\nfunction of applied current for models with varying D. In the limit of small Dthe exponent\napproaches the constant value \f= 2 consistent with previous uniaxial macrospin results8,10,15.\nForD > 0, the exponent \fdepends nonlinearly on the applied current intensity. Only for values\nD\u0018D0do we notice that in the limit of small applied currents, \f!1 as suggested by similar\nenergy di\u000busion studies from the literature11,12. For intermediate values D0> D > 0 the low\ncurrent limit of \fcan be obtained analytically (67). On the other hand, in the limit I!I1\nCthe\nexponent\fcan be shown to diverge for all non-zero values of D.\nUnder such an approximation, the resulting theory predicts that the geometries involved\nin\ruence the respective dynamics in very precise ways. We found that the angular tilt\nbetween spin polarization and uniaxial anisotropy axes factors into the dynamics only as a\ntrivial rescaling of the applied current. Similarly, the relative orientation of the hard axis\nis predicted to play no role in the switching dynamics under conditions in which constant-\nenergy orbit averaging (CEOA) is a valid assumption.\nThe main parameter characterizing the di\u000berent dynamical regimes is the ratio Dbetween\nthe hard and easy axis anisotropies. For a speci\fc value of D, two critical currents ( I1\nC=\n(D+ 2)=2,I0\nC= (2=\u0019)p\nD(D+ 1)) are found to exist. The relative magnitude of these\ncritical currents depends nonlinearly on D. ForD>D 0'5:09, we showed that stable limit\ncycle magnetization precessions appear in well-de\fned current ranges; transitions between\nthese stable limit cycles proceeds through thermal activation. When D < D 0, limit cycles\n25generally do not appear and the switching dynamics become qualitatively similar to those\nof a uniaxial macrospin model.\nThe CEOA procedure requires that the energy di\u000busion due to spin torque and thermal\nnoise take place on a timescale much larger than the conservative precessional timescale of\nthe model. This allows us to establish general conditions for the validity of CEOA dynamics.\nWe test our \fndings numerically by solving the full macrospin evolution equations and\nanalyzing their applied current vs. mean switching time curves for various angular con\fgu-\nrations. Our techniques are found to be applicable in explaining ranges of applied currents\nsuch that:I\u001dI0\nCwhere any thermally activated process proceeds along nearly precessional\ntrajectories, and I=I0\nC\u001c1 + 1=(8\u000bp\nD) such that spin torque intensity is minimized along\nthe deterministic \row. These conditions do not generally hold in large Dmodels, but do for\nsmallD. In the limit D!0 we recover known results for the uniaxial macrospin model. Our\ntechnique also appears to be suitable for studying thermally activated switching in models\nwithD<D 0.\nFinally, Friedlin-Wentzell theory was employed to analytically study the exponential scal-\ning dependence between mean switching time and applied current in thermally activated\nscenarios. The exact analytical scaling was reduced to quadratures and an analytical ap-\nproximation was suggested. The resulting analytical scaling dependence to the standard\nform log(\u001c)/\u0018(1\u0000I=IC)\fand the current dependence of the exponent \fwas studied.\nThe exponent \fwas found to depend nonlinearly on the applied current intensity, similar\nto14. In the uniaxial macrsopin limit D!0, the constant \f= 2 result was recovered.\nForD6= 0,\fwas found to depend on both the applied current intensity and the precise\nvalue ofD.\nAcknowledgments\nThe authors would like to acknowledge A. MacFadyen and J. Z. Sun for useful discussions\nand comments leading to this paper. This research was supported by NSF-DMR-100657,\nNSF-DMR-1309202 and PHY0965015.\n\u0003Electronic address: daniele.pinna@nyu.edu\n261J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2J. A. Katine, F. J. Albert, and R. A. Buhrman, Phys. Rev. Lett. 84, 31493152 (2000).\n3L. Berger, Phys. Rev. B 54, 9353 (1996).\n4B.Ozyilmaz, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. Lett. 93, 176604\n(2004).\n5A. Brataas, A. D. Kent and H. Ohno, Nature Materials 11, 372 (2012).\n6W. F. Brown, Phys. Rev. 135, 5 (1963).\n7T. Taniguchi and H. Imamura, Phys. Rev. B 85, 18440 (2012).\n8D. Pinna, Aditi Mitra, D. L. Stein and A. D. Kent, arXiv:1205.6509 (2012).\n9D. Pinna, D. L. Stein, and A. D. Kent, arXiv:1210.7675 (2012).\n10T. Taniguchi and H. Imamura,Phys. Rev. B 83, 054432 (2011).\n11K. Newhall and E. Vanden-Eijnden, arXiv:1210.6253 (2012).\n12D. M. Apalkov and P. B. Visscher, Phys. Rev. B 72, 180405R (2005).\n13G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems\n(Elsevier, Oxford, UK, 2009).\n14T. Taniguchi, Y. Utsumi, M. Marthaler, D. S. Golubev and H. Imamura, Phys. Rev. B 87,\n054406 (2013).\n15D. Pinna, D. L. Stein, and A. D. Kent, arXiv:1210.7682 (2012).\n16I. Karatsas and S. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. (Springer-Verlag,\nNew York, 1997).\n17H. Nguyen, GPU Gems 3 (Addison-Wesley Professional, 2007).\n18P. L'Ecuyer, Operations Research 44, 5 (1996), pp. 816-822.\n19W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in C: The\nart of scienti\fc computing (Cambridge University Press, 1992).\n20J. Z. Sun, Phys. Rev. B 62, 1 (2000).\n21I. N. Krivorotov, N. C. Emley, A. G. F. Garcia, J. C. Sankey, S. I. Kiselev, D. C. Ralph and R.\nA. Buhrman, Phys. Rev. Lett. 93, 166603 (2004).\n22M. I. Freidlin and A. D. Wentzell Random Perturbations of Dynamical Systems (Springer, New\nYork, 1991).\n23E. T. Whittaker, G. N. Watson, A Course in Modern Analysis 4th Ed. (Cambridge University\nPress, Cambridge, UK 1927).\n2724E. Wong and M. Zakai, Ann. Math. Statist. 36(1965), p. 1560.\n25In modeling thermal e\u000bects through stochastic contributions in a system like ours, we are inter-\nested in considering the white noise limit of a potentially colred noise process24. Stratonovich\ncalculus is then to be preferred over ^Ito calculus as ascertained by the Wong-Zakai theorem.\n26We can allow ourselves the freedom to switch between expressions involving \rand\u000f.\r2=\nj\u000fj(D+1)\nD+j\u000fjis a monotonically increasing function of j\u000fjwith the convenient property that j\u000fj=\n1!\r= 1 andj\u000fj= 1!\r= 1. As such limits written in terms of \rand\u000fare equivalent.\nAppendix A: Elliptic Integral Identities\nThe identities\ne(x)\u0011xE(1\u00001\nx2) = E(x) (A1)\nk(x)\u00111\nxK(1\u00001\nx2) = K(x) (A2)\ncan be immediately proven by considering the de\fning di\u000berential equation satis\fed by\nK(x), namely23:\n@xK(x) =E(x)\u0000(1\u0000x)K(x)\n2x(1\u0000x): (A3)\nComputing then explicitely the derivative of k(x) using de\fnition (A1) and rearranging,\none \fnds that:\n@xk(x) =e(x)\u0000(1\u0000x)k(x)\n2x(1\u0000x); (A4)\nin other words they satisfy the same di\u000berential relation.\n28"    },    {        "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": "1307.7427v1.Theoretical_Study_of_Spin_Torque_Oscillator_with_Perpendicularly_Magnetized_Free_Layer.pdf",        "content": "arXiv:1307.7427v1  [cond-mat.mes-hall]  29 Jul 20131\nTheoretical Study of Spin-Torque Oscillator with\nPerpendicularly Magnetized Free Layer\nTomohiro Taniguchi, Hiroko Arai, Hitoshi Kubota, and Hiros hi Imamura∗\nSpintronics Research Center, AIST, Tsukuba, Ibaraki 305-8 568, Japan\nAbstract—The magnetization dynamics of spin torque oscilla-\ntor (STO) consisting of a perpendicularly magnetized free l ayer\nand an in-plane magnetized pinned layer was studied by solvi ng\nthe Landau-Lifshitz-Gilbert equation. We derived the anal ytical\nformula of the relation between the current and the oscillat ion\nfrequency of the STO by analyzing the energy balance between\nthe work done by the spin torque and the energy dissipation du e\nto the damping. We also found that the field-like torque break s\nthe energy balance, and change the oscillation frequency.\nIndex Terms —spintronics, spin torque oscillator, perpendicu-\nlarly magnetized free layer, the LLG equation\nI. INTRODUCTION\nSPIN torque oscillator (STO) has attracted much attention\ndue to its potential uses for a microwave generator and a\nrecording head of a high density hard disk drive. The self-\noscillation of the STO was first discovered in an in-plane\nmagnetized giant-magnetoresistive (GMR) system [1]. Afte r\nthat, the self-oscillation of the STO has been observed not\nonly in GMR systems [2]-[6] but also in magnetic tunnel\njunctions (MTJs) [7]-[11]. The different types of STO have\nbeen proposedrecently; for example,a point-contactgeome try\nwith a confinedmagneticdomainwall [12]-[14]whichenables\nusto controlthe frequencyfroma few GHz to a hundredGHz.\nRecently, Kubota et al.experimentally developed the MgO-\nbased MTJ consisting of a perpendicularly magnetized free\nlayer and an in-plane magnetized pinned layer [15],[16]. Th ey\nalso studied the self-oscillation of this type of MTJ, and\nobserved a large power ( ∼0.5µW) with a narrow linewidth\n(∼50MHz) [17].These results are great advancesin realizing\nthe STO device.However,the relationbetweenthe currentan d\nthe oscillation frequency still remains unclear. Since a pr ecise\ncontrol of the oscillation frequency of the STO by the curren t\nis necessary for the application, it is important to clarify the\nrelation between the current and the oscillation frequency .\nIn this paper, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency\nof the STO consisting of the perpendicularly magnetized\nfree layer and the in-plane magnetized pinned layer. The\nderivation is based on the analysis of the energy balance\nbetween the work done by the spin torque and the energy\ndissipation due to the damping. We found that the oscillatio n\nfrequencymonotonicallydecreases with increasing the cur rent\nby keeping the magnetization in one hemisphere of the free\nlayer. The validity of the analytical solution was confirmed\nby numerical simulations. We also found that the field-like\n∗Corresponding author. Email address: h-imamura@aist.go. jppmelectron (I>0)z\nxy\nspin torquedamping damping spin torque\nFig. 1. Schematic view of the system. The directions of the sp in torque and\nthe damping during the precession around the z-axis are indicated.\ntorque breaks the energy balance, and change the oscillatio n\nfrequency. The shift direction of the frequency, high or low ,\nis determined by the sign of the field-like torque.\nThis paper is organized as follows. In Sec. II, the current\ndependence of the oscillation frequency is derived by solvi ng\nthe Landau-Lifshitz-Gilbert (LLG) equation. In Sec. III, t he\neffect of the field-like torque on the oscillation behaviour is\ninvestigated. Section IV is devoted to the conclusions.\nII. LLG STUDY OF SPIN TORQUE OSCILLATION\nThe system we consider is schematically shown in Fig.\n1. We denote the unit vectors pointing in the directions\nof the magnetization of the free and the pinned layers as\nm= (sinθcosϕ,sinθsinϕ,cosθ)andp, respectively. The\nx-axis is parallel to pwhile the z-axis is normal to the film\nplane. The variable θofmis the tilted angle from the z-axis\nwhileϕis the rotation angle from the x-axis. The current I\nflows along the z-axis, where the positive current corresponds\nto the electron flow from the free layer to the pinned layer.\nWe assume that the magnetization dynamics is well de-\nscribed by the following LLG equation:\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt.(1)\nThe gyromagneticration and the Gilbert damping constant ar e\ndenotedas γandα, respectively.The magnetic field is defined\nbyH=−∂E/∂(Mm), where the energy density Eis\nE=−MHapplcosθ−M(HK−4πM)\n2cos2θ.(2)\nHere,M,Happl, andHKare the saturation magnetization,the\napplied field along the z-axis, and the crystalline anisotropy\nfield along the z-axis, respectively. Because we are interested\nin the perpendicularly magnetized system, the crystalline\nanisotropy field, HK, should be larger than the demagneti-\nzation field, 4πM. Since the LLG equation conserves the2\nI = 1.2 ~ 2.0 (mA)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\ncurrent (mA)34567(b)\nfrequency (GHz) \n1.2 1.4 1.6 1.8 2.0\nFig. 2. (a) The trajectories of the steady state precession o f the magne-\ntization in the free layer with various currents. (b) The dot s represent the\ndependence of the oscillation frequency obtained by numeri cally solving the\nLLG equation. The solid line is obtained by Eqs. (8) and (9).\nnormofthe magnetization,the magnetizationdynamicscan b e\ndescribed by a trajectory on an unit sphere. The equilibrium\nstates of the free layer correspond to m=±ez. In following,\nthe initial state is taken to be the north pole, i.e., m=ez. It\nshould be noted that a plane normal to the z-axis, in which θ\nis constant, corresponds to the constant energy surface.\nThe spin torque strength, Hsin Eq. (1), is [18]-[20]\nHs=/planckover2pi1ηI\n2e(1+λmx)MSd, (3)\nwhereSanddare the cross section area and the thickness\nof the free layer. Two dimensionless parameters, ηandλ\n(−1< λ <1), determine the magnitude of the spin polariza-\ntion and the angle dependence of the spin torque, respective ly.\nAlthough the relation among η,λ, and the material parameters\ndepends on the theoretical models [20]-[22], the form of Eq.\n(3) is applicable to both GMR system and MTJs. In particular,\nthe angle dependence of the spin torque characterized by λis\na key to induce the self-oscillation in this system.\nFigure 2 (a) shows the steady state precession of the mag-\nnetization in the free layer obtained by numerically solvin g\nEq. (1). The values of the parameters are M= 1313emu/c.c.,\nHK= 17.9kOe,Happl= 1.0kOe,S=π×50×50nm2,\nd= 2.0nm,γ= 17.32MHz/Oe, α= 0.005,η= 0.33,\nandλ= 0.38, respectively [17]. The self-oscillation was\nobservedforthe current I≥1.2mA.Althoughthe spintorque\nbreaks the axial symmetry of the free layer along the z-axis,\nthe magnetization precesses around the z-axis with an almost\nconstanttilted angle. Thetilted angle fromthe z-axisincreases\nwith increasing the current; however, the magnetization st ays\nin the northsemisphere( θ < π/2). The dotsin Fig. 2 (b) show\nthe dependence of the oscillation frequency on the current. As\nshown, the oscillation frequencymonotonicallydecreases with\nincreasing the current magnitude.\nLet us analytically derive the relation between the current\nand the oscillation frequency. Since the self-oscillation occurs\ndue to the energysupply into the free layer by the spin torque ,\nthe energy balance between the spin torque and the damping\nshould be investigated. By using the LLG equation, the time\nderivative of the energy density Eis given by dE/dt=Ws+\nWα, where the work done by spin torque, Ws, and the energydissipation due to the damping, Wα, are respectively given by\nWs=γMHs\n1+α2[p·H−(m·p)(m·H)−αp·(m×H)],\n(4)\nWα=−αγM\n1+α2/bracketleftBig\nH2−(m·H)2/bracketrightBig\n. (5)\nBy assuming a steady precession around the z-axis with a\nconstant tilted angle θ, the time averages of WsandWαover\none precession period are, respectively, given by\nWs=γM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ,(6)\nWα=−αγM\n1+α2[Happl+(HK−4πM)cosθ]2sin2θ.(7)\nThe magnetization can move from the initial state to a point\nat which dE/dt= 0. Then, the current at which a steady\nprecession with the angle θcan be achieved is given by\nI(θ) =2αeλMSd\n/planckover2pi1ηcosθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg−1\n×[Happl+(HK−4πM)cosθ]sin2θ.(8)\nThe corresponding oscillation frequency is given by\nf(θ) =γ\n2π[Happl+(HK−4πM)cosθ].(9)\nEquations (8) and (9) are the main results in this section.\nThe solid line in Fig. 2 (b) shows the current dependence of\nthe oscillation frequency obtained by Eqs. (8) and (9), wher e\nthe good agreement with the numerical results confirms the\nvalidity of the analytical solution. The critical current f or the\nself-oscillation, Ic= limθ→0I(θ), is given by\nIc=4αeMSd\n/planckover2pi1ηλ(Happl+HK−4πM).(10)\nThe value of Icestimated by using the aboveparametersis 1.2\nmA, showing a good agreement with the numerical simulation\nshown in Fig. 2 (a). The sign of Icdepends on that of λ,\nand the self-oscillation occurs only for the positive (nega tive)\ncurrent for the positive (negative) λ. This is because a finite\nenergy is supplied to the free layer for λ/negationslash= 0, i.e.,Ws>0. In\nthe case of λ= 0, the average of the work done by the spin\ntorque is zero, and thus, the self-oscillation does not occu r.\nIt should be noted that I(θ)→ ∞in the limit of θ→π/2.\nThis means the magnetization cannot cross over the xy-plane,\nand stays in the north hemisphere ( θ < π/2). The reason\nis as follows. The average of the work done by spin torque\nbecomes zero in the xy-plane (θ=π/2) because the direction\nof the spin torque is parallel to the constant energy surface .\nOn the other hand, the energy dissipation due to the damping\nis finite in the presence of the applied field [21]. Then,\ndE/dt(θ=π/2) =−αγMH2\nappl/(1 +α2)<0, which\nmeansthe dampingmovesthe magnetizationto the northpole.\nThus, the magnetization cannot cross over the xy-plane. The\ncontrollable range of the oscillation frequency by the curr ent\nisf(θ= 0)−f(θ=π/2) =γ(HK−4πM)/(2π), which is\nindependent of the magnitude of the applied field.3\nSince the spin torque breaks the axial symmetry of the\nfree layer along the z-axis, the assumption that the tilted\nangle is constant used above is, in a precise sense, not valid ,\nand thez-component of the magnetization oscillates around\na certain value. Then, the magnetization can reach the xy-\nplane and stops its dynamics when a large current is applied.\nHowever,thevalue ofsuch currentis morethan15mA forour\nparameter values, which is much larger than the maximum of\nthe experimentallyavailable current. Thus, the above form ulas\nwork well in the experimentally conventional current regio n.\nContrary to the system considered here, the oscillation be-\nhaviour of an MTJ with an in-plane magnetized free layer and\na perpendicularly magnetized pinned layer has been widely\ninvestigated [23]-[26]. The differences of the two systems\nare as follows. First, the oscillation frequency decreases with\nincreasing the current in our system while it increases in\nthe latter system. Second, the oscillation frequency in our\nsystem in the large current limit becomes independent of the\nz-component of the magnetization while it is dominated by\nmz= cosθin the latter system. The reasons are as follows.\nIn our system, by increasing the current, the magnetization\nmoves away from the z-axis due to which the effect of the\nanisotropy field on the oscillation frequency decreases, an d\nthe frequency tends to γHappl/(2π), which is independent\nof the anisotropy. On the other hand, in the latter system,\nthe magnetization moves to the out-of-plane direction, due\nto which the oscillation frequency is strongly affected by t he\nanisotropy (demagnetization field).\nThe macrospin model developed above reproduces the ex-\nperimentalresultswith the freelayerof2nmthick[17],for ex-\nample the current-frequency relation, quantitatively. Al though\nonlythezero-temperaturedynamicsisconsideredinthispa per,\nthe macrospin LLG simulation at a finite temperature also\nreproduces other properties, such as the power spectrum and\nits linewidth, well. However, when the free layer thickness\nfurther decreases, an inhomogeneousmagnetization due to t he\nroughnessattheMgOinterfacesmayaffectsthemagnetizati on\ndynamics: for example, a broadening of the linewidth.\nIII. EFFECT OF FIELD -LIKE TORQUE\nThe field-like torque arises from the spin transfer from the\nconductionelectrons to the local magnetizations,as is the spin\ntorque. When the momentum average of the transverse spin of\ntheconductionelectronsrelaxesinthefreelayerveryfast ,only\nthe spin torque acts on the free layer [19]. On the other hand,\nwhen the cancellation of the transverse spin is insufficient ,\nthe field-like torque appears. The field-like torque added to\nthe right hand side of Eq. (1) is\nTFLT=−βγHsm×p, (11)\nwhere the dimensionless parameter βcharacterizes the ratio\nbetween the magnitudes of the spin torque and the field-like\ntorque. The value and the sign of βdepend on the system\nparameters such as the band structure, the thickness, the\nimpurity density, and/or the surface roughness [22],[27]- [29].\nThe magnitude of the field-like torque in MTJ is much larger\nthan that in GMR system [30],[31] because the band selectionmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(b)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(c) (d)\n0 0.2 0.1\ntime (μs)0\n-0.5 \n-1.00.51.0mzβ=0 β=0.5\nβ=-0.5\nβ=-0.5\nβ=0.5β=0\nFig. 3. The magnetization dynamics from t= 0with (a) β= 0, (b)\nβ= 0.5, andβ=−0.5. The current magnitude is 2.0mA. (d) The time\nevolutions of mzfor various β.\nduringthe tunnelingleads to an insufficient cancellation o f the\ntransverse spin by the momentum average.\nIt should be noted that the effective energy density,\nEeff=E−βM/planckover2pi1ηI\n2eλMSdlog(1+λmx),(12)\nsatisfying −γm×H+TFLT=−γm×[−∂Eeff/(Mm)],\ncan be introduce to describe the field-like torque. The time\nderivative of the effective energy, Eeff, can be obtained by\nreplacing the magnetic field, H, in Eqs. (4) and (5) with\nthe effective field −∂Eeff/∂(Mm) =H+βHsp. Then, the\naverage of dEeff/dtover one precession period around the\nz-axis consists of Eq. (6), (7), and the following two terms:\nW′\ns=βγM\n1+α2/parenleftbigg/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n,\n(13)\nW′\nα=−αγM\n1+α2/parenleftbiggβ/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n−2αβγM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ.\n(14)\nThe constant energy surface of Eeffshifts from the xy-plane\ndue to a finite |β|(≃1), leading to an inaccuracy of the\ncalculation of the time average with the constant tilted ang le\nassumption. Thus, Eqs. (13) and (14) are quantitatively val id\nfor only |β| ≪1. However, predictions from Eqs. (13) and\n(14) qualitatively show good agreements with the numerical\nsimulations, as shown below.\nFor positive β,W′\nsis also positive, and is finite at θ=π/2.\nThus,dEeff/dt(θ=π/2)can be positive for a sufficiently\nlarge current. This means, the magnetization can cross over\nthexy-plane, and move to the south semisphere ( θ > π/2).\nOn the other hand, for negative β,W′\nsis also negative. Thus,\ntheenergysupplybythespintorqueissuppressedcomparedt o4\ncurrent (mA)34567frequency (GHz) \n1.2 1.4 1.6 1.8 2.02\n1\n0β=0\nβ=0.5β=-0.5\nFig. 4. The dependences of the oscillation frequency on the c urrent for\nβ= 0(red),β= 0.5(orange), and β=−0.5(blue), respectively.\nthecaseof β= 0.Then,arelativelylargecurrentisrequiredto\ninduce the self-oscillation with a certain oscillation fre quency.\nAlso, the magnetization cannot cross over the xy-plane.\nWe confirmed these expectations by the numerical simu-\nlations. Figures 3 (a), (b) and (c) show the trajectories of\nthe magnetization dynamics with β= 0,0.5, and−0.5\nrespectively,while thetime evolutionsof mzareshownin Fig.\n3 (d). The current value is 2.0 mA. The current dependences\nof the oscillation frequency are summarized in Fig. 4.\nIn the case of β= 0.5>0, the oscillation frequency is low\ncompared to that for β= 0because the energy supply by the\nspin torque is enhanced by the field-like torque, and thus, th e\nmagnetization can largely move from the north pole. Above\nI= 1.9mA, the magnetizationmovesto the southhemisphere\n(θ > π/2), and stops near θ≃cos−1[−Happl/(HK−4πM)]\nin the south hemisphere, which corresponds to the zero fre-\nquency in Fig. 4.\nOn the other hand, in the case of β=−0.5<0, the\nmagnetization stays near the north pole compared to the case\nofβ= 0, because the energy supply by the spin torque is\nsuppressed by the field-like torque. The zero frequency in\nFig. 4 indicates the increase of the critical current of the s elf-\noscillation. Compared to the case of β= 0, the oscillation\nfrequency shifts to the high frequency region because the\nmagnetization stays near the north pole.\nIV. CONCLUSIONS\nIn conclusion, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency of\nSTO consisting of the perpendicularly magnetized free laye r\nand the in-plane magnetized pinned layer. The derivation is\nbased on the analysis of the energy balance between the work\ndone by the spin torque and the energy dissipation due to the\ndamping.The validityof the analyticalsolutionwas confirm ed\nby numerical simulation. We also found that the field-like\ntorque breaks the energy balance, and changes the oscillati on\nfrequency. The shift direction of the frequency, high or low ,\ndepends on the sign of the field-like torque ( β).ACKNOWLEDGMENT\nThe authors would like to acknowledge T. Yorozu, H.\nMaehara, A. Emura, M. Konoto, A. Fukushima, S. Yuasa, K.\nAndo, S. Okamoto, N. Kikuchi, O. Kitakami, T. Shimatsu, K.\nKudo, H. Suto, T. Nagasawa, R. Sato, and K. Mizushima.\nREFERENCES\n[1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C, Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature, vol.425,\npp.380-383, 2003.\n[2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva,\nPhys. Rev. Lett. , vol.92, pp.027201, 2004.\n[3] J. C. Sankey, I. N. Krivorotov, S. I. Kiselev, P. M. Baraga nca, N. C, Em-\nley, R. A. Buhrman, and D. C. Ralph, Phys. Rev. B , vol.72, pp.224427,\n2005.\n[4] I. N. Krivorotov, N. C, Emley, R. A. Buhrman, and D. C. Ralp h,Phys.\nRev. B, vol.77, pp.054440, 2008.\n[5] W. H. Rippard, M. R. Pufall, M. L. Schneider, K. Garello, a nd\nS. E. Russek J. Appl. Phys. , vol.103, pp.053914, 2008.\n[6] J. Sinha, M. Hayashi, Y. K. Takahashi, T. Taniguchi, M. Dr apeko,\nS. Mitani, and K. Hono, Appl. Phys. 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Buhrman,\nand D. C. Ralph, Nat. Phys. , vol.4, pp.67-71, 2008."    },    {        "title": "1308.0192v1.Inverse_Spin_Hall_Effect_in_nanometer_thick_YIG_Pt_system.pdf",        "content": "1 Inverse Spin Hall Effect in nanometer -thick  YIG/Pt system  \nO. d’Allivy Kelly1, A. Anane1*, R. Bernard1, J. Ben Youssef2, C. Hahn3, A-H. Molpeceres1, C. \nCarrétéro1, E. Jacquet1, C. Deranlot1, P. Bortolotti1, R. Lebourgois4, J-C. Mage1, G. de Loubens3, O. \nKlein3, V. Cros1 and A. Fert1 \n1Unité Mixte de Physique CNRS/Thales and Université Paris -Sud, 1 avenue Augustin Fresnel, \nPalaiseau, France  \n2Université de Br etagne Occidentale, LMB -CNRS, Brest, France  \n3Service de Physique de l’Etat Condensé, C EA/CNRS, Gif-sur-Yvette, France  \n4Thales Research and Technology, 1 avenue Augustin Fresnel, Palaiseau, France  \n \nKey words:  YIG, FMR, Inverse Spin Hall Effect, spin waves  \n \nAbstract:  \nHigh  quality nanometer -thick (20 nm , 7 nm and 4 nm) epitaxial YIG films have been grown on GGG \nsubstrates using pulsed laser deposition. The Gilbert damping coefficient for the 20 nm thick films is \n2.3 x 10-4 which is the lowest value reported for sub -micrometric thick films. We demonstrate Inverse \nspin Hall effect (ISHE) detection of propagating spin waves using Pt. The amplitu de and the lineshape \nof the ISHE voltage correlate  well to the increase of the Gilbert damping when decreasing thickness of \nYIG. Spin Hall effect based loss -compensation experiments have been conducted but no change in the \nmagnetization dynamics could be d etected.  \n \n \n \n* Contact author :  \nAbdel madjid Anane  \nabdelmadjid.anane@thalesgroup.com  \n  2  \nAmong all magnetic materials, Yttrium Iron Garnet Y 3Fe5O12 (YIG) has been the one that had the most \nprominent role in understanding high frequency magnetization dynamics. Because of its unique \nproperties, bulk YIG crystal was the prototypal material for ferromagnetic resonance (FMR) studies in \nthe mid -twentieth ce ntury. The attractive properties of YIG include: high Curie temperature , ultra low \ndamping (the lowest among all materials at room temperature), electrical insulation, high chemical \nstability and easy synthesis in single crystalline form. Micrometer thick films of YIG were first grown \nusing liquid phase epitaxy (LPE)1, and paved the way for the emergence of a large variety of \nmicrowave devices for high -end analogue electronic applic ations throughout the 1970’s  2. \nMore recently, the interest in emerging large -scale integrated circuit technologies for beyond CMOS \napplications has fostered  new paradigms for data processing . Many of them are based on state \nvariables other than the electron charge and may eventually allow for unforeseen functionalities. \nCoding the information in a spin wave (SW) is among the most promising routes under investigation  \nand has been referred to as magnonics  3. Exciting and detecting spin waves has been ma inly achieved  \nthrough inductive coupling with radiofrequenc y (rf) anten nas but this technology remains \nincompatible with large scale integration4. Disruptive  solution s merging magnonics and spintronics \nhave been recently proposed  where spin transfer torque  (STT)  and magnetoresistive effects  would be \nused to couple to the SW s. For instance, using a STT -nano -oscillator  in a nanocontact geometry,  \ncoherent SWs emission in Ni81Fe19 (Py) thin metallic layer has been recently demonstrated and probed \nby micro -focused  Brilloui n light scattering (BLS)  5.  \nYIG is often considered as the best medium for SW propagation because of its very small  Gilbert \ndamping coefficient (2x10-5 for bulk YIG) . Being an electrical insulator, electron mediated angular \nmomentum transfer can only  occur at the  interface between YIG and a metallic layer . In that context , \nmetals with  large Spin Orbit Coupling (SOC)  like Pt  where  a pure spin current can be generated \nthrough Spin Hall Effect (SHE) 6 have been used to excite7 or amplify8,9  propagating SWs  through \nloss compensation  in YIG . Moreover, d etection of SW  can be achie ved using the  Inverse Spin Hall \nEffect (ISHE) . In ISHE , the flow of a pure spin current from the YIG into the large SOC metal \ngenerates a dc voltage . The ISHE voltage is proportional to the Spin Hall angle () and the effective \nspin mixing conductance  ( ) that is in play in the physics of spin pumping . As for the SOC \nmaterials, u p to recently, mainly Pt has been used , however it can be observed that  other 5 d heavy \nmetals such as Ta 10, W 11 or CuBi 12 are also very promising . \nAs the  amount of angular momentum transferred from (to) the YIG magnetic film per unit volume \nscales with 1/  (where t is the YIG thickness) , it is necessary to reduce the YIG thickness as much as \npossible while keeping its magnetic properties . Indeed , the threshold current density in the SOC metal \nfor the macrospin  mode excitation is expressed as13 :  (Eq. 1) where  is the spin \nHall angle, the electron charge,  the gyromagnetic ratio ,  the FMR frequency and  the YIG’s 3 saturation magnetization . Furthermore, a better understanding of the physics involved in spin \nmomentum transfer at the YIG/metal interface would be achieved by reducing the YIG film thickness \nbelow the exchange length (~ 10 nm)  14 . Up to now, sub micrometer -thick YIG films have been \nmainly grown by  LPE but the ultimate thickn ess are around  200 nm15. To further reduce the \nthickness, other growth  methods are to be considered . Pulsed laser deposition (PLD) is the most \nversatile technique for oxide films epitax y. Several  groups have worked on PLD grown YIG16,17,18,19 \nbut it is only recently that the films quality is approaching that of LPE20,21. \nIn this letter , we present PLD growth of ultrathin YIG films with various thickness (20 nm, 7 nm and 4 \nnm) on Gadolinium Gallium  Garnet  (GGG)  (111) substrates . Structural and magnetic  characterizations  \nand FMR  measurements  demonstrate the high quality of our nanometer -thick YIG films  comparable to \nstate of the art LPE films . The growth has been performed using a frequency tripled (\n = 355 nm) \nNd:YAG laser  and a stochiometric polycrystalline YIG pellet . The pulse rate was 2.5  Hz and the \nsubstrate -target distance was 4 4 mm. Prior to the YIG deposition, the GGG substrate  is annealed  at \n700°C under an oxygen pressure of 0.4 mbar. Growth temperature is then set to be 650 °C, and oxygen \npressure to 0.25 mbar. After the film deposition , samples are cooled down to room temperat ure under \n300 mbar of O 2. The YIG thickness is measured for each sample  using X -ray reflectometry which \nyield a precision better than 0.3 nm. The surface morphology and roughness have been studied by \natomic force microscopy ( AFM ). RMS roughness has been measured over 1 µm2  ranges between 0.2 \nnm and 0.3 nm  for all films  (Fig 1a) . As often with PLD growth, d roplets  are present  on the film \nsurface, here the ir lateral sizes  are below 100 nm and their density is very low (~ 0.1 µm-2). X-Ray \nDiffraction  (XRD)  spectr a using Cu K\n 1 radiation  show that the growth is along the (111) direction . \nOnly peaks characteristic of YIG and the GGG substrate are observed ( see Fig 1b ). The YIG lattice \nparameter is very close to that of the substrate and can only be resolved ev entually at large diffraction \nangle s. For the 20 nm YIG  film (Fig 1b ,1a), refinement using EVA software on the 888 reflection \nyields a cubic lattice parameter of 1.2459 nm, to be compared to 1.2376  nm for the bulk YIG22. For \nthinner films (between 4 and 15 nm) , it was not possible to distinguish the YIG peaks from those of \nthe substrate  (Fig 1d, 1e). This  sharp variation of the XRD spectra  with respect to the film thickness  \ntends to point  towards a critical thickness  for strain relaxation. It is however worth noting that the \ncubic lattice parameter of the 20 nm thick film is larger than the bulk lattice parameter  but also of that \nto the GGG substrate (1.2383 nm) . A slight o ff-stoichiometry (either oxygen vaca ncies or cation \ninterstitials)  is probably at the origin of this observation . Pole figure  measurements  have been \nperformed to gain insight s into the in -plane crystal structure, but it was not possible to resolve , at this \nstage,  the film s peaks from the substrate peaks  from w hich we infer that  the growth is epitaxial and the \nfilm single crystalline . \nFrom SQUID magnetometry with in -plane  magnetic field , we measure  a magnetization of 4\nMs = \n2100 G \n 50 G at room temperature  for both the 20 and 7 nm films . This value is independently  4 confirmed by out -of plan e FMR resonance  while the tabulated bulk value  for YIG  is 4\nMs = 1760 G. \nA similar increase of the PLD grown YIG magnetization have been reported and attributed to an  off-\nstoichiome try19. The coercitive  fields are extremely small , about 0.2 Oe  (which is the experimental \nresolution ) and the saturation field is 5 Oe. There is no evidence for in -plane magnetic anisotropy . The \noverall magnetic signature is that of an ultra -soft material . Note that for the thinnest films (4 nm) we \nmeasure a decrease of the saturation  magnetization  to roughly 1700 G . Finally, we emphasize that the \nstructural and magnetic properties of the samples are well reproducible with respect to the elaboration \nconditions.  \nFMR fields and linewidths were measured at frequencies in the range 1 -40 GHz using high sensitive \nwideband r esonance spectrometer with a nonresonant microstrip transmission line.  The FMR is \nmeasured via the derivative of microwave power absorption using a small rf exciting field. Resonance \nspectra were recorded with the applied static magnetic field  oriented in plane . During the magnetic \nfield sweeps , the amplitude of the modulation field was appreciably smaller than the FMR linewidth. \nThe amplitude of the excit ation field h rf is about 1 mOe , which corresponds to the linear response \nregime. A phase -sensitive detector with lock -in detection was used. The field derivative of the \nabsorbed power  is proportional to the field derivative of the imaginary part of the rf susceptibility: \n (where  and \n″ is the imaginary part of the susceptibility of the uniform \nmode) . Typical resonance curve s are  plotted in Fig. 2a 2b . In Fig. 2c, we  show the frequency \ndependence of the peak -to-peak linewidth  for three different  YIG thicknesses , i.e., 20, 7 and 4 nm . As \nfor the 20 nm  YIG film, we find a linear dependence of the FMR linewidth with rf frequency  while for \nthe thinnest films , we do find an almost linear increase in the low frequency range (< 12 GHz) and \nthen a saturation  of the linewidth with frequency . Such qualitative difference depending on the \nthickness is reminiscent of the qualitative difference discussed earlier in the X -ray diffraction data  (Fig \n1). The linear dependence  of the resonance linewidth  is expected with in the frame of the Landau -\nLifshitz Gilbert equation and allow for a straightforward  calculation  of the intrinsic Gilbert damping \ncoefficient ( for the 20 nm thick film). The zero frequency intercept of the fitting line , \nusually referred to a s the extrinsic lin ewidth  23,24, is found to be  \nH0  =1.4 Oe. We emphasize that our \nvalue for the  intrinsic damping on the 20 nm thick film is among the best ever reported  independ ently \nof the growth technique  and is only outperformed by the 1.3 µm film used by Y. Kajiwara  et al. 7. As \nfor the extrinsic damping, our value s are still a bit larger than those obtained for 200 nm thick films \ngrown by  LPE (\n H0= 0.4 Oe) 10. The saturation of the linewidth with increasing  excitation  frequency  \nobserved for  thinnest films ( t = 7 and 4 nm) is usually  ascribed to two -magnon s scattering  due to the \ninterfaces 25. An estimation of the intrinsic damping in such thin films is  thus not correct. \nNevertheless , and only for the sake of comparison, considering frequencies under 6 GHz , we can \nrough ly estimate the low frequency Gilbert damping  to be  1.610-3 for the 7 nm and 3.810-3 the 4nm \nYIG films . However, it is  worth mentioning that for those  two thinnest films;  samples  sliced  from the 5 same substrate  can give different linewidths ( up to a factor of 3) with for some of them up to 2 \nabsorption s lines. This observation point s to a slight lateral non-homogeneity  in the chemical \ncomposition24. The data presented in figure 2 are tho se of the best samples showing a single absorption \nline. For the 20  nm thick films , all samples have only one resonance line and the dispersion of \nlinewidths is within 5%.   \nIn order to characterize the conversion of propagating SWs in YIG into a charge current in a normal \nadjacent metal with large SOC , we perform ISHE detection of SW  with the strip geometry used by \nChumak et al.26. In our sample design (see Fig 3a) , SWs  excitation is achieved using a patterned 100 \nµm wide  Au stripline  antenna  whereas the ISHE voltage is measured on a 13 nm thick Pt strip ( 0.2 \nmm x  5 mm ) located at  100 µm away from the Au stripe  and parallel to it . The metallic Pt strip is \ndeposited using dc magnetron sputtering  and lift -off. Prior to  the Pt deposition , an in -situ O2/Ar-\nplasma is used to  remove  the photo -resist residue s and increase the ISHE voltage.  This cleaning step \nhas been shown recently to improve the ISHE signal by one order of magnitude27,28. Measurement s of \nthe ISHE signal is performed either using a lock -in (with a 5 kHz TTL modulation  of the rf power ) or \na nano -voltmeter.  In order to increase the output ISHE signa l, we chose a specific configuration with a \nmagnetic field at 45° from  to the SW propagation direction  (cf Fig. 3a). This configuration  has the \nadvantage of provid ing a good coupling of the YIG film  to the rf field under the antenna while still \nhaving a sig nificant spin polarization  () that is orthogonal  to the measured ISHE electrical field . We \nshould point out  that our choice of magnetic field direction implies that the propagating SWs are \nneither Damon -Eshbach modes  where k \n M nor backward volume modes where  k // M.  \nIn Fig 3b, we display the  ISHE voltage (without any geometrical correction) as a function of the in \nplane magnetic field  measured  on a 20 nm thick YIG under a 10 mW rf excitation at 1 GHz. The sign \nof voltage peak (occurring at magnetic fields that resonantly excite the magnetization) reverses when \nthe applied field is reversed as expected from the ISHE symmetry 29 : where  \nis the pure spin current that flows through the YIG/Pt interface , is the spin polarization vector \nparallel to the dc magnetization direction  and  is the unit vector parallel to the Pt strip.  A close -up \non the ISHE signal lineshape for the different thickness es is plotted in Fig . 3c 3d 3e. In accordance \nwith FMR study, the linewidth increases with decreasing YIG thickness. The spectral  lineshap es are \nalmost symmetrical confirming previous reports on YIG films  grown using LPE  30. The ISHE \nmaximum voltage decreases when the film thickness  is decrease d. Going from 20 nm to 4 nm this \ndecrease is as large as two order s of magn itudes  and correlates to the increase of the Gilbert damping . \nA decrease of the ISHE signal when increasing damping is expected.  Indeed, a s we are in the weak \nexcitation regime, the ISHE voltage is expected to scale with 1/\n2, see for instance supplementary  \nmate rials of Ref [7]. If we consider the Gilbert damping obt ained from  FMR on the bar e YIG samples,  \na more dramatic decrease of the I SHE signal  than the one observed is expected.  In fact here, one \nshould consider the effective damping of the Pt/YIG stack as spin -pumping will increase the YIG 6 effective damping under the Pt strip. We should ag ain point out  that our measurement geometry relies  \non propagating SWs  and therefor e they are subject to an exponential decay with distance . The length \nscale of this exponential decay is different for the three  thicknesses  owing to the difference in the \ndamping parameter ; hence a quantitative interpretation of the voltage amplitude is to be avoided here.  \nWe thus succeeded  to grow high quality ultrathin YIG film and demonstrated that a n efficient  spin \nangular momentum transfer from the YIG film toward the Pt layer . The nex t objective is to induce an \nmodification on the effective YIG damping coefficient via interfacial spin injection using SHE , as it \nhas been  reported in Py/Pt31,32 system . Using YIG/Pt,  Kajiwara et al.  have shown that even without rf \nexcitation, spin injection induced by  a dc current in Pt can generate propagating SWs ; the threshold  \ncurrent density has been estimated to be 4.4 x  108 A.m-2. Such unexpectedly low value has been \nattributed to the presence of an easy –axis surface anisotropy13. Hence, w e have performed experiments  \nwhere  a large  dc bias current is applied on the Pt electrode  while pumping spin waves  in the 20 nm \nthick YIG film  with TTL modulated rf excitation field . The VISHE linewidth  is measured at the TTL \nmodulation frequency using a lock -in. We  expected to increase or decrease  this linewidth  depending \non the dc current  polarity  but w e have not been able to see any sizable effect even for current densities \nas large as 6 x 109 Am-2, within a 0.2 Oe resolution, the measured linewidth remains absolutely \nunchanged . Theory  predicts that th e threshold dc current for the onset of magnetic excitations scales \nwith the Gilbert  damping and the thickness  of the ferromagnetic insulator  (Eq. 1), the smaller the \n product the lower  the threshold current for SW excitation  is. In Kajiwara et al. ’s experiment  \n nm; in our case  nm (considering the spin pumping contribution  to the \ndamping ). We therefore applie d a dc current that is roughly 2  orders of magnitude  larger than the \nexpected threshold current . One possible explanation  is that in our films , surface anisotropy is absent \nand therefore the pumped angular momentum is spread over many excitations modes33. Further \ninvestigations are under progress to clarify this point.  \nIn summary,  we have fabricated PLD grown YIG on GGG (111) substrates with t hickness as low as 4 \nnm. We present here a comparative study on three  different thicknesses : 4, 7 and 20 nm . The Gilbert \ndamping coefficient for the 20 nm thick films is 2.3 x 10-4 which is the lowest value reported for sub -\nmicrometric thick films. We demonstrate ISHE detection of SWs for ultra thin YIG film. The \namplitudes of the ISHE voltage correlate s well to the increase of the Gilbert damping when decreasing \nthickness. Owing to extremely low product  of the 20 nm film that is almost 10 time s smaller \nthan the one reported by Kajiwara et al.  we expected to observe  compensation of the damping by spin \ncurrent injection through the SHE  but our preliminary results on the VISHE  linewidth did not reveal any \neffect on the magnetization dynamics.  \n \nAcknowledgement:  \nThis work has been supported by ANR -12-ASTR -0023 Trinidad   7  \nReferences:  \n \n1 J. E. Mee, J. L. Archer, R.H. Meade, and T.N. Hamilton,  Applied Physics Letters  10 \n(1967).  \n2 J. P. Castera,  Journal of Applied Physics 55 (6), 2506 (1984).  \n3 A. Khitun, M. Q. Bao, and K. L. Wang,  Journal of Physics D -Applied Physics 43 \n(26), 10 (2010).  \n4 V. Vlaminck and M. Bailleul,  Physical Review B 81 (1) (2010).  \n5 Vladislav E. Demidov, Sergei Urazhdin, and Sergej O. Demokritov,  Nature  Materials \n9 (12), 984 (2010).  \n6 J. E. Hirsch,  Physical Review Letters 83 (9), 1834 (1999).  \n7 Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, \nH. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh,  Nature 464 (7286),  \n262 (2010).  \n8 Zihui Wang, Yiyan Sun, Mingzhong Wu, Vasil Tiberkevich, and Andrei Slavin,  \nPhysical Review Letters 107 (14) (2011).  \n9 E. Padron -Hernandez, A. Azevedo, and S. M. Rezende,  Applied Physics Letters 99 \n(19), 3 (2011).  \n10 C. Hahn, G. de Loubens,  O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef,  \nPhysical Review B 87 (17) (2013).  \n11 C. F. Pai, L. Q. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman,  Applied \nPhysics Letters 101 (12) (2012).  \n12 Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deran lot, H. X. Yang, M. Chshiev, T. Valet, \nA. Fert, and Y. Otani,  Physical Review Letters 109 (15) (2012).  \n13 J. Xiao and G. E. W. Bauer,  Physical Review Letters 108 (21), 4 (2012).  \n14 Hujun Jiao and Gerrit E. W. Bauer,  Physical review letters 110 (21), 217602 (2013).  \n15 V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef,  Physical Review B 86 \n(13), 6 (2012).  \n16 P. C. Dorsey, S. E. Bushnell, R. G. Seed, and C. Vittoria,  Journal of Applied Physics \n74 (2), 1242 (1993).  \n17 Y. Dumont, N. Keller, O. Popova, D. S. Schmool, F. Gendron, M. Tessier, and M. \nGuyot,  Journal of Magnetism and Magnetic Materials 272, E869 (2004).  8 18 Y. Krockenberger, H. Matsui, T. Hasegawa, M. Kawasaki, and Y. Tokura,  Applied \nPhysics Letters 93 (9), 3  (2008).  \n19 S. A. Manuilov, S. I. Khartsev, and A. M. Grishin,  Journal of Applied Physics 106 \n(12) (2009).  \n20 Y. Y. Sun, Y. Y. Song, H. C. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Z. Wu, \nH. Schultheiss, and A. Hoffmann,  Applied Physics Letters 101 (15), 5 (2012).  \n21 M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, \nH. Huebl, S. Geprags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. M. \nSchmalhorst, G. Reiss, L. M. Shen, A. Gupta, Y. T. Chen, G. E. W. Bauer, E. Sait oh, \nand S. T. B. Goennenwein,  Physical Review B 87 (22), 15 (2013).  \n22 M. A. Gilleo and S. Geller,  Physical Review 110 (1), 73 (1958).  \n23 D. L. Mills and S. M. Rezende,  Spin Dynamics in Confined Magnetic Structures Ii \n87, 27 (2003).  \n24 B. Heinrich, C. B urrowes, E. Montoya, B. Kardasz, E. Girt, Young -Yeal Song, Yiyan \nSun, and Mingzhong Wu,  Physical Review Letters 107 (6) (2011).  \n25 R. Arias and D. L. Mills,  Physical Review B 60 (10), 7395 (1999).  \n26 A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. \nTiberkevich, and B. Hillebrands,  Applied Physics Letters 100 (8), 3 (2012).  \n27 C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y. Y. Song, \nand M. Z. Wu,  Applied Physics Letters 100 (9), 4 (2012).  \n28 M. B. Jungfle isch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands,  \n(arXiv:1302.6697  [cond -mat.mes -hall], 2013).  \n29 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,  Applied Physics Letters 88 (18) \n(2006).  \n30 T. Tashiro, R. Takahashi, Y. Kajiwara, K. Ando, H. Nakaya ma, T. Yoshino, D. \nKikuchi, and E. Saitoh, presented at the Conference on Spintronics V, San Diego, CA, \n2012 (unpublished).  \n31 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh,  \nPhysical Review Letters 101 (3), 4 (2008).  \n32 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. \nSchmitz, and S. O. Demokritov,  Nature Materials 11 (12), 1028 (2012).  \n33 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. \nMancoff, M. A. Yar, a nd J. Akerman,  Nature Nanotechnology 6 (10), 635 (2011).  9 Figure Captions :  \nFigure 1 : \n(color online). (a) 1 µm x 1µm AFM surface topography of a 20nm YIG film on GGG (111) \n(RMS roughness=0.23  nm). (b) XRD  : \n-2\n scan of the same film using Cu -K\n1 radiation. \nThis spectrum shows that the growth is along the (111) direction with no evidence of parasitic \nphases. The YIG peak are best resolved at high diffraction angle, a zoom around the 888 GGG \npeak is shown in panel (c). \n(c) (d) (e) XRD diffraction s pectrum centered on GGG (888) reflection for different YIG \nthicknesses: 20, 7 and 4 nm. For the 2 thinnest films, the YIG peak is masked by the substrate \npeak . The cubic lattice parameter of the 20 nm thick YIG film (obtained from 888 peak) is \nslightly lar ger than that of the substrate ( aYIG=1.2459 nm , aGGG=1.2383  nm) \n \nFigure 2 : \n(color online) ( a) (b). FMR absorption derivative spectra of 20 and 4 nm thick YIG films at an \nexcitation frequency of 6 GHz.  \n(c) rf excitation frequency dependence of FMR absorption linewidth measured on different \nYIG film thicknesses with an in -plane oriented static field. The black continuous line is a \nlinear fit on the 20  nm thick film from w hich a Gilbert damping coefficient of 2. 3 x 10-4 can \nbe inferred  ( . The damping of the 7 nm and 4 nm films is \nsignificantly larger but must off all the frequency dependence is not linear (see text for \ndiscussion).   \n \nFigure 3 : \n(a) Schematic illustration of the experimental setup: spin waves are excited through the YIG \nwaveguide with a microstrip antenna. The detection is performed by measuring ISHE voltage \ndc signal on a ~200µm wide Pt stripe. (b) External dc magnetic field dependen ce of ISHE \nvoltage measured on Pt electrode showing the polarity inversion when the magnetic field is \nreversed. The peaks occur at the FMR conditions as verified through S 11 spectroscopy on the \nAu antenna (not shown here).  \n(c). (d). (e). ISHE voltage for d ifferent YIG film thicknesses around the resonance magnetic \nfield; microwave frequency is f=3GHz. The peaks can be fitted to a lorentzian shape and the \nextracted linewidth are respectively:  19.2 , 13.2 and 4.6 Oe. The dramatic increase of the \nISHE signal w ith thickness is discussed in the text.   10  \n \nFig 1  \n \n \n  \n11  \nFig 2  \n  \n12  \nFig 3  \n"    },    {        "title": "1308.0450v2.Spin_pumping_damping_and_magnetic_proximity_effect_in_Pd_and_Pt_spin_sink_layers.pdf",        "content": "arXiv:1308.0450v2  [cond-mat.mes-hall]  5 Apr 2016Spin pumping damping and magnetic proximity effect in Pd and P t spin-sink layers\nM. Caminale,1,2,∗A. Ghosh,2,†S. Auffret,2U. Ebels,2K. Ollefs,3F. Wilhelm,4A. Rogalev,4and W.E. Bailey1,5,‡\n1Fondation Nanosciences, F-38000 Grenoble, France\n2SPINTEC, Univ. Grenoble Alpes / CEA / CNRS, F-38000 Grenoble , France\n3Fakult¨ at f¨ ur Physik and Center for Nanointegration (CENI DE),\nUniversit¨ at Duisburg-Essen, 47057 Duisburg, Germany\n4European Synchrotron Radiation Facility (ESRF), 38054 Gre noble Cedex, France\n5Dept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: April 6, 2016)\nWe investigated the spin pumping damping contributed by par amagnetic layers (Pd, Pt) in both\ndirect and indirect contact with ferromagnetic Ni 81Fe19films. We find a nearly linear dependence\nof the interface-related Gilbert damping enhancement ∆ αon the heavy-metal spin-sink layer thick-\nnesses t Nin direct-contact Ni 81Fe19/(Pd, Pt) junctions, whereas an exponential dependence is o b-\nservedwhenNi 81Fe19and(Pd, Pt)areseparated by3nmCu. Weattributethequasi-l inear thickness\ndependence to the presence of induced moments in Pt, Pd near t he interface with Ni 81Fe19, quan-\ntified using X-ray magnetic circular dichroism (XMCD) measu rements. Our results show that the\nscattering of pure spin current is configuration-dependent in these systems and cannot be described\nby a single characteristic length.\nI. INTRODUCTION\nAs a novel means of conversion between charge- and\nspin-currents,spinHallphenomenahaverecentlyopened\nup new possibilities in magneto-electronics, with poten-\ntial applications in mesocale spin torques and electrical\nmanipulation of domain walls1–9. However, several as-\npects of the scattering mechanisms involved in spin cur-\nrent flow across thin films and interfaces are not entirely\nunderstood. Fundamental studies of spin current flow in\nferromagnet/non-magnetic-meal (F/N) heterostructures\nin the form of continuous films have attempted to iso-\nlate the contributions of interface roughness, microstruc-\nture and impurities10–12. magnet/non-magnetic-meal\n(F/N) heterostructures in the form of continuous films\nhave attempted to isolate the contributions of interface\nroughness, microstructure and impurities10–12. Proto-\ntypical systems in this class of studies are Ni 81Fe19/Pt\n(Py/Pt)3,5–7,13–18andNi 81Fe19/Pd(Py/Pd)8,11,14,16,19,20\nbilayers. Inthesesystems, PtandPdareemployedeither\nas efficient spin-sinks or spin/charge current transform-\ners, in spin pumping and spin Hall experiments, respec-\ntively. Pd and Pt are metals with high paramagnetic\nsusceptibility and when placed in contact with a ferro-\nmagnetic layer (eg. Py, Ni, Co or Fe) a finite magnetic\nmoment is induced at the interface by direct exchange\ncoupling21–24.\nThe role of the magnetic proximity effect on interface\nspin transport properties is still under debate. Zhang et\nal.25havereportedthat induced magnetic momentsin Pt\nand Pd films in direct contact with Py correlate strongly\nreduced spin Hall conductivities. This is ascribed to a\nspin splitting of the chemical potential and on the energy\ndependence of the intrinsic spin Hall effect. In standard\nspin pumping theory26, possible induced moments in N\nare supposed to be a priori included in calculations of\nthe spin-mixing conductance g↑↓of a F|N interface27,28,which tends to be insensitive to their presence.\nRecent theoretical works, on the other hand, propose\nthe need of a generalized spin pumping formalisms in-\ncluding spin flip and spin orbit interaction at the F |N\ninterface, in order to justify discrepancies between exper-\nimental and calculated values of mixing conductance29,30.\nAt present, it is still an open issue whether and how\nproximity-induced magnetic moments in F/N junctions\narelinked to the varietyofthe spin-transportphenomena\nreported in literature10,17,31.\nHere, we present an experimental study of the pro-\ntotypical systems: Py/Pd, Pt and Py/Cu/Pd, Pt het-\nerostructures. The objective of our study is to address\nthe role of proximity induced magnetic moments in spin\npumpingdamping. Tothisend, weemployedtwocomple-\nmentary experimental techniques. X-ray magnetic circu-\nlar dichroism (XMCD) is an element sensitive technique\nwhich allows us to quantify any static proximity-induced\nmagnetic moments in Pt and Pd. Ferromagnetic reso-\nnance(FMR) measurementsprovideindirectinformation\non the spin currents pumped out the Py layer by the pre-\ncessing magnetization, through the characterization of\nthe Pd, Pt thickness dependence of the interface-related\nGilbert damping α. In Fig. 3(Sec.IIIB), comparative\nmeasurements in Py/Cu/N and Py/N structures show a\nchange of the N thickness dependence of ∆ α(tN) from\nan exponential to a linear-like behavior. A change in\n∆α(tN) indicates a transformation in the spin scattering\nmechanism occurring at the interface, ascribed here to\nthe presence of induced moments in directly exchange\ncoupled F/N systems. Theoretical works predicted a\ndeviation from a conventional N-thickness dependence\nwhen interface spin-flip scattering is considered in the\npumping model29,30, howeverno functional form waspro-\nvided. For Py/N systems, we find that the experimental\nthickness dependence cannot be described by standard\nmodels16,26,32, but rather a linear function reproduces2\nthe data to a better degree of accuracy, by introducing\na different characteristic length. We speculate that the\nspatial extent of spin current absorption in F/N systems\nshows an inverse proportionality to interfacial exchange\ncoupling energy, obtained from XMCD, as proposed be-\nfore for spin polarized, decoupled interfaces in F 1/Cu/F 2\nheterostructures14.\nII. EXPERIMENT\nThe heterostructures were fabricated by DC mag-\nnetron sputtering on ion-cleaned Si/SiO 2substrates in\nthe form of substrate/seed/multilayer/cap stacks, where\nTa(5nm)/Cu(5nm) bilayerwasemployedas seed. Ta/Cu\nis employed to promote <111>growth in Py and subse-\nquent fcc layers (Pd, Pt), and Ta is known to not affect\nthe dampingstrongly17,32,33. Different stacksweregrown\nasmultilayer for each measurement.\nFor FMR measurements, we have multilayer =\nPy(tF)/N(tN), Py(tF)/Cu(3nm)/N( tN) with N = Pd,\nPt; an Al(3nm) film, oxidized in air, was used as\ncap. The smallest N layer thickness tNdeposited is\n0.4nm, the maximum interdiffusion length observed\nfor similar multilayers34. Samples with multilayer =\nPy(tN)/Cu(3nm) and no sink layer were also fabricated\nas reference for evaluation of the Gilbert damping en-\nhancement due tothe Pd orPtlayer. The tN-dependence\nmeasurementsofFMRweretakenforPythicknesses tF=\n5and 10nm. Results from the tF= 10nm data set are\nshown in Appendix A. Measurements of the FMR were\ncarried out at fixed frequency ωin the 4-24 Ghz range,\nby means of an in-house apparatus featuring an external\nmagnetic field up to 0.9T parallel to a coplanar waveg-\nuide with a broad center conductor width of 350 µm.\nFor XMCD measurements, given the low X-ray ab-\nsorption cross-sectionpresented by Pt and Pd absorption\nedges, a special set of samples was prepared, consisting\nof 20 repeats per structure in order to obtain sufficiently\nhighsignal-to-noiseratio. Inthiscase,wehave multilayer\n= [Py(5nm)/N] 20, with N = Pd(2.5nm) and Pt(1nm);\nCu(5nm)/Py(5nm)/Al(3nm) was deposited as cap. The\nPt and Pd thicknesses were chosen to yield a damping\nenhancement equal about to half of the respective satu-\nration value (as it will be shown later), i.e. a thicknesses\nfor which the F/N interface is formed but the damp-\ning enhancement is still increasing. XMCD experiments\nwere carried out at the Circular Polarization Beamline\nID-12 of the European Synchrotron Radiation Facility\n(ESRF)35. Measurementsweretakenintotalfluorescence\nyield detection mode, at grazing incidence of 10◦, with\neither left or right circular helicity of the photon beam,\nswitching a 0.9T static magnetic field at each photon en-\nergy value (further details on the method are in Ref.22).\nNo correction for self-absorption effects is needed; how-\never XMCD spectra measured at the L 2,3edges of Pd\nhave to be corrected for incomplete circular polarization\nrate of monochromatic X-rays which is 12% and 22% at/s49/s49/s46/s53/s52 /s49/s49/s46/s53/s54 /s49/s49/s46/s53/s56 /s49/s49/s46/s54/s48 /s49/s49/s46/s54/s50 /s49/s51/s46/s50/s54 /s49/s51/s46/s50/s56 /s49/s51/s46/s51/s48 /s49/s51/s46/s51/s50 /s49/s51/s46/s51/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s116/s32/s88/s65/s83\n/s32/s65/s117/s32/s88/s65/s83\n/s80/s100/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s80/s116\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s51/s46/s49/s52 /s51/s46/s49/s54 /s51/s46/s49/s56 /s51/s46/s50/s48 /s51/s46/s50/s50 /s51/s46/s51/s48 /s51/s46/s51/s50 /s51/s46/s51/s52 /s51/s46/s51/s54 /s51/s46/s51/s56/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s100/s32/s88/s65/s83\n/s32/s65/s103/s32/s88/s65/s83/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s45/s48/s46/s48/s57/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\nFIG. 1. (Color online) X-ray absorption (XAS, left axis) and\nmagnetic circular dichroism (XMCD, right axis) spectra at\nthe L-edges of Pt (top panel) and Pd (bottom panel) for\n[Py(5nm)/Pt(1nm)] 20and [Py(5nm)/Pd(2.5nm)] 20multilay-\ners. The dashed traces represent XAS spectra at L-edge of\nAg and Au used as background of Pd and Pt, respectively, to\nextract the values of induced magnetic moment reported in\nTab.I.\nL3and L2, respectively. The circular polarization rate is\nin excess of 95 % at the L 2,3edges of Pt.\nIII. RESULTS AND ANALYSIS\nIn order to study how the proximity-induced mag-\nnetic moments may affect the absorption of spin-currents\nthrough interfaces, the static moment induced in Pt, Pd\nlayers in direct contact with ferromagnetic Py in char-\nacterized first, by means of XMCD. The value of the\ninduced moment extracted for the two Py/N systems is\nusedtoestimatetheinterfacialexchangeenergyactingon\nthe two paramagnets. Afterwards, the dynamic response\nofthe magnetization is addressedby FMR measurements\nin Py/N(direct contact) and Py/Cu/N(indirect contact)\nheterostructures. From FMR measurements carried out\non both configurations as a function of N thickness, the\ndamping enhancement due to the presence of the spin-\nsink layers Pt and Pd is obtained from the frequency-\ndependence of the FMR linewidth. The relation between\nthe static induced moment and the spin pumping damp-\ning is discussed by comparing the results of the direct\nwith indirect contact systems.3\n-0.06-0.04-0.020.00XMCD\n11.5911.5811.5711.5611.55\nPhoton Energy (keV)1.0\n0.5\n0.0\n3210tCu (nm) Area\n 1 nm Cu\n 0.5 nm Cu\n 0 nm Cu\nFIG. 2. (Color online) XMCD spectra at the L 3edge of Pt\nfor [Py(5nm)/Cu(t Cu)/Pt(1nm)] 15, with t Cu= 0, 0.5 and\n1nm. As inset the ares of the peak is plot as a function of Cu\nthickness.\nA. XMCD: Probing the induced magnetic moment\nIn Fig.1we report X-ray absorption (XAS) and mag-\nnetic circular dichroism (XMCD) spectra at the L2,3\nedges of Pt (top panel) and Pd (bottom panel) taken on\nPy(5nm)/Pt(1nm) |20and Py(5nm)/Pd(2.5nm) |20, re-\nspectively. Rather intense XMCD signals have been de-\ntected at both Pt and Pd L 2,3edges, showing unambigu-\nously that a strong magnetic moment is induced by di-\nrect exchange coupling at the Py |N interface. The static\ninduced moment is expected to be ferromagnetically cou-\npled with the magnetization in Py21. From the integrals\nof XMCD spectra, the induced magnetic moment on the\nPt, Pd sitesis determined byapplying the sum rulesasin\nRef.22(and references therein). In Py/Pd(2.5nm) |20, Pd\natoms bear a moment of 0.12 µB/at, averaged over the\nwhole volume of the volume, with an orbital-to-spinratio\nmL/mS= 0.05. In Py/Pt(1nm) |20, a magnetic moment\n0.27µB/at is found on Pt, comparable to that reported\nfor Ni/Pt epitaxial multilayers23, with a relatively high\norbitalcharacter mL/mS= 0.18, ascomparedwithPdin-\nduced moment. The large difference in volume-averaged\ninduced moment per atom comes from the different film\nthickness, hence volume, for Pt and Pd. Assuming that\nthe induced magnetic moment is confined to the first\natomic layers at the interface with Py23,24, one could\nestimate 0.32 µB/at for Pd and 0.30 µB/at for Pt36.\nWhen a3nm thick Cu interlayeris introducedbetween\nPy and N, a two ordersofmagnitude smaller induced mo-\nment (0.0036 µB/at) was found for 2.5nm Pd22, while Pt\nshowed an XMCD signal of the order of the experimen-\ntal sensitivity, ∼0.5·10−3µB/at. In Fig. 2XMCD\nspectra at the L 3edge of Pt are shown for Cu interlayer\nthicknesses 0, 0.5 and 1nm. For 0.5nm Cu the integral\nof XMCD signal at the L 3edge shrinks to 30%, while\nfor 1nm it is reduced to zero within experimental error.\nThis result could be explained either by a 3d growth of\nthe Cu layer, allowing a fraction of the Pt layer to be in\ndirect contact with Py for Cu coverages of 0.5nm, or byNχmol37S37N0abulktN/angbracketleftM/angbracketrightMiJex\n[cm3/mol][1/eV·at][nm][nm][µB/at][µB/at][meV]\n10−4\nPd5.5±0.29.30.83±0.030.3892.50.1160.3242\nPt1.96±0.13.70.74±0.040.3921.00.270.30109\nTABLE I. Spin-sink layer N properties in Py/N heterostruc-\ntures: experimental molar susceptibility χmolat 20◦C; den-\nsity of states N0calculated from tabulated χmol; Stoner pa-\nrameter S from Ref.37; bulk lattice parameter a; layer thick-\nnessestN; volume averaged induced magnetic moment /angbracketleftM/angbracketright\nfrom XMCD measurement in Fig. 1; interface magnetic mo-\nmentMi36; Py|N interfacial exchange energy per interface\natomJex(Eq.1).\ndiffusion of magnetic Ni atoms in Cu on a scale shorter\nthan 1nm. The film then becomes continuous, and at\n1nm coverage, no direct exchange coupling takes place\nbetween Py and Pt layers. For FMR measurements pre-\nsented in the followingsection, a 3nm thick Cu interlayer\nis employed, reducing also any possible indirect exchange\ncoupling.\nFrom the values of induced moments in Pd and Pt,\nwe can make a step forward and estimate the interfacial\nexchange coupling energies for the two cases. Equating\ninteratomic exchange energy Jexand Zeeman energy for\nan interface paramagnetic atom, we have (see Appendix\nB1for the derivation)\nJex=1\n2/angbracketleftM/angbracketright\nµBN0StN\nti(1)\nwhere/angbracketleftM/angbracketrightis the thickness-averaged paramagnetic\nmoment, N0is the single-spin density of states (in\neV−1at−1), S is the Stoner factor and ti= 2∗a/√\n3\nis the polarized interface-layer thickness36. The 1/2fac-\ntor accounts for the fact that in XMCD measurements\nthe N layer has both interfaces in contact with F. Un-\nder the simplifying assumption that all the magnetic mo-\nment is confined to the interface N layer and assuming\nexperimental bulk susceptibility parameters for χv, we\nobtainJPd\nex= 42 meV for Pd and JPt\nex= 109 meV for\nPt (results and properties are summarized in Tab. I).\nHere the difference in estimated Jex, despite roughly\nequalMi, comes from the larger Stoner factor S for\nPd. A stronger interfacial exchange energy in Pt de-\nnotes a stronger orbital hybridization, yielding possibly\na higher orbital character of the interfacial magnetic mo-\nment in the ferromagnetic Py counterpart21. For com-\nparison, we consider the interatomic exchange param-\netersJexin ferromagnetic Py and Co, investigated in\nRef.14.Jexis estimated from the respective Curie tem-\nperatures TC, through Jex≃6kBTC/(m/µB)2, wherem\nis the atomic moment in µB/at (see Appendix B2). Ex-\nperimental Curie temperatures of 870K and 1388K give\nJCo\nex=293meV for Co and JPy\nex=393meV for Py, which\nare of the same order of the value calculated for Pt (de-\ntails about calculation in Appendix B2).4\n1.0\n0.5\n0.0\n14121086420\ntPt (nm) Py/Pt\n Py/Cu/Pt12\n10\n8\n6\n4\n2\n0\n468\n12468\n10\ntPt (nm) Py/Pt\n tc = 2.4 nm\n Py/Cu/Pt\n λα = 1.8 nm\n1.0\n0.5\n0.0∆α / ∆α0\n14121086420\ntPd (nm) Py/Pd\n Py/Cu/Pd6\n5\n4\n3\n2\n1\n0∆α (x103)\n468\n12468\n10\ntPd (nm) Py/Pd\n tc = 5.0 nm\n Py/Cu/Pd\n λα = 5.8 nma) b)\nc) d)\nFIG. 3. (Color online) Damping enhancement ∆ α, due to\npumped spin current absorption, as a function of thickness tN\nfor Py(5nm)/N and Py(5nm)/Cu(3nm)/N heterostructures,\nwith N = Pd( tN) (panels a,c), Pt( tN) (panels b,d). Solid\nlines result from a fit with exponential function (Eq. 2) with\ndecay length λα. Dashed lines represents instead a linear-\ncutoff behavior (Eq. 3) fortN< tc. Please notice in panels a,\nc the x-axis is in logarithmic scale. In panels b, c the dampin g\nenhancement is normalized to the respective saturation val ue\n∆α0.\nIn the following, the effect of these static induced mo-\nments on the spin pumping damping of the heterostruc-\ntures characterized will be discussed.\nB. FMR: damping enhancement\nThe main result of our work is now shown in Figure\n3. In Fig. 3the damping enhancement ∆ αis plotted as\na function of the spin-sink layer thickness tN, for Py/Pd,\nPy/Cu/Pd (panels a, c) and Py/Pt, Py/Cu/Pt (panels\nb, d). The enhancement ∆ αis compared with the damp-\ningαof a reference structure Py(5nm)/Cu, excluding\nthe sink layer N. Each value of αresults from established\nanalysis of the linewidth of 11 FMR traces13,14, employ-\ning ag-factor equal to 2.09 as a constant fit parameter\nfor all samples.\nIn Py/Cu/N systems (Fig. 3, green square markers),\n∆αrises with increasing tNthickness to similar satura-\ntion values ∆ α0= 0.0027, 0.0031 for Pd and Pt, but\nreachedondifferentlengthscales,giventhedifferentchar-Ng↑↓\neff(Py|Cu/N) λαg↑↓\neff(Py|N)tc\n[nm−2][nm] [nm−2][nm]\nPd 7.2 5.8±0.2145.0±0.3\nPt 8.3 2.4±0.1321.8±0.2\nTABLE II. Mixing conductance values extracted from the\ndamping enhancement ∆ αat saturation in Fig. 3, and re-\nspective length scales (see text for details).\nacteristic spin relaxation lengths of the two materials.\nFrom the saturation value, an effective mixing conduc-\ntanceg↑↓\neff(Py|Cu/N) = 7 .2−8.3nm−2is deduced in the\nframeworkof standardspin pumping picture13,17,19, with\nPy saturation magnetization µ0Ms= 1.04T. The fact\nthat the spin-mixing conductance is not material depen-\ndent indicates that similar Cu |N interfaces are formed.\nThe thickness dependence is well described by the expo-\nnentialfunction14,20\n∆α(tN) = ∆α0(1−exp(−2tN/λN\nα)) (2)\nas shown by the fit in Fig. 3a-b (continuous line). As\na result, exponential decay lengths λPt\nα= 1.8nm and\nλPd\nα= 5.8nm are obtained for Pt and Pd, respectively.\nWhen the Pt, Pd spin-sink layers come into direct con-\ntact with the ferromagnetic Py, the damping enhance-\nment ∆α(tN) changes dramatically. In Py/N systems\n(Fig.3a-b, trianglemarkers),the damping saturationval-\nues become ∆ αPt\n0= 0.0119 and ∆ αPd\n0= 0.0054 for Pt\nand Pd, respectively a factor ∼2 and∼4 larger as com-\npared to Py/Cu/N. Within the spin-pumping descrip-\ntion, a largerdamping enhancement implies a largerspin-\ncurrentdensitypumped outoftheferromagnetacrossthe\ninterface and depolarized in the sink.\nIn Py/N heterostructures, because of the magnetic\nproximity effect, few atomic layers in N are ferromagnet-\nically polarized, with a magnetic moment decaying with\ndistance from the Py |N interface. The higher value of\ndamping at saturation might therefore be interpreted as\nthe result of a magnetic bi-layer structure, with a thin\nferromagnetic N characterized by high damping αN\nhigh\ncoupled to a low damping αF\nlowferromagnetic Py38. To\ninvestigate whether damping is of bi-layer type, or truly\ninterfacial, in Fig 4we show the tFthickness dependence\nof the damping enhancement ∆ α, for a Py( tF)/Pt(4 nm)\nseries of samples. The power law thickness dependence\nadheres very closely to t−1\nF, as shown in the logarithmic\nplot. The assumption of composite damping for syn-\nchronous precession, as ∆ α(t1) = (α1t1+α2t2)/(t1+t2),\nshown here for t2= 0.25nm and 1.0nm, cannot follow\nan inverse thickness dependence over the decade of ∆ α\nobserved. Damping is therefore observed to have a pure\ninterfacial character.\nIn this case, the mixing conductances calculated from\nthe saturation values are g↑↓\neff(Py|Pd) = 14nm−2and\ng↑↓\neff(Py|Pt) = 32nm−2. From ab initio calculations\nwithin a standard spin-pumping formalism in diffusive\nfilms10,29, it is found g↑↓\neff(Py|Pd) = 23nm−2for Pd and5\n10 3 4 56789 20 30 40\ntF (nm)110\n2345678920∆α(10−3)Py(tF)/Pt(4)\ninterfacial, K=0.055\nbilayer, t2=0.25 nm\n             t2=1.0 nm\n26101520 30\ntF (nm)051015202530α(10−3)Py(t)/Pt(4nm)\nPy(t)\nFIG. 4. Logarithmic plot of the damping enhancement ∆ α\n(triangle markers) as a function of the Py layer thickness tF,\nin Py(t F)/Pt(4nm). Solid and dashed lines represents, re-\nspectively, fits according to the spin pumping ( interfacial )\nmodel ∆ α=Kt−1\nFand to a αlow(tF)/αhigh(t2)bilayermodel,\nwitht2= 0.25,1.0nm.Inset: Gilbert damping αfor Py(t F)\n(square markers) and Py(t F)/Pt(4nm) (round markers).\ng↑↓\neff(Py|Pt) = 22nm−2for Pt. Theoretical spin mixing\nconductance from a standard picture does reproduce the\nexperimental order of magnitude, but it misses the 2.3\nfactor between the Py |Pt and Py |Pd interfaces. Beyond\na standard pumping picture, Liu and coworkers29intro-\nduce spin-flipping scattering at the interface and calcu-\nlate from first principles, for ideal interfaces in finite dif-\nfusive films: g↑↓\neff(Py|Pd) = 15nm−2, in excellent agree-\nment with the experimental value here reported for Pd\n(Tab.II), andg↑↓\neff(Py|Pt) = 25nm−2. Zhang et al.10\nsuggest an increase up to 25% of the mixing conductance\ncan be obtained by introducing magnetic layers on the\nPt side. The results here reported support the emerging\nidea that a generalized model of spin pumping including\nspin-orbit coupling and induced magnetic moments at\nF|N interfaces may be required to describe the response\nof heterostructures involving heavy elements.\nThe saturation value of damping enhancement at ∆ α0\nas a function of the Cu interlayer thickness is shown in\nFig.5to follow the same trend of the XMCD signal\n(dashed line), reported from Fig. 2. Indeed, it is found\nthat the augmented ∆ α0in Py/N junctions is drasti-\ncally reduced by the insertion of 0.5nm Cu at the Py |N\ninterface17, and the saturation of the Py/Cu/N configu-\nration is already reached for 1nm of Cu interlayer. As\nsoonasacontinuousinterlayerisformedandnomagnetic\nmoment is induced in N, ∆ α0is substantially constant\nwith increasing Cu thickness.\nThe N-thickness dependence of ∆ α(tN) in Py/N sys-\ntems before saturation is addressed in the following. At\nvariance with the Py/Cu/N case, the thickness depen-\ndence of ∆ αis not anymore well described by an expo-1.0\n0.5\n0.0\n3210\ntCu (nm)1.0\n0.5\n0.0L3 XMCD area (norm.)Pt(3nm)\n XMCD Pt\n3210\ntCu (nm)1.0\n0.5\n0.0∆α0 (norm.)Pd(7nm)\n XMCD Pt\nFIG. 5. (Color online) Normalized damping enhancement ∆ α\n(left axis), due to spin pumping, as a function of interlayer\nthickness tCufor Py(5nm)/Cu( tCu)/N heterostructures, with\nN = Pd(7nm), N = Pt(3nm). The dashed line represents the\nXMCD signal (right axis) reported from inset in Fig. 2.\nnential behavior, as an exponential fit (with exponential\ndecay length as only free fit parameter) fails to repro-\nduce the increase of ∆ αtowards saturation (solid lines\nin Fig.3a-b). More rigorous fitting functions employed\nin spin pumping experiments, within standard spin trans-\nport theory16,26,32, cannot as well reproduce the experi-\nmental data (see Appendix. A). It is worth mentioning\nthat the same change of trend between the two configu-\nrations was observed for the same stacks with a 10nm\nthick Py layer (data shown in Appendix A, Fig.7). A\nchange of the functional dependence of ∆ αontNre-\nflects a change in the spin-depolarization processes the\npumped spin current undergoes, as for instance shown in\nRef.30when interfacial spin-orbit coupling is introduced\nin the spin-pumping formalism. Experimentally, a linear\nthickness dependence with sharp cutoff has been shown\nto characterize spin-current absorption in spin-sink lay-\ners exhibiting ferromagnetic order at the interface, as re-\nported for F 1/Cu/F 2(tF2) junctions with F = Py, Co,\nCoFeB14. Given the presence of ferromagnetic order in\nN at the interface of F/N structures, the data are tenta-\ntively fit with a linear function\n∆α= ∆α0tN/tN\nc (3)\nThis linear function better reproduce the sharp rise of\n∆α(dashed lines in Fig. 3a-b) and gives cutoff thick-\nnessestPt\nc= 2.4±0.2nm and tPd\nc= 5.0±0.3nm for Pt\nand Pd, respectively. The linearization is ascribed to the\npresence of ferromagnetic order in the paramagnetic Pd,\nPt spin-sink layers at the interface with the ferromag-\nnetic Py. The linear trend extends beyond the thickness\nfor which a continuous layer is already formed (less than\n1nm), and, especially for Pd, far beyond the distance\nwithin the non-uniform, induced moment is confined (up\nto0.9nm). InRef.14, thecutoff tcinF/Cu/Fheterostruc-\nturesis proposedto be on the orderofthe transversespin\ncoherence length λJin ferromagnetically ordered layers.\nλJcan be expressed in terms of the exchange splitting6\n6\n5\n4\n3\n2\n1\n0tc(nm)\n2520151050\n1/Jex(eV-1)PtPd\nPy\nCo\nFIG. 6. Effect of direct exchange strength on length scale of\nspin current absorption. Cutoff thickness tcextracted from\nthe ∆α(tN) data in Fig. 3as a function of reciprocal interfa-\ncial exchange energy 1 /Jexextracted from XMCD in Fig. 1.\nLabels are given in terms of Jex. The Co and Py points are\nfrom Ref.14.\nenergyJex,\nλJ=hvg\n2Jex(4)\nwherevgis the electronic group velocity at the Fermi\nlevel. This form, found from hot-electron Mott\npolarimetry1, is expressed equivalently for free electrons\nasπ/|k↑−k↓|, which is a scaling length for geometrical\ndephasing in spin momentum transfer2. Electrons which\nenter the spin-sink at E Fdo so at a distribution of angles\nwith respect to the interface normal, traverse a distribu-\ntionofpathlengths, andprecessbydifferentangles(from\nminority to majority or vice versa ) before being reflected\nback into the pumping ferromagnet. For a constant vg,\nit is therefore predicted that tcis inversely proportional\nto the exchange energy Jex.\nIn Figure 6we plot the dependence of the cutoff thick-\nnesstN\ncupontheinverseoftheestimatedexchangeenergy\nJex(Tab.I),asextractedfromtheXMCDmeasurements.\nA proportionality is roughly verified, as proposed for the\ntransverse spin coherence length across spin polarized in-\nterfaces. Under the simplistic assumption that tc=λJ,\nfrom the slope of the line we extract a Fermi velocity\nof∼0.1·106m/s (Eq.4), of the order of magnitude ex-\npectedforthematerialsconsidered39,40. Thesedatashow\nthat, up to a certain extent, length scale for spin-current\nscattering shares common physical origin in ferromag-\nnetic layers and paramagnetic heavy-metals, such as Pd\nand Pt, under the influence of magnetic proximity effect.\nThis unexpected results is observed in spite of the fact\nthat F 1/Cu/F 2and F/N systems present fundamental\ndifferences. In F/N structure, the induced moment in N\nis expected to be directly exchange coupled with the fer-\nromagneticcounterpart. Whereasin F 1/Cu/F 2, themag-\nnetic moment in F 2(off-resonance) are only weakly cou-\npled with the precession occurring in F 1(in-resonance),\nthrough spin-orbit torque and possible RKKY interac-tion. Magnetization dynamics in N might therefore be\nexpected with its own pumped spin current, albeit, to\nthe best of our knowledge, no experimental evidence\nof a dynamic response of proximity induced moments\nwas reported so far. From these considerations and the\nexperimental findings, counter-intuitively the proximity-\ninduced magnetic moments appear not to be involved in\nthe production of spin current, but rather to contribute\nexclusively with an additional spin-depolarization mech-\nanism at the interface.\nIV. CONCLUSIONS\nWe have investigated the effect of induced magnetic\nmoments in heavy metals at Py/Pt and Py/Pd inter-\nfaces on the absorption of pumped spin currents, by\nanalyzing ferromagnetic resonance spectra with varying\nPt, Pd thicknesses. Static, proximity-induced magnetic\nmoments amount to 0.32 and 0.3 µB/atom in Pd and\nPt, respectively, at the interface with Py, as deduced\nfrom XMCD measurements taken at the L 2,3edges. We\nhave shown that when the proximity induced moment\nin Pt and Pd is present, an onset of a linear-like thick-\nness dependence of the damping is observed, in con-\ntrast with an exponential trend shown by Py/Cu/Pd\nand Py/Cu/Pt systems, for which no induced moment\nis measured. These results point to the presence of an\nadditional spin-flip process occurringat the interface and\nto a change of the character of spin current absorption\nin the ultrathin Pd and Pt paramagnets because of the\ninterfacial spin polarization. The range of linear increase\nis proposed to be inversely proportional to the interfa-\ncial exchange energy in Py/Pt and Py/Pd, inferred from\nXMCD data.\nWEB acknowledges the Universit´ e Joseph Fourier and\nFondation Nanosciences for his research stay at SPIN-\nTEC. This work was supported in part by the U.S. NSF-\nECCS-0925829 and the EU EP7 NMP3-SL-2012-280879\nCRONOS. MC is financed by Fondation Nanosciences.\nAppendix A: N-thickness dependence\nIn order to confirm the results presented in the\nmanuscript, additional sample series with thicker Py\nlayer were fabricated and measured. The experimental\nresults for 10nm thick Py layer are shown in Fig.s 7and\n8forPdandPt, respectively. Wehavepresentedthedata\nhere, rather than including them with the other plots in\nFigure3, to keep the figures from being overcrowded. As\nexpected when doubling the ferromagnet thickness, the\nsaturation values ∆ α0are about half of those measured\nfor 5nm Py (Fig. 3). Confirming the data presented in\nthe manuscript, it is observedagaina changeof thickness\ndependence of ∆ α(tN), fromexponential for Py/Cu/N\n(solid lines; Eq. 2,λα= 4.8nm and 1.4nm for Pd and Pt7\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPd (nm)Py/Pd\nPy/Cu/Pd\n Eq. 3\n,  Eq. 2\n [16, 32] - ρPd=1.4E-7\n [16, 32] - ρPd->ρ(tPd)\nFIG. 7. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPd\nfor Py(10nm)/Pd and Py(10nm)/Cu(3nm)/Pd heterostruc-\ntures. Solid lines result from a fit with exponential functio n\n(Eq.2) with decay λα. Dashed lines represents instead a\nlinear-cutoff behavior (Eq. 3) fortPd< tc. Short-dash and\npoint-dash traces are fit to the data, employing equations\nfrom standard spin transport theory (see text for details)16,32.\nIn bottom panel, ∆ αis normalized to the respective satura-\ntion value.\nrespectively) to linear-like for Py/N (dashed lines; Eq. 3,\ntc= 5.3nm and 2nm for Pd and Pt respectively).\nThe experimental data are also fitted with a set of\nequations derived from standard theory of diffusive spin\ntransport16,26,32, describing the the dependence of ∆ α\non the thickness of adjacent metallic layers (either N or\nCu/N in our case) as follow\n∆α=γ¯h\n4πMstFMg↑↓\n1+g↑↓/gx\next(A1)\nwith (Eq. 7 in Ref.16, and Eq. 6 in Ref.32)\ngN\next=gNtanhtN/λN\nsd\ngCu/N\next=gCugCucothtN/λN\nsd+gNcothtCu/λCu\nsd\ngCucothtN/λN\nsdcothtCu/λCu\nsd+gN(A2)\nwheregx=σx/λx\nsd,σxandλx\nsdare the electrical conduc-\ntivity and spin diffusion length of the non magnetic layer\nx. For the thin Cu layer, we used a resistivity ρCu=\n1×107Ωm and a spin diffusion length λCu\nsd= 170nm32.\nFor the Pt and Pd layers, two fitting models in which the\nconductivity of the films is either constant or thickness\ndependent areconsidered, as recently proposed by Boone\nand coworkers16. The values of conductivity, as taken di-\nrectly from Ref.16, will influence the spin diffusion length\nλN\nsdand spin mixing conductance g↑↓resulting from the\nfit, but will not affect the conclusions drawn about the\noverall trend. When a constant resistivity is used (short-\ndash, blue lines), the model basically corresponds to the6\n5\n4\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPt (nm) Py/Pt\n Py/Cu/Pt\n Eq. 3\n, Eq. 2\n [16, 32] - ρPt=1.7E-7\n [16, 32] - ρPt->ρ(tPt)\nFIG. 8. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPtfor\nPy(10nm)/Pt and Py(10nm)/Cu(3nm)/Pt heterostructures.\nSolid lines result from a fit with exponential function (Eq. 2)\nwith decay λα. Dashed lines represents instead a linear-cutoff\nbehavior (Eq. 3) fortPt< tc. Short-dash and point-dash\ntraces are fit to the data, employing equations from standard\nspin transport theory (see text for details)16,32. In bottom\npanel, ∆ αis normalized to the respective saturation value.\nsimple exponential function in Eq. 2. It nicely repro-\nduces the data in the indirect contact case (Py/Cu/N)\nfor both Pd (Fig. 7) and Pt (Fig. 8), but it fails to fit\nthe direct contact (Py/N) configuration. When a thick-\nness dependent resistivity of the form ρN=ρb\nN+ρs\nN/tNis\nused (dash-point, cyan lines)16, in Py/Cu/N systems, no\nsignificant difference with the other functions is observed\nfor Pt, while for Pd a deviation from experimental trend\nis observed below 1.5nm. In Py/Nsystems, the fit better\ndescribes the rise at thicknesses shorter than the charac-\nteristic relaxation length, while deviates from the data\naround the saturation range.\nModels from standard spin transport theory cannot\nsatisfactorily describe the experimental data for the di-\nrect contact Py/N systems. For this reason a different\nmechanismforthe spin depolarizationprocesseshasbeen\nproposed, considering the presence of induced magnetic\nmoments in N in contact with the ferromagnetic layer.\nAppendix B: Interfacial interatomic exchange\n1. Paramagnets\nWe will show estimates for exchange energy based\non XMCD-measured moments in [Py/(Pt, Pd)] repeatsu-\nperlattices. Calculations of susceptibility are validated\nagainst experimental data for Pd and Pt. Bulk suscepti-\nbilities will be used to infer interfacial exchange parame-8\ntersJi\nex.\na. Pauli susceptibility For an itinerant electron sys-\ntem characterized by a density of states at the Fermi\nenergyN0, if an energy ∆ Esplits the spin-up and spin-\ndown electrons, the magnetization resulting from the\n(single-spin) exchange energy ∆ Eis\nM=µB/parenleftbig\nN↑−N↓/parenrightbig\n= 2µBN0S∆E(B1)\nwhereN0is the density of states in # /eV/at,Sis the\nStoner parameter, and 2∆ Eis the exchange splitting in\neV. Moments are then given in µB/at. Solving for ∆ E,\n∆E=M\n2µBN0S(B2)\nIf the exchange splitting is generated through the ap-\nplication of a magnetic field, ∆ E=µBH,\nµBH=M\n2µBN0S(B3)\nand the dimensionless volume magnetic susceptibility\ncan be expressed\nχv≡M\nH= 2µ2\nBN0S (B4)\nIn this expression, the prefactor can be evaluated\nthrough\nµ2\nB= 59.218 eV ˚A3(B5)\nso with N0[=]/eV/at, χvtakes units of volume per\natom, and is then also called an atomic susceptibility, in\ncm3/at, as printed in Ref37.\nb. Molar susceptiblity Experimentalvaluesaretabu-\nlated as molar susceptibilities. The atomic susceptibility\nχvcanbe contrastedwith the masssusceptibility χmand\nmolar susceptibility χmol\nχmass=χv\nρχmol=ATWT\nρχv(B6)\nwhere ATWT is the atomic weight (g/mol) and ρis\nthe density (g/cm3). These have units of χmass[=]cm3/g\nandχmol[=]cm3/mol. The molar susceptibility χmolis\nthen\nχmol= 2µ2\nBN0NAS (B7)\nin cm3/mol, where µBis the Bohr magneton, and\n2N0S=χmol\nNAµ2\nB(B8)Eq.B8providesaconvenentmethodtoestimateexper-\nimental unknowns, the density of states N0and Stoner\nparameter S, from measurements of χmol.\nExample: for Pd, the low-temperature measurement\n(differentfromtheroom-temperaturemeasurementinTa-\nbleI) isχmol∼7.0×10−4cm3/mol. In the denomina-\ntor, (NAµ2\nB) = 2.622 ×10−6Ry·cm3/mol, The value\n2N0Sconsistent with the experiment is 266/(Ry-at) or\n19.6/(eV-at). For the tabulated measurement of S= 9.3,\nthe inferred density of states is then N0= 1.05/eV/at.\nc. Interfacial exchange We canassumethat the Zee-\nman energy per interface atom is equal to its exchange\nenergy, through the Heisenberg form\nM2\np\nχvVat= 2Ji\nexsfsp (B9)\nwhereMpis the magnetization of the paramagnet,\nwith the atomic moment of the paramagnet mpin terms\nof its per-atom spin sp,\nMp=mp\nVatmp= 2µBsp (B10)\nVatis the volume of the paramagnetic site, sf,pare the\nper-atom spin numbers for the ferromagnetic and para-\nmagnetic sites, and Ji\nexis the (interatomic) exchange en-\nergy acting on the paramagnetic site from the ferromag-\nneticlayersonthe othersideoftheinterface. Interatomic\nexchangeenergyhasbeen distinguished fromintraatomic\n(Stoner) exchange involved in flipping the spin of a single\nelectron. Rewriting Eq B9,\nM2\np\nχvVat= 2Ji\nexsfMp\n2µBVat (B11)\nifsf= 1/2, appropriate for 4 πMs∼10 kG,\nJi\nex= 2µBMp\nχv(B12)\nand substituting for χvthrough Eq B4,\nJi\nex=Mp\nµBN0S(B13)\nIn the XMCD experiment, we measure the thickness-\naveraged magnetization as < M > in a [F/N]nsuper-\nlattice. We make a simplifying assumption that the ex-\nchange acts only on nearest-neighbors and so only the\nnear-interface atomic layer has a substantial magnetiza-\ntion. We can then estimate Mpfrom< M >through\n< M > t p= 2Mpti (B14)9\nwheretiis the polarized interface-layer thickness of\nN36. Since the interface exists on both sides of the N\nlayer, 2tiis the thickness in contact with F. Finally,\nJex=1\n2< M >\nµBN0Stp\nti(B15)\nThe exchange energy acting on each interface atom,\nfrom all neighbors, is JPt\nex= 109 meV for Pt and\nJPd\nex= 42 meV for Pd. Per nearest neighbor for an\nideal F/N(111) interface, it is JPy|Pt= 36 meV and\nJPy|Pd= 14 meV. Per nearest neighbor for an inter-\nmixed interface (6 nn), the values drop to 18 meV and 7\nmeV, respectively.\nSince explicit calculations for these systems are not\nin the literature, we can compare indirectly with theo-\nretical values. Dennler41showed that at a (3 d)F/(4d)N\ninterface (e.g. Co/Rh), there is a geometrical enhance-\nment in the moment induced in Nper nearest-neighbor\nofF. The 4d Natoms near the Finterface have larger\ninduced magnetic moments per nn of Fby a factor of\nfour. Specific calculations exist of JF|N(per neighbor)\nfor dilute Co impurities in Pt and dilute Fe impurties in\nPd42.JFe−Pd∼3 meV is calculated, roughly indepen-\ndent of composition up to 20% Fe. If this value is scaled\nup by a factor of four, to be consistent with the inter-\nface geometry in the XMCD experiment, it is ∼12 meV,\ncomparable with the value for Pd, assuming intermixing.\nTherefore the values calculated have the correct order of\nmagnitude.\n2. Ferromagnets\nThe Weiss molecular field,\nHW=βMs (B16)\nwhereβis a constant of order 103, can be used to give\nan estimate of the Curie temperature, as\nTC=µBgJJ(J+1)\n3kBHW (B17)\nDensity functional theory calculations have been used\ntoestimatethemolecularfieldrecently42,43; forspintype,\ntheJ(J+1) term is substutited with < s >2, giving an\nestimate of\nTC=2< s >2J0\n3kB(B18)where< s >is the number of spins on the atom as in\nEqB10; see the text by St¨ ohr and Siegmann44.< s >\ncan be estimated from m=1.07µBfor Py and 1.7 µBfor\nCo, respectively. Then\nJ0≃6kBTC\n(m/µB)2(B19)\nwith experimental Curie temperatures of 870 and 1388\nK, respectively, gives estimates of J0= 293 meV for Co\nandJ0= 393 meV for Py.\nNote that there is also a much older, simpler method.\nKikuchi45has related the exchange energies to the Curie\ntemperature for FCC lattices through\nJ= 0.247kBTC (B20)\nTaking 12 NN, 12 Jgives a total energy of 222 meV for\nPy (870 K) and 358 meV for FCC Co (1400K), not too\nfar off from the DFT estimates.\nd. Other estimates TheJ0exchange parameter is\ninteratomic, describing the interaction between spin-\nclusters located on atoms. Reversing the spin of one of\nthese clusters would change the energy J0. The Stoner\nexchange ∆ is different, since it is the energy involved in\nreversing the spin of a single electron in the electron sea.\nGenerally ∆ is understood to be greater than J0because\nit involvesmore coloumbrepulsion; interatomic exchange\ncan be screened more easily by spelectrons.\nThis exchange energy is that which is measured by\nphotoemissionandinversephotoemission. Measurements\nare quite different for Py and Co. Himpsel40finds an\nexchange splitting of ∆ = 270 meV for Py, which is not\ntoo far away from the Weiss J0value. For Co, however,\nthe value is between 0.9 and 1.2 eV, different by a factor\nof four. For Co the splitting needs to be estimated by a\ncombination of photoemission and inverse photoemission\nbecause the splitting straddles EF.46.\nFor comparison with the paramagnetic values of Ji\nex,\nwe use the J0estimates, since they both involve a bal-\nance between Zeeman energy (here in the Weiss field)\nand Heisenberg interatomic exchange. 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Lett. 64, 1059\n(1990)."    },    {        "title": "1308.3578v1.Quantum_Gilbert_Varshamov_Bound_Through_Symplectic_Self_Orthogonal_Codes.pdf",        "content": "arXiv:1308.3578v1  [cs.IT]  16 Aug 2013Quantum Gilbert-Varshamov Bound Through\nSymplectic Self-Orthogonal Codes\nLingfei Jin and Chaoping Xing\nAbstract—It is well known that quantum codes can be con-\nstructed through classical symplectic self-orthogonal co des. In\nthis paper, we give a kind of Gilbert-Varshamov bound for\nsymplectic self-orthogonal codes firstand thenobtain theG ilbert-\nVarshamov bound for quantum codes. The idea of obtaining the\nGilbert-Varshamov bound for symplectic self-orthogonal c odes\nfollows from counting arguments.\nIndex Terms —Symplectic self-orthogonal, Quantum Gilbert-\nVarshamov bound, Symplectic distance.\nI. INTRODUCTION\nInthe pastfew years,thetheoryof quantumerror-correctin g\ncodes have been developed rapidly. Various constructions a re\ngiven through classical coding. However, it is still a great\nchallenge to construct good quantum codes. Shor and Steane\ngave the first construction of quantum codes through classic al\nself-orthogonalcodes. Subsequently,constructionsof qu antum\ncodes through classical codes with certain self-orthogona lity\nhave been extensively studied and investigated. For instan ce,\nquantum codes can be obtained from Euclidean, Hermitian\nself-orthogonal or Symplectic self-orthogonal codes (see [1],\n[2], [6], [11], [12], [13]).\nAs in the classical coding theory, the quantum Gilbert-\nVarshomov(GV,forshort)boundisalsoabenchmarkforgood\nquantum codes. The first quantum GV bound was obtained in\n[3]forthebinarycase.Oneyearlater,AshikhminandKnill[ 2]\ngeneralized the binary quantum GV bound to the q-ary case.\nIn 2004, Feng and Ma [11] derived a finite version of Gilbert-\nVaranmovboundfor classical Hermitian self-orthogonalco des\nand then applied to quantum codes to obtain a finite version\nof quantum GV bound.\nIn 1998, Macwilliams and Sloane [17] showed through\ncounting arguments that binary self-dual codes can achieve\nthe GV bound for classical case. In this paper, we use the\ncounting arguments as well to show that classical symplecti c\nself-orthogonal codes can achieve the GV bound and then\napply to quantum codes to obtain the asymptotic quantum GV\nbound.\nOurpaperis organizedasfollows.We first recallsome basic\nnotations and results of symplectic slef-orthogonal codes in\nSection II. In section III, we present a kind of GV bound for\nsymplectic self-orthogonal codes through counting argume nts.\nIn section IV, we apply our result obtained in Section III to\nquantum codes and derive the asymptotic quantum GV bound.\nII. PRELIMINARIES\nWith the development of classical error-correcting codes,\npeople have extensively studied the Euclidean inner produc tand investigated the Euclidean self-orthogonal codes. How -\never, due to applications to quantum codes in recent years,\nother inner products such as Hermitian and symplectic inner\nproductshaveattractedresearchersinthisarea andmanyin ter-\nsecting results have been obtained already. In this section , we\nintroduce some basic results and notations about symplecti c\ninner products and symplectic slef-orthogonal codes.\nLetFqbe a finite field with qelements, where qis\na prime power. For four vectors a= (a1,...,a n),b=\n(b1,...,b n),a′= (a′\n1,...,a′\nn),b′= (b′\n1,...,b′\nn)∈Fn\nq, the\nsymplectic inner product /a\\}⌊ra⌋ketle{t,/a\\}⌊ra⌋ketri}htSis defined by\n/a\\}⌊ra⌋ketle{t(a|b),(a′|b′)/a\\}⌊ra⌋ketri}htS=/a\\}⌊ra⌋ketle{ta,b′/a\\}⌊ra⌋ketri}htE−/a\\}⌊ra⌋ketle{tb,a′/a\\}⌊ra⌋ketri}htE,\nwhere/a\\}⌊ra⌋ketle{t,/a\\}⌊ra⌋ketri}htEis defined as the ordinary dot inner product (or\nEuclidean inner product). For an Fq-linear code CinF2n\nq,\ndefine the symplectic dual of Cby\nC⊥S={(x|y) :/a\\}⌊ra⌋ketle{t(x|y),(a|b)/a\\}⌊ra⌋ketri}htS= 0for all(a|b)∈C}.\nIt is easy to show that dimFq(C) + dim Fq(C⊥S) = 2n. A\ncodeCis said symplectic self-orthogonal if C⊆C⊥S, and\nself-dual if C=C⊥S.\nFor an vector (a|b) = (a1,...,a n|b1,...,b n)∈F2n\nq, the\nsymplectic weight is defined by :\nwtS(a|b) =|{i: (ai,bi)/\\e}atio\\slash= (0,0),i= 1...n}|\nFor two vectors (a|b),(a′|b′)∈Fq2n, the symplectic distance\nis defined by :\ndS((a|b),(a′|b′)) = wt S(a−a′|b−b′).\nThe symplectic minimumdistance of a linear code C∈F2n\nq\nis defined by\ndS(C) =: min{wtS(a|b) : (a|b)∈C−{(0|0)}}.\nThen it is straightforwardto verify that a [2n,k]-linear code\nCalso satisfies the symplectic Singleton bound:\nk+2dS(C)≤2n+2.\nIII. GILBERT-VARSHAMOV BOUND FOR SYMPLECTIC\nSELF-ORTHOGONAL CODES\nIn the previous section, we saw the symplectic Singleton\nbound already. Similarly, we can derive the symplectic GV\nbound. However, for application to quantum codes, we are\ninterested in a GV type bound for symplectic self-orthogona l\ncodes. The main goal of this section is to derive such a bound\nthrough counting argument.\nFirst, we have the following simple but useful observation.Lemma 3.1: Everyvectorin F2n\nqisorthogonalto itself with\nsymplectic inner product.\nLemma 3.2: The number of symplectic self-orthogonal\ncodes of length 2nand dimension kin the vector space F2n\nq\nis given by\n(q−1)k−1(q2n−2k+2−1)(q2n−2k+4−1)...(q2n−1)\n(qk−1)(qk−1−1)...(q−1).\n(III.1)\nProof:For convenience, denote by Akthe number of\nsymplectic self-orthogonal codes of length 2nand dimension\nkand denote by C2n,ka symplectic self-orthogonal code\nof length 2nand dimension koverFq. Then for k≤n,\nC2n,k−1can be extended to C2n,kby adding a vector u∈\nC⊥S\n2n,k−1\\C2n,k−1. Thus, we can obtain |C⊥S\n2n,k−1/C2n,k−1|−\n1 =q2n−2(k−1)−1distinct symplectic self-orthogonal codes\nof length 2nand dimension kfromC2n,k−1through this way.\nOn the other hand, by the above argument, we know\nthat every symplectic self-orthogonal code of length 2nand\ndimension k−1is contained in a symplectic self-orthogonal\ncode of length 2nand dimension k. Since every subspace of\ndimension k−1ofC2n,kis symplectic self-orthogonal and\nthere are (qk−1)/(q−1)subspaces of dimension k−1in\nC2n,k, we get a recursive formula\nAk=(q−1)(q2n−2(k−1)−1)\nqk−1Ak−1\nfork= 2,...n. The desired result follows from the above\nrecursive formula and the fact that A1=q2n−1\nq−1.\nLemma 3.3: Given a nonzero vector u∈F2n\nq, the number\nof symplectic self-orthogonalcodes containing uof length 2n\nand dimension kis given by\n(q−1)k−1(q2n−2k+2−1)(q2n−2k+4−1)...(q2n−2−1)\n(qk−1−1)(qk−2−1)...(q−1).\n(III.2)\nProof:Similarly, we denote by C2n,k(u)a symplectic\nself-orthogonalcode containing uof length 2nand dimension\nkoverFqand denote by Bk(u)the number of such C2n,k(u).\nUsing the similar argumentsas in lemma 3.2, we can establish\nthe following recursive formula for Bk(u),\nBk(u) =(q−1)(q2n−2(k−1)−1)\nqk−1−1Bk−1(u).\nThe desired result follows from the above recursive formula\nand the fact that B1(u) = 1.\nTheorem 3.4 (GV bound): For1≤k≤nandd≥1, there\nexists a[2n,k,d]symplectic self-orthogonal code over Fqif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i<q2n−1\nqk−1.\nProof:AsBk(u)is independent of u, we denote by Bk\nthe number given in (III.2).\nFirst, the numberof nonzerovectors uof symplectic weight\nless than dis given by\nV(2n,d) =:i/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i.On the other hand, there are at most/summationtext\nwtS(u)<dBk(u) =\nV(2n,d)Bksymplectic self-orthogonal codes with symplectic\ndistance less than d. Therefore,there is at least one symplectic\nself-orthogonal code with symplectic distance at least dif\nAk> V(2n,d)Bk. The desired result follows from this\ninequality and Lemmas 3.2 and 3.3.\nTheorem 3.4 shows existence of symplectic self-orthogonal\ncodes with good symplectic distance. However, to con-\nstruct good quantumcodes throughsymplectic self-orthogo nal\ncodes, we have to control symplectic dual distance for a give n\nsymplectic self-orthogonal code.\nLemma 3.5: For a given nonzero vector u∈F2n\nq, the\nnumberEkof symplectic self-orthogonal codes Cof length\n2nand dimension ksuch that C⊥Scontainusatisfies the\nfollowing recursive formula\nEk=(q−1)(q2n−2(k−1)−1−1)\nqk−1Ek−1+(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n(III.3)\nfor anyk≥2, whereBkis the quantity defined in (III.2).\nProof:LetWbe the symplectic dual space of /a\\}⌊ra⌋ketle{tu/a\\}⌊ra⌋ketri}ht. Then\nC⊥Scontainsuif and only if Cis a subspace of W. Thus,\nEkstands for the number of symplectic self-orthogonal codes\nCof length 2nand dimension ksuch that C⊆W. We denote\nbyDkthe number of symplectic self-orthogonal codes Cof\nlength2nand dimension ksuch that u∈C⊆W. We also\ndenote by Fkthe number of symplectic self-orthogonal codes\nCof length 2nand dimension ksuch that u/\\e}atio\\slash∈C⊆W. Then\nit is easy to see that Dk=BkandDk+Fk=Ek.\nWithout confusion, we denote by C2n,ka symplectic self-\northogonal code of length 2nand dimension koverFqsuch\nthatC⊆W. Then for k≤n,C2n,k−1can be extended\ntoC2n,kby adding a vector v∈W∩(C⊥S\n2n,k−1\\C2n,k−1).\nThus, we can obtain |C⊥S\n2n,k−1/C2n,k−1|−1 =q2n−2(k−1)−1\n(orq2n−2(k−1)−1−1, respectively) distinct symplectic self-\northogonal codes of length 2nand dimension kfromC2n,k−1\nthrough this way if u∈C2n,k−1(or ifu/\\e}atio\\slash∈C2n,k−1,\nrespectively).\nOn the other hand, by the above argument, we know\nthat every symplectic self-orthogonal codes of length 2nand\ndimension k−1is contained in a symplectic self-orthogonal\ncodes of length 2nand dimension k. Since every subspace of\ndimension k−1ofC2n,kis symplectic self-orthogonal and\nthere are (qk−1)/(q−1)subspaces of dimension k−1in\nC2n,k, we get a recursive formula\nqk−1\nq−1Ek= (q2n−2(k−1)−1)Dk−1+(q2n−2(k−1)−1−1)Fk−1.\nThe desired reclusive formula follows.\nCorollary 3.6: LetEkstands for the same number defined\nin Lemma 3.5. Then one has\nEk≤k(q−1)k−1(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)(III.4)\nfor allk≥1.\nProof:We know that E1=q2n−1−1\nq−1. So it is true for\nk= 1.Now assume that the result is also true for k−1. Then by\nLemma 3.5, we have\nEk=(q−1)(q2n−2(k−1)−1−1)\nqk−1Ek−1+\n(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n≤(k−1)(q−1)k−2(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)+\n(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n= (k−1)(q−1)k−2(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)+\n(q−1)kq2n−2k+1(q2n−2k+4−1)···(q2n−2−1)\n(qk−1)···(q−1)\n≤k(q−1)k−1(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1).\nThis completes the proof.\nBy using the same arguments in the proof of Theorem 3.4. we\nobtain the following result.\nCorollary 3.7: For1≤k≤nandd≥1, there exists\na[2n,k]symplectic self-orthogonal code CoverFqwith\ndS(C⊥S)≥dif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nIV. APPLICATION TO QUANTUM GILBERT-VARSHAMOV\nBOUND\nFor a quantum code Q, we denote by n(Q),K(Q),d(Q)\nlength,dimensionand minimumdistance of Q, respectively.A\nfundamental domain UQ\nq⊆[0,1]×[0,1]is defined as follows.\nDefinition 4.1: UQ\nqis a set which is consisted of ordered\npairs(δ,R)∈[0,1]×[0,1]such that there exists a family of\nq-ary quantum codes {Qi}∞\ni=1satisfying\nn(Qi)→ ∞,R= lim\ni→∞logqK(Qi)/n(Qi),δ= lim\ni→∞d(Qi)/n(Qi).\nAs in the classical coding theory, determining the domain UQ\nq\nis one of the central asymptotic problem for quantum coding\ntheory. One can imagine that it is very hard to completely\ndetermine UQ\nq. Nevertheless, some bounds on UQ\nqhave been\ngiven by many researchers [2], [9], [10]. A useful descripti on\nofUQ\nqthrough a function αQ\nqwas given by Feng-Ling-Xing\n[10]:\nthere exists a function αQ\nq(δ),δ∈[0,1], such that\nUQ\nqis the union of the domain\n{(δ,R)∈R2: 0≤R < αQ\nq(δ),0≤δ≤1}\nwith some points on the boundary αQ\nq(δ).\nThus, determining UQ\nqis almost equivalent to determining the\nfunctionαQ\nq(δ).\nTo apply our results in the previous section, let us establis h\na connection between symplectic self-orthogonal codes and\nquantum codes.Lemma 4.2: [2] IfCis aq-ary symplectic self-orthogonal\n[2n,k]code, then there exists a q-ary[[n,n−k,d]]quantum\ncode with d=dS(C⊥S).\nThus, by combining Corollary 3.7 with Lemma 4.2, we\nimmediately get the following result.\nCorollary 4.3: For1≤k≤nandd≥1, there exists a\nq-ary[[n,n−k,d]]quantum code Cif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nFinally we are ready to derive the quantum GV bound.\nTheorem 4.4:\nαQ\nq(δ)≥1−δlogq(q+1)−Hq(δ).\nwhereHq(x)is theq-ary entropy function xlogq(q−1)−\nxlogqx−(1−x)logq(1−x).\nProof:Fixδ∈(0,1). For every n≥1/δ, denote by d\nthe quantity ⌊δn⌋. Thend/ntends toδwhenntends to∞.\nPut\nk=/floorleftBigg\nlogq/parenleftBiggd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i/parenrightBigg\n+logqk/floorrightBigg\n+1.\nThen we have\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i<qk\nk</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nHence, we have a q-ary quantum [[n,n−k,d]]-code by\nCorollary 4.3. Therefore, one has\nn−k\nn= 1−k\nn→1−δlogq(q+1)−Hq(δ).\nThe proof is completed.\nREFERENCES\n[1] S. A. Aly, A. Klappenecker and P. K. Sarvepalli, “On quant um and\nclassical BCH codes,” IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183–\n1188, Mar. 2007.\n[2] A. Ashikhmin and E. Knill, “Nonbinary quantum stablizer codes,”IEEE\nTrans. Inf. Theory, vol. 47, no. 7, pp. 3065–3072, Nov. 2001.\n[3] A. Ashikhmin, A. M. Barg, E. Knill and S. N. Litsyn, “Quant um error\ndetection II: Bounds,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 789–\n800, May. 2000.\n[4] A. Ashikhmin and S. Litsyn, “Upper bounds on the size of qu antum\ncodes,”IEEE Trans. Inf. Theory, vol. 45, no. 4, pp. 1206–1215, May\n1999.\n[5] A. Ashikhmin, S. Litsyn and M. A. Tsfasman, ”Aymptotical ly good\nquantum codes,” Phys. Rev. A, vol. 63, no. 3, p. 032311, Mar. 2 001\n[6] J. Bierbrauer and Y. Edel, “Quantum twisted codes,” J. Comb. Designs,\nvol. 8, pp. 174–188, 2000.\n[7] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane ,\n“Quantum error correction via codes over GF(4),” IEEE Trans. Inf.\nTheory,vol. 44, no. 4, pp. 1369–1387, July 1998.\n[8] H. Chen, S. Ling and C. Xing, “Quantum codes from concaten ated\nalgebraic-geometric codes,” IEEE Trans. Inf. Theory, vol. 51, no. 8, pp.\n2915–2920, Aug. 2005.\n[9] H. Chen, S. Ling, and C. P. Xing, ”Aymptoticially good qua ntum\ncodes exceeding the Ashikhmin-Litsyn-Tsfasman bound,” IEEE. Trans.\nInform. Theory, vol. 47, no. 5, pp. 2055-2058, Jul. 2001.\n[10] K. Q. Feng, S. Ling and C. P. Xing, ”Aymptotic Bounds on Qu antum\nCodes From Algebraic Geometry Codes,” IEEE. Trans. Inform. Theory,\nvol. 52, no. 3, pp. 986-991, Mar. 2006.[11] K. Q. Feng and Z. Ma, “A finite Gilbert-Varshmov Bound for pure\nstabilizer quantum codes,” IEEE. Trans. Inform. Theory, vol. 50, no. 12,\npp. 3323-3325, Dec. 2004.\n[12] M.Grassland T.Beth,“Quantum BCHcodes,” Internation al Symposium\non Theoretical Electrical Engineering, Magdeburg, 1999, p p. 207-212.\n[13] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, “Nonbinary\nstablizer codes over finite fields,” IEEE. Trans. Inform. Theory, vol. 52,\nno. 11, pp. 4892–4914, Nov. 2006.\n[14] E.Knilland R. Laflamme,“Atheory ofquantum error-corr ecting codes,”\nPhys. Rev. A, vol. 55, no. 2, pp. 900–911, 1997.\n[15] Z. Li, L. J. Xing and X. M. Wang, “A family of asymptotical ly good\nquantum codes based on code concatenation,” IEEE. Trans. Inform.\nTheory,vol. 55, no. 8, pp.3821–3824, Aug. 2009.\n[16] S. Ling and C. P. Xing, Coding Theory – A First Course, Cambridge\nUniversity Press, 2004.\n[17] F. J. Macwilliams and N. J. Sloane, ”Good Self-Dual code s Exists,”\nDiscrete Mathematics, vol. 3, no. 1,2,3, pp. 153-162, Sep, 1 972.\n[18] E. M. Rains, “Nonbinary quantum codes,” IEEE. Trans. Inform. Theory,\nvol. 45, no. 6, pp.1827–1832, Sept. 1999.\n[19] J. H. Silverman, The Arithmetic of Elliptic Curves , Springer, New York,\n1986.\n[20] A. M. Steane, “Enlargement of Calderbank-Shor-Steane quantum\ncodes,”IEEE. Trans. Inform. Theory, vol. 45, no. 7, pp. 2492–2495,\nNov. 1999.\n[21] M. A. Tsfasman and S. G. Vl ˘adut, ”Algebraic-Geometric Codes. Ams-\nterdam,” The Netherlands:Kluwer, 1991."    },    {        "title": "1308.3787v1.Thickness_and_power_dependence_of_the_spin_pumping_effect_in_Y3Fe5O12_Pt_heterostructures_measured_by_the_inverse_spin_Hall_effect.pdf",        "content": "arXiv:1308.3787v1  [cond-mat.mes-hall]  17 Aug 2013Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt\nheterostructures measured by the inverse spin Hall effect\nM. B. Jungfleisch,1,∗A. V. Chumak,1A. Kehlberger,2V. Lauer,1\nD. H. Kim,3M. C. Onbasli,3C. A. Ross,3M. Kl¨ aui,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany\n3Department of Materials Science and Engineering, MIT, Camb ridge, MA 02139, USA\n(Dated: September 17, 2018)\nThe dependence of the spin-pumpingeffect on the yttrium iron garnet (Y 3Fe5O12, YIG) thickness\ndetected by the inverse spin Hall effect (ISHE)has been inves tigated quantitatively. Due to the spin-\npumping effect driven by the magnetization precession in the ferrimagnetic insulator Y 3Fe5O12film\na spin-polarized electron current is injected into the Pt la yer. This spin current is transformed into\nelectrical charge current by means of the ISHE. An increase o f the ISHE-voltage with increasing film\nthickness is observed and compared to the theoretically exp ected behavior. The effective damping\nparameter of the YIG/Pt samples is found to be enhanced with d ecreasing Y 3Fe5O12film thickness.\nThe investigated samples exhibit a spin mixing conductance ofg↑↓\neff= (7.43±0.36)×1018m−2and\na spin Hall angle of θISHE= 0.009±0.0008. Furthermore, the influence of nonlinear effects on the\ngenerated voltage and on the Gilbert damping parameter at hi gh excitation powers are revealed. It\nis shown that for small YIG film thicknesses a broadening of th e linewidth due to nonlinear effects at\nhighexcitation powers is suppressedbecause of alack ofnon linear multi-magnon scatteringchannels.\nWe have found that the variation of the spin-pumping efficienc y for thick YIG samples exhibiting\npronounced nonlinear effects is much smaller than the nonlin ear enhancement of the damping.\nI. INTRODUCTION\nThe generation and detection of spin currents have at-\ntracted much attention in the field of spintronics.1,2An\neffective method for detecting magnonic spin currents is\nthe combination of spin pumping and the inverse spin\nHall effect (ISHE). Spin pumping refers to the genera-\ntion of spin-polarized electron currents in a normal metal\nfrom the magnetization precession in an attached mag-\nnetic material.3,4These spin-polarized electron currents\nare transformed into conventional charge currents by the\nISHE, which allows for a convenient electric detection of\nspin-wave spin currents.5–7\nAfter the discovery of the spin-pumping effect in fer-\nrimagnetic insulator (yttrium iron garnet, Y 3Fe5O12,\nYIG)/non-magnetic metal (platinum, Pt) heterosystems\nby Kajiwara et al.7, there was rapidly emerging inter-\nest in the investigation of these structures.6–13Since\nY3Fe5O12is an insulator with a bandgap of 2.85 eV14no\ndirect injection of a spin-polarized electron current into\nthe Pt layer is possible. Thus, spin pumping in YIG/Pt\nstructures can only be realized by exchange interaction\nbetween conduction electrons in the Pt layer and local-\nized electrons in the YIG film.\nSpin pumping into the Pt layer transfers spin angular\nmomentum from the YIG film thus reducing the mag-\nnetization in the YIG. This angular momentum transfer\nresults in turn in an enhancement of the Gilbert damp-\ning of the magnetization precession. The magnitude of\nthe transfer of angular momentum is independent of the\nferromagnetic film thickness since spin pumping is an in-\nterface effect. However, with decreasing film thickness,\nthe ratio between surface to volume increases and, thus,the interface character of the spin-pumping effect comes\ninto play: the deprivation of spin angular momentum be-\ncomes notable with respect to the precession ofthe entire\nmagnetizationinthe ferromagneticlayer. Thus, theaver-\nage damping for the whole film increases with decreasing\nfilm thicknesses. It is predicted theoretically3and shown\nexperimentallyinferromagneticmetal/normalmetalhet-\nerostructures (Ni 81Fe19/Pt) that the damping enhance-\nment due to spin pumping is inversely proportional to\nthe thickness of the ferromagnet.15,16\nSince the direct injection of electrons from the insula-\ntor YIG into the Pt layer is not possible and spin pump-\ning is an interface effect, an optimal interface quality is\nrequired in order to obtain a high spin- to charge cur-\nrent conversion efficiency.17,18Furthermore, Tashiro et\nal. have experimentally demonstrated that the spin mix-\ning conductance is independent of the YIG thickness in\nYIG/Pt structures.11Recently, Castel et al. reported\non the YIG thickness and frequency dependence of the\nspin-pumpingprocess.19Incontrasttoourinvestigations,\nthey concentrate on rather thick ( >200 nm) YIG films,\nwhich are much thicker than the exchange correlation\nlength in YIG20–22and thicker than the Pt thickness.\nThus, the YIG film thickness dependence in the nanome-\nter range is still not addressed till now.\nIn this paper, we report systematic measurements of\nthe spin- to charge-current conversion in YIG/Pt struc-\ntures as a function of the YIG film thickness from 20 nm\nto 300 nm. The Pt thickness is kept constant at 8.5 nm\nfor all samples. We determine the effective damping as\nwell as the ISHE-voltage as a function of YIG thickness\nand find that the thickness plays a key role. From these\ncharacteristics the spin mixing conductance and the spin2\nFIG. 1. (Color online) (a) Schematic illustration of the ex-\nperimental setup. (b) Dimensions of the structured Pt layer\non the YIG films. The Pt layer was patterned by means of\noptical lithography and ion etching. (c) Scheme of combined\nspin-pumping process and inverse spin Hall effect.\nHall angle are estimated. The second part of this paper\naddresses microwave power dependent measurements of\nthe ISHE-induced voltage UISHEand the ferromagnetic\nresonance linewidth for varying YIG film thicknesses.\nThe occurrence of nonlinear magnon-magnon scattering\nprocessesonthe widening ofthe linewidth aswell astheir\ninfluence on the spin-pumping efficiency are discussed.\nII. SAMPLE FABRICATION AND\nEXPERIMENTAL DETAILS\nIn Fig.1(a) a schematic illustration of the investigated\nsamples is shown. Mono-crystalline Y 3Fe5O12samples of\n20, 70, 130, 200 and 300 nm thickness were deposited by\nmeans of pulsed laser deposition (PLD) from a stoichio-\nmetric target using a KrF excimer laser with a fluence of\n2.6J/cm2andarepetition rateof10Hz.23In orderto en-\nsure epitaxial growth of the films, single crystalline sub-\nTABLE I. Variation of saturation magnetization MSand\nGilbert damping parameter α0as a function of the YIG film\nthickness. Results are obtained using a VNA-FMR measure-\nment technique.\ndYIG(nm)MS(kA/m) α0(×10−3)\n20 161.7 ±0.2 2.169 ±0.069\n70 176.4 ±0.1 0.489 ±0.007\n130 175.1 ±0.2 0.430 ±0.015\n200 176.4 ±0.1 0.162 ±0.008\n300 176.5 ±0.1 0.093 ±0.007FIG. 2. Original Gilbert damping parameter α0measured by\nVNA-FMR technique. The increased damping at low sample\nthicknesses is explained by an enhanced ratio between surfa ce\nto volume, which results in an increased number of scatterin g\ncenters and, thus, in an increased damping. The inset shows\nthe saturation magnetization as a function of the YIG film\nthickness dYIG. The error bars are not visible in this scale.\nstratesofgadoliniumgalliumgarnet(Gd 3Ga5O12, GGG)\nin the (100) orientation were used. We achieved opti-\nmal deposition conditions for a substrate temperature\nof 650◦C±30◦C and an oxygen pressure of 6.67 ×10−3\nmbar. Afterwards, each film was annealed ex-situ at\n820◦C±30◦C by rapid thermal annealing for 300 s un-\nder a steady flow of oxygen. This improves the crystal-\nlographic order and reduces oxygen vacancies. We deter-\nmined the YIG thickness by profilometer measurements\nandthecrystallinequalitywascontrolledbyx-raydiffrac-\ntion(XRD). InordertodepositPtontothesamples,they\nwere transferred at atmosphere leading to possible sur-\nface adsorbates. Therefore, the YIG film surfaces were\ncleaned in-situ by a low power ion etching before the\nPt deposition.17We used DC sputtering under an ar-\ngon pressure of 1 ×10−2mbar at room temperature to\ndeposit the Pt layers. XRR measurements yielded a Pt\nthickness of 8.5 nm, which is identical for every sample\ndue to the simultaneously performed Pt deposition. The\nPt layer was patterned by means of optical lithography\nand ion etching. In order to isolate the Pt stripes from\nthe antenna we deposited a 300 nm thick square of SU-8\nphotoresist on the top. A sketch of the samples and the\nexperimental setup is shown in Fig. 1(a), the dimensions\nof the structured Pt stripe are depicted in Fig. 1(b).\nIn order to corroborate the quality of the fabricated\nYIG samples, we performed ferromagnetic resonance\n(FMR) measurements using a vector network analyzer\n(VNA).25Since the area deposited by Pt is small com-\npared to the entire sample size, we measure the damp-\ningα0of the bare YIG by VNA (this approach results\nin a small overestimate of α0), whereas in the spin-\npumping measurement we detect the enhanced damp-\ningαeffof the Pt covered YIG films. The VNA-FMR\nresults are summarized in Tab. Iand in Fig. 2. Appar-\nently, the 20 nm sample features the largest damping3\nofα20nm\n0= (2.169±0.069)×10−3. With increasing film\nthickness α0decreasesto α300nm\n0= (0.093±0.007)×10−3.\nThere might be two reasons for the observed behavior:\n(1) The quality of the thinner YIG films might be worse\ndue to the fabrication process by PLD. (2) For smaller\nYIG film thicknesses, the ratio between surface to vol-\nume increases. Thus, the two-magnon scattering pro-\ncess at the interface is more pronounced for smaller film\nthicknesses and gives rise to additional damping.26The\nVNA-FMR technique yields the saturation magnetiza-\ntionMSfor the YIG samples (see inset in Fig. 2and\nTab.I). The observed values for MSare larger than the\nbulkvalue,27,28butinagreementwiththevaluesreported\nfor thin films.29The general trend of the film thickness\ndependence of MSis in agreement with the one reported\nin Ref.29,30and might be associated with a lower crystal\nquality after the annealing.\nThespin-pumpingmeasurementsfordifferentYIGfilm\nthicknesses were performed in the following way. The\nsamples were magnetized in the film plane by an exter-\nnal magnetic field H, and the magnetization dynamics\nwas excited at a constant frequency of f= 6.8 GHz by\nan Agilent E8257D microwave source. The microwave\nsignals with powers Pappliedof 1, 10, 20, 50, 100, 250 and\n500mW wereapplied to a 600 µm wide 50Ohm-matched\nCumicrostripantenna. Whiletheexternalmagneticfield\nwas swept, the ISHE-voltage UISHEwas recorded at the\nedges of the Pt stripe using a lock-in technique with an\namplitudemodulationatafrequencyof500Hz, aswellas\nthe absorbed microwave power Pabs. All measurements\nwere performed at room temperature.\nIII. THEORETICAL BACKGROUND\nThe equations describing the ferromagnetic resonance,\nthe spin pumping and the inverse spin Hall effect are\nprovided in the following and used in the experimental\npart of this paper.\nA. Ferromagnetic resonance\nIn equilibrium the magnetization Min aferromagnetic\nmaterial is aligned along the bias magnetic field H. Ap-\nplying an alternating microwave magnetic field h∼per-\npendicularly to the external field Hresults in a torque\nonMand causes the magnetic moments in the sample to\nprecess (see also Fig. 1(a)). In ferromagnetic resonance\n(FMR)themagneticfield Handtheprecessionfrequency\nffulfill the Kittel equation31\nf=µ0γ\n2π/radicalbig\nHFMR(HFMR+MS), (1)\nwhereµ0is the vacuum permeability, γis the gyromag-\nnetic ratio, HFMRis the ferromagnetic resonance field\nandMSis the saturation magnetization (experimentallyobtained values of MSfor our samples can be found in\nTab.I).\nThe FMR linewidth ∆ H(full width at half maximum)\nis related to the Gilbert damping parameter αas16,18,27\nµ0∆H= 4πfα/γ. (2)\nB. Spin pumping\nBy attaching a thin Pt layer to a ferromagnet, the\nresonance linewidth is enhanced,3which accounts for an\ninjection of a spin current from the ferromagnet into the\nnormal metal due to the spin-pumping effect (see illus-\ntration in Fig. 1(c)). In this process the magnetization\nprecession loses spin angular momentum, which gives\nrise to additional damping and, thus, to an enhanced\nlinewidth. The effective Gilbert damping parameter αeff\nfor the YIG/Pt film is described as16\nαeff=α0+∆α=α0+gµB\n4πMSdYIGg↑↓\neff,(3)\nwhereα0is the intrinsic damping of the bare YIG film,\ngis the g-factor, µBis the Bohr magneton, dYIGis the\nYIG film thickness and g↑↓\neffis the real part of the ef-\nfective spin mixing conductance. The effective Gilbert\ndamping parameter αeffis inversely proportional to the\nYIG film thickness dYIG: with decreasing YIG thickness\nthe linewidth and, thus, the effective damping parameter\nincreases.\nWhen the system is resonantlydriven in the FMR con-\ndition, a spin-polarized electron current is injected from\nthe magnetic material (YIG) into the normal metal (Pt).\nInaphenomenologicalspin-pumpingmodel, theDCcom-\nponent of the spin-current density jsat the interface, in-\njected in y-direction into the Pt layer (Fig. 1(c)), can be\ndescribed as15,16,32\njs=f/integraldisplay1/f\n0¯h\n4πg↑↓\neff1\nM2\nS/braceleftBig\nM(t)×dM(t)\ndt/bracerightBig\nzdt,(4)\nwhereM(t) is the magnetization. {M(t)×dM(t)\ndt}zis the\nz-component of {M(t)×dM(t)\ndt}, which is directed along\nthe equilibrium axis of the magnetization (see Fig. 1(c)).\nDue to spin relaxation in the normal metal (Pt) the\ninjected spin current jsdecays along the Pt thickness\n(y-direction in Fig. 1(c)) as15,16\njs(y) =sinhdPt−y\nλ\nsinhdPt\nλj0\ns, (5)\nwhereλis the spin-diffusion length in the Pt layer. From\nEq. (4) one can deduce the spin-current density at the\ninterface ( y= 0)15\nj0\ns=g↑↓\neffγ2(µ0h∼)2¯h(µ0MSγ+/radicalbig\n(µ0MSγ)2+16(πf)2)\n8πα2\neff((µ0MSγ)2+16(πf)2).\n(6)4\nFIG. 3. (Color online) ISHE-induced voltage UISHEas a func-\ntion of the magnetic field Hfor different YIG film thicknesses\ndYIG. Applied microwave power Papplied= 10 mW, ISHE-\nvoltage for the 20 nm thick sample is multiplied by a factor\nof 5.\nSincej0\nsis inverselyproportionalto α2\neffandαeffdepends\ninversely on dYIG(Eq. (3)), the spin-current density at\nthe interface j0\nsincreases with increasing YIG film thick-\nnessdYIG.\nC. Inverse spin Hall effect\nThe Pt layer acts as a spin-current detector and trans-\nforms the spin-polarized electron current injected due to\nthe spin-pumping effect into an electrical charge current\nby means of the ISHE (see Fig. 1(c)) as6,7,12,15,16\njc=θISHE2e\n¯hjs×σ, (7)\nwhereθISHE,e,σdenote the spin Hall angle, the elec-\ntron’s elementary charge and the spin-polarization vec-\ntor, respectively. Averaging the charge-current density\nover the Pt thickness and taking into account Eqs. ( 4) –\n(7) yields\n¯jc=1\ndPt/integraldisplaydPt\n0jc(y)dy=θISHEλ\ndPt2e\n¯htanh/parenleftbigdPt\n2λ/parenrightbig\nj0\ns.(8)\nTaking into account Eqs. ( 3), (6) and (8) we calcu-\nlate the theoretically expected behavior of IISHE=A¯jc,\nwhereAis the cross section of the Pt layer. Ohm’s law\nconnects the ISHE-voltage UISHEwith the ISHE-current\nIISHEviaUISHE=IISHE·R, whereRis the electric resis-\ntance of the Pt layer. Rvaries between 1450 Ω and 1850\nΩ for the different samples.\nIV. YIG FILM THICKNESS DEPENDENCE OF\nTHE SPIN-PUMPING EFFECT DETECTED BY\nTHE ISHE\nIn Fig.3the magnetic field dependence of the gener-\nated ISHE-voltage UISHEas a function of the YIG filmthickness is shown. Clearly, the maximal voltage UISHE\nat the resonance field HFMRand the FMR linewidth ∆ H\nvary with the YIG film thickness. The general trend\nshows, that the thinner the sample the smaller is the\nmagnitude of the observed voltage UISHE. At the same\ntime the FMR linewidth increases with decreasing YIG\nfilm thickness.\nIn the following the ISHE-voltage generated by spin\npumping is investigated as a function of the YIG film\nthickness. For these investigations we have chosen a\nrather small exciting microwave power of 1 mW. Thus,\nnonlinear effects like the FMR linewidth broadening due\nto nonlinear multi-magnon processes can be excluded\n(such processes will be discussed in Sec. V). Sec.IVA\ncovers the YIG thickness dependent variation of the en-\nhanced damping parameter αeff. From these measure-\nments the spin mixing conductance g↑↓\neffis deduced. In\nSec.IVBwe focus on the maximal ISHE-voltage driven\nby spin pumping as a function of the YIG film thickness.\nFinally, the spin Hall angle θISHEis determined.\nA. YIG film parameters as a function of the YIG\nfilm thickness\nAs described in Sec. IIIB, the damping parameter is\nenhanced when a Pt layeris deposited onto the YIG film.\nThisenhancementisinvestigatedasafunctionoftheYIG\nfilm thickness: the effective Gilbert damping parameter\nαeff(see Eq. ( 3)) is obtained from a Lorentzian fit to the\nexperimental data depicted in Fig. 3and Eq. ( 2). The\nresultisshowninFig. 4. With decreasingYIG film thick-\nness the linewidth and, thus, the effective damping αeff\nincreases. This behavior is theoretically expected: ac-\ncording to Eq. ( 3)αeffis inversely proportional to dYIG.\nSince the Pt film is grown onto all YIG samples simulta-\nFIG. 4. (Color online) Enhanced damping parameter αeff\nof the YIG/Pt samples obtained by spin-pumping measure-\nments. The red solid curve shows a fit to Eq. (3) taking the\nFMR measured values for MSand a constant value for g↑↓\neff\ninto account. Papplied= 1 mW. The error bars for the mea-\nsurement points at higher sample thicknesses are not visibl e\nin this scale.5\nFIG. 5. (Color online) (a) ISHE-voltage UISHEas a function\nof the YIG film thickness dYIG. The black line is a linear in-\nterpolation as a guide tothe eye. (b) Corresponding thickne ss\ndependent charge current IISHE. The red curve shows a fit to\nEqs. (6), (7), (8) with theparameters g↑↓\neff=(7.43±0.36)×1018\nm−2andθISHE= 0.009±0.0008. The applied microwave\npower used is Papplied= 1 mW.\nneously, the spin mixing conductance g↑↓\neffat the interface\nis considered to be constant for all samples.11Assum-\ningg↑↓\neffas constant and taking the saturation magne-\ntizationMSobtained by VNA-FMR measurements (see\nFig.2and Tab. I) into account, a fit to Eq. ( 3) yields\ng↑↓\neff= (7.43±0.36)×1018m−2. The fit is depicted as a\nred solid line in Fig. 4.\nB. YIG thickness dependence of the ISHE-voltage\ndriven by spin pumping\nFig.5(a) shows the maximum voltage UISHEat the\nresonance field HFMRas a function of the YIG film\nthickness. UISHEincreases up to a YIG film thickness\nof around 200 nm when it starts to saturate (in the\ncase of an applied microwave power of P applied= 1\nmW). The corresponding charge current IISHEis shown\nin Fig.5(b). The observed thickness dependent behavior\nis in agreement with the one reported for Ni 81Fe19/Pt16\nand for Y 3Fe5O12/Pt.11With increasing YIG film thick-\nness the generated ISHE-current increases and tends to\nsaturate at thicknesses near 200 nm (Fig. 5(b)). Accord-\ning to Eq. ( 3), (6) and (8) it isIISHE∝j0\ns∝1/α2\neff∝\n(α0+c/dYIG)−2, wherecis a constant. Therefore, theISHE-current IISHEincreases with increasing YIG film\nthickness dYIGand goes into saturation at a certain YIG\nthickness.\nFrom Eqs. ( 3), (6) and (8) we determine the expected\nbehavior of IISHE=A¯jcand compare it with our ex-\nperimental data. In order to do so, the measured values\nforMS(see Tab. I), the original damping parameter α0\ndetermined by VNA-FMR measurements at 1 mW (see\nTab.I) and the enhanced damping parameter αeffob-\ntained by spin-pumping measurements at a microwave\npower of 1 mW (see Fig. 4) are used. The Pt layer thick-\nness isdPt= 8.5 nm and the microwave magnetic field\nis determined to be h∼= 3.2 A/m for an applied mi-\ncrowavepower of 1 mW using an analytical expression.24\nThe spin-diffusion length in Pt is taken from literature as\nλ= 10 nm33,36and the damping parameter is assumed\nto be constant as α0= 6.68×10−4, which is the aver-\nage of the measured values of α0. The fit is shown as a\nred solid line in Fig. 5(b). We find a spin Hall angle of\nθISHE= 0.009±0.0008,which is in agreementwith litera-\nture values varying in a range of 0.0037- 0.08.33–35Using\nFIG. 6. (Color online) (a) YIG thickness dependence of the\nISHE-voltage driven by spin pumping for microwave powers\nin the range between 1 and 500 mW. The general thickness\ndependent behavior is independent of the applied microwave\npower. The error bars for the measurement at lower mi-\ncrowave powers are not visible in this scale. (b) Deviation\nof the ISHE-voltage from the linear behavior with respect to\nthe measured voltage U500mW\nISHE. The inset shows experimental\ndata for a YIG film thickness dYIG= 20 nm and the theoret-\nically expected curve. The error bars of the 20 nm and the\n70 nm samples are not visible in this scale.6\nthe fit we estimate the saturation value of the generated\ncurrent. Although we observe a transition to saturation\nat sample thicknesses of 200 – 300 nm, we find that ac-\ncording to our fit, 90% of the estimated saturation level\nof 5 nA is reached at a sample thickness of 1.2 µm.\nV. INFLUENCE OF NONLINEAR EFFECTS ON\nTHE SPIN-PUMPING PROCESS FOR VARYING\nYIG FILM THICKNESSES\nIn order to investigate nonlinear effects on the spin-\npumping effect for varying YIG film thicknesses, we per-\nformed microwavepower dependent measurements of the\nISHE-voltage UISHEas function of the film thickness\ndYIG. For higher microwavepowersin the rangeof 1 mW\nto 500 mW we observe the same thickness-dependent\nbehavior of the ISHE-voltage as in the linear case\n(Papplied= 1 mW, discussed in Sec. IVB): Near 200 nm\nUISHEstarts to saturate independently of the applied mi-\ncrowave power, as it is shown in Fig. 6(a). Furthermore,\nit is clearly visible from Fig. 6(a) that for a constant\nfilm thickness the spin pumping driven ISHE-voltage in-\ncreases with increasing applied microwave power. At\nhigh microwave powers the voltage does not grow lin-\nearly and saturates. Fig. 6(b) shows the deviation of\nthe ISHE-voltage ∆ UISHEfrom the linear behavior with\nrespect to the measured value of U500mW\nISHEat the excita-\ntion power Papplied= 500 mW. In order to obtain the\nrelation between UISHEandPappliedfor each YIG film\nthickness dYIGthe low power regime up to 20 mW is\nfitted by a linear curve and extrapolated to 500 mW.\nTheinsetin Fig. 6(b) showsthe correspondingviewgraph\nfor the case of the 20 nm thick sample. As it is visible\nfrom Fig. 6(b), the deviation from the linear behavior\nis drastically enhanced for larger YIG thicknesses. For\nthe thin 20 nm and 70 nm samples we observe an almost\nlinear behavior between UISHEandPappliedover the en-\ntire microwavepower range, whereas for the thicker sam-\nples the estimated linear behavior and the observed non-\nlinear behavior differ approximately by a factor of 2.5\n(Fig.6(b)). We observe an increase of the ISHE-voltage\nas well as an broadening of the FMR linewidth with in-\ncreasing microwave power. In Fig. 7(a) the normalized\nISHE-voltage UISHEas function of the external magnetic\nfieldHis shown for different microwave powers Papplied\nin the range of 1 mW to 500 mW (YIG film thickness\ndYIG= 300 nm). The linewidth tends to be asymmet-\nric at higher microwave powers. The shoulder at lower\nmagnetic field is widened in comparison to the shoulder\nat higher fields. The reason for this asymmetry might\nbe due to the formation of a foldover effect,37,38due to\nnonlinear damping or a nonlinear frequency shift.39,40\nThe results of the damping parameter αeffobtained\nby microwave power dependent spin-pumping measure-\nments are depicted in Fig. 7(b). It can be seen, that\nwith increasing excitation power the Gilbert damping for\nthicker YIG films is drastically increased. To present\nFIG. 7. (Color online) (a) Illustration of the linewidth bro ad-\nening at higher excitation powers. The normalized ISHE-\nvoltage spectra are shown as a function of the magnetic\nfieldHfor different excitation powers. Sample thickness:\n300 nm. (b) Power dependent measurement of the damp-\ning parameter αefffor different YIG film thicknesses dYIGob-\ntained by a Lorentzian fit to the ISHE-voltage signal. The\nerror bars are omitted in order to provide a better readabil-\nity of the viewgraph. (c) Nonlinear damping enhancement\n(α500mW\neff−α1mW\neff)/α1mW\neffas a function of the YIG film thick-\nnessdYIG. Due to a reduced number of scattering channels to\nother spin-wave modes for film thicknesses below 70 nm, the\ndamping is only enhanced for thicker YIG films with increas-\ning applied microwave powers. The error bars of the 200 nm\nand the 300 nm samples are not visible on this scale.\nthis result more clearly the nonlinear damping enhance-\nment (α500mW\neff−α1mW\neff)/α1mW\neffis shown in Fig. 7(c). The\ndampingparameterat asamplethicknessof20nm α20nm\neff7\nFIG. 8. (Color online) Dispersion relations calculated for each sample thickness taking into account the measured valu es of the\nsaturation magnetization MS(see Tab. I). Backward volume magnetostatic spin-wave mode s as well as magnetostatic surface\nspin-wave modes (in red) and the first perpendicular standin g thickness spin-wave modes are depicted (in black and gray) .\n(a)–(e) show the dispersion relations for the investigated sample thicknesses of 20 nm – 300 nm.\nis almost unaffected by a nonlinear broadening at high\nmicrowave powers. With increasing film thickness the\noriginal damping α1mW\neffatPapplied= 1 mW increases by\na factor of around 3 at Papplied= 500 mW. This factor\nis very close to the value of the deviation of the ISHE-\nvoltage from the linear behavior (Fig. 6(b)).\nThis behavior can be attributed to the enhanced prob-\nability of nonlinear multi-magnon processes at larger\nsample thicknesses: In order to understand this, a fun-\ndamental understanding of the restrictions for multi-\nmagnon scattering processes can be derived from the en-\nergy and momentum conservation laws:\nN/summationdisplay\ni¯hωi=M/summationdisplay\nj¯hωjandN/summationdisplay\ni¯hki=M/summationdisplay\nj¯hkj,(9)\nwhere the left/right sum of the equations runs over the\ninitial/final magnons with indices i/j which exist be-\nfore/after the scattering process, respectively.41–43The\nmost probable scattering mechanism in our case is the\nfour-magnon scattering process with N= 2 and M=\n2.43In Eq. (9) the wavevector ki/jand the frequency ωi/j\nare connected by the dispersion relation 2 πfi/j(ki/j) =\nωi/j(ki/j). The calculated dispersion relations are shown\nin Fig.8(backward volume magnetostatic spin-wave\nmodes with a propagation angle /negationslash(H,k) = 0◦as well\nas magnetostatic surface spin-wave modes /negationslash(H,k) =\n90◦).44For this purpose, the measured values of MS\n(Tab.I) for each sample are used. In the case of the\n20 nm sample thickness, the first perpendicular standing\nspin-wave mode (thickness mode) lies above 40 GHz, thesecond above 120 GHz. Thus, the nonlinear scattering\nprobability obeying the energy- and momentum conser-\nvation is largely reduced. This means magnons cannot\nfind a proper scattering partner and, thus, multi-magnon\nprocesses are prohibited or at least largely suppressed.\nWith increasing film thickness the number of standing\nspin-wavemodesincreasesand, thus, thescatteringprob-\nability grows. As a result, the scattering of spin waves\nfrom the initially excited uniform precession (FMR) to\nother modes is allowed and the relaxation of the original\nFMR mode is enhanced. Thus, the damping increases\nand we observe a broadening of the linewidth, which is\nequivalent to an enhanced Gilbert damping parameter\nαeffat higher YIG film thicknesses (see Fig. 7).\nIn orderto investigatehowthe spin-pumping efficiency\nis affected by the applied microwave power, we measure\nsimultaneously the generated ISHE-voltage UISHEand\nthe transmitted ( Ptrans), as well as the reflected ( Prefl)\nmicrowave power, which enables us to determine the\nabsorbed microwave power Pabs=Papplied−(Ptrans+\nPrefl).17Since the 300 nm sample exhibits a strong non-\nlinearity(largedeviationfromthelinearbehavior(Fig. 6)\nand large nonlinear linewidth enhancement (Fig. 7)), we\nanalyze this sample thickness. In Fig. 9the normalized\nabsorbed microwave power Pnorm=Pabs/PPapplied=1mW\nabs\nand the normalized ISHE-voltage in resonance Unorm=\nUISHE/UPapplied=1mW\nISHEare shown as a function of the ap-\nplied power Papplied. Both curves tend to saturate at\nhigh microwave powers above 100 mW. The absorbed\nmicrowave power increases by a factor of 110 for applied\nmicrowave powers in the range between 1 and 500 mW,8\nFIG. 9. (Color online) Normalized absorbed power Pnorm=\nPabs/PPapplied=1mW\nabs (black squares) and normalized ISHE-\nvoltageUnorm=UISHE/UPapplied=1mW\nISHE (red dots) for varying\nmicrowave powers Papplied. The inset illustrates the indepen-\ndence of the spin-pumping efficiency UISHE/PabsonPapplied.\nYIG thickness illustrated: 300 nm. Error bars of the low\npower measurements are not visible in this scale.\nwhereas the generated voltage increases by a factor of\n80. The spin-pumping efficiency UISHE/Pabs(see inset\nin Fig.9) varies within a range of 30% for the differ-\nent microwave powers Pappliedwithout clear trend. Since\nthe 300 nm thick film shows a nonlinear deviation of the\nISHE-voltageby afactorof2.3 (Fig. 6(b)) and the damp-\ning is enhanced by a factor of 3 in the same range of\nPapplied(Fig.7(c)), we conclude that the spin-pumping\nprocess is only weakly dependent on the magnitude of\nthe applied microwave power (see inset in Fig. 9). In our\nprevious studies reported in Ref.12,13we show that sec-\nondary magnons generated in a process of multi-magnon\nscattering contribute to the spin-pumping process and,\nthus, the spin-pumping efficiency does not depend on the\napplied microwave power.\nVI. SUMMARY\nThe Y 3Fe5O12thickness dependence of the spin-\npumping effect detected by the ISHE has been inves-\ntigated quantitatively. It is shown that the effective\nGilbert damping parameter of the the YIG/Pt sam-\nples is enhanced for smaller YIG film thicknesses, which\nis attributed to an increase of the ratio between sur-\nface to volume and, thus, to the interface character of\nthe spin-pumping effect. We observe a theoretically\nexpected increase of the ISHE-voltage with increasing\nYIG film thickness tending to saturate above thick-\nnesses near 200 – 300 nm. The spin mixing conductance\ng↑↓\neff= (7.43±0.36)×1018m−2as well as the spin Hall an-gleθISHE= 0.009±0.0008 are calculated and are found\nto be in agreement with values reported in the literature\nfor our materials.\nThe microwave power dependent measurements reveal\nthe occurrence of nonlinear effects for the different YIG\nfilm thicknesses: for low powers, the induced voltage\ngrows linearly with the power. At high powers, we ob-\nserve a saturation of the ISHE-voltage UISHEand a de-\nviation by a factor of 2.5 from the linear behavior. The\nmicrowave power dependent investigations of the Gilbert\ndamping parameter by spin pumping show an enhance-\nment by a factor of 3 at high sample thicknesses due\nto nonlinear effects. This enhancement of the damping\nis due to nonlinear scattering processes representing an\nadditional damping channel which absorbs energy from\nthe originally excited FMR. We have shown that the\nsmaller the sample thickness, the less dense is the spin-\nwave spectrum and, thus, the less nonlinear scattering\nchannels exist. Hence, the smallest investigated sample\nthicknesses (20 and 70 nm) exhibit a small deviation of\nthe ISHE-voltage from the linear behavior and a largely\nreduced enhancement of the damping parameter at high\nexcitation powers. Furthermore, we have found that the\nvariation of the spin-pumping efficiencies for thick YIG\nsamples which show strongly nonlinear effects is much\nsmaller than the nonlinear enhancement of the damping.\nThis is attributed to secondary magnons generated in a\nprocessofmulti-magnonscatteringthatcontributetothe\nspin pumping. It is shown, that even for thick samples\n(300 nm) the spin-pumping efficiency is only weakly de-\npendent on the applied microwave power and varies only\nwithin a range of 30% for the different microwave powers\nwithout a clear trend.\nOur findings provide a guideline to design and create\nefficient magnon- to charge current converters. Further-\nmore, the results are also substantial for the reversed\neffects: the excitation of spin waves in thin YIG/Pt bi-\nlayers by the direct spin Hall effect and the spin-transfer\ntorque effect.45\nVII. ACKNOWLEDGMENTS\nWe thank G.E.W. Bauer and V.I. Vasyuchka for valu-\nable discussions. 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We study the exponential stability of constant steady state\nofisentropic compressible Eulerequationwith dampingon Tn. Thelocal\nexistence of solutions is based on semigroup theory and some commuta-\ntor estimates. We propose a new method instead of energy esti mates to\nstudy the stability, which works equally well for any spatia l dimensions.\n1.Introduction\nWe consider the compressible Euler equation with frictiona l damping in\na periodic box [0 ,1]n, which takes the form\n(1.1)\n\nρt+∇·(ρU) = 0,\n(ρU)t+∇·(ρU⊗U)+∇P=−αρU,\n(ρ,U)(x,0) = (ρ0,U0)(x),\n(ρ,U)(t,···,xi,···) = (ρ,U)(t,···,xi+1,···)∈Rn+1, t≥0,/integraldisplay\n[0,1]nρ0dx=ρ >0.\nOur goal is show the exponential stability of the steady stat e (ρ,0).\nSuch a system occurs in the mathematical modeling of compres sible flow\nthrough a porous medium. Here the unknowns ρ,UandPdenote the den-\nsity, velocity and pressure, respectively. The constant α >0 models friction.\nWe assume the flow is a polytropic perfect gas, then P(ρ) =P0ργ, where\nγ >1 is the adiabatic gas exponent. To keep our exposition clean (but not\nloss of generality), we will take P0=1\nγ,α= 1 and ρ= 1 starting from\nsection 2.\nThere exist extensive literatures in past decades about com pressible Euler\nequation subject to various initial and initial-boundary c onditions. Both\nclassical and weak solutions are constructed and their long time behavior\nare investigated. For the Cauchy problem, we refer the reade rs to [8, 19,\n21, 22] and references therein for the existence of small smo oth solutions; to\n[3, 5, 20] for L∞solutions. For large time behavior of solutions, we refer to\n[12, 14, 23] for small smooth solutions and [6, 24] and refere nces therein for\nweak solutions. In the direction of initial-boundary value problems, we refer\nto[9, 10, 13] for small solutions and[17] for L∞solutions. For non-isentropic\n12 NAN LU LEHIGH UNIVERSITY\nflows, see [15, 16] and references therein. It is also worth to mention some\nrecent work [1, 2, 11] for hyperbolic systems with partial di ssipations in\nseveral spatial dimensions.\nWith thenotation introducedinsection 2, ourmainresultis thefollowing:\nMain Theorem. Lets >n\n2. There exists ǫ⋆>0andδ >0such that if\n/ba∇dblρ0−ρ/ba∇dbls+1+/ba∇dblU0/ba∇dbls+1< ǫ⋆, there exists a unique solution with initial value\n(ρ0,U0)such that\n(ρ,U)∈C0([0,∞),Xs+1)∩C1([0,∞),Xs),\nwhich also satisfies\n/ba∇dblρ(t,·)−ρ/ba∇dbls+1+/ba∇dblU(t,·)/ba∇dbls+1≤C(/ba∇dblρ0−ρ/ba∇dbls+1+/ba∇dblU0/ba∇dbls+1)e−δt.\nThe result in this paper is not new (especially in low dimensi ons) for\nresearchers working on compressible Euler equations. Howe ver, the method\nwe use is new in the literature. Moreover, the proof is simple r and shorter\ncomparedtoearlier works andappliestoany spatial dimensi onsequally well.\nEnergy estimates is the standard approach for analyzing glo bal existence\nand asymptotic behavior of partial differential equations. H owever, such\nmethod gets a little tediously long when the spatial dimensi on increases,\nbecause the function spaces needs to more regular in order to make the\nequation well-posed. The main contribution of this paper is contained in\nsubsection3.3, whereweproposeanewmethodtoobtain thede cay property\nofsmall solutions. Ourstrategy istomakeachangeof coordi natesothat the\nsolution lives on a subspace under such coordinate system. U nlike the usual\ncoordinate system using a fixed frame, we define ours by using a moving\nframe which depends on the solution itself. Under such movin g frame, we\ncan decompose the solution into two parts and discover the de cay property.\nThe rest of this paper is organized as follows. In Section 2, w e introduce\nsome notations and rewrite (1.1) in a dynamical system form. In Section 3,\nwe present the proof of the Main Theorem.\n2.Set Up\nThroughout this paper, we use Hsto denote Sobolev space of periodic\nfunctions equipped with norm /ba∇dbl·/ba∇dbls, i.e.,\n/ba∇dblf/ba∇dbls/defines(/summationdisplay\n|α|≤s/ba∇dblDαf/ba∇dbl2\nL2)1\n2,\nwhereαis a multi-index.\nFor any vector valued function F= (f1,···,fm) : [0,1]n−→Rm,\n/ba∇dblF/ba∇dbls/definesn/summationdisplay\ni=1/ba∇dblfi/ba∇dbls.COMPRESSIBLE EULER WITH DAMPING ON TORUS 3\nFor anys∈Z, we define Banach spaces\nXs/defines{(f,g)/vextendsingle/vextendsinglef∈Hs,g∈ ⊕nHs},˜Xs/defines{(f,g)∈Xs/vextendsingle/vextendsingle/integraldisplay\n[0,1]nf(x)dx= 0},\nwhich are equipped with norms\n/ba∇dbl(f,g)/ba∇dbls/defines/ba∇dblf/ba∇dbls+/ba∇dblg/ba∇dbls.\nFrom now on, we will assume P0=1\nγ,α= 1 and ρ= 1. Let θ=γ−1\n2and\nσ=ρθ−1\nθ. Then the system (1.1) can be written as\n(2.1)\n\n/parenleftbigg\nσ\nU/parenrightbigg\nt=/parenleftbigg\n0−∇·\n−∇ −1/parenrightbigg/parenleftbigg\nσ\nU/parenrightbigg\n−/parenleftbigg\nU·∇θσ∇·\nθσ∇U·∇/parenrightbigg/parenleftbigg\nσ\nU/parenrightbigg\n,\n(σ,U)(x,0) = (σ0,U0)(x),/integraldisplay\n[0,1]nσ0dx=σ.\nIt is clear that classical solutions of (1.1) and (2.1) are eq uivalent through\nthe transformation σ=ρθ−1\nθ. Let\nA/defines/parenleftbigg\n0−∇·\n−∇ −1/parenrightbigg\n, Bσ,U/defines−/parenleftbigg\nU·∇θσ∇·\nθσ∇U·∇/parenrightbigg\nand (2.1) can be written abstractly as\n(2.2) ( σ,U)t= (A+Bσ,U)(σ,U).\nFor any bounded linear operator T∈L(X,Y), we use |T|L(X,Y)to denote\nthe operator norm. We will drop the subscript if the context c auses no\nconfusion. For any Banach space Z, we use Bδ(Z) to the denote the ball\ncentered at the origin of radius δinZ.\n3.Proof\nIn this section, we give the proof of the Main Theorem. We spli t this\nsection into three subsections. In subsection 3.1, we show t hatAgener-\nates a semigroup T(t) andA+Bσ,Ugenerates an evolutionary operator\nMσ,Ufor suitably chosen ( σ,U), respectively. To make our presentation\nself-contained, we prove local well-posedness in subsecti on 3.2. In subsec-\ntion 3.3, we analyze the asymptotic behavior of local soluti ons.\n3.1.Semigroup and Evolutionary Operator. In this subsection, we\nprovide estimates on the semigroup and the evolutionary ope rator gener-\nated byAandA+Bσ,UonXs. We begin with some properties of A.\nLemma 3.1. The linear operator Agenerates a strongly continuous semi-\ngroupT(t)onXsfor every s. Moreover, there exists K≥1such that for\nt≥0,\n(3.1) |T(t)|L(Xs,Xs)≤K ,|T(t)|L(˜Xs,˜Xs)≤Ke−1\n2t.4 NAN LU LEHIGH UNIVERSITY\nProof.SinceAis a closed, densely defined linear operator and is the sum of\nan anti-selfadjoint and a bounded linear operator, the stan dard semigroup\ntheory implies Agenerates a C0-semigroup T(t) onXs. LetˆA(ξ) be the\nsymbol of A, namely,\nˆA(ξ) =/parenleftbigg\n0−iξT\n−iξ−I/parenrightbigg\n, ξ∈Zn.\nThe above matrix has eigenvalues −1 with multiplicity n−1 andλ1=\n−1\n2+1\n2/radicalbig\n1−4|ξ|2,λ2=−1\n2−1\n2/radicalbig\n1−4|ξ|2. Forξ∈Zn/{0}, we choose an\northonormal basis as follows\nV1=1/radicalbig\n|ξ|2+|λ2|2/parenleftbigg\n−iλ2\nξ/parenrightbigg\n, V2=/parenleftbigg\nh0\nh1/parenrightbigg\n=\ni|ξ|2\nλ2/radicalbigg\n|ξ|4\n|λ2|2+|ξ|2\nξ/radicalbigg\n|ξ|4\n|λ2|2+|ξ|2\n,\nVi=/parenleftbigg\n0\nηi/parenrightbigg\n, ηi∈ {Cξ}⊥,1≤i≤n−1.\nLetR(ξ) = (V1,···,Vn+1) and we have R(ξ) is unitary, namely, R−1(ξ) =\nR⋆(ξ) =Rt(ξ). Consequently,\nR−1(ξ)ˆA(ξ)R(ξ)\n=\nh0 ht\n1\niλ2√\n|ξ|2+|λ2|2ξt√\n|ξ|2+|λ2|2\n0 ηt\n1......\n0 ηn−1\n/parenleftbigg\n0−iξt\n−iξ−1/parenrightbigg\nh0−iλ2√\n|ξ|2+|λ2|20···0\nh1ξ√\n|ξ|2+|λ2|2η1···ηn−1\n\n=\n−iht\n1·ξh0−ih0ξt·h1−1|h1|2−ht\n1·ξλ2−ih0|ξ|2−1ht\n1·ξ√\n|ξ|2+|λ2|2\n−i|ξ|2h0+(λ2−1)ξt·h1 √\n|ξ|2+|λ2|2(λ2−λ2−1)\n|ξ|2+|λ2|20\n0 −In−1\n\n=\nλ20\n−1λ10\n0−In−1\n/definesB(ξ).\nIt follows that\netB(ξ)=\neλ2t0\n−eλ2t−eλ1t\nλ2−λ1eλ1t\ne−tIn−1\n.\nWe note that for ξ∈Zn/{0}, there exists K≥1 independent of ξsuch that\n|eλ2t−eλ1t\nλ2−λ1| ≤Ke−1\n2t,COMPRESSIBLE EULER WITH DAMPING ON TORUS 5\nwhich implies\n|etˆA(ξ)|L(Cn+1,Cn+1)=|etR−1(ξ)B(ξ)R(ξ)|=|R−1(ξ)etB(ξ)R(ξ)| ≤Ke−1\n2t.\nForξ= 0, we have ˆA(0) =/parenleftbigg\n0 0\n0−1/parenrightbigg\n. Therefore, for t≥0,\n(3.2) |T(t)|L(Xs,Xs)≤K ,|T(t)|L(˜Xs,˜Xs)≤Ke−1\n2t.\n/square\nProposition 3.2. Lets >n\n2andǫ⋆>0be a sufficiently small constant.\nFor any (σ,U)such that /ba∇dbl(σ,U)/ba∇dblC0([0,T],Xs+1)< ǫ⋆,A+Bσ,Ugenerates an\nevolutionary system Mσ,U(t,τ)onXs, which satisfies\n(3.3) |Mσ,U(t,τ)|L(Xs,Xs)≤e(Cǫ⋆+1)(t−τ)for0≤τ≤t≤T,\nwhereConly depends on sandn.\nProof.For any ( σ1,U1)∈Xs+1, we have\n< Bσ,U/parenleftbigg\nσ1\nU1/parenrightbigg\n,/parenleftbigg\nσ1\nU1/parenrightbigg\n>Hs\n=−/summationdisplay\n|α|≤s/integraldisplay\n[0,1]nDα/parenleftbigg\nU·∇σ1+θσ∇·U1\nθσ∇σ1+U·∇U1/parenrightbigg\n·Dα/parenleftbigg\nσ1\nU1/parenrightbigg\ndx\n=−/integraldisplay\n[0,1]n(U·Ds∇σ1+θσDs∇·U1)Dsσ1+(θσDs∇σ1+U·Ds∇U1)·DsU1dx+l.o.t\n=/integraldisplay\n[0,1]n∇·U(1\n2(Dsσ1)2)+∇(θσ)·(Dsσ1DsU1)+∇·U(1\n2U1·U1)dx+l.o.t\n≤C/ba∇dblD(σ,U)/ba∇dblL∞/ba∇dbl(σ1,U1)/ba∇dbl2\ns≤Cǫ⋆/ba∇dbl(σ1,U1)/ba∇dbl2\ns,\nwhereConly depends on sandn. This means Bρ,U−Cǫ⋆is dissipative.\nLet˜A=/parenleftbigg\n0−∇·\n−∇0/parenrightbigg\n, i.e.,˜A=A+/parenleftbigg\n0 0\n0In×n/parenrightbigg\n. Since ˜Ais anti-\nselfadjoint, it generates a unitary group U(t) fort∈R. Note that\n/ba∇dbl(Bρ,U−Cǫ⋆)(σ1,U1)/ba∇dbls\n≤/ba∇dblU·∇σ1+θσ∇·U1/ba∇dbls+/ba∇dblθσ∇σ1+U·∇U1/ba∇dbls+Cǫ⋆/ba∇dbl(σ1,U1)/ba∇dbls\n≤C(/ba∇dbl(σ,U)/ba∇dblL∞+/ba∇dblD(σ,U)/ba∇dblL∞)(/ba∇dbl∇σ1/ba∇dbls+/ba∇dbl∇·U1/ba∇dbls)+Cǫ⋆/ba∇dbl(σ1,U1)/ba∇dbls\n≤Cǫ⋆/ba∇dblA(σ1,U1)/ba∇dbls+Cǫ⋆/ba∇dbl(σ1,U1)/ba∇dbls.\nTherefore, by Corollary 3.3.3 in [18], ˜A+Bρ,U−Cǫ⋆generates a family of\ncontractions on Xsfor small ǫ⋆. Since/parenleftbigg\n0 0\n0−In×n/parenrightbigg\nis bounded with norm\nequal to 1. Consequently, A+Bρ,U−Cǫ⋆generates an evolutionary operator\nMσ,Uwith satisfies (3.3). /square6 NAN LU LEHIGH UNIVERSITY\n3.2.Local existence. In this subsection, we prove the local existence of\nsmall initial data though a contraction mapping argument. R ecall that we\nobtain the evolutionary system Mσ,U(t,τ)∈L(Xs,Xs) for small ( σ,U)∈\nXs+1. Therefore, we need to show such operator also maps Xs+1toXs+1.\nThis fact can be proved by the following commutator estimate . The main\nidea is developed by Kato in [7], which has wide applications in studying\nlocal well-posedness of quasi-linear equations. Let\nSf(x)/defines/summationdisplay\nξ∈Zne2πix·ξ(1+|ξ|2)1\n2ˆf(ξ),\nwhich is an isomorphism between Xs+1andXssuch that |S|=|S−1|= 1.\nLemma 3.3. Lets >n\n2and(σ,U)∈Xs+1, then\nS(A+Bσ,U)S−1−(A+Bσ,U)∈L(Xs,Xs).\nProof.It is clear that Acommutes with S. Thus, we only need to show\nSBσ,US−1−Bσ,U∈L(Xs,Xs). The Fourier transform of the difference\noperator has kernel\n(/hatwiderSBσ,US−1−ˆBσ,U)(ξ,η)\n=/parenleftbig\n(1+|ξ|2)1\n2−(1+|η|2)1\n2/parenrightbig/parenleftbiggˆU(ξ−η)·(iη)θˆσ(ξ−η)(iηT)\nθˆσ(ξ−η)(iη)ˆU(ξ−η)·(iη)/parenrightbigg\n(1+|η|2)−1\n2.\nNote that|η|√\n1+|η|2≤1 and\n/vextendsingle/vextendsingle(1+|ξ|2)1\n2−(1+|η|2)1\n2/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/integraldisplay1\n0d\ndp(1+|pξ+(1−p)η|2)1\n2dp/vextendsingle/vextendsingle\n≤|ξ−η|/integraldisplay1\n0|pξ+(1−p)η|\n(1+|pξ+(1−p)η|2)1\n2dp≤ |ξ−η|.\nTherefore, we have\n(3.4)|/hatwiderSBσ,US−1−ˆBσ,U| ≤C/summationdisplay\nη|ξ−η|(|ˆU(ξ−η)|+|ˆσ(ξ−η)|)\n≤C/ba∇dbl(σ,U)/ba∇dbls+1,\nwhich completes the proof. /square\nConsequently, for any y∈Xs+1, letx=Syand we have\n/ba∇dblMσ,U(t,τ)y/ba∇dbls+1=/ba∇dblS−1SMσ,US−1x/ba∇dbls+1≤ |SMσ,U(t,τ)S−1|L(Xs,Xs)/ba∇dbly/ba∇dbls+1.\nTherefore, if /ba∇dbl(σ,U)/ba∇dblC0([0,T],Xs+1)< ǫ⋆, by (3.4), we have\n(3.5) |Mσ,U(t,τ)|L(Xs+1,Xs+1)≤e(2Cǫ⋆+1)(t−τ),\nwhereCdepends only on sandn.\nDefine the following space\nZT/defines{(σ,U)∈C([0,T],Xs)/vextendsingle/vextendsingle/vextendsingle(σ(0),U(0)) = (σ0,U0),/ba∇dbl(σ,U)/ba∇dblC0([0,T],Xs+1)≤ǫ⋆},COMPRESSIBLE EULER WITH DAMPING ON TORUS 7\nequipped with the norm /ba∇dbl(σ,U)/ba∇dblZ/defines/ba∇dbl(σ,U)/ba∇dblC0([0,T],Xs). For any ( σ0,U0)∈\nXs+1and (σ,U)∈Z, we let\nF(σ,U,σ0,U0)(t) =Mσ,U(t,0)(σ0,U0).\nProposition 3.4. Lets >n\n2andǫ⋆>0such that (3.3)holds. Then for any\n/ba∇dbl(σ0,U0)/ba∇dbls+1≤ǫ⋆\n2e, the mapping F(·,·,ρ0,U0)has a unique fixed point in\nZTfor some T >0. Consequently, for small initial data, (2.1)has a unique\nsolution\n(3.6) ( σ,U)∈C([0,T],Xs+1)∩C1([0,T],Xs).\nProof.LetT=1\n2Cǫ⋆+1<1, where Cdepends on sandnappearing in\nProposition 3.2. For /ba∇dbl(ρ,U)/ba∇dblC0([0,T],Xs+1)≤ǫ⋆, by (3.3) and (3.5), we have\n(3.7)/ba∇dblF(σ,U,σ0,U0)/ba∇dblZ≤/ba∇dblF(σ,U,σ0,U0)/ba∇dblC0([0,T],Xs+1)\n≤e(2Cǫ⋆+1)Tǫ⋆\n2e< ǫ⋆,\nwhichmeans F(·,·,σ0,U0)mapsBǫ⋆(C0([0,T],Xs+1))andBǫ⋆(Z)intothem-\nselves, respectively.\nFor any ( σ1,U1) and (σ2,U2) inZ, we have\n/ba∇dblF(σ1,U1,σ0,U0)−F(σ2,U2,σ0,U0)/ba∇dblZ\n= sup\nt∈[0,T]/ba∇dbl/integraldisplayt\n0Mσ1,U1(t,τ)(Bσ1,U1(τ)−Bσ2,U2(τ))Mσ2,U2(τ,0)(ρ0,U0)dτ/ba∇dbls\n≤sup\nt∈[0,T]/integraldisplayt\n0e(cǫ⋆+1)(t−τ)/ba∇dbl(σ1−σ2,U1−U2)(τ)/ba∇dblse(2cǫ⋆+1)τ/ba∇dbl(σ0,U0)/ba∇dbls+1dτ\n≤e(2Cǫ⋆+1)T/ba∇dbl(σ0,U0)/ba∇dbls+1/ba∇dbl(σ1−σ2,U1−U2)/ba∇dblZ≤1\n2/ba∇dbl(σ1−σ2,U1−U2)/ba∇dblZ,\nwhich implies Fis a contraction on Bǫ⋆(Z). Therefore, Fhas a unique fixed\npoint in Z. To prove the fixed point is in Xs+1, we note that all iterations\n(σn,Un)/definesF(σn−1,Un−1,σ0,U0) are bounded in Xs+1due to (3.7). Conse-\nquently, there exists a subsequence njsuch that ( σnj,Unj) converges weakly\ninXs+1and thus strongly in Xs. The limit of such subsequence is exactly\nthe fixed point of F. Therefore, the local solution of (2.1) is also in Xs+1\nand (3.6) holds for T=1\n2Cǫ⋆+1and all/ba∇dbl(ρ0,U0)/ba∇dbls+1≤ǫ⋆\n2e. /square\n3.3.Asymptotic Behavior. Inthissubsection, weintroduceanewmethod\nto study the asymptotic behavior of local solutions obtaine d in the pre-\nvious subsection. The method is motivated by the following o bservation.\nFrom Lemma 3.1, we see that the phase space Xscan be decomposed into\ntwo subspaces ˜Xsand its orthogonal complement generated by the vector\n(σ,U) = (1,0), which are invariant under the semigroup S(t). Moreover,\nthe semigroup has exponential decay on ˜Xs. Intuitively, for small ( σ,U),\nthe linear operator Bσ,Uis a small perturbation relative to Aand thus we\nexpect a similar dynamical picture for Mσ,U. It turns out given any local8 NAN LU LEHIGH UNIVERSITY\nsolution ( σ(t),U(t)), we can define a codimension one linear subspace E2(t)\nfor each t, which is isomorphic and close to ˜Xs, such that\n(3.8) Xs=E1⊕E2(t),\nfor 0≤τ≤t≤T. HereE1={(σ,U) = (c,0),c∈R}. By conjugating the\nflow from E2(τ) onto˜Xsand showing that the solution projected along E1\ndirection can be slaved by the projection along E2(t) direction , we discover\nthe exponential decay property of the solution.\nWe begin with the construction of E2(t) in the following Lemma. We will\nusePfto denote the average of fover [0,1]n, i.e.,Pf=/integraltext\n[0,1]nf dx.\nLemma 3.5. Lets >n\n2. There exists δ >0such that for all /ba∇dbl(σ,U)/ba∇dbls<\nδ, there exist operators (L1,L2) :Bδ(Xs)−→L(Xs,R)such that for all\n(σ1,U1)∈ˆXs+1,\n(3.9) ( A+Bσ,U)/parenleftbigg\nL1σ1+L2U1+σ1\nU1/parenrightbigg\n=/parenleftbigg\nL1˜σ1+L2˜U1+ ˜σ1\n˜U1/parenrightbigg\n,\nwhere\n˜σ1= (I−P)(U·∇(L1σ1+L2U1+σ1)+(1+θσ)∇·U1),\n˜U1= (1+θσ)∇(L1σ1+L2U1+σ1)+U1+U·∇U1.\nProof.We note (3.9) is equivalent to\n(3.10)0 =G(L1,L2;σ,U)(σ1,U1)\n/definesP(U·∇(L1σ1+L2U1+σ1)+(1+θσ)∇·U1)\n−L1/parenleftbig\n(I−P)U·∇(L1σ1+L2U1+σ1)+(1+θσ)∇·U1/parenrightbig\n−L2/parenleftbig\n(1+θσ)∇(L1σ1+L2U1+σ1)+U1+U·∇U1/parenrightbig\n=P(U·∇σ1+θσ∇·U1)−L1/parenleftbig\n(I−P)U·∇σ1\n+(1+θσ)∇·U1/parenrightbig\n−L2/parenleftbig\n(1+θσ)∇σ1+U1+U·∇U1/parenrightbig\n.\nConsider Gas a mapping from ˜X−s×Xsto˜X−(s+1). It is obvious that\nG(0,0;0,0) = 0 and\nDL1,L2G(0,0;0,0)(˜L1,˜L2) = (˜L1,˜L2)/parenleftbigg\n0−∇·\n−∇ −I/parenrightbigg\n.\nSince/parenleftbigg\n0−∇·\n−∇ −I/parenrightbigg\nis an isomorphism between ˜Xrand˜Xr−1for anyr,\nthe implicit function theorem implies there exists δ >0 such that for any\n/ba∇dbl(ρ,U)/ba∇dbls< δ, there exists a unique pair of operators ( L1,L2)∈L(ˆXs,R)\nsuchthat (3.9) (or equivalently (3.10)) holds. Moreover, ( L1,L2) aresmooth\nin (σ,U) and\n(3.11)L1(0,0) = 0, L2(0,0) = 0,|D(L1,L2)|C0(Bδ(˜Xs),L(˜Xs,˜X−s))≤C1δ,\nwhereC1is independent of ( σ,U). /squareCOMPRESSIBLE EULER WITH DAMPING ON TORUS 9\nWe extend L1to a linear functional on Hs(still denoted by L1) by setting\n(3.12) L1f=L1(I−P)f , f∈Hs.\nConsequently, we have\n(3.13) L2\n1= 0, L1L2= 0,(I+L1)−1=I−L1,\nand\n(3.14)/parenleftbigg\nI+L1L2\n0I/parenrightbigg−1\n=/parenleftbigg\nI−L1−L2\n0I/parenrightbigg\n.\nLet (σ(t),U(t)) be a local solution of (2.1) and we define\nE2(t) ={/parenleftbigg\nI+L1(σ(t),U(t))L2(σ(t),U(t))\n0 I/parenrightbigg/parenleftbigg\nσ1\nU1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle(σ1,U1)∈˜Xs}.\nRecall that E1={(σ,U) = (c,0),c∈R}. It is clear that\nXs=E1⊕E2(t).\nWe then decompose ( σ(t),U(t)) dynamically as\n(3.15)/parenleftbigg\nσ(t)\nU(t)/parenrightbigg\n=/parenleftbigg\nc(t)\n0/parenrightbigg\n+/parenleftbigg\nI+L1L2\n0I/parenrightbigg/parenleftbigg\n(I−P)σ(t)\nU(t)/parenrightbigg\n.\nPlugging such decomposition into (2.1), we have\n/parenleftbigg\nc\n0/parenrightbigg\nt+/parenleftbigg\nI+L1L2\n0I/parenrightbigg\nt/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n+/parenleftbigg\nI+L1L2\n0I/parenrightbigg/parenleftbigg\n(I−P)σ\nU/parenrightbigg\nt\n=(A+Bσ,U)/parenleftbigg\nI+L1L2\n0I/parenrightbigg/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n.\nWe note that/parenleftbigg\nI+L1L2\n0I/parenrightbigg\nt/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n=/parenleftbigg\n∂tL1∂tL2\n0 0/parenrightbigg/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n∈E1.\nBy the fact E1∩E2(t) ={0}and (3.9), we obtain\n(3.16) ct(t) =−∂tL1(I−P)σ−∂tL2U,\nand\n(3.17)/parenleftbigg\nI+L1L2\n0I/parenrightbigg/parenleftbigg\n(I−P)σ\nU/parenrightbigg\nt\n=(A+Bσ,U)/parenleftbigg\nI+L1L2\n0I/parenrightbigg/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n.\nNext we conjugate flows from E2(t) onto˜Xs. Let\n/parenleftbigg\nσ1\nU1/parenrightbigg\n=/parenleftbigg\n(I−P)σ\nU/parenrightbigg\n=/parenleftbigg\nI−L1−L2\n0I/parenrightbigg/parenleftbigg\nσ−c\nU/parenrightbigg\n∈˜Xs+1.\nA straightforward computation shows (3.17) is equivalent t o\n(3.18)/parenleftbigg\nσ1\nU1/parenrightbigg\nt= (A+Bc+(I+L1)σ1+L2U1,U1+˜Bσ1,U1)/parenleftbigg\nσ1\nU1/parenrightbigg\n,10 NAN LU LEHIGH UNIVERSITY\nwhere˜Bσ1,U1= (˜Bi,j) fori,j= 1,2 and\n˜B1,1=L1U1·∇+L2(1+θ(c+(I+L1)σ1+L2U1))∇,\n˜B1,2=L1θ(c+(I+L1)σ1+L2U1)∇·+L1∇·+L2(I+U1·∇),\n˜B2,1=0,˜B2,2= 0.\nSince we already have a local solution ( σ,U), the linear operator on the\nright hand side of (3.18) generates an evolutionary system ˜Mσ1,U1(t,τ) on\n˜Xs, which also maps ˜Xs+1to˜Xs+1.\nLemma 3.6. Under the hypothesis in Proposition 3.2, there exists ˜K >0\nsuch that the evolutionary system ˜M(t,τ)satisfies\n(3.19) |˜M(t,τ)|L(ˆXs+1,ˆXs+1)≤˜Ke−1\n4(t−τ)for0≤τ≤t≤T.\nProof.From the second estimate in (3.1), we can deduce for small ǫ⋆and\nanyλ >−1\n4,\n/ba∇dbl(λ−(A+Bc+(I+L1)σ1+L2U1,U1+˜Bσ1,U1))−1/ba∇dblL(˜Xs,˜Xs)\n≤/ba∇dbl(I+(Bc+(I+L1)σ1+L2U1,U1+˜Bσ1,U1)(λ−A)−1)−1/ba∇dbl/ba∇dbl(λ−A)−1/ba∇dblL(˜Xs,˜Xs)\n≤2K\nλ+1\n2≤4K.\nBy Theorem V.1.11 in [4], there exists ˜K >0 such that (3.19) holds. The\nproof is completed. /square\nThe above lemma implies ( σ1,U1) decays as long as ( c,σ1,U1) are small.\nThus, if we can bound cin terms of ( σ1,U1), the local solution can be\nextended to a global one, which also satisfies the decay estim ate. To con-\nfirm this fact, we first note that if ρsatisfies (1.1), then ρ−1 satisfies the\nPoincare's inequality, namely,\n/ba∇dblρ−1/ba∇dblL2≤C/ba∇dbl∇ρ/ba∇dblL2.\nAccording to the definition of σ=ρθ−1\nθ, we also have\n(3.20) /ba∇dblσ/ba∇dblL2≤C/ba∇dbl∇σ/ba∇dblL2.\nFrom (3.16), we can write the solution c(t) =c(σ1,U1)(t), where we recall\n(σ1,U1) = ((I−P)σ,U). By the standard ODE theory, c(·,·) is smooth in\n(σ1,U1). We claim that c(0,0) = 0. If it is not true, by (3.16),\nct= 0 =⇒c(t) =c,\nwhich contradicts to (3.20). Consequently, there exists Cindependent of\n(σ1,U1) such that\n|c(σ1,U1)(t)| ≤C(/ba∇dblσ1(t)/ba∇dbls+1+/ba∇dblU1(t)/ba∇dbls+1).\nUsing such estimate and Lemma 3.6, we conclude /ba∇dblσ1/ba∇dbls+1+/ba∇dblU1/ba∇dbls+1decays\nexponentially. Therefore, the Hs+1norm of ( σ,U) also decay exponentially.\nThe proof of the Main Theorem is completed.COMPRESSIBLE EULER WITH DAMPING ON TORUS 11\nReferences\n[1] S. Bianchini, B. Hanouzet, R. 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Pazy, Semigroups of linear operators and applications to partial differential equa-\ntions, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279\npp.\n[19] T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D\ncompressible Euler equations with damping , Comm. Partial Differential Equations 28\n(2003) 795-816.\n[20] Z. Tan, G. Wu, Large time behavior of solutions for compressible Euler equ ations with\ndamping in R3, J. Differential Equations 252 (2012), no. 2, 1546-1561.\n[21] W. Wang, T. Yang, The pointwise estimates of solutions for Euler equations wi th\ndamping in multi-dimensions , J. Differential Equations 173 (2001) 410-450.\n[22] W. Wang, T. Yang, Stability and Lpconvergence of planar diffusion waves for 2-D\nEuler equations with damping , J. Differential Equations 242 (1) (2007) 40-71.12 NAN LU LEHIGH UNIVERSITY\n[23] H.J. Zhao, Convergence to strong nonlinear diffusion waves for solution s of p-system\nwith damping , J. Differential Equations 174 (2001) 200-236.\n[24] C.J. Zhu, Convergence rates to nonlinear diffusion waves for weak entro py solutions\nto p-system with damping , Sci. China Ser. A 46 (2003) 562-575.\n14 East Packer Avenue, Department of Mathematics, Lehigh Un iversity,\nBethlehem, PA, 18015\nE-mail address :nal314@lehigh.edu"    },    {        "title": "1309.5523v4.Patterns_formation_in_axially_symmetric_Landau_Lifshitz_Gilbert_Slonczewski_equations.pdf",        "content": "arXiv:1309.5523v4  [math.AP]  30 Mar 2017Pattern formation in axially symmetric\nLandau-Lifshitz-Gilbert-Slonczewski equations\nOctober 2, 2018\nC. Melcher1& J.D.M. Rademacher2\nAbstract\nThe Landau-Lifshitz-Gilbert-Slonczewski equation describes mag netization dynamics\nin the presence of an applied field and a spin polarized current. In the case of axial sym-\nmetry and with focus on one space dimension, we investigate the eme rgence of space-time\npatterns in the form of wavetrainsand coherent structures, wh ose local wavenumbervaries\nin space. A major part of this study concerns existence and stabilit y of wavetrains and of\nfront- and domain wall-type coherent structures whose profiles a symptote to wavetrains\nor the constant up-/down-magnetizations. For certain polarizat ion the Slonczewski term\ncan be removed which allows for a more complete charaterization, inc luding soliton-type\nsolutions. Decisive for the solution structure is the polarization par ameteras well as size of\nanisotropy compared with the difference of field intensity and curre nt intensity normalized\nby the damping.\n1 Introduction\nThis paper concerns the analysis of spatio-temporal patter n formation for the axially sym-\nmetric Landau-Lifshitz-Gilbert-Slonczewski equation fo r which the applied magnetic field and\ncurrent are aligned with or orthogonal to the material aniso tropy. In one space dimension we\nthus consider\n∂tm=m×/bracketleftBig\nα∂tm−∂2\nxm+(µm3−h)ˆe3+β\n1+ccpm3m׈e3/bracketrightBig\n(1)\nasamodelforthemagnetization dynamics m=m(x,t)∈S2(i.e.misadirectionfield)driven\nby an external field h=hˆe3and current j=β\n1+ccpm3ˆe3with polarization parameter ccp∈\n(−1,1). The parameters α >0 andµ∈Rare the Gilbert damping factor and the anisotropy\nconstant, respectively. A brief overview of thephysical ba ckground and interpretation of terms\nis given below in Section 2.\nThe constant up- or down-magnetization states m=±ˆe3are always steady states of (1)\nand magnetic domain walls spatially separating these state s are of major interest. While\nthe combination of field and current excitations gives rise t o a variety of pattern formation\nphenomena, see e.g. [5, 14, 15, 19, 23], not much mathematica lly rigorous work is available so\nfar, in particular for the dissipative case α >0 that we consider. The case of axial symmetry\nis not only particularly convenient from a technical perspe ctive. It offers at the same time\nvaluable insight in the emergence of space-time patterns an d displays strong similarities to\nbetter studied dynamical systems such as real and complex Gi nzburg-Landau equations. In\n1Lehrstuhl I f¨ ur Mathematik and JARA-FIT, RWTH Aachen Unive rsity, 52056 Aachen, Germany,\nmelcher@rwth-aachen.de\n2Fachbereich 3 – Mathematik, Universit¨ at Bremen, Postfach 33 04 40, 28359 Bremen, Germany,\nrademach@math.uni-bremen.de\n1m1m2m3\n(a) (b)\nFigure 1: Illustration of a wavetrain profile m(ϕ) (a) in the 2-sphere showing their constant\naltitude and (b) as a space-time plot of, e.g., m2. In (a) the thick arrow represents m=\n(m,m3), the thin line its trajectory as a function of ϕ=kx−ωt.\nthis framework we examine the existence and stability of wav etrain solutions of (1), i.e.,\nsolutions of the form\nm(x,t) =ei(kx−ωt)m0,\nwhere the complex exponential acts on m0∈S2by rotation about the ˆe3-axis (cf. Figure 1\nfor an illustration). In the special case ccp= 0 it turns out that (1) can be transformed to the\nvariational LLG-equation with β= 0: in a rotating frame about the m3-axis with frequency\n−β/αthe current dependent term vanishes and hchanges to h−β/α; see§2.1. This allows\nfor a complete characterization of wavetrains and their L2-stability.\nWe also investigate the existence of coherent structure sol utions which are locally in space\nof wavetrain form\nm(x,t) =eiϕ(x,t)m0(x−st) where ϕ(x,t) =φ(x−st)+Ωt (2)\nsuch that m0(ξ) = (sin θ(ξ),0,cosθ(ξ)). Samples are plotted in Figures 2, 3, 4. In the\nvariational case ccp= 0 we completely characterize the existence of small amplit ude coherent\nstructuresandstationary ( s= 0) ones, whichinfactcorrespondtostandingwaves intheab ove\nrotating frame. Throughthecoherent structureviewpoint w e recover a family of ‘homogenous’\ndomain wall type solutions of arbitrary velocity, having no azimuthal profile, i.e., constant φ\nand thus vanishing local wavenumber d ϕ/dx. For general ccpand large speeds, |s| ≫1,\nwe prove existence of a family of more general front-type coh erent structures with nontrivial\nlocal wavenumbers, which can also form a spatial interface b etween±ˆe3and wavetrains. The\nanalysis of these kinds of solutions is inspired by and bears similarities with that of the real\nand complex Ginzburg-Landau equations.\nMore specifically, the parameter space for existence of wave trains and coherent structures\nis largely organized by the stability of the equilibrium sta tesm=±ˆe3. The nature of\nbifurcations that we find motivates the following notions to organize the parameter space of\n(1): We refer to parameters as being\n•‘supercritical’ if ±ˆe3are both unstable\n•‘subcritical’ if ±ˆe3have different stability\n•‘subsubcritical’ if ±ˆe3are both stable.\n20102030405060/Minus0.50.00.5\nxm2\n050100150200/Minus0.8/Minus0.400.4\nxm2\n/Minus0.8/Minus0.4 0 0.4/Minus0.50.00.5\nm1m2\n/Minus0.8/Minus0.6/Minus0.4/Minus0.20.00.2/Minus0.8/Minus0.400.4\nm1m2\n(a) (b)\nFigure 2: Plots of coherent structure profiles for supercrit ical anisotropy with µ= 7,h−Ω =\n−1,Ω =β/α,ccp= 0, and first integral C= 1, cf. (43). (a) A quasi-periodic solution that\nmaps to a solution with period ≈16 in the reduced equations (44). (b) The same solution\ntype with period 200 closer to a soliton-type solution with w avetrain as its asymptotic state.\nFrom a physical viewpoint µandαare material specific, while h,βare control parameters. In\nour exposition we choose µas a primary parameter and speak of super-, sub- or subsubcri tical\nanisotropy; one may also choose βorhat the price of less convenient conditions.\nOur results may be summarized somewhat informally as follow s.\nThe up- and down-magnetization equilibria ±ˆe3.(Lemma 1) Let β±:=β/(1±ccp).\nThe constant state m=ˆe3is strictly stable if and only if µ < h−β+/αandm=−ˆe3\nif and only if µ <−(h−β−/α). Instabilities are of Hopf-type for the essential spectru m\nwith onset via spatially homogenous modes of frequency β/α. In other words, stability of\n±ˆe3changes when the difference of signed anisotropy ±µand the force balance h−β±\nα, of\nmagnetic field strength minus the ratio of current-polariza tion intensity and damping factor,\nchanges sign. This corresponds to the well known instabilit y threshold in the more broadly\nstudied ODE for solutions that are homogeneous in space. Not ably, for ccp= 0 the anisotropy\nis subsubcritical precisely for −µ >|h−β/α|(‘easy-axis’), and supercritical precisely for\nµ >|h−β/α|(‘easy-plane’).\nFast and small amplitude coherent structure. (Theorems 6, 7 and corollaries) For each\nsufficiently large speed there exists a family of front-type c oherent structures parametrized by\ntheazimuthal frequency. Theirprofilesconnect ±ˆe3witheach otheror, if therearewavetrains,\nthere are fronts connecting these and/or ±ˆe3in the order of altitudes. An example is plotted\nin Figure 3. Small amplitude coherent structures are of fron t type and, for ccp= 0, exist only\nfor super- and subcritical anistropy.\n3−1 −0.5 0 0.5 1−101\nm1m3\n−1 −0.5 0 0.5 1−101\nm1m2\n0 100 200 30000.511.522.53\nxθ, q\n(a) (b)\nFigure 3: Profile of a ‘fast’ front connecting the wavetrain a nd the unstable −ˆe3computed\nwith the coherent structure ODE guided by the asymptotic pre diction of equation (35). Here\nµ= 1,h= 0.5,s= 5,Ω = 2,ccp= 0 and the asymptotic wavetrain on the left has wavenumber\nk= Ω/s= 0.4, and is spectrally stable.\nWavetrains. In the case ccp= 0 (Theorems 2, 4) for each wavenumber k∈Rat most one\nwavetrain exists, and moreover:\n1.Supercritical anisotropy: Wavetrains exist precisely for kwith|k|>/radicalbig\nµ+|h−β/α|or\n0≤ |k|</radicalbig\nµ−|h−β/α|. There is (explicitly known) k∗∈(0,/radicalbig\nµ−|h−β/α|such\nthat all wavetrains with |k|< k∗are stable and sideband unstable for |k|> k∗.\n2.Subcritical anisotropy: Wavetrain exists precisely for kwith|k|>/radicalbig\nµ+|h−β/α|, but\nare all unstable.\n3.Subsubcritical anisotropy: Wavetrains exist for all k, but are all unstable.\nThe overall picture for wavetrains of (1) with ccp= 0 can be viewed as a combination of\nthose in a supercritical and a subcritical real Ginzburg-La ndau equation; see Figure 11.\nFor general ccp∈(−1,1) additional effects are (1) a nontrivial nonlinear dispersi onrelation\nω(k) with nonzero group velocitiesd\ndkω(k), (2) the occurrence of ‘hyperbolic’ and ‘elliptic’\nbifurcation points of wavetrains and (3) coexistence of sta ble wavetrains and stable ±ˆe3.\nWavetrains for k2> µare always unstable (Theorems 2, 3), but for ccp/ne}ationslash= 0 wavetrains are\npotentially convectively but not absolutely unstable, tho ugh we do not investigate this here.\nDomain walls for ccp= 0.(Theorem 5) For any µ <0 there exists a family of fronts whose\nspatial profiles connect ±ˆe3withθ′=√−µsin(θ), and that are ‘homogeneous’ in the sense\nthatq≡0 so there is no azimuthal profile. They corresponds to well kn own domains walls of\nthe LLG-equation3. Here we readily locate these within the coherent structure framework.\nStationary coherent structures for ccp= 0.(Theorems 8, 9)\n1.Supercritical anisotropy: For fixed parameters there exist various stationary coheren t\nstructures ( s= 0) including ‘homogeneous’ ones (cf. Figure 4). An interes ting case of\n3After acceptance of the present manuscript for publication , we found these were also obtained in [12].\n4−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm3\n−15−10−5051015−1−0.500.51\nxm3\n−15−10−5051015−1−0.500.51\nxm3\n(a) (b) (c)\nFigure 4: Snapshots of sample homogeneous coherent structu res with spatially periodic pro-\nfiles, having d φ/dξ=q= 0. Compare Figure 12. Here ccp= 0,α= 1 and Ω = βis arbitrary.\n(a) Near a pair of domains walls ( µ=−1,h= 10−4−β). (b) Near an upward ‘phase slip\nsoliton’ with plateaus near the oscillation at m3=h−β(µ= 1,h= 0.8−β), and (c) near an\nupward-downward pair of such solitons ( µ= 1,h= 0.8−β).\nthe latter is a symmetric pair of ‘phase slip’ soliton-type c oherent structures, whose\nspatially asymptotic states are the same spatially homogen eous oscillation ( k= 0), but\nthe intermediate profile crosses either ˆe3or−ˆe3, so that the asymptotic states differ\nazimuthally by 180◦. There also exists a non-homogeneous soliton-type solutio n with\nasymptotic state being a wavetrain (cf. Figure 2).\n2.Sub- and subsubcritical anisotropy: All stationary coherent structures have periodic\nprofiles except a homogeneous phase slip soliton with spatia lly asymptotic state ±ˆe3for\nsgn(h−β/α) =±1.\nHigher space dimensions. The model for Nspace dimensions has the second derivative\nwith respect to xin (1) replaced by a Laplace operator/summationtextN\nj=1∂2\nxj. Wavetrain type solutions\nare then of the form\nm(x,t) =m∗(k·x−ωt),\nwherek= (k1,...,kN). Notably, for kj= 0, 2≤j≤Nthese are solutions from one space\ndimension extended trivially (constant) in the additional directions.\nConveniently, the rotation symmetry (gauge invariance in t he Ginzburg-Landau context),\nmeans that the analyses of ±ˆe3and these wavetrains is already covered by that of the one-\ndimensional case: the linearization is space-independent and therefore there is no symmetry\nbreaking due to different kj. Indeed, all relevant quantities are rotation symmetric, d epending\nonly onk2=/summationtextN\nj=1k2\njorℓ2=/summationtextN\nj=1ℓ2\nj, whereℓ= (ℓ1,...,ℓN) is the Fourier wavenumber vec-\ntor of the linearization. In particular, the instabilities occur simultaneously for all directions.\n5Concerning coherent structures, in higher space dimension the defining equation (see (33)\nbelow) turns into an elliptic PDE in general. The analysis in this paper only covers the trivial\nconstant extension into higher dimensions.\nThis paper is organized as follows. In Section 2, the terms in the model equation (1) and\nits well-posedness are discussed. Section 3 concerns the st ability of the trivial steady states\n±ˆe3and in§4 existence and stability of wavetrains are analyzed. Secti on 5 is devoted to\ncoherent structures.\nAcknowledgement. JR has been supported in part by the NDNS+ cluster of the Dutch\nScience Fund (NWO). We thank the anonymous reviewers for sug gestions that helped improve\nthe manuscript, and Lars Siemer as well as Ivan Ovsyannikov f or their critical reading.\n2 Review of Landau-Lifshitz-Gilbert-Slonczewski equatio ns\nThe classical equation of dissipative magnetization dynam ics, the Landau-Lifshitz-Gilbert\nequation [11, 22] for unit vector fields m=m(x,t)∈S2,\n∂tm=m×(α∂tm−γheff).\nfeatures a damped precession of maround the effective field heff=−δE(m), i.e., minus the\nvariational derivative of the interaction energy E=E(m). The gyromagnetic ratio γ >0 is\na parameter which appears as the typical precession frequen cy. By rescaling time, one can\nalways assume γ= 1. The Gilbert damping factor α >0 is a constant that can be interpreted\ndynamically as the inverse of the typical relaxation time. I t is useful to take into account that\nthere are several equivalent forms of LLG. Elementary algeb raic manipulations taking into\naccount that −m×m×ξ=ξ−(m·ξ)myield the so-called Landau-Lifshitz form\n(1+α2)∂tm=−m×(αm×heff+heff), (3)\nintroduced in the original work [22]. In case α >0, the energy E(m) is not conserved but is\na Lyapunov functional, i.e., more precisely (recall heff=−δE(m))\nd\ndtE(m(t)) =−α/bardbl∂tm(t)/bardbl2or equivalentlyd\ndtE(m(t)) =−α\n1+α2/bardblm×heff/bardbl2.\nGilbert damping enables the magnetization to approach (spi ral down to) a steady state, i.e.\nsatisfying m×heff= 0 (Browns equation), as t→ ∞.\nSpin-torque interaction. The system can be driven out of equilibrium conventionally b y\nan external magnetic field hwhich appears as part of the effective field. In modern spin-\ntronic applications, magnetic systems are excited by spin p olarized currents (with direction of\npolarization ˆep∈S2) giving rise to a spin torque\nm×m×jwherej=βˆep\n1+ccpm·ˆep, (4)\nwhich has been introduced in [2, 34]. Here, the parameters β >0 andccp∈(−1,1) depend\non the intensity of the current and ratio of polarization [4] . Typically we have ˆep=ˆe3. Ac-\ncordingly, the modified Landau-Lifshitz-Gilbert equation , also called Landau-Lifshitz-Gilbert-\nSlonczewski equation (LLGS), reads\n∂tm=m×(α∂tm−heff+m×j). (5)\n6One may extend the notion of effective field to include current i nteraction by letting\nHeff=heff−m×j,\nwhere the second term is usually called Slonczewski term. In this framework (5) can also be\nwritten in the form (3) with heffreplaced by Heff. Observe, however, that the Slonczewski\nterm(andhence Heff)isingeneralnon-variational andthattheenergyisnolong eraLyapunov\nfunctional. Introducing the potential Ψ( m) =β\nccpln(1+ccpm·ˆep) ofj(forccp/ne}ationslash= 0) reveals\ntheskew variational structure\nm×[α∂tm+δE(m)] =−m×m×[∂tm+δΨ(m)],\nsee[6]. Inthemicromagnetic model theunderlyinginteract ion energies areintegral functionals\ninmcontaining in particular exchange (Dirichlet) interactio n, dipolar stray-field interaction,\ncrystal anisotropy and Zeeman interaction with external ma gnetic field, see e.g. [16]. In this\npaper we shall mainly focus on the spatially one-dimensiona l situation and consider energies\nof the form\nE(m) =1\n2/integraldisplay/parenleftbig\n|∂xm|2+µm2\n3/parenrightbig\ndx−/integraldisplay\nh·mdx. (6)\nHere,h∈R3is a constant applied magnetic field. The parameter µ∈Rfeatures easy plane\nanisotropy for µ >0andeasy axis anisotropyfor µ <0, respectively, accordingtoenergetically\npreferred subspaces. This term comprises crystalline and s hape anisotropy effects. Shape\nanisotropy typically arises from stray-field interactions which prefer magnetizations tangential\nto the sample boundaries. Hence µ >0 corresponds to a thin-film perpendicular to the ˆe3-\naxis whereas µ <0 corresponds to a thin wire parallel to the ˆe3-axis. The effective field\ncorresponding to (6) reads\nheff=∂2\nxm−µm3ˆe3+h. (7)\nWith the choices h=hˆe3and ˆep=ˆe3, the Landau-Lifshitz-Gilbert-Slonczewski equation\n(5) exhibits the aforemented rotation symmetry about the ˆe3-axis. The presence of a spin\ntorquem×m×jexerted by a constant current may induce switching between m agnetization\nstates or magnetization oscillation [3, 4, 7]. For the latte r effect, the energy supply due to\nthe electric current compensates the energy dissipation du e to damping enabling a stable\noscillation, called precessional states . In applications the typical frequency is in the range of\nGHz, so that a precessional state would basically act as a mic rowave generator. In the class\nof spatially homogeneous states, precessional states are p eriodic orbits with m3=const.and\nof constant angular velocity β/αwhenccp= 0. It is more subtle, however, to understand the\noccurrence and stability of spatially non-homogeneous pre cessional states. This is the theme\nof this paper.\nExtensions and related work. There is a wealth of literature studying the dynamics of\nrelated Landau-Lifschitz models with and without damping a nd axial symmetry and including\neffects other than spin-torque interaction and as general ref erence we mention the book [6] as\nwell as the review article [21]. More specifically, spatiall y non-trivial states and their stabililty\nhave been considered in [18], where the spin-torque part of t he effective field is replaced by\na demagnetization term solving Maxwell’s equation. Also co upled nano-oscillators of LLGS\ntype have been considered widely, e.g. recently in [33, 35]. Recently, for a situation without\naxial symmetry, Turing patterns of spin states have been num erically found in [23].\n7Non-symmetric variants of our equation (1) have been used e. g. in the description of\nthe field driven motion of a flat domain wall connecting antipo dal steady states m3=±1\nasx1→ ±∞. A prototypical situation is the field driven motion of a flat B loch wall in an\nuniaxial the bulk magnet governed by\n∂tm=m×/parenleftbig\nα∂tm−∂2\nxm+µ1m1ˆe1+(µ3m3−h)ˆe3/parenrightbig\n. (8)\nIn this case µ1>0> µ3, whereµ1corresponds to stray-field and µ3to crystalline anisotropy.\nExplicit traveling wave solutions were obtained in unpubli shed work by Walker, see e.g. [16],\nand reveal interesting effects such as the existence of a termi nal velocity (called Walker ve-\nlocity) and the notion of an effective wall mass. A mathematica l account on Walker’s explicit\nsolutions and investigations on their stability, possible extensions to finite layers and curved\nwalls can be found e.g. in [9, 25, 30]. Observe that our axiall y symmetric model is obtained\nin the limit µ1ց0. On the other hand, the singular limit µ3→+∞leads to trajectories\nconfined to the {m3= 0}plane (equator map), and can be interpreted as a thin-film lim it.\nIn suitable parameter regimes it can be shown that the limit e quation is a dissipative wave\nequation governing the motion of N´ eel walls [8, 24, 26].\nWell-posedness of LLGS. It is well-known that Landau-Lifshitz-Gilbert equations a nd its\nvariants have the structure of quasilinear parabolic syste ms. In the specific case of (1), one\nhas the extended effective field Heff=heff−m×j, more precisely\nHeff=∂2\nxm−f(m) where f(m) = (µm3−h)ˆe3+β\n1+ccpm3m׈e3.(9)\nHence the corresponding Landau-Lifshitz form (3) of (1) rea ds\n(1+α2)∂tm=−m×/bracketleftBig\n∂2\nxm−f(m)/bracketrightBig\n−αm×m×/bracketleftBig\n∂2\nxm−f(m)/bracketrightBig\n. (10)\nTaking into account\nm×∂2\nxm=∂x(m×∂xm) and −m×m×∂2\nxm=∂2\nxm+|∂xm|2m,(11)\nvalid for msufficiently smooth and |m|= 1, one sees that (10) has the form\n∂tm=∂x(A(m)∂xm)+B(m,∂xm) (12)\nwith analytic functions A:R3→R3×3andB:R3×R3→R3such that A(m) is uniformly\nelliptic for α >0, in fact\nξ·A(m)ξ=α\n1+α2|ξ|2for allξ∈R3.\nWell-posedness results for α >0 can now be deduced from techniques based on higher\norder energy estimates as in [27, 28] or maximal regularity a nd interpolation as in [29]. In\nparticular, weshallrelyonresultsconcerningperturbati onsofwavetrains, travelingwaves, and\nsteady states. Suppose m∗=m∗(x,t) is a smooth solution of (1) with bounded derivatives\nup to all high orders (only sufficiently many are needed) and m0:R→S2is such that\nm0−m∗(·,0)∈H2(R). Then there exist T >0 and a smooth solution m:R×(0,T)→S2\nof (1) such that m−m∗∈C0([0,T);H2(R))∩C1([0,T);L2(R)) with\nlim\ntց0/bardblm(t)−m0/bardblH2= 0 and lim\ntրT/bardblm(t)−m∗(·,t)/bardblH2=∞ifT <∞.\n8The solution is unique in its class and the flow map depends smo othly on initial conditions\nand parameters.\nGiven the smoothness of solutions, we may compute pointwise ∂t|m|2= 2m·∂tm, so that\nfor|m|= 1 the cross product form of the right hand side of (10) gives ∂t|m|2= 0. Hence, the\nset of unit vector fields, {|m|= 1}, is an invariant manifold of (12) consisting of the solution s\nto (1) that we are interested in.\nIn addition to well-posedness, also stability and spectral theory for (12), see, e.g., [29],\ncarry over to (1). In particular, the computations of L2-spectra in the following sections are\njustified and yield nonlinear stability for strictly stable spectrum and nonlinear instability for\nthe unstable (essential) spectrum.\nLandau-Lifshitz-Gilbert-Slonczewski versus complex Ginzburg-Landau equations.\nStereographic projection of (1) yields\n(α+i)ζt=∂2\nxζ−2¯ζ(∂xζ)2\n1+|ζ|2+µ(1−|ζ|2)ζ\n1+|ζ|2−(h+iβ)ζwhereζ=m1+im2\n1+m3,\nvalid for magnetizations avoiding the south pole.\nStudying LLG-type equations via stereographic projection has a long history and has\nbeen employed in several of the aforementioned references, see, e.g., the review [21] and the\nreferences therein.mThere is also a global connection betw een LLG and CGL in the spirit\nof the classical Hasimoto transformation [13], which turns the (undamped) Landau-Lifshitz\nequation in one space dimension ( heff=∂2\nx) into the focussing cubic Schr¨ odinger equation\n[20, 36]. The idea is to disregard the customary coordinates representation and to introduce\ninstead a pull-back frame on the tangent bundle along m. In the case of µ=β=h= 0, i.e.\nheff=∂xm, this leads to\n(α+i)Dtu=D2\nxu (13)\nwhereu=u(x,t) is the complex coordinate of ∂xmin the moving frame representation,\nandDxandDtare covariant derivatives in space and time giving rise to cu bic and quintic\nnonlinearities, see [27, 28] for details.\n2.1 Symmetry and the variational structure for ccp= 0\nThe aforementioned rotation symmetry of (1) about the m3-axis of all terms manifests as\nan equivariance of the right hand side of (12) with respect to any such rotation Rϕ: let\nm=Rϕ/tildewiderm, then\n∂x(A(m)∂xm)+B(m,∂xm) =Rϕ(∂x(A(/tildewiderm)∂x/tildewiderm)+B(/tildewiderm,∂x/tildewiderm)).\nFor a rotating frame m=eiΩtm, write the rotation about the ˆe3-axis asRΩtand note\nthat time derivatives become ∂tm=RΩt(ΩR∗\nΩtR′\nΩt/tildewiderm+∂t/tildewiderm), where R∗\nΩtR′\nΩtm=m׈e3.\nTherefore, having ccp= 0, (1) is also an equation for /tildewidermwith the parameter βchanged to\nβ+αΩ(from∂tmwithin the brackets) and htoh+Ω(from∂tmon the left hand side).\nIn other words, for ccp= 0, changing spin torque current has the same effect as changin g the\nmagnetic field.\n9Choosing Ω=−β/αyieldsβ= 0 in (1), which is therefore variational with respect to\nthe energy (6) as discussed above. This has strong structura l consequences for the coherent\nstructures (2) and allows for a (largely) complete characte rization. In particular, it turns\nout that the existence of coherent structures that are stati onary (s= 0), but not necessarily\ntime independent, requires their superimposed azimuthal f requency Ω to satisfy Ω = β/α; see\n§5.3. The reduction to β= 0 thus implies Ω = 0 and therefore turns the stationary coher ent\nstructures into standing waves and hence to time-independe nt solutions.\n3 Hopf instabilities of the steady states m=±ˆe3\nAs a starting point and to motivate the subsequent analysis o f more complex patterns, let\nus consider the stability of the constant magnetizations ±ˆe3. It is well-known that a Hopf\nbifurcation of these states occurs in the ODE associated to ( 1) in the absence of diffusion,\nthat is, for spatially constant solutions. In the following , we account in addition for spatial\ndependence.\nUse the shorthand β±=β/(1±ccp). Substituting m=±ˆe3+δn+o(δ), where n=\n(n,0)∈TmS2, into (1) gives, at order δ, the linear equation\n∂tn= (±µ−h)n׈e3±ˆe3×(α∂tn−∂2\nxn+β±n׈e3),\nwhich may be written in complex form as\n∂tn±β±n=i/parenleftbig\nα∂tn−∂2\nxn−(h∓µ)n/parenrightbig\n.\nIts eigenvalue problem diagonalizes in Fourier space (for x) and yields the matrix eigenvalue\nproblem/vextendsingle/vextendsingle/vextendsingle/vextendsingle±β±−λΛ\n−Λ±β±−λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,\nwhere Λ = ±µ−h∓αλ∓ℓ2withℓthe Fourier wave number. The determinant reads\n(±β±−λ)2=−Λ2⇔ ±β±−λ=σiΛ, σ∈ {±1}.\nConsidering real and imaginary parts this leads to\n(1+α2)Re(λ) =±β±−α(ℓ2±h−µ) =α(µ∓(h−β±/α)−ℓ2)\nIm(λ) =σ(∓Re(λ)+β±/α),\nso that the maximal real part has ℓ= 0. At criticality, where Re( λ) = 0 the imaginary\nparts are ±β±/α, which (if nonzero) corresponds to a so-called Hopf-instab ility of the (purely\nessential) spectrum and we expect the emergence of oscillat ing solutions whose frequency at\nonset isβ±/α[31]. Since the critical mode has ℓ= 0, the onset of instability coincides with\nthe aforementioned Hopf-bifurcation of the ODE associated with diffusionless (1).\nThe formulas for real- and imaginary parts immediately give the results mentioned in §1\nand\nLemma 1 The constant state m=ˆe3is (strictly) L2-stable if and only if µ < µ+:=h−β+/α\nandm=−ˆe3if and only if µ < µ−:=−(h−β−/α). Instabilities are of Hopf-type for the\nessential spectrum and have frequency β/α.\nForccp= 0the anisotropy is subsubcritical precisely for −µ >|h−β/α|, and supercritical\nprecisely for µ >|h−β/α|.\n104 Wavetrains\nTo exploit the rotation symmetry about the ˆe3-axis, we change to polar coordinates in the\nplanar components m= (m1,m2) of the magnetization m= (m,m3). With m=rexp(iϕ)\nequation (1) changes to\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggr2∂tϕ\n∂tm3/parenrightbigg\n=/parenleftbigg∂x(r2∂xϕ)\n∂2\nxm3+|∂xm|2m3/parenrightbigg\n+r2/parenleftbiggβ/(1+ccpm3)\nh−µm3/parenrightbigg\n,(14)\nwhere\n|∂xm|2= (∂xr)2+r2(∂xϕ)2+(∂xm3)2andr2+m2\n3= 1.\nThis can be seen as follows. In view of (3), with heffreplaced by the extended effective field\nHeff=heff−m×jas in (9) and taking into account (11), (1) reads\nα∂tm+m×∂tm=∂2\nxm+(h−µm3)ˆe3+/parenleftbig\n|∂xm|2+µm2\n3−hm3/parenrightbig\nm−β\n1+ccpm3m׈e3.\nThe third component of the above equation is the second compo nent of (14), whereas the first\ncomponent of (14) is obtained upon inner multiplication by m⊥= (m⊥,0) = (ieiϕ,0) and\ntaking into account that m׈e3=−m⊥.\nTherotation symmetryhasturnedintotheshiftsymmetry ϕ/ma√sto→ϕ+const. Infullspherical\ncoordinates m=/parenleftbigeiϕsinθ\ncosθ/parenrightbig\n, (14) further simplifies to\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggsinθ∂tϕ\n−∂tθ/parenrightbigg\n=/parenleftbigg2cosθ∂xθ∂xϕ+sinθ∂2\nxϕ\n−∂2\nxθ+sinθcosθ(∂xϕ)2/parenrightbigg\n+sinθ/parenleftbiggβ/(1+ccpcos(θ))\nh−µcosθ/parenrightbigg\n.(15)\n4.1 Existence of wavetrains\nWavetrains are solutions of the form m(x,t) =m∗(kx−ωt), where kis referred to as the\nwavenumber and ωas the frequency. A natural type of wavetrains are relative e quilibria with\nrespect tothephaseshiftsymmetry forwhich ϕ=kx−ωtandm3,rareconstant. SeeFigure1\nfor an illustration.\nTheorem 1 Wavetrains with frequency ωand wavenumber kare in one-to-one correspon-\ndence to solutions of\nΓ(ω,k) :=ccpαω(ω+h)−(β+αω)(k2−µ) = 0,\nunder the constraint |(ω+h)/(k2−µ)| ≤1. In particular, for each kthere are at most two\nvalues of ωthat yield a wavetrain, and for each ω/ne}ationslash=−β/αthere is at most one value of k2\nthat gives a wavetrain, unless ccpαω(ω+h) = 0forω=−β/α. Moreover, for |ccp|<1,\n(a)ω/ne}ationslash= 0,k2/ne}ationslash=µ,sgn(ω) =−sgn(β)andω∈[min{−β±/α},max{−β±/α}].\n(b) As|k| → ∞we have ω→ −β/αandm3→0.\n(c) Bifurcations of k∼0fromk= 0for fixed ωare unfolded for increasing µ.\n11/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.55\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.57\nFigure 5: Plots of wavetrain locations in the ( k,ω)-plane when parameters pass through\nan elliptic bifurcation point. Labels are the values of µ. Other parameters are fixed at\nccp= 0.5,h= 2,α= 1,β= 2.1sothat ω±=−β±,β+= 1.4,β−= 4.2andµ+= 0.6,µ−= 2.2.\nTherefore, µ= 1.55,1.57 are both subcritical with ˆe3stable and −ˆe3unstable. Shaded regions\nhave|m3|>1. The elliptic point lies at µsn≈1.558,ωsn=−2.558.\n(d) Bifurcations for m/ne}ationslash=±ˆe3occur atω=ωsn:=β±√\nβ(β−4αh)\n2αwith|m3|<1if|1\nccp−h\nµ|<2\nand are either:\n... a hyperbolic point for k/ne}ationslash= 0, ifh=β/αand then ω=−h,k2=−ccph+µ,\n... a hyperbolic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) = 1,\n... an elliptic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) =−1.\nIn the following we discuss the existence problem and thereb y prove each statement of the\ntheorem.\nThe terms elliptic and hyperbolic refers to the use in [32] an d will be explained below.\nNote that the latter two bifurcation types do not occur for ccp= 0. That case is considered\nin detail in §4.2. We note that Γ depends on konly through k2−µso thatµ≥0 is the same\nasµ= 0 up to change in wavenumber and solutions for fixed ωcan increase |k|only through\nincreasing µ.\nSubstituting the wavetrain ansatz into (15) yields the alge braic equations\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbigg−sin(θ)ω\n0/parenrightbigg\n=/parenleftbigg0\nsin(θ)cos(θ)k2/parenrightbigg\n+sin(θ)/parenleftbiggβ/(1+ccpcos(θ))\nh−µcos(θ)/parenrightbigg\n.\nThus either θ≡0 modπor (recall |ccp|<1)\n−αω=β\n1+ccpm3,−ω= (k2−µ)m3+h. (16)\nIn the first case we have r= 0, which corresponds to the constant upward or downward\nmagnetizations, ( r,m3) = (0,±1) with unspecified kandω. In the second case, we notice\naside that absence of dissipation ( α= 0) requires absence of current ( β= 0) and there is a\ntwo-parameter set of wavetrains given by the second equatio n. The case we are interested in\nisα >0 and then β= 0 requires ω= 0, and this falls into the special case ccp= 0.\n12/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.55\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.64\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.65\nFigure 6: Analogue of Figure 5 with fixed parameters as there, whenµpasses through a\nhyperbolic bifurcation point at k= 0 with µsn≈0.641,ωsn=−1.641. Note that between\nµ= 0.55 andµ= 0.64 the upper branch enters the region |m3| ≤1 at a bifurcation of in this\ncaseˆe3atµ=µ+= 0.6. Therefore, µ= 0.64,0.65 are supercritical with ±ˆe3both unstable.\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ1.99\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ2\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ2.001\nFigure 7: Analogue of Figure 5 when parameters pass through h yperbolic bifurcation points\nwithk/ne}ationslash= 0 so that h=β/α. Notably this cannot be unfolded by variation of µ. Labels are\nthe values of βand other parameters are fixed at ccp= 0.5,h= 2,α= 1,µ= 1.2. This is\nsubcritical with ˆe3unstable since µ+= 2(1−β/3),µ−= 2(β−1) and 1.2∈(µ+,µ−).\nIn the generic case β,ccp/ne}ationslash= 0, the first equation implies that ω≈0 is not possible for\n|m3| ≤1 and we obtain\nm3=−1\nccp/parenleftbiggβ\nαω+1/parenrightbigg\n, m 3=−ω+h\nk2−µ(17)\nwhere for k2=µwe have ω=−hand the first equation holds.\nEliminating m3and rearranging terms gives the existence condition in term s ofωandk\nas zeros of Γ( ω,k) as in the theorem. For ω/ne}ationslash=−β/αthis gives k2as a quadratic function\nofωinverse to the nonlinear dispersion relation ω(k). The exceptional ω=−β/αoccurs\nprecisely when h=β/αand implies m3= 0, i.e. a solution on the equator (or ccp= 0).\nThe strict monotonicity of Γ in kaway from k= 0 also means that upon parameter change\nnew solution branches can emerge only through local extrema of Γ atk= 0, i.e., an ‘elliptic’\npoint. Specifically, this occurs if at a critical point ∂2\nkΓ(ω,0) = 2(β+αω) has the same sign\n13as∂2\nωΓ(ω,0) = 2αccpand, e.g. Γ(0 ,0) =βµvaries.\nIn particular, Γ( ω,k) =∂ωΓ(ω,k) = 0 occurs at\nαω2+β(2ω+h) = 0 (18)\nand is a fold point of wavetrains (in the form of homogeneous o scillations) with fixed k. For\nk= 0 these have at frequency and parameters (recall ccp/ne}ationslash= 0)\nωsn=−ccph+µ\n2ccp,4βµccp=α(µ+ccph)2. (19)\nNote that |m3|<1 fork= 0 is by (17) equivalent to |ωsn+h|<|µ|which yields |1\nccp−h\nµ|<2.\nAt such critical point we also have\n∂2\nkΓ(ω,0) =−α\nccp(ccph−µ)(ccph+µ), (20)\nso that the relative size of ccphandµdetermines whether such a bifurcation point is elliptic\nor hyperbolic in the language of [31]. In terms of critical pa rameters, substituting ω=ωsn+˜ω\nand, for instance µ=µsn+ ˜µto unfold with µand other parameters fixed we obtain\n˜µ=k2−ccpα\nβ+αωsn˜ω2(21)\nwhich gives the options of hyperbola or ellipses for level se ts. Hyperbolic points are saddle\npoints of Γ and at such points the connectivity of existing br anches changes. For k/ne}ationslash= 0 this\noccurs in particular, if h=β/αwhen the two branches of Γ = 0 are the line ω=−hfor any\nkandω=1\nccp(k2−µ).\nWe plot examples of these situations in Figures 5, 6, 7.\nMore globally, since Γ is quadratic in ωthere are at most two solutions for each kand by\nstrict monotonicity in k2, away from h=β/α, there is at most one solution for each ωor the\nwhole line ω=−β/α. The only complication is the constraint |m3| ≤1 – dispersion curves\ntouch the boundary m3= 1 at bifurcations of ±ˆe3, which were studied in §3. The figures\nillustrate the essential scenarios.\n4.2 Existence in the case ccp= 0\nIn this case the existence conditions can be conveniently wr itten as\nω=−β\nα(22)\ncos(θ) =h−β/α\nµ−k2,(µ/ne}ationslash=k2). (23)\nAs expected from the variational structure in rotating coor dinates discussed in §2.1, all wave-\ntrains oscillate with frequency given by the ratio of applie d current and dissipation. In par-\nticular, in this case the natural representation of wavetra ins is that ( θ,k)-plane rather than\n(ω,k) as above.\nAn involution symmetry involving parameters is\n(h−β/α,θ)→(β/α−h,θ+π), (24)\n14/Minus4/Minus20240Π2Π\nkΘ\n/Minus4/Minus20240Π2Π\nkΘ\n/Minus4/Minus20240Π2Π\nkΘ\n(a) (b) (c)\nFigure 8: Plots of equilibrium locations in the ( k,m3)-plane including the trivial equilibria\n±ˆe3plotted with thick line if stable (when not intersecting wav etrain parameters). Compare\nFigure 9. See (23). (a) supercritical (easy plane) anisotro py (µ= 1,h−β/α= 1/2), (b)\nsubcritical anisotropy ( µ= 1,h−β/α= 2), where no homogeneous oscillations ( k= 0) exist,\nand (c) subsubcritical (easy axis) anisotropy µ=−1,h−β/α= 0.9.\nsupercritical\nsubcritical\nsubsubcritical−|h−β/α||h−β/α|\nσˆe3stable±ˆe3unstable\n±ˆe3stableµ\nk\nFigure 9: Sketch of existence region (shaded) in the ( k,µ)-plane, with boundary given by (25)\nforσ:= sgn(h−β/α)/ne}ationslash= 0. The sign of σdetermines which of ±ˆe3is stable in the subcritical\nrange.\nso that the sign of h−β/αis irrelevant for the qualitative picture.\nSolvability of (23) requires that |µ−k2|>|h−β/α|(unlessr= 0), so that only for super-\nand subsubcritical anisotropy,\n|µ|>|h−β/α|, (25)\nthere exist wavetrains with wavenumber in an interval aroun dk= 0. In other words, non-\ntrivial spatially nearly homogeneous oscillations requir e sufficiently small (in absolute value)\ndifferencebetweenappliedmagneticfieldandoscillationfre quency(ratioofappliedcurrentand\ndissipation). Thetransitionintothisregimegoesviathe‘ Hopf’instability from §3. Combining\n(25) with Lemma 1 and straightforward analysis of (23) gives the following lemma. The three\ntypes of solution sets are plotted in Figure 8. Clearly, the s olution sets are symmetric with\nrespect to the signs of kandθ, respectively.\nTheorem 2 There are three types of wavetrain parameter sets solving (23):\n151. For supercritical anisotropy there is one connected comp onent of wavetrain parameters\nincluding k= 0, and two connected components with unbounded |k|, each with constant\nsign ofk.\n2. For subcritical anisotropy there are two connected compo nents with unbounded |k|, each\nwith constant sign of k.\n3. For subsubcritical anisotropy there are two connected co mponents, each a graph over the\nk-axis.\nThe Hopf-type instabilities of ±ˆe3noted in §3 at the transition from sub- to supercritical\nanisotropy is a supercritical bifurcation in the sense that solutions emerge at the loss of stability\nof the basic solution, here ±ˆe3, while that from sub- to subsubcritical is subcritical in th e sense\nthat solutions emerge at the gain of stability.\nProof. Let us consider the existence region of wavetrains in wavenu mber-parameter space.\nFrom (23), r= 0 atθ≡0 modπgives the boundary for nontrivial amplitude,\nµ=k2±(h−β/α), (26)\nas a pair quadratic parabolas in ( k,µ)-space. The solution set in this projection is sketched in\nFigure9. Remark that this set is non-empty for any parameter setα,β,h∈Rof (1). However,\nas in the general case not all wavenumbers are possible due to the geometric constraint.\nNotably, the existence region consists of two disjoint sets , one contained in {µ >0}with\nconvex boundary and one extending into {µ <0}with concave boundary.\n4.3 Stability of wavetrains\nIn this section we discuss spectral stability of wavetrains and in summary we obtain the\nfollowing result.\nTheorem 3\n(a) Wavetrains bifurcating from ±ˆe3atk= 0are stable if µ±>max{0,ccpβ±/α}, which\nimplies supercritical bifurcation, i.e., for increasing µ. They are unstable if µ±<\nmax{0,ccpβ±/α}, which means subcritical bifurcation, i.e., decreasing µ.\n(b) Forµ >0there isk∗>0such that precisely the wavetrains with wavenumber |k|< k∗\nandµ > k2+αω2ccp/βare sideband stable. These are fully spectrally stable if βccp≥0\norα≥0is sufficiently small. A wavetrain and ˆe3or−ˆe3can be simultaneously stable\nonly for ccp/ne}ationslash= 0.\n(c) For each kat most one wavetrain can be stable. Wavetrains with r∼0andk/ne}ationslash= 0are\nunstable. All wavetrains with k2> µare unstable.\n(d) Near the hyperbolic or elliptic bifurcation points at k= 0the sideband stable wavetrains\nlie in a sector that is to leading order bounded by |ω−ωsn|=S|k|whose opening angle\nless than πand which includes ωwith a selected sign of ω−ωsn.\n16/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.57\n/Minus1.0/Minus0.50.00.51.0/Minus2.1/Minus2.0/Minus1.9/Minus1.8/Minus1.7/Minus1.6/Minus1.5/Minus1.4\nkΩ0.64\n/Minus1.0/Minus0.50.00.51.0/Minus2.4/Minus2.2/Minus2.0/Minus1.8/Minus1.6\nkΩ1.99\n/Minus1.0/Minus0.50.00.51.0/Minus2.10/Minus2.05/Minus2.00/Minus1.95/Minus1.90\nkΩ2.001\nFigure 10: Dispersion curves and stability regions. Wavetr ains on the dispersion curves in the\ngreen shaded region are unstable due to unstable eigenvalue ofA(0,0) using (29), and in the\nred shaded region due to unstable sideband only, using ˜λ′′\n0(0) with k2from Γ = 0. Parameters\nare as in (a): Figure 5; (b): Figure 6; (c),(d): Figure 7. Only the wavetrains near k= 0 are\nstable (except for the lower branch in (b)).\nRemark 1 Item (a) should be compared with the Hopf instabilities discu ssed in§3. The case\nccp= 0is simplest: the constraint simply means that wavetrains bi furcating at the subsub- to\nsubcritical transition are unstable. For general ccpthe condition also accounts for interaction\nwith folds.\nConcerning item (b), notably in the easy axis case µ <0all wavetrains are unstable. The\ncondition ccpβ≥0is not sharp. However, we do not know whether a ‘Hopf’ instabi lity can\noccur for otherwise stable wavetrains if ccpβ <0is sufficiently large negative. Item (d) is an\nanalog to the results in [32].\nIn order to study spectral stability of wavetrains we consid er the comoving frame y=\nx−cphtwithcph=ω/kthe wavespeed so that the wavetrain is an equilibrium of (14) . For\nconvenience, time is rescaled to t= (1+α2)˜t. The explicit formulation of (14) then reads\n∂˜t/parenleftbiggϕ\nm3/parenrightbigg\n=/parenleftbiggα(∂y(r2∂yϕ)/r2+˜β(m3))+(∂2\nym3+|∂ym|2m3)/r2+h−µm3+cph∂yϕ\nα(∂2\nym3+|∂ym|2m3+r2(h−µm3))−∂y(r2∂yϕ)−r2˜β(m3)+cph∂ym3/parenrightbigg\n,\n(27)\nwherer2= 1−m2\n3and˜β(m3) :=β/(1+ccpm3).\nLetF= (F1,F2)tdenote the right hand side of (27). Wavetrains have constant randm3\nso that quadratic terms in their derivatives can be discarde d for the linearization LofFin a\nwavetrain, and from |∂ym|2onlyr2(∂yϕ)2is relevant. The components of Lare\n∂ϕF1=α∂2\ny+cph∂y+2km3∂y\n∂m3F1=r−2∂2\ny+k2−µ−2αkm3r−2∂y+α˜β′(m3)\n∂ϕF2= 2αkm3r2∂y−r2∂2\ny\n∂m3F2=α∂2\ny+αr2(k2−µ)+cph∂y−2αm3(m3k2+h−µm3)+2m3(k∂y+˜β(m3))\n−r2˜β′(m3) =α∂2\ny+αr2(k2−µ)+cph∂y+2m3k∂y−r2˜β′(m3),\nwhere the last equation is due to (17) and ˜β′(m3) =−ccpβ/(1+ccpm3)2. Since all coefficients\nare constant, the eigenvalue problem\nLu=λu\n17is solved by the characteristic equation arising from an exp onential ansatz u= exp(νy)u0,\nwhich yields the matrix\nA(ν,cph) :=/parenleftbiggαν2+(cph+2km3)ν−r−2ν(−ν+2αkm3)+k2−µ+α˜β′(m3)\nr2ν(−ν+2αkm3)αν2+(cph+2km3)ν+αr2(k2−µ)−r2˜β′(m3)/parenrightbigg\n.\nThe characteristic equation then reads\ndcph(λ,ν) :=|A(ν,cph)−λ|=|A(ν,0)−(λ−νcph)|=d0(λ−cphν,ν)\nd0(λ,ν) =λ2−t(ν)λ+d(ν) (28)\nwith trace and determinant of A(ν,0)\nt(ν) := trA(ν,0) = 2ν(αν+2km3)+αr2(k2−µ)−r2˜β′(m3)\nd(ν) := detA(ν,0) = (1+ α2)ν/parenleftBig\nν(ν2+4k2m2\n3+r2(k2−µ))−2r2˜β′(m3)km3/parenrightBig\n= (1+α2)ν/parenleftBigg\nν/parenleftBigg\nν2+(3k2+µ)(˜β(m3)/α−h)2\n(k2−µ)2+k2−µ/parenrightBigg\n−2r2˜β′(m3)km3/parenrightBigg\n.\nIn the last equation (23) was used.\nThe characteristic equation is also referred to as the complex (linear) dispersion relation .\nThespectrum of L, for instance in L2(R), consists of solutions for ν= iℓand is purely essential\nspectrum (in the sense that λ−Lis not a Fredholm operator with index zero). Indeed, setting\nν= iℓcorresponds to Fourier transforming in ywith Fouriermode ℓ. Note that the solution\nd(0,0) = 0 stems from spatial translation symmetry in y. For the same reason the real part\nof solutions λof (28) for any given ν∈iRdoes not depend on cph, which means that spectral\nstability is independent of cphand is therefore completely determined by d0(λ,iℓ) = 0.\nAs a first observation concerning stability we note that at r= 0,|m3|= 1 the solutions λ±\ntod0(λ,iℓ) = 0 have Re( λ±) =−ℓ(ℓ±2k) whose maximum is k2. Hence, wavetrains at (and\nthus near) r= 0 are unstable for k/ne}ationslash= 0.\nFold stability. First note the eigenvalues A(0,0) are 0 and\nτ(k2) :=t(0) =r2(α(k2−µ)−˜β′(m3))\nso that a ‘fold instability’ occurs precisely for r= 0 (compare §3) orα(k2−µ) =˜β′(m3).\nWe readily compute that the latter corresponds to the critic al points in (18). Note that\n˜β′(m3) =−α2ω2ccp/β. In particular, for the loops of wavetrains emerging from an elliptic\npoint, the upper and lower ω-values have opposite signs of τ(0); specifically the lower is stable\nifβccp>0, cf. Figure 5.\nAt the bifurcation points of ±ˆe3(see§3) where r= 0, we have µ=µ±andω=−β±/α\nwhich yields Theorem 3(a) except for the super-/subcritica lity. To see this recall that ±ˆe3\ndestabilize always through increasing µ. Whether wavetrains with k= 0 emerge from ±ˆe3\ndepends on the sign of ∂µm3. From (16) we find µ=h\nm3−β\nαm3(1+ccpm3)and compute\n∂m3µ(±1) =±τ(0)\nαr2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm3=±1.\n18Hence, fold-stability implies ∂µm3(µ+)<0 or∂µm3(µ−)>0, respectively, so that increasing\nµyields|m3|<1 and thus emergence of solutions.\nMore generally, sign changes of τ(k2) correspond to fold points and a curve of spectrum\ncrosses the origin. We plot some fold stability boundaries i n Figure 10. Using τ(k2) we have\nthat wavetrains with µ < k2+ccpα\nβω2are unstable. In particular, for ccpβ≥0 all wavetrains\nwithk2> µare unstable. Using the existence condition we may also writ e this condition\nindependent of kas\nω(β(2ω+h)+αω2)\nβ(β+αω)<0, (29)\nwhich explains the changes in the fold stability indicator a tω=−β/αin the figure.\nComing back to k= 0, at the bifurcation points of ±ˆe3(see§3) where r= 0, we have\nµ=µ±. Fold-stability is then µ > αc cpω2/βwhich holds if\n±/parenleftbigg\nh−β±\nα/parenrightbigg\n>βccp\nα(1±ccp)2⇔ ±h >β\nα2ccp±1\n(1±ccp)2,\nas noted in Theorem 3 item (a).\nWe next check the other possible marginal stability configur ations case by case.\nSideband instability. A sideband instability occurs when the curvature of the curv e of\nessential spectrum attached to the origin changes sign so th at the essential spectrum extends\ninto positive real parts. Let ˜λ0(ℓ) denote the curve of spectrum of A(iℓ,0) attached to the\norigin, that is ˜λ0(0) = 0, and let′denote the differentiation with respect to ℓ. Derivatives of\n˜λ0can be computed by implicit differentiation of d0(λ,iℓ) =λ2−t(iℓ)+d(iℓ) = 0.\nThis gives ˜λ′\n0(0) = id′(0)\nt(0)=−i˜β′(m3)2(1+α2)km3\nα(k2−µ)−˜β′(m3)therefore\n˜λ′′\n0(0) =−2d′(d′−t′t)+d′′t2\nt3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nℓ=0=: Λ(k2)2(1+α2)r4\nτ3(k2),\nwhere the ω-dependence is suppressed. Some calculations yield Λ as a cu bic polynomial in\nK=k2given by Λ = a3K3+a2K2+a1K+a0with\na3=−α2(4−3r2),\na2=α(2˜β′r2+αµ(8−5r2)),\na1=−α2µ2(4−r2)−4(1−r2)α2˜β′(m3)2−(˜β′(m3)2+4α˜β′(m3)µ)r2,\na0=µ(˜β′(m3)+αµ)2r2.\nNotably, a0=µτ(0)2and also a3<0 sincer∈(0,1). A wavetrain with wavenumber kis\ntherefore (strictly) sideband stable precisely when Λ( k2)τ(k2)<0 andsideband unstable for\nΛ(k2)τ(k2)>0. As mentioned above, τ(k2)>0 for large enough k2so that all such (already\nunstable) wavetrains are also sideband unstable as a3<0 holds always.\nSinceτ(k2)>0 is the unstable fold case, we next assume τ(k2)<0 so that sideband\nstability is precisely Λ( k2)>0.\n19Homogeneous oscillations. Fork= 0 we obtain ˜λ′′\n0(0) = Λ(0)2(1+α2)r4\nτ3(0)=−2(1+α2)µ\nτ(0),\nso thatµ >0 is required for sideband stability (given fold stability τ(0)<0).\nLet us study this situation near the Hopf instability of ±ˆe3, whereµ≈µ±=±(h−β±/α)\nwithµ > µ±andω=β±/α. Hence, sideband stability of the emerging wavetrains requ ires\nh > β+/αorh < β−/α, that is, µ±>0. Compare Theorem 3(a).\nNear homogeneous. To leading order Λ( k2) = 0 isa1k2+a0=O(k4) and via (21) we have\na1=a1(˜ω,k2),a0=a0(˜ω,k2), where at bifurcation a0(0) =∂ωa0(0) = 0. Upon expanding we\ntherefore find sideband instabilities to leading order at\n˜ω2=a1(0)−∂k2a0(0)\n∂˜ω2a0(0)k2+O(|k|3+|˜ω|3),\nwhere∂˜ω2a0(0) =µsnr2\nsn/parenleftBig\n2\nβα2ωsnccp/parenrightBig2\nand∂k2a0(0) =αµsnr2\nsnwithrsn= 1−/parenleftBig\nωsn+h\nµ/parenrightBig2\nther-coordinate of the wavetrain at the bifurcation point. Some algebra yields a1(0) =\n−4α(1+α2)(h+ωsn)2.\nIn accordance with the results in [31], this means that wavet rains in a sector in the ( ω,k)-\nplane near the fold point are sideband stable, while wavetra ins outside this sector are sideband\nunstable. We expect that the opening angle of this sector can be changed while keeping\nthe dispersion curves essentially fixed. Here we do not pursu e this further, but note that\nsince sideband instabilities do occur and the stable region cannot include the fold points,\nthe prefactor of k2is positive and ˜ ωhas a selected sign. The sideband boundaries for some\nexamples are plotted in Figure 10.\nGeneral wavetrains. Concerning the sign of Λ in general, Λ(0) = a0has the sign of\nµand thus all wavetrains for µ <0 are sideband unstable. For µ >0 we have Λ( µ) =\n−4α2(˜β′(m3))2m2\n3µ <0, so a sign change occurs at some Ksb∈(0,µ), which implies sideband\ninstability for k2≥µ. Moreover, µ >0 andccpβ≤0 imply a1<0 anda2>0. Since\na3<0 this means both roots of Λ′, (2a2±/radicalbig\n4(a2\n2−3a3a1))/(6a3) lie at negative Kand so Λ\nis monotone decreasing for all K >0. Therefore Ksbis the unique sideband instability in this\ncase.\nOn the other hand, for a1>0 the roots of Λ′have opposite signs so that due to the sign\nchange in the interval (0 ,µ) the positive one must be a local maximum so that also in this\ncaseKsbis the unique sideband instability. Moreover, in all cases w avetrains with k2> µare\nunstable since Λ <0 in this range.\nHopf instability. A Hopf instability occurs when the essential spectrum touch es the imag-\ninary axis at nonzero values. In particular, there is γ/ne}ationslash= 0 so that d0(iγ,iℓ) = 0. At k= 0\nwe have Im( d0(iγ,iℓ)) =γ(2αℓ2−τ(0)) so that in the fold stable case τ(0)<0 there is real ℓ\nforγ/ne}ationslash= 0. Recall τ(0) =−(˜β′(m3)+αµ)r2. Therefore, there is no (relevant) Hopf instability\nneark= 0.\nMore generally, solving Im( d0(iγ,iℓ)) forγand substituting the result into Re( d0(iγ,iℓ))\nwe obtain up to a factor (1+ α2)ℓ2\nℓ2+(µ−k2)r2+4α2m2\n3k2G(ℓ2), G(L) =−4L2−4L(k2−µ)r2+(˜β′(m3)2+(k2−µ)2)r4\n(2αL−τ(k2))2\n20/Minus4/Minus20240Π2Π\nkΘsupercritical\nsubcritical\nsubsubcriticalµ\nk\n(a) (b)\nFigure 11: Analogues of Figures 8(a) and 9 with stable range o f wavetrains in (a) bold line,\nand in (b) the dark shaded region.\nwhere all terms except possibly G(L) are positive in the interesting range µ > k2. Note that\nfor sufficiently small α >0 there is no root besides ℓ= 0 and thus no Hopf and in fact no\nsideband instability. But there seems to be no satisfying ex plicit bound. (While the same\nseems to occur for m3∼0, such wavetrains have k2> µand are thus unstable.)\nHowever, Gis nondecreasing for ˜β′(m3)≤0, i.e.,ccpβ≥0, since then\nG′(L) =−4˜β′(m3)r22L+(µ−k2−α˜β′(m3))r4\n(2αL−τ(k2))3≥0,\nin the fold stable case τ(k2)≤0. Thus, besides ℓ=γ= 0, there is at most one solution\nd0(iγ,iℓ) = 0 for ccpβ≥0, which rules out a Hopf instability as this requires two suc h\nsolutions. We do not know whether or not Hopf instabilities c an occur for general αand\nccpβ <0.\nTuring instability. A Turing instability occurs when the spectrum touches the or igin for\nnonzero ℓ, that is, there is ℓ/ne}ationslash= 0 so that d0(0,iℓ) = 0, which means det A(iℓ,0) = 0. Since\nIm(detA(iℓ,0)) =−2(1 +α2)˜β′(m3)ℓkm3r2and our previous considerations already cover\nzeros of this, such instabilities do not occur.\nThis exhausts the list of possible marginal stability.\n4.4 Stability of wavetrains for ccp= 0\nForccp= 0 the wavetrain frequency ωis independent of the wavenumber kin (22) so that\nthe phase velocity ω/kof all wavetrains is −β/(αk) and the group velocity d ω/dk, which\ndescribes the motion of perturbations by localized wave pac kets, vanishes for all wavetrains.\nDue to Theorem 3 destabilizations of stable wavetrains can o nly occur through a unique\nsideband instability |k|=k∗>0. This value can be explicitly determined since d′(0) = 0 and\n21d′′(0) =−2(1+α2)((3K+µ)r2−4K) so that ˜λ′′\n0(0) = 0 at\nk2\n∗=µr2\n4−3r2.\nTaking into account that k2=µfor a wavetrain can occur only if h=β/αwe thus have\nTheorem 4 Consider ccp= 0. All wavetrains whose wavenumber ksatisfies |k|> k∗are\nunstable. For µ >0wavetrains with wavenumber |k|< k∗are spectrally stable, while those\nwith|k|> k∗are unstable. In case h/ne}ationslash=β/αa sideband instability occurs at k=±k∗. There\nis no secondary instability for k2< µ. Nontrivial spectrally stable wavetrains exist only for\nsupercritical anisotropy, |h−β/α|< µ.\nThe overall picture for wavetrains of (1) with ccp= 0 is thus a combination of the scenarios\nfrom a supercritical and a subcritical real Ginzburg-Landa u equation ∂tA=∂2\nxA+ ˜µA∓\nA|A|2, A(x,t)∈C, which describes the dynamics near pattern forming Turing i nstabilities\nand possesses the gauge-symmetry A→eiϕA. The interested reader is referred to the review\n[1] and the references therein.\n5 Coherent structures\nThe coexistence of wavetrains and constant magnetizations raises the question how these\ninteract. In this section we study solutions that have spati ally varying local wave number. In\nparticular, we consider solutions that spatially connect w avetrains or ±ˆe3in a coherent way.\nIn order to locate such solutions induced by symmetry we make the ansatz\nξ=x−st\nϕ=φ(ξ)+Ωt\nθ=θ(ξ),(30)\nwith constant s,Ω. Solutions of (14) of this form are generalized travelling waves to (1)\nwith speed sthat have a superimposed oscillation about ˆe3with frequency Ω. This ansatz\nis completely analogous to that used in the aforementioned s tudies of the real and complex\nGinzburg-Landau equations [1].\nLet¯β(θ) :=˜β(cos(θ)) =β/(1 +ccpcos(θ)) denote the ( β,ccp)-dependent term of (15).\nSubstituting ansatz (30) into (15) with′= d/dξandq=φ′gives, after division by sin( θ), the\nODEs\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggsin(θ)(Ω−sq)\nsθ′/parenrightbigg\n=/parenleftbigg2cos(θ)θ′q+sin(θ)q′\n−θ′′+sin(θ)cos(θ)q2/parenrightbigg\n+sin(θ)/parenleftbigg¯β(θ)\nh−µcos(θ)/parenrightbigg\n,(31)\non the cylinder ( θ,q)∈S1×R, which is the same as {(m3,r,q)∈R3:m2\n3+r2= 1}.\nWavetrains. Steady states with vanishing ξ-derivative of θandqhaveϕ=q(x−st)+Ωt\nand thus correspond to the wavetrains discussed in §4 with wavenumber k=qand frequency\nω=sq−Ω. Hence, the ansatz (30) removes all wavetrains whose waven umber and frequency\ndo not lie on the line {ω=sk−Ω}in (ω,k)-space.\n22We may visualize this by drawing the line ω=sk−Ω into the wavetrain existence and\nstability plots in the ( ω,k)-plane such as Figure 10. Specifically, for s/ne}ationslash= 0, steady states of\n(31) with m3/ne}ationslash=±1 are wavetrains with wavenumber qfor which there exists θsuch that\nq=Ω−¯β(θ)/α\ns. (32)\nIn particular, for ccp= 0 we have constant ¯β(θ) =βso that equilibria of (31) (other than\n±ˆe3) have uniquely selected wavenumber q. Hence, heteroclinic solutions to (31) for ccp= 0\ncan only be domain walls connecting ±ˆe3or connect one of ±ˆe3to a wavetrain.\nCoherent structure ODEs. Writing (31) as an explicit ODE gives\nθ′=p\np′= sin(θ)/parenleftbig\nh+(q2−µ)cos(θ)−(Ω−sq)/parenrightbig\n−αsp\nq′=α(Ω−sq)−β/(1+ccpcos(θ))−s+2cos(θ)q\nsin(θ)p,(33)\nwhose study is the subject of the following sections. For lat er use we also note the ‘desingu-\nlarization’ by the (singular) coordinate change p= sin(θ)˜pso that ˜p′=p′/sin(θ)−˜p2cos(θ),\nwhich gives\nθ′= sin(θ)˜p\n˜p′=h+(q2−µ)cos(θ)−(Ω−sq)−αs˜p−cos(θ)˜p2\nq′=α(Ω−sq)−β/(1+ccpcos(θ))−(s+2cos(θ)q)˜p.(34)\nHence, (33) is equivalent to (34) except at zeros of sin( θ). Inparticular, for ˜ p= 0the equilibria\nof (34) with sin( θ)/ne}ationslash= 0 are those of (33), but θ=nπ,n∈Zare invariant subspaces which\nmay contain equilibria with ˜ p/ne}ationslash= 0.\nNext, we first consider various moving heteroclinic coheren t structures with s/ne}ationslash= 0 and for\nccp= 0 give a complete analysis of stationary coherent structur es (s= 0). Coherent structures\nalso emerge near the elliptic and hyperbolic wavetrain bifu rcations that arise from fold points\nforccp/ne}ationslash= 0. However, the detailed analysis of this case is beyond the scope of this paper.\n5.1 Homogeneous domain walls\nClassical domain walls connect antipodal equilibria at x=±∞. For the model equation (8)\nexplicit (Walker) solutions are known to exist below a criti cal fieldh. These solutions exhibit a\ntilting of the azimuthal angle ϕ=const.in order to balance precessional forces. An analogue\nsituation arises in our context when q=q′= 0. In this case we have ϕ= Ωtand therefore no\nspatially varying azimuthal profile.4\nTheorem 5 Non-equilibrium coherent structure solutions with q≡0fors/ne}ationslash= 0and|ccp|<1\nexist for µ <0andccp= 0orβ= 0only, and have Ω =h+αβ\n1+α2,s2=−(β−αh)2\nµ(1+α2)2. They\nare oscillating heteroclinic fronts connecting ±ˆe3that solve θ′=σ√−µsin(θ)whereσ=\nsgn(s(αh−β)). The family of such fronts is smooth and extends to s= 0, whereh= Ω =β/α,\nand fronts exist for both signs of σ.\n4After acceptance of the present manuscript for publication , we found that the sufficiency of ccp= 0 for\nexistence of such domains walls in Theorem 5 is contained in [ 12].\n23Proof.Recall¯β(θ) =β/(1+ccpcos(θ)) sod\ndξ¯β(θ) =ccpβsin(θ)θ′/(1+ccpcos(θ))2.\nSuppose a solution ( θ,p,q) to (33) has q≡0, so also q′≡0. Then the third equation of\n(33) fors/ne}ationslash= 0 yields, using the first equation, θ′=αΩ−¯β(θ)\nssin(θ). Differentiation gives\nθ′′=αΩ−¯β(θ)\nscos(θ)θ′−ccpβsin2(θ)\ns(1+ccpcos(θ))2θ′\n= sin(θ)αΩ−¯β(θ)\ns2/parenleftbigg\n(αΩ−¯β(θ))cos(θ)−ccpβ(1−cos2(θ))\n(1+ccpcos(θ))2)/parenrightbigg\n.\nOn the other hand, the second equation of (33) requires\nθ′′= sin(θ)(h−µcos(θ)−Ω−α(αΩ−¯β(θ))).\nFirst consider ccp= 0 orβ= 0 so that ¯β(θ) =βand these two right hand sides for θ′′simplify.\nEquating them and comparing the coefficients of cos( θ)j,j= 0,1, yields h= Ω+α(αΩ−β)\nandµ=−/parenleftBig\nαΩ−β\ns/parenrightBig2\n, which means ( αΩ−β)/s=σ√−µforσ= sgn(s(αΩ−β)). Taken\ntogether, the parameter conditions can be equivalently cas t as the equations for Ω, s2andσ\nin the theorem statement.\nHence, for ccp= 0 orβ= 0 these parameter choices and θ′=σ√−µsin(θ) are necessary\nconditions for q≡0. As a scalar equation, the only non-trivial and bounded sol utions are\nheteroclinic orbits between equilibria. Taking Ω = β/α+s˜µfor some ˜ µ/ne}ationslash= 0 gives a smooth\nparametrization up to s= 0 and σ= sgn(˜µ).\nConversely, for ccp= 0 orβ= 0 and these choices of parameters, any ( θ,p,q)(ξ) where\nθ(ξ) satisfies θ′=σ√−µsin(θ),p=θ′andq≡0 is a solution to (33).\nIt remains to show that if β/ne}ationslash= 0 then ccp= 0 is necessary for a non-trivial solution\nwithq≡0: Subtracting the two right hand sides for θ′′from above and multiplication with\n(1+ccpcos(θ))3gives a polynomial in cos( θ) of degree four. A straightforward computation\nshows that the 4th order term to vanish requires µ=−α2Ω2/s2. Using this the coefficients\najof cos(θ)j,j= 0,1,2,3, in this polynomial can be computed as\na0=−h+(1+α2)Ω−αβ+(β−αΩ)βccp/s2,\na1= 3ccp((1+α2)Ω−h)+β2/s2−αβ(2ccp+(2+c2\ncp)Ω/s2),\na2=c2\ncp(3((1+α2)Ω−h)−αβ)−3αβΩccp/s2,\na3=c3\ncp((1+α2)Ω−h)−αβΩc2\ncp/s2.\nWe first solve a0= 0 trivially for hand proceed with somewhat tedious, but straightforward\ncalculations: substituting this hintoa1= 0 (which is linear in Ω) we solve for Ω, which uses\n|ccp|<1. Substituting the resulting h,Ω gives\na2=ccpβ\n2s2/parenleftbig\n3β(c2\ncp−1)+αccps2/parenrightbig\n,\nso that for a2= 0 either ccp= 0 (since β/ne}ationslash= 0; note that then also a3= 0) ors2= 3βc2\ncp−1\nαccp. In\nthe latter case, substituting the previous h,Ω and this s2intoa3would give a3=αβc3\ncp/3/ne}ationslash= 0.\nHence,ccp= 0 is necessary as claimed.\n24Remark 2\n1. Fors= 0further coherent structure solutions exist, but not as doma in walls. See\nTheorem 8 below.\n2. In§5.2.1 we find fast domain walls and fronts that have non-trivi alq.\n3. Numerical simulations suggest that these domain walls ar e dynamically stable solutions\nin the subsubcritical case. They are unstable in the subcriti cal case |h−β/α|>−µ,\nwhich occurs for large |s|, since then either ˆe3or−ˆe3is unstable.\n4. The profile of these domain walls depends only on the paramet erµ. In particular, the\nsubfamily parameterized by hhas arbitrarily large speed but constant shape, though the\noscillation frequency Ωdepends on h.\n5.2 Moving front-type coherent structures\nUsing the desingularized system (34), we prove existence of some non-stationary coherent\nstructures of front-type, spatially connecting wavetrain s or±ˆe3.\n5.2.1 Near the fast limit |s| ≫1\nTheorem 6 For any bounded set of (α,β,µ,Ω0,Ω1)andccp∈(−1,1)there exists s0>0and\nneighborhood UofM0:={θ∈[0,π],˜p=q−Ω1= 0}such that for all |s| ≥s0the following\nholds for (34)withΩ = Ω 0+Ω1s. The heteroclinic orbits of (34)inUform a smooth family\nin the parameter s−1for each sign of s, which reverses their orientation. These heteroclinics\nand are in one-to-one correspondence with those of the ODE\nd\ndηθ=−α\n1+α2sin(θ)/parenleftbigg¯β(θ)\nα−h+(Ω2\n1−µ)cos(θ)/parenrightbigg\n, (35)\non the spatial scale η=ξ/s, which also gives the θ-profile to leading order in s−1. Moreover,\nfor such a heteroclinic orbit (θh,˜ph,qh)(ξ)withθσ:= lim ξ→σ∞θh(ξ)∈ {0,π}forσ= 1or\nσ=−1, theq-limit is/are\nlim\nξ→σ∞qh(ξ) = Ω1−1\ns/parenleftbiggh+α¯β(θσ)−σµ\n1+α2−Ω0/parenrightbigg\n+O(s−2). (36)\nIn particular, for Ω1/ne}ationslash= 0or(1+α2)Ω0/ne}ationslash=h+α¯β(θσ)−σµlocal wavenumbers are nontrivial:\nqh/ne}ationslash≡0.\nBefore proving the theorem we note the consequences of this f or coherent structures and\ndomains walls in in (33) and (1).\nCorollary 1 The heteroclinic solutions of Theorem 6 are in one-to-one corr espondence with\nheteroclinic solutions to (33)and thus heteroclinic coherent structures in (1)that lie in Uand\nconnectθ= 0,πor a wavetrain with r/ne}ationslash= 0. Forθ∈(0,π)all properties carry over to (33)\nwith the bijection given by p= sin(θ)˜p.\n25Proof.Recall that (34) and (33) are equivalent for θ∈(0,π). Since the limit of the vector\nfield of (33) along such a heteroclinic from (34) is zero by con struction in all cases. Hence, for\neach of the heteroclinic orbits in (34) of Theorem 6, there ex ist a heteroclinic orbit in (33) in\nthe sense of the corollary statement.\nCorollary 2 For any ‘bandgap’ parameter set of (33)such that there exist no wavetrains\nsatisfying (32)for any|s|> s1, for some s1>0, there exist fast domain wall type coherent\nstructures spatially connecting ±ˆe3for all sufficiently large velocity |s|.\nProof. Choosing Ω 1=kthere are by assumption no equilibria in (35) besides ±ˆe3, which\nare therefore connected by a heteroclinic orbit. Theorem 6 t hen implies the claim.\nSuch ‘bandgaps’ occur in particular if µ >0 for Ω2\n1∼µ.\nRemark 3\n1. Concerning stability, Lemma 1 and Theorem 4 imply that for ccp= 0the domain walls\nconnecting ±ˆe3might be stable in the subsubcritical case only since otherw ise one of the\nasymptotic states is unstable: the unique wavetrain with θ∈(0,π)in the subsubcritical\ncase and ˆe3or−ˆe3in the sub- and supercritical cases. However, it may be that s ome\nfronts are stable in a suitable weighted sense as invasion fr onts into an unstable state.\n2. For increasing speeds these solutions are decreasingly l ocalized, hence far from a sharp\ntransition.\n3. The uniqueness statement in the corollaries is limited, si nce in the (θ,p,q)-coordinates\nthe neighborhood Ufrom the theorem is ‘pinched’ near θ= 0,π: a uniform neighborhood\nin(θ,˜p)has a sinus-shaped boundary in (θ,p).\nRemark 4 Part of the family homogeneous domains walls from Theorem 5, w hereccp= 0, is\na continuation to smaller |s|of homogeneous ( q≡0) fronts in the family of Theorem 6. The\nlatter are decreasingly localized, which requires in the fo rmer that√µ=O(s). Specifically,\nµ=−(β−αh)2\ns2(1+α2)2andΩ0=h+αβ\n1+α2,Ω1= 0in the heteroclinics of Theorem 6. Then µ→0as\ns2→ ∞so thatµ= 0in the leading order equation (35)and in(36)µis removed from the\norders−1term. Since σs√−µ=−(β−αΩ0)andβ−αΩ =−α\n1+α2/parenleftBig\nh−β\nα/parenrightBig\nindeed(35)equals\nthe equation in Theorem 5. In particular, (36)is consistent with q≡0.\nFinally, remark that the ODE (35) is the spatial variant of th e temporal heteroclinic\nconnection in (15): setting all space derivatives to zero, t heθ-equation of (15) reads\n−∂tθ=α\n1+α2sin(θ)/parenleftbigg\nh−¯β(θ)\nα−µcos(θ)/parenrightbigg\n,\nwhich is (35) with µreplaced by Ω2\n1−µand up to possible direction reversal. Since Ω 1=q\non the slow manifold M0(i.e. at leading order), the reduced flow equilibria reprodu ce the\nwavetrain existence condition (23). This kind of relation b etween temporal dynamics and fast\ntravelling waves holds formally (but in general not rigorou sly) for any evolution equation in\none space dimension. Here the symmetry makes the temporal OD E scalar.\n26Proof (Theorem 6) Let us set s=ε−1so that the limit to consider is ε→0. Since we will\nrescale space with εandε−1this means sign changes of sreflect the directionality of solutions.\nThe existence proof relies on a geometric singular perturba tion argument and we shall use\nthe terminology from this theory, see [10, 17], and also some times suppress the ε-dependence\nofθ,˜p,q.\nUpon multiplying the ˜ p- andq-equations of (34) by εwe obtain the, for ε/ne}ationslash= 0 equivalent,\n‘slow’ system\nθ′= sin(θ)˜p\nε˜p′=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2)\nεq′=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(37)\nSettingε= 0 gives the algebraic equations\nA/parenleftbigg˜p\nq/parenrightbigg\n=−Ω1/parenleftbigg−1\nα/parenrightbigg\n,whereA=−/parenleftbiggα−1\n1α/parenrightbigg\n.\nSince det A= 1+α2>0 the unique solution is ˜ p=q−Ω1= 0 and thus the ‘slow manifold’\nisM0as defined in the theorem, with ‘slow flow’ given by\nθ′= sin(θ)˜p.\nSince ˜p= 0 atε= 0,M0is a manifold (a curve) of equilibria at ε= 0, so that the slow\nflow is in fact ‘superslow’ and will be considered explicitly below. Since the slow manifold is\none-dimensional (and persists for ε >0 as shown below) it suffices to consider equilibria for\nε >0. These lie on the one hand at θ=θ0∈ {0,π}, if\nA/parenleftbigg˜p\nq/parenrightbigg\n+Ω1/parenleftbigg−1\nα/parenrightbigg\n+εF(˜p,q) = 0, F(˜p,q) :=/parenleftbiggh+σ(q2−µ)−Ω0−σ˜p2\nαΩ0−¯β(θ0)−2σq˜p/parenrightbigg\n,\nwhereσ= cos(θ0)∈ {−1,1}. Since det A=−(1 +α2)<0 the implicit function theorem\nprovides a locally unique curve of equilibria (˜ pε,qε) for sufficiently small ε, where\nd\ndε/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nε=0/parenleftbigg˜pε\nqε/parenrightbigg\n=−A−1F(0,0) =−A−1/parenleftbiggh−σµ−Ω0\nαΩ0−β±/parenrightbigg\n.\nThis proves the claimed location of asymptotic states.\nOn the other hand, for θ/ne}ationslash= 0 system (34) is equivalent to (33). From the previous consi d-\nerations of equilibria (=wavetrains) we infer that the uniq ue equilibria in an ε-neighborhood\nofM0are those at θ=θ0, (˜p,q) = (˜pε,qε) together with the possible additional θ-values of\nwavetrains, where kis now replaced by q=ε(Ω0−¯β(θ)/α).\nTowards the persistence of M0as a perturbed invariant manifold for |ε|>0, let us switch\nto the ‘fast’ system by rescaling the time-like variable to ζ=ξ/ε. With˙θ= dθ/dζetc., this\ngives\n˙θ=εsin(θ)˜p\n˙˜p=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2)\n˙q=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(38)\n27Note that M0is (also) a manifold of equilibria at ε= 0 in this system and the linearization of\n(38) inM0for transverse directions to M0is given by A. Since the eigenvalues of A,−α±i,\nare away from the imaginary axis, M0is normally hyperbolic and therefore persists as an\nε-close invariant one-dimensional manifold Mε, smooth in εand unique in a neighborhood of\nM0. See [10]. The aforementioned at least two and at most three e quilibria lie in Mε, and,\nMεbeing one-dimensional, these must be connected by heterocl inic orbits.\nFor the connectivity details it is convenient to derive an ex plicit expression of the leading\norder flow. We thus switch to the superslow time scale η=εξand setp= ˜p/ε,q= (q−Ω1)/ε,\nwhich changes (37) to (subdindex ηmeans d/dη)\nθη= sin(θ)p\nεpη=−αp+q+h+(Ω2\n1−µ)cos(θ)−Ω0+εcos(θ)(ε(q2−p2)+2qΩ1)\nεqη=−p−αq+αΩ0−¯β(θ)−2εcos(θ)(εqp+Ω1p).(39)\nAtε= 0, solving the algebraic equations for ( p,q) gives\n/parenleftbiggp\nq/parenrightbigg\n=−A−1/parenleftbiggh+(Ω2\n1−µ)cos(θ)−Ω0\nαΩ0−¯β(θ)/parenrightbigg\n=1\n1+α2/parenleftbiggαh+α(Ω2\n1−µ)cos(θ)−¯β(θ)\n(1+α2)Ω0−h−(Ω2\n1−µ)cos(θ)−α¯β(θ)/parenrightbigg\nwhose first component gives pso that the leading order superslow flow on the invariant man-\nifold is indeed given by (35).\n5.2.2 The case of small amplitudes\nIn this section we consider small amplitude coherent struct ures, which means qmust lie near\na bifurcation point of wavetrains. Here we focus on the inter section points of the solution\ncurves from (16) with θ=θ0= 0,π, which gives\ncos(θ0)/parenleftbig\nq2−µ/parenrightbig\n=¯β(θ0)\nα−h. (40)\nForccp= 0 this is possible for super- and subcritical anisotropy on ly, compare Figure 8.\nWe show that these intersection points are pitchfork-type b ifurcations in (34) that give\nrise to front-type coherent structures. As in the previous s ection, we locate such solutions in\n(33) from an analysis of (34).\nIt is convenient to write (16) in terms of m3= cos(θ) so equilibria of (34) with ˜ p= 0 solve\n˜Γ(m3) :=˜β(m3)\nα−/parenleftbig\nq2(m3)−µ/parenrightbig\nm3−h= 0,\nwhereq(m3) is the selected qfrom (32). Recall ˜β(m3) =β\n1+ccpm3.\nTheorem 7 Consider θ=θ0∈ {0,π}and setm0\n3:= cos(θ0). Suppose that parameters of (34)\nare such that s/ne}ationslash= 0,˜Γ(m0\n3) = 0and∂m3˜Γ(m0\n3)/ne}ationslash= 0. Then the equilibrium point (θ0,0,q(m0\n3))\nof(34)undergoes a pitchfork bifurcation upon any perturbation of horµ.\nMore precisely, let Sε= (αε,βε,hε,µε,Ωε,ccp(ε),sε),ε∈(−ε0,ε0)for some ε0>0, be\na curve in the parameter space of (34)with|ccp(ε)|<1,αε>0,sε/ne}ationslash= 0and such that S0\nsatisfies ˜Γ(m0\n3) = 0and˜γ:=∂m3˜Γ(m0\n3)/ne}ationslash= 0.\n28Then(34)has a curve of equilibria (θ0,˜pε,qε), with possibly adjusted ε0, such that ˜p0=\n0,q0=q(m0\n3)and two equilibria with θ/ne}ationslash=θ0bifurcate from (θ0,0,q(m0\n3))for increasing ε\nif, with parameters Sε,m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0>0. The bifurcating equilibria are connected to\n(θ0,˜pε,qε)by heteroclinic orbits which converge to (θ0,˜pε,qε)asℓξ→ ∞forℓ=−m0\n3˜pε.\nSpecifically, this occurs if hε=h0−m0\n3˜γεorµε=µ0+˜γεandℓ= ˜γs, or ifβε=β0+m0\n3˜γε\nandℓ=−(2q(m0\n3)m0\n3+s)˜γ, with all other parameters fixed in each case.\nAnalogously to Corollary 1 we have\nCorollary 3 The heteroclinic solutions of Theorem 7 are in one-to-one corr espondence with\nheteroclinic solutions to (33), connecting to θ≡0orθ=πinU. Bounded solutions for\nθ/ne}ationslash∈ {0,π}are also in one-to-one correspondence.\nProof (Theorem 7) Note that ˜Γ(m3) = 0 with |m3|<1 is equivalent to (and if |m3|= 1\nsufficient for) the existence of an equilibrium of (34) at θwith cos( θ) =m3,q=q(m3) from\n(40) and ˜ p= 0. Assuming ˜ γ=∂m3˜Γ(m0\n3)/ne}ationslash= 0 and ∂ν˜Γ(m0\n3)/ne}ationslash= 0 forν=horν=µimplies\nexistence of a locally unique curve of equilibria m3(ν) that transversely crosses m0\n3. The case\nof parameters Sεis analogous with a curve m3(ε), where ∂εm3(0) =−∂ε˜Γ(m0\n3)/∂m3˜Γ(m0\n3)|ε=0\nhaving the sign of −m0\n3means bifurcation of two equilibria for ε >0.\nItremainstoshowthatthecentermanifoldassociatedtothe bifurcationisone-dimensional,\nand to obtain the directionality of heteroclinics.\nFor the former it suffices to show that the linearization at the bifurcation point has only\na simple eigenvalue on the imaginary axis, namely at zero. Th e linearization of (34) in any\npoint with ˜ p= 0,θ=θ0gives the 3 ×3 matrix\n˜A=\n00 0\n0\n0B\n, B=/parenleftbigg−αs 2qm0\n3+s\n−(s+2qm0\n3)−αs/parenrightbigg\n,\nwhich has a kernel with eigenvector (1 ,0,0)t. The remaining eigenvalues are those of B,\nwhich are −αs±(s+m0\n32q)i. Since these lie off the imaginary axis for s/ne}ationslash= 0 there is indeed\nat most one simple zero eigenvalue on the imaginary axis. Thi s implies the existence of a\none-dimensional center manifold which includes all equili bria and heteroclinic connections\nnear (θ0,0,q(m0\n3)) for nearby parameters. Equilibria in the symmetry plane {m3=m0\n3}are\nsolutions of (˜ p′,q′) =:G(˜p,q) = 0 with Ggiven by (34). Since DG(0,q(m0\n3)) =Bis invertible\nwe obtain a curve ( θ0,˜pε,qε) for parameters at Sεas claimed.\nTheuniquenessofbifurcatingequilibriaoneithersideoft hesymmetryplaneandinvariance\nof the one-dimensional center manifold implies existence a nd local uniqueness of heteroclinic\nconnections for sgn( ε) = sgn( m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0). In order to determine the directionality of\nthese, a perturbation in the kernel gives θ′= sin(θ0+δ)˜p=m0\n3δ˜p+O(δ2) so that for m0\n3˜pε<0\nthe equilibrium at ( θ0,˜pε,qε) is stable in the center manifold, and unstable for reversed sign.\nSince existence of heteroclinics requires sgn( ε) = sgn(m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0) this implies stability\nifℓ:=−˜γ∂ε˜Γ(m0\n3)∂ε˜p|ε=0<0 and thus convergence to ( θ0,˜pε,qε) asℓξ→ ∞.\nForhε=h0−m0\n3˜γεwith otherwise fixed parameters ∂ε˜Γ(m0\n3) =m0\n3˜γso heteroclinics\nexist for ε >0. On the other hand, ∂ε˜pε|ε=0is the first component −m0\n3˜γαsdet(B) of\n−B−1∂hG(0,q(m0\n3))(−m0\n3˜γ), where αdet(B)>0. Hence, ℓ=s˜γas claimed. The cases\nε=µ,βare determined analogously using ∂µ˜Γ(m0\n3) =m0\n3,∂µG(0,q(m0\n3) = (−m0\n3,0) and\n∂β˜Γ(m0\n3) = ((1+ ccpm0\n3)α)−1>0,∂βG(0,q(m0\n3) = (0,−((1+ccpm0\n3)α)−1).\n290Π 2Π/Minus2/Minus1012\nΘΘ'supercritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'subcritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'subsubcritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'domainwalls\n(a) (b) (c) (d)\nFigure 12: Phase plane streamplots of (42) with Mathematica . (a)-(c) have h−Ω = 1/2.\n(a) supercritical anisotropy (here µ= 1), (b) subcritical (here µ= 0), (c) subsubcritical (here\nµ=−1), (d) subsubcritical case that allows for standing domain walls,h= Ω,µ=−1.\n5.3 Stationary coherent structures for ccp= 0\nIn this section we consider the case s= 0 (which does not imply time-independence) and\nccp= 0 (which will imply integrability), so that equations (33) reduce to\nθ′′= sin(θ)/parenleftbig\nh−Ω+(q2−µ)cos(θ)/parenrightbig\nq′=αΩ−β−2cot(θ)θ′q.(41)\nIn case Ω /ne}ationslash=β/αthere are no equilibria and it will be shown at the end of this s ection\nthat there are no coherent structure-type solutions in that case. Recall from §2.1 that for\nccp= 0 we may choose coordinates so that β= 0, which means Ω = 0 and thus stationary\ncoherent structures are turned into standing waves. Howeve r, we choose not to remove the\nparameter βin order to emphasize the typically oscillatory nature of so lutions to (1) and for\nconsistency in parameter relations. Nevertheless, the sym metries and integrals that we will\nfind are consequences of this reducibility.\nFor Ω = β/α, system (41) is invariant under the reflection q→ −qso that{q= 0}is an\ninvariant plane which separates the three dimensional phas e space. In particular, there cannot\nbe connections between equilibria (=wavetrains) with oppo site signs of q, that is, sign reversed\nspatial wavenumbers .\n5.3.1 Homogeneous solutions ( q= 0)\nSolutions in the invariant set {q= 0}have the form m(ξ) =r(ξ)exp(itΩ) and (31) turns into\na second order ODE on the circle {m2\n3+r2= 1}. The ODE for θfrom (41) is given by the\nnonlinear pendulum equation\nθ′′= sin(θ)(h−Ω−µcos(θ)), (42)\nwhich is invariant under θ→ −θand is Hamiltonian with potential energy\nP0(θ) = cos(θ)(h−Ω−µ\n2cos(θ)).\nThe symmetry (24) applies and we therefore assume in the foll owing that Ω = β/α < h.\n30We plot the qualitatively different vector fields of (42) in Fig ure 12 and some profiles\nin Figure 4. Coherent structure solutions are completely ch aracterized via the figure, which\nwe formulate next explicitly for the original PDE with m= (m,m3),m=reiϕ,r= sin(θ),\nm3= cos(θ). Homoclinic profiles may beinterpreted as (dissipative) s olitons. Theheteroclinic\nconnections in item 2(a) can be viewed as (dissipative) soli tons with ‘phase slip’.\nTheorem 8 Lets= 0andΩ =β/αand consider solutions to (14)of the form (30)withϕ\nconstant in ξ, i.e.,q= 0. These oscillate in time pointwise about the ˆe3-axis with frequency\nΩ =β/α. Assume without loss of generality, due to (24), thath >Ω.\n1. Subcritical anisotropy h−Ω>|µ|>0. There exist no nontrivial wavetrains with q= 0,\nand the coherent structure solutions with q= 0are a pair of homoclinic profiles to ˆe3, and\nthree one-parameter families of periodic profiles, one boun ded and two semi-unbounded.\nThe homoclinic profiles each cross once through −ˆe3in opposite θ-directions. The limit\npoints of the bounded curve of periodic profiles are −ˆe3and the union of homoclinic\nprofiles. Each of the homoclinics is the limit point of one of t he semi-unbounded families,\neach of which has unbounded θ-derivatives. The profiles from the bounded family each\ncross−ˆe3once during a half-period, the profiles of the unbounded fami ly cross both ±ˆe3\nduring one half-period.\n2. Suppose super- or subsubcritical anisotropy |µ|> h−Ω. There exists a wavetrain with\nk= 0, which is stable in the supercritical case ( µ >0) and unstable in the subsubcritical\ncase (µ <0). In(42)this appears in the form of two equilibria being the symmetri c pair\nof intersection points of the wavetrain orbit and a meridian on the sphere, phase shifted\nbyπinϕ-direction. Details of the following can be read off Figure 12 analogous to item\n1.\n(a) In the supercritical case ( µ > h−Ω) the coherent structure solutions with q= 0\nare two pairs of heteroclinic connections between the wavet rain and its phase shift,\nand four curves of periodic profiles; two bounded and two semi -unbounded.\n(b) In the subsubcritical case ( µ < h−Ω) the coherent structure solutions with q= 0are\ntwo pairs of homoclinic connections to ±ˆe3, respectively, and five curves of periodic\nprofiles, three bounded and two semi-unbounded.\n3. The degenerate case h= Ω,µ <0is the only possibility for profiles of stationary coherent\nstructures to connect between ±ˆe3, which then come in a pair as in the corresponding\npanel of Figure 12. The remaining coherent structures with q= 0are analogous to the\nsupercritical case with ±ˆe3and the pair of wavetrain and its phase shift interchanged.\nProof.As for wavetrains discussedin §4, thecondition cos( θ) = (h−Ω)/µyields theexistence\ncriterion |h−Ω|<|µ|for an equilibrium to (42) in (0 ,π). The derivative of the right hand\nside of (42) at θ= 0 ish−Ω−µ, which dictates the type of all equilibria and only saddles\ngenerate heteroclinic or homoclinic solutions.\nIt remains to study the connectivity of stable and unstable m anifolds of saddles, which is\ngiven by the difference in potential energy P0(θ). Since P0(0)−P0(π) = 2(h−Ω) the claims\nfollow.\n31/Minus4/Minus20240Π2Π\nkΘ0Π 2Π/Minus0.8/Minus0.400.4\nΘ\n0Π 2Π/Minus0.400.4\nΘ\n(a) (b)\nFigure 13: (a) Figure 11(a) with solutions to (43) for θ∈(0,π) andC= 0.1 (thick solid line),\nandC= 0.4 (thick dashed line). (b) Upper panel: potential P(θ) for parameters as in (a) and\nC= 0.1. Lower panel: same with C= 0.4.\n5.3.2 Non-homogeneous solutions ( q/ne}ationslash= 0)\nIn order to study (41) for q/ne}ationslash= 0, we note the following first integral. Since Ω = β/α, the\nequation for qcan be written as\n(log|q|)′=−2(log|sin(θ)|)′,\nand therefore explicitly integrated. With integration con stantC= sin(θ(0))2|q(0)|this gives\nq=C\nsin(θ)2. (43)\nSubstituting this into the equation for θyields the nonlinear pendulum\nθ′′= sin(θ)(h−Ω−µcos(θ))+C2cos(θ)\nsin(θ)3, (44)\nwith singular potential energy\nP(θ) =P0(θ)+1\n2C2cot(θ)2.\nThe energy introduces barriers at multiples of πso that solutions for q/ne}ationslash= 0 cannot pass ±ˆe3,\nand asCincreases the energy landscape becomes qualitatively inde pendent of Ω ,h,µ.\nAsanimmediateconsequenceoftheenergybarriersandthefa ctthatonlystablewavetrains\nare saddle points we get\nTheorem 9 Lets= 0andΩ =β/α. Consider solutions to (14)of the form (30). Intersec-\ntions in(q,θ)-space of the curve Cgiven by (43)with the wavetrain existence curves Wfrom\n(23)(withkreplaced by q) are in one-to-one correspondence with equilibria of (41).\n32Consider supercritical anisotropy 0≤h−Ω< µand assume Cis such that Ctransversely\nintersects the component of Wwhich intersects {q= 0}. Then the intersection point with\nsmallerq-value corresponds to a spectrally stable wavetrain, and in (41)there is a pair of\nhomoclinic solutions to this wavetrain. All other intersec tion points are unstable wavetrains.\nAll other non-equilibrium solutions of the form (30)withs= 0are periodic in ξ, and this\nis also the case for all other parameter settings.\nThe homoclinic orbit is a soliton-type solution to (1) with a symptotic state a wavetrain\n(cf. Figure 2).\nNotably, the tangential intersection of CandWis at the sideband instability.\nIn Figure 13 we plot an illustration in case of supercritical anisotropy µ >Ω−h >0. In\nthe upper panel of (b) the local maxima each generate a pair of homoclinic solutions to the\nstable wavetrain it represents. The values of qthat it visits lie on the bold curve in (a), whose\nintersections with the curve of equilibria are the local max ima and minima. The lower panel\nin (b) has C= 0.4 and the local maxima disappeared. All solutions are period ic and lie on\nthe thick dashed curve in (a).\n5.3.3 Absence of wavetrains ( Ω/ne}ationslash=β/α)\nIn this case the first integral Q= log(|q|sin(θ)2) is monotone,\nQ′=Ω−β/α\nq/ne}ationslash= 0,\nand therefore |q|is unbounded as ξ→ ∞orξ→ −∞. In view of (44), we also infer that θ\noscillates so that there are no relevant solutions of the for m (30).\nReferences\n[1]Aranson, I., and Kramer, L. The world of the complex Ginzburg-Landau equation.\nRev. Mod. Phys. 74 (2002), 99–143.\n[2]Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current.\nPhys. Rev. 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Theoretical and Mathematical\nPhysics 38 , 1 (1979), 17–23.\n35"    },    {        "title": "1309.7483v1.High_efficiency_GHz_frequency_doubling_without_power_threshold_in_thin_film_Ni81Fe19.pdf",        "content": "arXiv:1309.7483v1  [cond-mat.mtrl-sci]  28 Sep 2013High-efficiency GHz frequency doubling without power thresh old in thin-film Ni 81Fe19\nCheng Cheng1and William E. Bailey1\nMaterials Science and Engineering Program, Department of A pplied\nPhysics and Applied Mathematics, Columbia University, New York,\nNY 10027\nWe demonstrate efficient second-harmonic generation at moderat e input power for\nthin film Ni 81Fe19undergoing ferromagnetic resonance (FMR). Powers of the gene r-\natedsecond-harmonicareshowntobequadraticininputpower, wit hanupconversion\nratio three orders of magnitude higher than that demonstrated in ferrites1, defined\nas ∆P2ω/∆Pω∼4×10−5/W·Pω, where ∆ Pis the change in the transmitted rf\npower and Pis the input rf power. The second harmonic signal generated exhibit s\na significantly lower linewidth than that predicted by low-power Gilbert damping,\nand is excited without threshold. Results are in good agreement with an analytic,\napproximate expansion of the Landau-Lifshitz-Gilbert (LLG) equa tion.\n1Nonlinear effects in magnetizationdynamics, apart frombeing offun damental interest1–4,\nhave provided important tools for microwave signal processing, es pecially in terms of fre-\nquency doubling and mixing5,6. Extensive experimental work exists on ferrites1,4,6, tradi-\ntionally used in low-loss devices due to their insulating nature and narr ow ferromagnetic\nresonance (FMR) linewidth. Metallic thin-film ferromagnets are of int erest for use in these\nand related devices due to their high moments, integrability with CMOS processes, and\npotential for enhanced functionality from spin transport; low FMR linewidth has been\ndemonstrated recently in metals through compensation by the spin Hall effect7. While some\nrecent work has addressed nonlinear effects8–10and harmonic generation11–13in metallic\nferromagnets and related devices14–16, these studies have generally used very high power\nor rf fields, and have not distinguished between effects above and b elow the Suhl instabil-\nity threshold. In this manuscript, we demonstrate frequency dou bling below threshold in\na metallic system (Ni 81Fe19) which is three orders of magnitude more efficient than that\ndemonstrated previously in ferrite materials1. The results are in good quantitative agree-\nment with an analytical expansion of the Landau-Lifshitz-Gilbert (L LG) equation.\nFor all measurements shown, we used a metallic ferromagnetic thin fi lm structure, Ta(5\nnm)/Cu(5 nm)/Ni 81Fe19(30 nm)/Cu(3 nm)/Al(3 nm). The film was deposited on an oxi-\ndized silicon substrate using magnetron sputtering at a base press ure of 2.0 ×10−7Torr. The\nbottom Ta(5 nm)/Cu(5 nm) layer is a seed layer to improve adhesion a nd homogeneity of\nthe film and the top Cu(3 nm)/Al(3 nm) layer protects the Ni 81Fe19layer from oxidation.\nA diagram of the measurement configuration, adapted from a basic broadband FMR setup,\nis shown in Fig.1. The microwave signal is conveyed to and from the sam ple through a\ncoplanar waveguide (CPW) with a 400 µm wide center conductor and 50 Ω characteristic\nimpedance, which gives an estimated rf field of 2.25 Oe rms with the inpu t power of +30\ndBm. We examined the second harmonic generation with fundamenta l frequencies at 6.1\nGHz and 2.0 GHz. The cw signal from the rf source is first amplified by a solid state am-\nplifier, then the signal power is tuned to the desirable level by an adj ustable attenuator.\nHarmonics of the designated input frequency are attenuated by t he bandpass filter to less\nthan the noise floor of the spectrum analyzer (SA). The isolator limit s back-reflection of\nthe filtered signal from the sample into the rf source. From our ana lysis detailed in a later\nsection of this manuscript, we found the second harmonic magnitud e to be proportional to\n2FIG. 1. Experimental setup and the coordinate system, θ= 45◦; see text for details. EM:\nelectromagnet; SA: spectrum analyzer. Arrows indicate the transmission of rf signal.\nthe product of the longitudinal and transverse rf field strengths , and thus place the center\nconductor of CPW at 45◦from H Bto maximize the Hrf\nyHrf\nzproduct. The rf signal finally\nreaches the SA for measurements of the power of both the funda mental frequency and its\nsecond harmonic.\nFig.2(a) demonstrates representative field-swept FMR absorptio n and the second har-\nmonic emission spectra measured by the SA as 6.1 GHz and 12.2 GHz pea k intensities as\na function of the bias field H B. We vary the input rf power over a moderate range of +4\n- +18 dBm, and fit the peaks with a Lorentzian function to extract t he amplitude and\nthe linewidth of the absorbed (∆ Pω) and generated (∆ P2ω) power. Noticeably, the second\nharmonic emission peaks have a much smaller linewidth, ∆ H1/2∼10 Oe over the whole\npower range, than those of the FMR peaks, with ∆ H1/2∼21 Oe. Plots of the absorption\nand emission peak amplitudes as a function of the input 6.1 GHz power, shown in Fig.2(b),\nclearly indicate a linear dependence of the FMR absorption and a quad ratic dependence\nof the second harmonic generation on the input rf power. Taking th e ratio of the radiated\nsecond harmonic power to the absorbed power, we have a convers ion rate of 3.7 ×10−5/W,\nas shown in Fig.2(c).\nSince the phenomenon summarized in Fig.2 is clearly not a threshold effe ct, we look into\nthe second-harmonic analysis of the LLG equation with small rf fields , which is readily de-\nscribed in Gurevich and Melkov’s text for circular precession relevan t in the past for low-M s\n3FIG. 2. Second harmonic generation with ω/2π= 6.1 GHz. a) left panel : 6.1 GHz input power\n+17.3 dBm; right panel : 6.1 GHz input power +8.35 dBm. b) amplitudes of the ω(FMR) and\ngenerated 2 ωpeaks as a function of input power Pω; right and top axes represent the data set\nin log-log plot (green), extracting the power index; c) rati o of the peak amplitudes of FMR and\nsecond harmonic generation as a function of the input 6.1 GHz power; green: log scale.\n4ferrites18. For metallic thin films, we treat the elliptical case as follows. As illustra ted in\nFig.1, the thin film is magnetized in the yzplane along /hatwidezby the bias field H B, with film-\nnormal direction along /hatwidex. The CPW exerts both a longitudinal rf field hrf\nzand a transverse\nrf field hrf\nyof equal strength. First consider only the transverse field hrf\ny. In this well es-\ntablished case, the LLG equation ˙m=−γm×Heff+αm×˙mis linearized and takes the\nform \n˙/tildewidermx\n˙/tildewidermy\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx\n/tildewidermy\n+\nγ/tildewiderhrf\ny\n0\n (1)\n, whereγis the gyromagnetic ratio, αis the Gilbert damping parameter, ωM≡γ4πMs, and\nωH≡γHz. Introducing first order perturbation to mx,yunder additional longitudinal hrf\nz\nand neglecting the second order terms, we have\n\n˙/tildewidermx+˙/tildewidest∆mx\n˙/tildewidermy+˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx+/tildewidest∆mx\n/tildewidermy+/tildewidest∆my\n+\nγ/tildewiderhrf\nz/tildewidermy\n−γ/tildewiderhrf\nz/tildewidermx\n+\nγ/tildewiderhrf\ny\n0\n(2)\nSubtracting (1) from (2) and taking/tildewiderhrfy,z=Hrf\ny,ze−iωt,/tildewidemx,y= (Hrf\ny/Ms)e−iωt/tildewideχ⊥,/bardbl(ω), the\nequation for the perturbation terms is\n\n˙/tildewidest∆mx\n˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidest∆mx\n/tildewidest∆my\n+Hrf\nzHrf\ny\nMse−i2ωt\nγ/tildewiderχ/bardbl(ω)\n−γ/tildewiderχ⊥(ω)\n(3)\nSinceχ⊥is one order of magnitude smaller than χ/bardbl, we neglect the term −γ/tildewiderχ⊥(ω).\nIn complete analogy to equation (1), the driving term could be viewed as an effective\ntransverse field of Hrf\nz(Hrf\ny/Ms)/tildewiderχ/bardbl(ω)e−i2ωt, and the solutions to equation (3) would be\n/tildewidest∆mx= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ⊥(2ω)e−i2ωt,/tildewidest∆my= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ/bardbl(2ω)e−i2ωt. We\ncan compare the power at frequency fand 2fnow that we have the expressions for\nboth the fundamental and second harmonic components of the pr ecessing M. The time-\naveraged power per unit volume could be calculated as /angbracketleftP/angbracketright= [/integraltext2π\nω\n0P(t)dt]/(2π/ω), P(t) =\n−∂U/∂t= 2M∂H/∂twhere only the transverse components of MandHcontribute\nto P(t). Using the expression for /angbracketleftP/angbracketright,MandH, we have Pω=ωH2\ny,rfχ(ω)′′\n/bardbland\nP2ω= 2ωH2\nz,rf(Hrf\ny/Ms)2|˜χ(ω)/bardbl|2χ(2ω)′′\n/bardbl, from which we conclude that under H Bfor FMR\nat frequency f=ω/(2π), we should see a power ratio\nP2ω/Pω= 2(Hrf\nz/Ms)2χ(ω)′′\n/bardblχ(2ω)′′\n/bardbl (4)\nWithMs= 844 Oe, α= 0.007 as measured by FMR for our Ni 81Fe1930 nm sample and\n2.25 Oe rf field amplitude at input power of 1 W for the CPW, we have a ca lculated 2 f/f\n5power ratio of 1.72 ×10−5/W, which is in reasonable agreement with the experimental data\n3.70×10−5/W as shown in Fig.2(c). To compare this result with the ferrite exper iment in\nref.[1], we further add the factor representing the ratio of FMR ab sorption to the input rf\npower, which is 3 .9×10−2in our setup. This leads to an experimental upconversion ratio of\n1.44×10−6/W in ref.[1]’s definition (∆ P2ω/Pω\nin2), compared with 7 .1×10−10/W observed\nin Mg 70Mn8Fe22O (Ferramic R-1 ferrite).\nExamining Eq.(4), we noticethatthereshouldbetwo peaksinthefield -swept 2femission\nspectrum: the first coincides with the FMR but with a narrower linewid th due to the term\n|˜χ(ω)/bardbl|2, and the second positioned at the H Bfor the FMR with a 2 finput signal due to\nthe term χ(2ω)′′\n/bardbl. The second peak should have a much smaller amplitude. Due to the fie ld\nlimit of our electromagnet, we could not reach the bias field required f or FMR at 12.2 GHz\nunder this particular configuration and continued to verify Eq.(4) a t a lower frequency of 2.0\nGHz. We carried out an identical experiment and analysis and observ ed an upconversion\nefficiency of 0.39 ×10−3/W for the 4.0 GHz signal generation at 2.0 GHz input, again in\nreasonable agreement with the theoretical prediction 1.17 ×10−3/W. Fig.3 demonstrates the\ntypical line shape of the4 GHzspectrum, in which the input 2 GHzpowe r being +18.9 dBm.\nA second peak at the H Bfor 4 GHz FMR is clearly visible with a much smaller amplitude\nand larger linewidth than the first peak, qualitatively consistent with Eq.(4). A theoretical\nline (dashed green) from equation (4) with fixed damping parameter α= 0.007 is drawn to\ncompare with the experimental data. The observed second peak a t the 2fresonance H B\nshows a much lower amplitude than expected. We contribute this diffe rence to the possible\n2fcomponent in the rf source which causes the 2 fFMR absorption. The blue line shows\nthe adjusted theoretical line with consideration of this input signal impurity.\nSummary : We have demonstrated a highly efficient frequency doubling effect in thin-\nfilm Ni 81Fe19for input powers well below the Suhl instability threshold. An analysis of\nthe intrinsically nonlinear LLG equation interprets the observed phe nomena quantitatively.\nThe results explore new opportunities in the field of rf signal manipula tion with CMOS\ncompatible thin film structures.\nWe acknowledge Stephane Auffret for the Ni 81Fe19sample. We acknowledge support\nfrom the US Department of Energy grant DE-EE0002892 and Natio nal Science Foundation\n6FIG. 3. 4 GHz generation with input signal at 2 GHz, +18.9 dBm. A second peak at the bias field\nfor 4 GHz FMR is clearly present; red dots: experimental data ; dashed green: theoretical; blue:\nadjusted theoretical with input rf impurity. See text for de tails.\nECCS-0925829.\nREFERENCES\n1W. P. Ayres, P. H. Vartanian, and J. L. Melchor, J. Appl. Phys. 27, 188 (1956)\n2N. Bloembergen and S. Wang, Phys. Rev. 93, 72 (1954)\n3H. Suhl, J. Phys. Chem. Solids. 1, 209 (1957)\n4J. D. Bierlein and P. M. Richards, Phys. Rev. B 1, 4342 (1970)\n5G. P. Ridrigue, J. Appl. Phys. 40, 929 (1969)\n6V. G. Harris, IEEE Trans. Magn. 48, 1075 (2012)\n7V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. B aither, G. Schmitz\nand S. O. Demokritov, Nature Mat. 11, 1028 (2012)\n8A. Berteaud and H. Pascard, J. Appl. Phys. 37, 2035 (1966)\n9T. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silv a, Phys. Rev. Lett.\n98, 207602 (2007)\n10H. M. Olson, P. Krivosik, K. Srinivasan, and C. E. Patton, J. Appl. Ph ys.102, 023904\n(2007)\n711M. Bao, A. Khitun, Y. Wu, J. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93,\n072509 (2008)\n12Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P . Krivosik, R. E. Camley,\nand Z. Celinski, Appl. Phys. Lett. 98, 042505 (2011)\n13J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Appl. Phys. Le tt.100, 102404\n(2012)\n14M. Yana, P. Vavassori, G. Leaf, F.Y. Fradin, and M. Grimsditch, J. M agn. Magn. Mater\n320, 1909 (2008)\n15V. E. Demidov, H. Ulrichs, S. Urazhdin, S. O. Demokritov, V. Besson ov, R. Gieniusz, and\nA. Maziewski, Appl. Phys. Lett. 99, 012505 (2011)\n16C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, and J. Q. Xia o, Appl. Phys.\nLett.99, 232506 (2011)\n17S. E. Bushnell, W. B. Nowak, S. A. Oliver, and C. Vittoria, Rev. Sci. In strum.63, 2021\n(1992)\n18A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves ( CRC, Boca\nRaton, 1996)\n8"    },    {        "title": "1310.7657v1.Observational_Study_of_Large_Amplitude_Longitudinal_Oscillations_in_a_Solar_Filament.pdf",        "content": "arXiv:1310.7657v1  [astro-ph.SR]  29 Oct 2013Nature of Prominences and their role in Space Weather\nProceedings IAU Symposium No. 300, 2014\nB. Schmieder, JM. Malherbe & S. Wu, eds.c/circlecopyrt2014 International Astronomical Union\nDOI: 00.0000/X000000000000000X\nObservational Study of Large Amplitude\nLongitudinal Oscillations in a Solar Filament\nKalman Knizhnik1,2,Manuel Luna3, Karin Muglach2,4\nHolly Gilbert2, Therese Kucera2, Judith Karpen2\n1Department of Physics and Astronomy\nThe Johns Hopkins University, Baltimore, MD 21218\nemail: kalman.knizhnik@nasa.gov\n2NASA/GSFC, Greenbelt, MD 20771, USA\n3Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, T enerife, Spain\n4ARTEP, Inc., Maryland, USA\nAbstract. On 20 August 2010 an energetic disturbance triggered damped large-amplitude lon-\ngitudinal (LAL) oscillations in almost an entire filament. I n the present work we analyze this\nperiodic motion in the filament to characterize the damping a nd restoring mechanism of the\noscillation. Our method involves placing slits along the ax is of the filament at different angles\nwith respect to the spine of the filament, finding the angle at w hich the oscillation is clearest,\nand fitting the resulting oscillation pattern to decaying si nusoidal and Bessel functions. These\nfunctions represent the equations of motion of a pendulum da mped by mass accretion. With\nthis method we determine the period and the decaying time of t he oscillation. Our preliminary\nresults support the theory presented by Luna and Karpen (201 2) that the restoring force of LAL\noscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism\nis the ongoing accumulation of mass onto the oscillating thr eads. Following an earlier paper, we\nhave determined the magnitude and radius of curvature of the dipped magnetic flux tubes host-\ning a thread along the filament, as well as the mass accretion r ate of the filament threads, via\nthe fitted parameters.\nKeywords. solar prominences, oscillations, magnetic structures\n1. Procedure\nLAL oscillations consist of periodic motions of the prominence thread s along the mag-\nnetic field that are disturbed by a small energetic event close to the filament (see Luna\net al. paper in this volume). Luna and Karpen (2012) argue that pro minence oscillations\ncan be modeled as a damped oscillating pendulum, whose equation of mo tion satis-\nfies a zeroth-order Bessel function. In their model, a nearby trig ger event causes quasi-\nstationary preexisting prominence threads sitting in the dips of the magnetic structure\nto oscillate back and forth, with the restoring force being the proj ected gravity in the\ntubes where the threads oscillate (e.g. Luna et al. (2012)). In this paper, we report pre-\nliminary results of comparisonsof observations of prominence oscilla tions with the model\npresented by Luna and Karpen (2012). More details will be available in the forthcoming\npaper by Luna et al. (2013).\nIn this analysis, we place slits along the filament spine and measure the intensity along\neach slit as a function of time. Fig. 1 (left) shows the filament in the AI A 171˚A filter\nwith the slits overlaid. Each slit is then rotated in increments of 0.5◦from 0◦to 60◦\nwith respect to the filament spine. We select the best slit according t o the following\ncriteria: (a) continuity of oscillations, (b) amplitude of the oscillation is maximized, (c)\nclear transition from dark to bright regions, (d) maximum number of cycles.\nThe oscillation for a representative slit is shown in Figure 1 (right), wh ich corresponds\nto the grey slit in Figure 1 (left). We identify the position of the cente r of mass of the\n12 K. Knizhnik, M. Luna, K. Muglach, H. Gilbert, T. Kucera, J. Karpen\nFigure 1. Left: Filament seen in AIA 171 with best slits overlaid. Right: An intensity distance–\ntime slit, showing an oscillation with the Bessel fit (white c urve) to equation (2.1) in Luna et\nal. (this volume). The sinusoidal fit was not as good as the Bes sel fit and is not shown.\nthread by finding the intensity minimum along the slit, indicated by black crosses in\nFigure 1 (right). These points are then fit to equation (2.1) of Luna et al. (this volume),\nand the resulting fit is shown in white.\n2. Results\nFitting our data to equation (2.1) of Luna et al. (this volume) yields va lues ofχ2\nranging between 1-13. Using equation (2.2) of Luna et al. (this volum e), we find the\naverageradiusofcurvatureofthe magneticfield dips that suppor t the oscillatingthreads.\nWe find it to be approximately 60 Mm. We also calculate a threshold value for the field\nitself that would allow it to support the observedthreads. Using equ ation (3.1) of Luna et\nal.(this volume), wefind anaveragemagneticfield of ∼20G, assumingatypicalfilament\nnumber density of 1011cm−3, in good agreement with measurements (e.g. Mackay et al.\n2010). On average, the oscillations form an angle of ∼25owith respect to the filament\nspine, and have a period of ∼0.8 hours. To explain the very strong damping mass must\naccrete onto the threads at a rate of about 60 ×106kg/hr.\n3. Conclusions\nWe conclude that the observedoscillationsarealongthe magneticfie ld, which formsan\nangle of∼25owith respect to the filament spine (Tandberg-Hanssen & Anzer, 19 70). We\nfind that both the curvature and the magnitude of the magnetic fie ld are approximately\nuniform on different threads. Both the Bessel and sinusoidal func tions are well fitted,\nindicating that mass accretion is a likely damping mechanism of LAL oscilla tions, and\nthat the restoring force is the projected gravity in the dips where the threads oscillate.\nThe mass accretion rate agrees with the theoretical value (Karpe n et al., 2006, Luna,\nKarpen, & DeVore, 2012).\nReferences\nKarpen, J. T., Antiochos, S. K., Klimchuk, J. A. 2006, ApJ, 63 7, 531\nLuna, M., Karpen, J. T., & Devore, C. R. 2012a, ApJ, 746, 30\nLuna, M., & Karpen, J. 2012, ApJ, 750, L1\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., this volume , 2014\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., ApJ, 2013,in prep.\nMackay, D., Karpen, J., Ballester, J., Schmieder, B., Aulan ier, G. 2010, Sp. Sci. Rev. , 151, 333\nTandberg-Hanssen, E. and Anzer, U. 1970, Solar Physics 15, 158T"    }]